empirical or theoretical research

Difference between Theoretical and Empirical Research

The difference between theoretical and empirical research is fundamental to scientific, scholarly research, as it separates the development of ideas and models from their testing and validation.

These two approaches are used in many different fields of inquiry, including the natural sciences, social sciences, and humanities, and they serve different purposes and employ different methods.

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What is theoretical research.

Theoretical research involves the development of models, frameworks, and theories based on existing knowledge, logic, and intuition.

It aims to explain and predict phenomena, generate new ideas and insights, and provide a foundation for further research.

Theoretical research often takes place at the conceptual level and is typically based on existing knowledge, data, and assumptions.

What is Empirical Research?

In contrast, empirical research involves collecting and analysing data to test theories and models.

Empirical research is often conducted at the observational or experimental level and is based on direct or indirect observation of the world.

Empirical research involves testing theories and models, establishing cause-and-effect relationships, and refining or rejecting existing knowledge.

Theoretical vs Empirical Research

Theoretical research is often seen as the starting point for empirical research, providing the ideas and models that must be tested and validated.

Theoretical research can be qualitative or quantitative and involve mathematical models, simulations, and other computational methods.

Theoretical research is often conducted in isolation, without reference to primary data or observations.

On the other hand, empirical research is often seen as the final stage in the scientific process, as it provides evidence that supports or refutes theoretical models.

Empirical research can be qualitative or quantitative, involving surveys, experiments, observational studies, and other data collection methods.

Empirical research is often conducted in collaboration with others and is based on systematic data collection, analysis, and interpretation.

It is important to note that theoretical and empirical research are not mutually exclusive and can often complement each other.

For example, empirical data can inform the development of theories and models, and theoretical models can guide the design of empirical studies.

The most valuable research combines theoretical and empirical approaches in many fields, allowing for a comprehensive understanding of the studied phenomena.

It is important to note that this table is not meant to be exhaustive or prescriptive but rather to provide a general overview of the main difference between theoretical and empirical research.

The boundaries between these two approaches are not always clear, and in many cases, research may involve a combination of theoretical and empirical methods.

What are the Limitations of Theoretical Research?

Assumptions and simplifications may be made that do not accurately reflect the complexity of real-world phenomena, which is one of its limitations. Theoretical research relies heavily on logic and deductive reasoning, which can sometimes be biased or limited by the researcher’s assumptions and perspectives.

Furthermore, theoretical research may not be directly applicable to real-world situations without empirical validation. Applying theoretical ideas to practical situations is difficult if no empirical evidence supports or refutes them.

Furthermore, theoretical research may be limited by the availability of data and the researcher’s ability to access and interpret it, which can further limit the validity and applicability of theories.

What are the Limitations of Empirical Research?

There are many limitations to empirical research, including the limitations of the data available and the quality of the data that can be collected. Data collection can be limited by the resources available to collect the data, accessibility to populations or individuals of interest, or ethical constraints.

The researchers or participants may also introduce biases into empirical research, resulting in inaccurate or unreliable findings.

Lastly, due to confounding variables or other methodological limitations, empirical research may be limited by the inability to establish causal relationships between variables, even when statistical associations are identified.

What Methods Are Used In Theoretical Research?

In theoretical research, deductive reasoning, logical analysis, and conceptual frameworks generate new ideas and hypotheses. To identify gaps and inconsistencies in the present understanding of a phenomenon, theoretical research may involve analyzing existing literature and theories.

To test hypotheses and generate predictions, mathematical or computational models may also be developed.

Researchers may also use thought experiments or simulations to explore the implications of their ideas and hypotheses without collecting empirical data as part of theoretical research.

Theoretical research seeks to develop a conceptual framework for empirically testing and validating phenomena.

What Methods Are Used In Empirical Research?

Methods used in empirical research depend on the research questions, type of data collected, and study design. Surveys, experiments, observations, case studies, and interviews are common methods used in empirical research.

An empirical study tests hypotheses and generates new knowledge about phenomena by systematically collecting and analyzing data.

These methods may utilize standardized instruments or protocols for data collection consistency and reliability. Statistical analysis, content analysis, or qualitative analysis may be used for the data collection type.

As a result of empirical research, the findings can inform theories, models, and practical applications.

Conclusion: Theoretical vs Empirical Research

In conclusion, theoretical and empirical research are two distinct but interrelated approaches to scientific inquiry, and they serve different purposes and employ different methods.

Theoretical research involves the development of ideas and models, while empirical research involves testing and validating these ideas.

Both approaches are essential to research and can be combined to provide a more complete understanding of the world.

• Dictionary.com. “ Empirical vs Theoretical “.
• PennState University Libraries. “ Empirical Research in the Social Sciences and Education “.
• William M. Landes and Richard A. Posner. “ Legal Precedent: A Theoretical and Empirical Analysis “, The Journal of Law and Economics, 1976.

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Theoretical Research  is a logical exploration of a system of beliefs and assumptions, working with abstract principles related to a field of knowledge.

• Essentially...theorizing

Empirical Research is    based on real-life direct or indirect observation and measurement of phenomena by a researcher.

• Basically... Collecting data by Observing or Experimenting

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Empirical Research: Defining, Identifying, & Finding

Defining empirical research, what is empirical research, quantitative or qualitative.

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Calfee & Chambliss (2005)  (UofM login required) describe empirical research as a "systematic approach for answering certain types of questions."  Those questions are answered "[t]hrough the collection of evidence under carefully defined and replicable conditions" (p. 43). 

The evidence collected during empirical research is often referred to as "data." 

Characteristics of Empirical Research

Emerald Publishing's guide to conducting empirical research identifies a number of common elements to empirical research: 

• A  research question , which will determine research objectives.
• A particular and planned  design  for the research, which will depend on the question and which will find ways of answering it with appropriate use of resources.
• The gathering of  primary data , which is then analysed.
• A particular  methodology  for collecting and analysing the data, such as an experiment or survey.
• The limitation of the data to a particular group, area or time scale, known as a sample [emphasis added]: for example, a specific number of employees of a particular company type, or all users of a library over a given time scale. The sample should be somehow representative of a wider population.
• The ability to  recreate  the study and test the results. This is known as  reliability .
• The ability to  generalize  from the findings to a larger sample and to other situations.

If you see these elements in a research article, you can feel confident that you have found empirical research. Emerald's guide goes into more detail on each element. 

Empirical research methodologies can be described as quantitative, qualitative, or a mix of both (usually called mixed-methods).

Ruane (2016)  (UofM login required) gets at the basic differences in approach between quantitative and qualitative research:

• Quantitative research  -- an approach to documenting reality that relies heavily on numbers both for the measurement of variables and for data analysis (p. 33).
• Qualitative research  -- an approach to documenting reality that relies on words and images as the primary data source (p. 33).

Both quantitative and qualitative methods are empirical . If you can recognize that a research study is quantitative or qualitative study, then you have also recognized that it is empirical study. 

Below are information on the characteristics of quantitative and qualitative research. This video from Scribbr also offers a good overall introduction to the two approaches to research methodology: 

Characteristics of Quantitative Research 

Researchers test hypotheses, or theories, based in assumptions about causality, i.e. we expect variable X to cause variable Y. Variables have to be controlled as much as possible to ensure validity. The results explain the relationship between the variables. Measures are based in pre-defined instruments.

Examples: experimental or quasi-experimental design, pretest & post-test, survey or questionnaire with closed-ended questions. Studies that identify factors that influence an outcomes, the utility of an intervention, or understanding predictors of outcomes. 

Characteristics of Qualitative Research

Researchers explore “meaning individuals or groups ascribe to social or human problems (Creswell & Creswell, 2018, p3).” Questions and procedures emerge rather than being prescribed. Complexity, nuance, and individual meaning are valued. Research is both inductive and deductive. Data sources are multiple and varied, i.e. interviews, observations, documents, photographs, etc. The researcher is a key instrument and must be reflective of their background, culture, and experiences as influential of the research.

Examples: open question interviews and surveys, focus groups, case studies, grounded theory, ethnography, discourse analysis, narrative, phenomenology, participatory action research.

Calfee, R. C. & Chambliss, M. (2005). The design of empirical research. In J. Flood, D. Lapp, J. R. Squire, & J. Jensen (Eds.),  Methods of research on teaching the English language arts: The methodology chapters from the handbook of research on teaching the English language arts (pp. 43-78). Routledge.  http://ezproxy.memphis.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=125955&site=eds-live&scope=site .

Creswell, J. W., & Creswell, J. D. (2018).  Research design: Qualitative, quantitative, and mixed methods approaches  (5th ed.). Thousand Oaks: Sage.

How to... conduct empirical research . (n.d.). Emerald Publishing.  https://www.emeraldgrouppublishing.com/how-to/research-methods/conduct-empirical-research .

Scribbr. (2019). Quantitative vs. qualitative: The differences explained  [video]. YouTube.  https://www.youtube.com/watch?v=a-XtVF7Bofg .

Ruane, J. M. (2016).  Introducing social research methods : Essentials for getting the edge . Wiley-Blackwell.  http://ezproxy.memphis.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=1107215&site=eds-live&scope=site .  

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Theory and Observation in Science

Scientists obtain a great deal of the evidence they use by collecting and producing empirical results. Much of the standard philosophical literature on this subject comes from 20 th century logical empiricists, their followers, and critics who embraced their issues while objecting to some of their aims and assumptions. Discussions about empirical evidence have tended to focus on epistemological questions regarding its role in theory testing. This entry follows that precedent, even though empirical evidence also plays important and philosophically interesting roles in other areas including scientific discovery, the development of experimental tools and techniques, and the application of scientific theories to practical problems.

The logical empiricists and their followers devoted much of their attention to the distinction between observables and unobservables, the form and content of observation reports, and the epistemic bearing of observational evidence on theories it is used to evaluate. Philosophical work in this tradition was characterized by the aim of conceptually separating theory and observation, so that observation could serve as the pure basis of theory appraisal. More recently, the focus of the philosophical literature has shifted away from these issues, and their close association to the languages and logics of science, to investigations of how empirical data are generated, analyzed, and used in practice. With this shift, we also see philosophers largely setting aside the aspiration of a pure observational basis for scientific knowledge and instead embracing a view of science in which the theoretical and empirical are usefully intertwined. This entry discusses these topics under the following headings:

1. Introduction

2.1 traditional empiricism, 2.2 the irrelevance of observation per se, 2.3 data and phenomena, 3.1 perception, 3.2 assuming the theory to be tested, 3.3 semantics, 4.1 confirmation, 4.2 saving the phenomena, 4.3 empirical adequacy, 5. conclusion, other internet resources, related entries.

Philosophers of science have traditionally recognized a special role for observations in the epistemology of science. Observations are the conduit through which the ‘tribunal of experience’ delivers its verdicts on scientific hypotheses and theories. The evidential value of an observation has been assumed to depend on how sensitive it is to whatever it is used to study. But this in turn depends on the adequacy of any theoretical claims its sensitivity may depend on. For example, we can challenge the use of a particular thermometer reading to support a prediction of a patient’s temperature by challenging theoretical claims having to do with whether a reading from a thermometer like this one, applied in the same way under similar conditions, should indicate the patient’s temperature well enough to count in favor of or against the prediction. At least some of those theoretical claims will be such that regardless of whether an investigator explicitly endorses, or is even aware of them, her use of the thermometer reading would be undermined by their falsity. All observations and uses of observational evidence are theory laden in this sense (cf. Chang 2005, Azzouni 2004). As the example of the thermometer illustrates, analogues of Norwood Hanson’s claim that seeing is a theory laden undertaking apply just as well to equipment generated observations (Hanson 1958, 19). But if all observations and empirical data are theory laden, how can they provide reality-based, objective epistemic constraints on scientific reasoning?

Recent scholarship has turned this question on its head. Why think that theory ladenness of empirical results would be problematic in the first place? If the theoretical assumptions with which the results are imbued are correct, what is the harm of it? After all, it is in virtue of those assumptions that the fruits of empirical investigation can be ‘put in touch’ with theorizing at all. A number scribbled in a lab notebook can do a scientist little epistemic good unless she can recruit the relevant background assumptions to even recognize it as a reading of the patient’s temperature. But philosophers have embraced an entangled picture of the theoretical and empirical that goes much deeper than this. Lloyd (2012) advocates for what she calls “complex empiricism” in which there is “no pristine separation of model and data” (397). Bogen (2016) points out that “impure empirical evidence” (i.e. evidence that incorporates the judgements of scientists) “often tells us more about the world that it could have if it were pure” (784). Indeed, Longino (2020) has urged that “[t]he naïve fantasy that data have an immediate relation to phenomena of the world, that they are ‘objective’ in some strong, ontological sense of that term, that they are the facts of the world directly speaking to us, should be finally laid to rest” and that “even the primary, original, state of data is not free from researchers’ value- and theory-laden selection and organization” (391).

There is not widespread agreement among philosophers of science about how to characterize the nature of scientific theories. What is a theory? According to the traditional syntactic view, theories are considered to be collections of sentences couched in logical language, which must then be supplemented with correspondence rules in order to be interpreted. Construed in this way, theories include maximally general explanatory and predictive laws (Coulomb’s law of electrical attraction and repulsion, and Maxwellian electromagnetism equations for example), along with lesser generalizations that describe more limited natural and experimental phenomena (e.g., the ideal gas equations describing relations between temperatures and pressures of enclosed gasses, and general descriptions of positional astronomical regularities). In contrast, the semantic view casts theories as the space of states possible according to the theory, or the set of mathematical models permissible according to the theory (see Suppe 1977). However, there are also significantly more ecumenical interpretations of what it means to be a scientific theory, which include elements of diverse kinds. To take just one illustrative example, Borrelli (2012) characterizes the Standard Model of particle physics as a theoretical framework involving what she calls “theoretical cores” that are composed of mathematical structures, verbal stories, and analogies with empirical references mixed together (196). This entry aims to accommodate all of these views about the nature of scientific theories.

In this entry, we trace the contours of traditional philosophical engagement with questions surrounding theory and observation in science that attempted to segregate the theoretical from the observational, and to cleanly delineate between the observable and the unobservable. We also discuss the more recent scholarship that supplants the primacy of observation by human sensory perception with an instrument-inclusive conception of data production and that embraces the intertwining of theoretical and empirical in the production of useful scientific results. Although theory testing dominates much of the standard philosophical literature on observation, much of what this entry says about the role of observation in theory testing applies also to its role in inventing, and modifying theories, and applying them to tasks in engineering, medicine, and other practical enterprises.

2. Observation and data

Reasoning from observations has been important to scientific practice at least since the time of Aristotle, who mentions a number of sources of observational evidence including animal dissection (Aristotle(a), 763a/30–b/15; Aristotle(b), 511b/20–25). Francis Bacon argued long ago that the best way to discover things about nature is to use experiences (his term for observations as well as experimental results) to develop and improve scientific theories (Bacon 1620, 49ff). The role of observational evidence in scientific discovery was an important topic for Whewell (1858) and Mill (1872) among others in the 19th century. But philosophers didn’t talk about observation as extensively, in as much detail, or in the way we have become accustomed to, until the 20 th century when logical empiricists transformed philosophical thinking about it.

One important transformation, characteristic of the linguistic turn in philosophy, was to concentrate on the logic of observation reports rather than on objects or phenomena observed. This focus made sense on the assumption that a scientific theory is a system of sentences or sentence-like structures (propositions, statements, claims, and so on) to be tested by comparison to observational evidence. It was assumed that the comparisons must be understood in terms of inferential relations. If inferential relations hold only between sentence-like structures, it follows that theories must be tested, not against observations or things observed, but against sentences, propositions, etc. used to report observations (Hempel 1935, 50–51; Schlick 1935). Theory testing was treated as a matter of comparing observation sentences describing observations made in natural or laboratory settings to observation sentences that should be true according to the theory to be tested. This was to be accomplished by using laws or lawlike generalizations along with descriptions of initial conditions, correspondence rules, and auxiliary hypotheses to derive observation sentences describing the sensory deliverances of interest. This makes it imperative to ask what observation sentences report.

According to what Hempel called the phenomenalist account , observation reports describe the observer’s subjective perceptual experiences.

… Such experiential data might be conceived of as being sensations, perceptions, and similar phenomena of immediate experience. (Hempel 1952, 674)

This view is motivated by the assumption that the epistemic value of an observation report depends upon its truth or accuracy, and that with regard to perception, the only thing observers can know with certainty to be true or accurate is how things appear to them. This means that we cannot be confident that observation reports are true or accurate if they describe anything beyond the observer’s own perceptual experience. Presumably one’s confidence in a conclusion should not exceed one’s confidence in one’s best reasons to believe it. For the phenomenalist, it follows that reports of subjective experience can provide better reasons to believe claims they support than reports of other kinds of evidence.

However, given the expressive limitations of the language available for reporting subjective experiences, we cannot expect phenomenalistic reports to be precise and unambiguous enough to test theoretical claims whose evaluation requires accurate, fine-grained perceptual discriminations. Worse yet, if experiences are directly available only to those who have them, there is room to doubt whether different people can understand the same observation sentence in the same way. Suppose you had to evaluate a claim on the basis of someone else’s subjective report of how a litmus solution looked to her when she dripped a liquid of unknown acidity into it. How could you decide whether her visual experience was the same as the one you would use her words to report?

Such considerations led Hempel to propose, contrary to the phenomenalists, that observation sentences report ‘directly observable’, ‘intersubjectively ascertainable’ facts about physical objects

… such as the coincidence of the pointer of an instrument with a numbered mark on a dial; a change of color in a test substance or in the skin of a patient; the clicking of an amplifier connected with a Geiger counter; etc. (ibid.)

That the facts expressed in observation reports be intersubjectively ascertainable was critical for the aims of the logical empiricists. They hoped to articulate and explain the authoritativeness widely conceded to the best natural, social, and behavioral scientific theories in contrast to propaganda and pseudoscience. Some pronouncements from astrologers and medical quacks gain wide acceptance, as do those of religious leaders who rest their cases on faith or personal revelation, and leaders who use their political power to secure assent. But such claims do not enjoy the kind of credibility that scientific theories can attain. The logical empiricists tried to account for the genuine credibility of scientific theories by appeal to the objectivity and accessibility of observation reports, and the logic of theory testing. Part of what they meant by calling observational evidence objective was that cultural and ethnic factors have no bearing on what can validly be inferred about the merits of a theory from observation reports. So conceived, objectivity was important to the logical empiricists’ criticism of the Nazi idea that Jews and Aryans have fundamentally different thought processes such that physical theories suitable for Einstein and his kind should not be inflicted on German students. In response to this rationale for ethnic and cultural purging of the German educational system, the logical empiricists argued that because of its objectivity, observational evidence (rather than ethnic and cultural factors) should be used to evaluate scientific theories (Galison 1990). In this way of thinking, observational evidence and its subsequent bearing on scientific theories are objective also in virtue of being free of non-epistemic values.

Ensuing generations of philosophers of science have found the logical empiricist focus on expressing the content of observations in a rarefied and basic observation language too narrow. Search for a suitably universal language as required by the logical empiricist program has come up empty-handed and most philosophers of science have given up its pursuit. Moreover, as we will discuss in the following section, the centrality of observation itself (and pointer readings) to the aims of empiricism in philosophy of science has also come under scrutiny. However, leaving the search for a universal pure observation language behind does not automatically undercut the norm of objectivity as it relates to the social, political, and cultural contexts of scientific research. Pristine logical foundations aside, the objectivity of ‘neutral’ observations in the face of noxious political propaganda was appealing because it could serve as shared ground available for intersubjective appraisal. This appeal remains alive and well today, particularly as pernicious misinformation campaigns are again formidable in public discourse (see O’Connor and Weatherall 2019). If individuals can genuinely appraise the significance of empirical evidence and come to well-justified agreement about how the evidence bears on theorizing, then they can protect their epistemic deliberations from the undue influence of fascists and other nefarious manipulators. However, this aspiration must face subtleties arising from the social epistemology of science and from the nature of empirical results themselves. In practice, the appraisal of scientific results can often require expertise that is not readily accessible to members of the public without the relevant specialized training. Additionally, precisely because empirical results are not pure observation reports, their appraisal across communities of inquirers operating with different background assumptions can require significant epistemic work.

The logical empiricists paid little attention to the distinction between observing and experimenting and its epistemic implications. For some philosophers, to experiment is to isolate, prepare, and manipulate things in hopes of producing epistemically useful evidence. It had been customary to think of observing as noticing and attending to interesting details of things perceived under more or less natural conditions, or by extension, things perceived during the course of an experiment. To look at a berry on a vine and attend to its color and shape would be to observe it. To extract its juice and apply reagents to test for the presence of copper compounds would be to perform an experiment. By now, many philosophers have argued that contrivance and manipulation influence epistemically significant features of observable experimental results to such an extent that epistemologists ignore them at their peril. Robert Boyle (1661), John Herschell (1830), Bruno Latour and Steve Woolgar (1979), Ian Hacking (1983), Harry Collins (1985) Allan Franklin (1986), Peter Galison (1987), Jim Bogen and Jim Woodward (1988), and Hans-Jörg Rheinberger (1997), are some of the philosophers and philosophically-minded scientists, historians, and sociologists of science who gave serious consideration to the distinction between observing and experimenting. The logical empiricists tended to ignore it. Interestingly, the contemporary vantage point that attends to modeling, data processing, and empirical results may suggest a re-unification of observation and intervention under the same epistemological framework. When one no longer thinks of scientific observation as pure or direct, and recognizes the power of good modeling to account for confounds without physically intervening on the target system, the purported epistemic distinction between observation and intervention loses its bite.

Observers use magnifying glasses, microscopes, or telescopes to see things that are too small or far away to be seen, or seen clearly enough, without them. Similarly, amplification devices are used to hear faint sounds. But if to observe something is to perceive it, not every use of instruments to augment the senses qualifies as observational.

Philosophers generally agree that you can observe the moons of Jupiter with a telescope, or a heartbeat with a stethoscope. The van Fraassen of The Scientific Image is a notable exception, for whom to be ‘observable’ meant to be something that, were it present to a creature like us, would be observed. Thus, for van Fraassen, the moons of Jupiter are observable “since astronauts will no doubt be able to see them as well from close up” (1980, 16). In contrast, microscopic entities are not observable on van Fraassen’s account because creatures like us cannot strategically maneuver ourselves to see them, present before us, with our unaided senses.

Many philosophers have criticized van Fraassen’s view as overly restrictive. Nevertheless, philosophers differ in their willingness to draw the line between what counts as observable and what does not along the spectrum of increasingly complicated instrumentation. Many philosophers who don’t mind telescopes and microscopes still find it unnatural to say that high energy physicists ‘observe’ particles or particle interactions when they look at bubble chamber photographs—let alone digital visualizations of energy depositions left in calorimeters that are not themselves inspected. Their intuitions come from the plausible assumption that one can observe only what one can see by looking, hear by listening, feel by touching, and so on. Investigators can neither look at (direct their gazes toward and attend to) nor visually experience charged particles moving through a detector. Instead they can look at and see tracks in the chamber, in bubble chamber photographs, calorimeter data visualizations, etc.

In more contentious examples, some philosophers have moved to speaking of instrument-augmented empirical research as more like tool use than sensing. Hacking (1981) argues that we do not see through a microscope, but rather with it. Daston and Galison (2007) highlight the inherent interactivity of a scanning tunneling microscope, in which scientists image and manipulate atoms by exchanging electrons between the sharp tip of the microscope and the surface to be imaged (397). Others have opted to stretch the meaning of observation to accommodate what we might otherwise be tempted to call instrument-aided detections. For instance, Shapere (1982) argues that while it may initially strike philosophers as counter-intuitive, it makes perfect sense to call the detection of neutrinos from the interior of the sun “direct observation.”

The variety of views on the observable/unobservable distinction hint that empiricists may have been barking up the wrong philosophical tree. Many of the things scientists investigate do not interact with human perceptual systems as required to produce perceptual experiences of them. The methods investigators use to study such things argue against the idea—however plausible it may once have seemed—that scientists do or should rely exclusively on their perceptual systems to obtain the evidence they need. Thus Feyerabend proposed as a thought experiment that if measuring equipment was rigged up to register the magnitude of a quantity of interest, a theory could be tested just as well against its outputs as against records of human perceptions (Feyerabend 1969, 132–137). Feyerabend could have made his point with historical examples instead of thought experiments. A century earlier Helmholtz estimated the speed of excitatory impulses traveling through a motor nerve. To initiate impulses whose speed could be estimated, he implanted an electrode into one end of a nerve fiber and ran a current into it from a coil. The other end was attached to a bit of muscle whose contraction signaled the arrival of the impulse. To find out how long it took the impulse to reach the muscle he had to know when the stimulating current reached the nerve. But

[o]ur senses are not capable of directly perceiving an individual moment of time with such small duration …

and so Helmholtz had to resort to what he called ‘artificial methods of observation’ (Olesko and Holmes 1994, 84). This meant arranging things so that current from the coil could deflect a galvanometer needle. Assuming that the magnitude of the deflection is proportional to the duration of current passing from the coil, Helmholtz could use the deflection to estimate the duration he could not see (ibid). This sense of ‘artificial observation’ is not to be confused e.g., with using magnifying glasses or telescopes to see tiny or distant objects. Such devices enable the observer to scrutinize visible objects. The minuscule duration of the current flow is not a visible object. Helmholtz studied it by cleverly concocting circumstances so that the deflection of the needle would meaningfully convey the information he needed. Hooke (1705, 16–17) argued for and designed instruments to execute the same kind of strategy in the 17 th century.

It is of interest that records of perceptual observation are not always epistemically superior to data collected via experimental equipment. Indeed, it is not unusual for investigators to use non-perceptual evidence to evaluate perceptual data and correct for its errors. For example, Rutherford and Pettersson conducted similar experiments to find out if certain elements disintegrated to emit charged particles under radioactive bombardment. To detect emissions, observers watched a scintillation screen for faint flashes produced by particle strikes. Pettersson’s assistants reported seeing flashes from silicon and certain other elements. Rutherford’s did not. Rutherford’s colleague, James Chadwick, visited Pettersson’s laboratory to evaluate his data. Instead of watching the screen and checking Pettersson’s data against what he saw, Chadwick arranged to have Pettersson’s assistants watch the screen while unbeknownst to them he manipulated the equipment, alternating normal operating conditions with a condition in which particles, if any, could not hit the screen. Pettersson’s data were discredited by the fact that his assistants reported flashes at close to the same rate in both conditions (Stuewer 1985, 284–288).

When the process of producing data is relatively convoluted, it is even easier to see that human sense perception is not the ultimate epistemic engine. Consider functional magnetic resonance images (fMRI) of the brain decorated with colors to indicate magnitudes of electrical activity in different regions during the performance of a cognitive task. To produce these images, brief magnetic pulses are applied to the subject’s brain. The magnetic force coordinates the precessions of protons in hemoglobin and other bodily stuffs to make them emit radio signals strong enough for the equipment to respond to. When the magnetic force is relaxed, the signals from protons in highly oxygenated hemoglobin deteriorate at a detectably different rate than signals from blood that carries less oxygen. Elaborate algorithms are applied to radio signal records to estimate blood oxygen levels at the places from which the signals are calculated to have originated. There is good reason to believe that blood flowing just downstream from spiking neurons carries appreciably more oxygen than blood in the vicinity of resting neurons. Assumptions about the relevant spatial and temporal relations are used to estimate levels of electrical activity in small regions of the brain corresponding to pixels in the finished image. The results of all of these computations are used to assign the appropriate colors to pixels in a computer generated image of the brain. In view of all of this, functional brain imaging differs, e.g., from looking and seeing, photographing, and measuring with a thermometer or a galvanometer in ways that make it uninformative to call it observation. And similarly for many other methods scientists use to produce non-perceptual evidence.

The role of the senses in fMRI data production is limited to such things as monitoring the equipment and keeping an eye on the subject. Their epistemic role is limited to discriminating the colors in the finished image, reading tables of numbers the computer used to assign them, and so on. While it is true that researchers typically use their sense of sight to take in visualizations of processed fMRI data—or numbers on a page or screen for that matter—this is not the primary locus of epistemic action. Researchers learn about brain processes through fMRI data, to the extent that they do, primarily in virtue of the suitability of the causal connection between the target processes and the data records, and of the transformations those data undergo when they are processed into the maps or other results that scientists want to use. The interesting questions are not about observability, i.e. whether neuronal activity, blood oxygen levels, proton precessions, radio signals, and so on, are properly understood as observable by creatures like us. The epistemic significance of the fMRI data depends on their delivering us the right sort of access to the target, but observation is neither necessary nor sufficient for that access.

Following Shapere (1982), one could respond by adopting an extremely permissive view of what counts as an ‘observation’ so as to allow even highly processed data to count as observations. However, it is hard to reconcile the idea that highly processed data like fMRI images record observations with the traditional empiricist notion that calculations involving theoretical assumptions and background beliefs must not be allowed (on pain of loss of objectivity) to intrude into the process of data production. Observation garnered its special epistemic status in the first place because it seemed more direct, more immediate, and therefore less distorted and muddled than (say) detection or inference. The production of fMRI images requires extensive statistical manipulation based on theories about the radio signals, and a variety of factors having to do with their detection along with beliefs about relations between blood oxygen levels and neuronal activity, sources of systematic error, and more. Insofar as the use of the term ‘observation’ connotes this extra baggage of traditional empiricism, it may be better to replace observation-talk with terminology that is more obviously permissive, such as that of ‘empirical data’ and ‘empirical results.’

Deposing observation from its traditional perch in empiricist epistemologies of science need not estrange philosophers from scientific practice. Terms like ‘observation’ and ‘observation reports’ do not occur nearly as much in scientific as in philosophical writings. In their place, working scientists tend to talk about data . Philosophers who adopt this usage are free to think about standard examples of observation as members of a large, diverse, and growing family of data production methods. Instead of trying to decide which methods to classify as observational and which things qualify as observables, philosophers can then concentrate on the epistemic influence of the factors that differentiate members of the family. In particular, they can focus their attention on what questions data produced by a given method can be used to answer, what must be done to use that data fruitfully, and the credibility of the answers they afford (Bogen 2016).

Satisfactorily answering such questions warrants further philosophical work. As Bogen and Woodward (1988) have argued, there is often a long road between obtaining a particular dataset replete with idiosyncrasies born of unspecified causal nuances to any claim about the phenomenon ultimately of interest to the researchers. Empirical data are typically produced in ways that make it impossible to predict them from the generalizations they are used to test, or to derive instances of those generalizations from data and non ad hoc auxiliary hypotheses. Indeed, it is unusual for many members of a set of reasonably precise quantitative data to agree with one another, let alone with a quantitative prediction. That is because precise, publicly accessible data typically cannot be produced except through processes whose results reflect the influence of causal factors that are too numerous, too different in kind, and too irregular in behavior for any single theory to account for them. When Bernard Katz recorded electrical activity in nerve fiber preparations, the numerical values of his data were influenced by factors peculiar to the operation of his galvanometers and other pieces of equipment, variations among the positions of the stimulating and recording electrodes that had to be inserted into the nerve, the physiological effects of their insertion, and changes in the condition of the nerve as it deteriorated during the course of the experiment. There were variations in the investigators’ handling of the equipment. Vibrations shook the equipment in response to a variety of irregularly occurring causes ranging from random error sources to the heavy tread of Katz’s teacher, A.V. Hill, walking up and down the stairs outside of the laboratory. That’s a short list. To make matters worse, many of these factors influenced the data as parts of irregularly occurring, transient, and shifting assemblies of causal influences.

The effects of systematic and random sources of error are typically such that considerable analysis and interpretation are required to take investigators from data sets to conclusions that can be used to evaluate theoretical claims. Interestingly, this applies as much to clear cases of perceptual data as to machine produced records. When 19 th and early 20 th century astronomers looked through telescopes and pushed buttons to record the time at which they saw a star pass a crosshair, the values of their data points depended, not only upon light from that star, but also upon features of perceptual processes, reaction times, and other psychological factors that varied from observer to observer. No astronomical theory has the resources to take such things into account.

Instead of testing theoretical claims by direct comparison to the data initially collected, investigators use data to infer facts about phenomena, i.e., events, regularities, processes, etc. whose instances are uniform and uncomplicated enough to make them susceptible to systematic prediction and explanation (Bogen and Woodward 1988, 317). The fact that lead melts at temperatures at or close to 327.5 C is an example of a phenomenon, as are widespread regularities among electrical quantities involved in the action potential, the motions of astronomical bodies, etc. Theories that cannot be expected to predict or explain such things as individual temperature readings can nevertheless be evaluated on the basis of how useful they are in predicting or explaining phenomena. The same holds for the action potential as opposed to the electrical data from which its features are calculated, and the motions of astronomical bodies in contrast to the data of observational astronomy. It is reasonable to ask a genetic theory how probable it is (given similar upbringings in similar environments) that the offspring of a parent or parents diagnosed with alcohol use disorder will develop one or more symptoms the DSM classifies as indicative of alcohol use disorder. But it would be quite unreasonable to ask the genetic theory to predict or explain one patient’s numerical score on one trial of a particular diagnostic test, or why a diagnostician wrote a particular entry in her report of an interview with an offspring of one of such parents (see Bogen and Woodward, 1988, 319–326).

Leonelli has challenged Bogen and Woodward’s (1988) claim that data are, as she puts it, “unavoidably embedded in one experimental context” (2009, 738). She argues that when data are suitably packaged, they can travel to new epistemic contexts and retain epistemic utility—it is not just claims about the phenomena that can travel, data travel too. Preparing data for safe travel involves work, and by tracing data ‘journeys,’ philosophers can learn about how the careful labor of researchers, data archivists, and database curators can facilitate useful data mobility. While Leonelli’s own work has often focused on data in biology, Leonelli and Tempini (2020) contains many diverse case studies of data journeys from a variety of scientific disciplines that will be of value to philosophers interested in the methodology and epistemology of science in practice.

The fact that theories typically predict and explain features of phenomena rather than idiosyncratic data should not be interpreted as a failing. For many purposes, this is the more useful and illuminating capacity. Suppose you could choose between a theory that predicted or explained the way in which neurotransmitter release relates to neuronal spiking (e.g., the fact that on average, transmitters are released roughly once for every 10 spikes) and a theory which explained or predicted the numbers displayed on the relevant experimental equipment in one, or a few single cases. For most purposes, the former theory would be preferable to the latter at the very least because it applies to so many more cases. And similarly for theories that predict or explain something about the probability of alcohol use disorder conditional on some genetic factor or a theory that predicted or explained the probability of faulty diagnoses of alcohol use disorder conditional on facts about the training that psychiatrists receive. For most purposes, these would be preferable to a theory that predicted specific descriptions in a single particular case history.

However, there are circumstances in which scientists do want to explain data. In empirical research it is often crucial to getting a useful signal that scientists deal with sources of background noise and confounding signals. This is part of the long road from newly collected data to useful empirical results. An important step on the way to eliminating unwanted noise or confounds is to determine their sources. Different sources of noise can have different characteristics that can be derived from and explained by theory. Consider the difference between ‘shot noise’ and ‘thermal noise,’ two ubiquitous sources of noise in precision electronics (Schottky 1918; Nyquist 1928; Horowitz and Hill 2015). ‘Shot noise’ arises in virtue of the discrete nature of a signal. For instance, light collected by a detector does not arrive all at once or in perfectly continuous fashion. Photons rain onto a detector shot by shot on account of being quanta. Imagine building up an image one photon at a time—at first the structure of the image is barely recognizable, but after the arrival of many photons, the image eventually fills in. In fact, the contribution of noise of this type goes as the square root of the signal. By contrast, thermal noise is due to non-zero temperature—thermal fluctuations cause a small current to flow in any circuit. If you cool your instrument (which very many precision experiments in physics do) then you can decrease thermal noise. Cooling the detector is not going to change the quantum nature of photons though. Simply collecting more photons will improve the signal to noise ratio with respect to shot noise. Thus, determining what kind of noise is affecting one’s data, i.e. explaining features of the data themselves that are idiosyncratic to the particular instruments and conditions prevailing during a specific instance of data collection, can be critical to eventually generating a dataset that can be used to answer questions about phenomena of interest. In using data that require statistical analysis, it is particularly clear that “empirical assumptions about the factors influencing the measurement results may be used to motivate the assumption of a particular error distribution”, which can be crucial for justifying the application of methods of analysis (Woodward 2011, 173).

There are also circumstances in which scientists want to provide a substantive, detailed explanation for a particular idiosyncratic datum, and even circumstances in which procuring such explanations is epistemically imperative. Ignoring outliers without good epistemic reasons is just cherry-picking data, one of the canonical ‘questionable research practices.’ Allan Franklin has described Robert Millikan’s convenient exclusion of data he collected from observing the second oil drop in his experiments of April 16, 1912 (1986, 231). When Millikan initially recorded the data for this drop, his notebooks indicate that he was satisfied his apparatus was working properly and that the experiment was running well—he wrote “Publish” next to the data in his lab notebook. However, after he had later calculated the value for the fundamental electric charge that these data yielded, and found it aberrant with respect to the values he calculated using data collected from other good observing sessions, he changed his mind, writing “Won’t work” next to the calculation (ibid., see also Woodward 2010, 794). Millikan not only never published this result, he never published why he failed to publish it. When data are excluded from analysis, there ought to be some explanation justifying their omission over and above lack of agreement with the experimenters’ expectations. Precisely because they are outliers, some data require specific, detailed, idiosyncratic causal explanations. Indeed, it is often in virtue of those very explanations that outliers can be responsibly rejected. Some explanation of data rejected as ‘spurious’ is required. Otherwise, scientists risk biasing their own work.

Thus, while in transforming data as collected into something useful for learning about phenomena, scientists often account for features of the data such as different types of noise contributions, and sometimes even explain the odd outlying data point or artifact, they simply do not explain every individual teensy tiny causal contribution to the exact character of a data set or datum in full detail. This is because scientists can neither discover such causal minutia nor would their invocation be necessary for typical research questions. The fact that it may sometimes be important for scientists to provide detailed explanations of data, and not just claims about phenomena inferred from data, should not be confused with the dubious claim that scientists could ‘in principle’ detail every causal quirk that contributed to some data (Woodward 2010; 2011).

In view of all of this, together with the fact that a great many theoretical claims can only be tested directly against facts about phenomena, it behooves epistemologists to think about how data are used to answer questions about phenomena. Lacking space for a detailed discussion, the most this entry can do is to mention two main kinds of things investigators do in order to draw conclusions from data. The first is causal analysis carried out with or without the use of statistical techniques. The second is non-causal statistical analysis.

First, investigators must distinguish features of the data that are indicative of facts about the phenomenon of interest from those which can safely be ignored, and those which must be corrected for. Sometimes background knowledge makes this easy. Under normal circumstances investigators know that their thermometers are sensitive to temperature, and their pressure gauges, to pressure. An astronomer or a chemist who knows what spectrographic equipment does, and what she has applied it to will know what her data indicate. Sometimes it is less obvious. When Santiago Ramón y Cajal looked through his microscope at a thin slice of stained nerve tissue, he had to figure out which, if any, of the fibers he could see at one focal length connected to or extended from things he could see only at another focal length, or in another slice. Analogous considerations apply to quantitative data. It was easy for Katz to tell when his equipment was responding more to Hill’s footfalls on the stairs than to the electrical quantities it was set up to measure. It can be harder to tell whether an abrupt jump in the amplitude of a high frequency EEG oscillation was due to a feature of the subjects brain activity or an artifact of extraneous electrical activity in the laboratory or operating room where the measurements were made. The answers to questions about which features of numerical and non-numerical data are indicative of a phenomenon of interest typically depend at least in part on what is known about the causes that conspire to produce the data.

Statistical arguments are often used to deal with questions about the influence of epistemically relevant causal factors. For example, when it is known that similar data can be produced by factors that have nothing to do with the phenomenon of interest, Monte Carlo simulations, regression analyses of sample data, and a variety of other statistical techniques sometimes provide investigators with their best chance of deciding how seriously to take a putatively illuminating feature of their data.

But statistical techniques are also required for purposes other than causal analysis. To calculate the magnitude of a quantity like the melting point of lead from a scatter of numerical data, investigators throw out outliers, calculate the mean and the standard deviation, etc., and establish confidence and significance levels. Regression and other techniques are applied to the results to estimate how far from the mean the magnitude of interest can be expected to fall in the population of interest (e.g., the range of temperatures at which pure samples of lead can be expected to melt).

The fact that little can be learned from data without causal, statistical, and related argumentation has interesting consequences for received ideas about how the use of observational evidence distinguishes science from pseudoscience, religion, and other non-scientific cognitive endeavors. First, scientists are not the only ones who use observational evidence to support their claims; astrologers and medical quacks use them too. To find epistemically significant differences, one must carefully consider what sorts of data they use, where it comes from, and how it is employed. The virtues of scientific as opposed to non-scientific theory evaluations depend not only on its reliance on empirical data, but also on how the data are produced, analyzed and interpreted to draw conclusions against which theories can be evaluated. Secondly, it does not take many examples to refute the notion that adherence to a single, universally applicable ‘scientific method’ differentiates the sciences from the non-sciences. Data are produced, and used in far too many different ways to treat informatively as instances of any single method. Thirdly, it is usually, if not always, impossible for investigators to draw conclusions to test theories against observational data without explicit or implicit reliance on theoretical resources.

Bokulich (2020) has helpfully outlined a taxonomy of various ways in which data can be model-laden to increase their epistemic utility. She focuses on seven categories: data conversion, data correction, data interpolation, data scaling, data fusion, data assimilation, and synthetic data. Of these categories, conversion and correction are perhaps the most familiar. Bokulich reminds us that even in the case of reading a temperature from an ordinary mercury thermometer, we are ‘converting’ the data as measured, which in this case is the height of the column of mercury, to a temperature (ibid., 795). In more complicated cases, such as processing the arrival times of acoustic signals in seismic reflection measurements to yield values for subsurface depth, data conversion may involve models (ibid.). In this example, models of the composition and geometry of the subsurface are needed in order to account for differences in the speed of sound in different materials. Data ‘correction’ involves common practices we have already discussed like modeling and mathematically subtracting background noise contributions from one’s dataset (ibid., 796). Bokulich rightly points out that involving models in these ways routinely improves the epistemic uses to which data can be put. Data interpolation, scaling, and ‘fusion’ are also relatively widespread practices that deserve further philosophical analysis. Interpolation involves filling in missing data in a patchy data set, under the guidance of models. Data are scaled when they have been generated in a particular scale (temporal, spatial, energy) and modeling assumptions are recruited to transform them to apply at another scale. Data are ‘fused,’ in Bokulich’s terminology, when data collected in diverse contexts, using diverse methods are combined, or integrated together. For instance, when data from ice cores, tree rings, and the historical logbooks of sea captains are merged into a joint climate dataset. Scientists must take care in combining data of diverse provenance, and model new uncertainties arising from the very amalgamation of datasets (ibid., 800).

Bokulich contrasts ‘synthetic data’ with what she calls ‘real data’ (ibid., 801–802). Synthetic data are virtual, or simulated data, and are not produced by physical interaction with worldly research targets. Bokulich emphasizes the role that simulated data can usefully play in testing and troubleshooting aspects of data processing that are to eventually be deployed on empirical data (ibid., 802). It can be incredibly useful for developing and stress-testing a data processing pipeline to have fake datasets whose characteristics are already known in virtue of having been produced by the researchers, and being available for their inspection at will. When the characteristics of a dataset are known, or indeed can be tailored according to need, the effects of new processing methods can be more readily traced than without. In this way, researchers can familiarize themselves with the effects of a data processing pipeline, and make adjustments to that pipeline in light of what they learn by feeding fake data through it, before attempting to use that pipeline on actual science data. Such investigations can be critical to eventually arguing for the credibility of the final empirical results and their appropriate interpretation and use.

Data assimilation is perhaps a less widely appreciated aspect of model-based data processing among philosophers of science, excepting Parker (2016; 2017). Bokulich characterizes this method as “the optimal integration of data with dynamical model estimates to provide a more accurate ‘assimilation estimate’ of the quantity” (2020, 800). Thus, data assimilation involves balancing the contributions of empirical data and the output of models in an integrated estimate, according to the uncertainties associated with these contributions.

Bokulich argues that the involvement of models in these various aspects of data processing does not necessarily lead to better epistemic outcomes. Done wrong, integrating models and data can introduce artifacts and make the processed data unreliable for the purpose at hand (ibid., 804). Indeed, she notes that “[t]here is much work for methodologically reflective scientists and philosophers of science to do in string out cases in which model-data symbiosis may be problematic or circular” (ibid.)

3. Theory and value ladenness

Empirical results are laden with values and theoretical commitments. Philosophers have raised and appraised several possible kinds of epistemic problems that could be associated with theory and/or value-laden empirical results. They have worried about the extent to which human perception itself is distorted by our commitments. They have worried that drawing upon theoretical resources from the very theory to be appraised (or its competitors) in the generation of empirical results yields vicious circularity (or inconsistency). They have also worried that contingent conceptual and/or linguistic frameworks trap bits of evidence like bees in amber so that they cannot carry on their epistemic lives outside of the contexts of their origination, and that normative values necessarily corrupt the integrity of science. Do the theory and value-ladenness of empirical results render them hopelessly parochial? That is, when scientists leave theoretical commitments behind and adopt new ones, must they also relinquish the fruits of the empirical research imbued with their prior commitments too? In this section, we discuss these worries and responses that philosophers have offered to assuage them.

If you believe that observation by human sense perception is the objective basis of all scientific knowledge, then you ought to be particularly worried about the potential for human perception to be corrupted by theoretical assumptions, wishful thinking, framing effects, and so on. Daston and Galison recount the striking example of Arthur Worthington’s symmetrical milk drops (2007, 11–16). Working in 1875, Worthington investigated the hydrodynamics of falling fluid droplets and their evolution upon impacting a hard surface. At first, he had tried to carefully track the drop dynamics with a strobe light to burn a sequence of images into his own retinas. The images he drew to record what he saw were radially symmetric, with rays of the drop splashes emanating evenly from the center of the impact. However, when Worthington transitioned from using his eyes and capacity to draw from memory to using photography in 1894, he was shocked to find that the kind of splashes he had been observing were irregular splats (ibid., 13). Even curiouser, when Worthington returned to his drawings, he found that he had indeed recorded some unsymmetrical splashes. He had evidently dismissed them as uninformative accidents instead of regarding them as revelatory of the phenomenon he was intent on studying (ibid.) In attempting to document the ideal form of the splashes, a general and regular form, he had subconsciously down-played the irregularity of individual splashes. If theoretical commitments, like Worthington’s initial commitment to the perfect symmetry of the physics he was studying, pervasively and incorrigibly dictated the results of empirical inquiry, then the epistemic aims of science would be seriously undermined.

Perceptual psychologists, Bruner and Postman, found that subjects who were briefly shown anomalous playing cards, e.g., a black four of hearts, reported having seen their normal counterparts e.g., a red four of hearts. It took repeated exposures to get subjects to say the anomalous cards didn’t look right, and eventually, to describe them correctly (Kuhn 1962, 63). Kuhn took such studies to indicate that things don’t look the same to observers with different conceptual resources. (For a more up-to-date discussion of theory and conceptual perceptual loading see Lupyan 2015.) If so, black hearts didn’t look like black hearts until repeated exposures somehow allowed subjects to acquire the concept of a black heart. By analogy, Kuhn supposed, when observers working in conflicting paradigms look at the same thing, their conceptual limitations should keep them from having the same visual experiences (Kuhn 1962, 111, 113–114, 115, 120–1). This would mean, for example, that when Priestley and Lavoisier watched the same experiment, Lavoisier should have seen what accorded with his theory that combustion and respiration are oxidation processes, while Priestley’s visual experiences should have agreed with his theory that burning and respiration are processes of phlogiston release.

The example of Pettersson’s and Rutherford’s scintillation screen evidence (above) attests to the fact that observers working in different laboratories sometimes report seeing different things under similar conditions. It is plausible that their expectations influence their reports. It is plausible that their expectations are shaped by their training and by their supervisors’ and associates’ theory driven behavior. But as happens in other cases as well, all parties to the dispute agreed to reject Pettersson’s data by appealing to results that both laboratories could obtain and interpret in the same way without compromising their theoretical commitments. Indeed, it is possible for scientists to share empirical results, not just across diverse laboratory cultures, but even across serious differences in worldview. Much as they disagreed about the nature of respiration and combustion, Priestley and Lavoisier gave quantitatively similar reports of how long their mice stayed alive and their candles kept burning in closed bell jars. Priestley taught Lavoisier how to obtain what he took to be measurements of the phlogiston content of an unknown gas. A sample of the gas to be tested is run into a graduated tube filled with water and inverted over a water bath. After noting the height of the water remaining in the tube, the observer adds “nitrous air” (we call it nitric oxide) and checks the water level again. Priestley, who thought there was no such thing as oxygen, believed the change in water level indicated how much phlogiston the gas contained. Lavoisier reported observing the same water levels as Priestley even after he abandoned phlogiston theory and became convinced that changes in water level indicated free oxygen content (Conant 1957, 74–109).

A related issue is that of salience. Kuhn claimed that if Galileo and an Aristotelian physicist had watched the same pendulum experiment, they would not have looked at or attended to the same things. The Aristotelian’s paradigm would have required the experimenter to measure

… the weight of the stone, the vertical height to which it had been raised, and the time required for it to achieve rest (Kuhn 1962, 123)

and ignore radius, angular displacement, and time per swing (ibid., 124). These last were salient to Galileo because he treated pendulum swings as constrained circular motions. The Galilean quantities would be of no interest to an Aristotelian who treats the stone as falling under constraint toward the center of the earth (ibid., 123). Thus Galileo and the Aristotelian would not have collected the same data. (Absent records of Aristotelian pendulum experiments we can think of this as a thought experiment.)

Interests change, however. Scientists may eventually come to appreciate the significance of data that had not originally been salient to them in light of new presuppositions. The moral of these examples is that although paradigms or theoretical commitments sometimes have an epistemically significant influence on what observers perceive or what they attend to, it can be relatively easy to nullify or correct for their effects. When presuppositions cause epistemic damage, investigators are often able to eventually make corrections. Thus, paradigms and theoretical commitments actually do influence saliency, but their influence is neither inevitable nor irremediable.

Thomas Kuhn (1962), Norwood Hanson (1958), Paul Feyerabend (1959) and others cast suspicion on the objectivity of observational evidence in another way by arguing that one cannot use empirical evidence to test a theory without committing oneself to that very theory. This would be a problem if it leads to dogmatism but assuming the theory to be tested is often benign and even necessary.

For instance, Laymon (1988) demonstrates the manner in which the very theory that the Michelson-Morley experiments are considered to test is assumed in the experimental design, but that this does not engender deleterious epistemic effects (250). The Michelson-Morley apparatus consists of two interferometer arms at right angles to one another, which are rotated in the course of the experiment so that, on the original construal, the path length traversed by light in the apparatus would vary according to alignment with or against the Earth’s velocity (carrying the apparatus) with respect to the stationary aether. This difference in path length would show up as displacement in the interference fringes of light in the interferometer. Although Michelson’s intention had been to measure the velocity of the Earth with respect to the all-pervading aether, the experiments eventually came to be regarded as furnishing tests of the Fresnel aether theory itself. In particular, the null results of these experiments were taken as evidence against the existence of the aether. Naively, one might suppose that whatever assumptions were made in the calculation of the results of these experiments, it should not be the case that the theory under the gun was assumed nor that its negation was.

Before Michelson’s experiments, the Fresnel aether theory did not predict any sort of length contraction. Although Michelson assumed no contraction in the arms of the interferometer, Laymon argues that he could have assumed contraction, with no practical impact on the results of the experiments. The predicted fringe shift is calculated from the anticipated difference in the distance traveled by light in the two arms is the same, when higher order terms are neglected. Thus, in practice, the experimenters could assume either that the contraction thesis was true or that it was false when determining the length of the arms. Either way, the results of the experiment would be the same. After Michelson’s experiments returned no evidence of the anticipated aether effects, Lorentz-Fitzgerald contraction was postulated precisely to cancel out the expected (but not found) effects and save the aether theory. Morley and Miller then set out specifically to test the contraction thesis, and still assumed no contraction in determining the length of the arms of their interferometer (ibid., 253). Thus Laymon argues that the Michelson-Morley experiments speak against the tempting assumption that “appraisal of a theory is based on phenomena which can be detected and measured without using assumptions drawn from the theory under examination or from competitors to that theory ” (ibid., 246).

Epistemological hand-wringing about the use of the very theory to be tested in the generation of the evidence to be used for testing, seems to spring primarily from a concern about vicious circularity. How can we have a genuine trial, if the theory in question has been presumed innocent from the outset? While it is true that there would be a serious epistemic problem in a case where the use of the theory to be tested conspired to guarantee that the evidence would turn out to be confirmatory, this is not always the case when theories are invoked in their own testing. Woodward (2011) summarizes a tidy case:

For example, in Millikan’s oil drop experiment, the mere fact that theoretical assumptions (e.g., that the charge of the electron is quantized and that all electrons have the same charge) play a role in motivating his measurements or a vocabulary for describing his results does not by itself show that his design and data analysis were of such a character as to guarantee that he would obtain results supporting his theoretical assumptions. His experiment was such that he might well have obtained results showing that the charge of the electron was not quantized or that there was no single stable value for this quantity. (178)

For any given case, determining whether the theoretical assumptions being made are benign or straight-jacketing the results that it will be possible to obtain will require investigating the particular relationships between the assumptions and results in that case. When data production and analysis processes are complicated, this task can get difficult. But the point is that merely noting the involvement of the theory to be tested in the generation of empirical results does not by itself imply that those results cannot be objectively useful for deciding whether the theory to be tested should be accepted or rejected.

Kuhn argued that theoretical commitments exert a strong influence on observation descriptions, and what they are understood to mean (Kuhn 1962, 127ff; Longino 1979, 38–42). If so, proponents of a caloric account of heat won’t describe or understand descriptions of observed results of heat experiments in the same way as investigators who think of heat in terms of mean kinetic energy or radiation. They might all use the same words (e.g., ‘temperature’) to report an observation without understanding them in the same way. This poses a potential problem for communicating effectively across paradigms, and similarly, for attributing the appropriate significance to empirical results generated outside of one’s own linguistic framework.

It is important to bear in mind that observers do not always use declarative sentences to report observational and experimental results. Instead, they often draw, photograph, make audio recordings, etc. or set up their experimental devices to generate graphs, pictorial images, tables of numbers, and other non-sentential records. Obviously investigators’ conceptual resources and theoretical biases can exert epistemically significant influences on what they record (or set their equipment to record), which details they include or emphasize, and which forms of representation they choose (Daston and Galison 2007, 115–190, 309–361). But disagreements about the epistemic import of a graph, picture or other non-sentential bit of data often turn on causal rather than semantical considerations. Anatomists may have to decide whether a dark spot in a micrograph was caused by a staining artifact or by light reflected from an anatomically significant structure. Physicists may wonder whether a blip in a Geiger counter record reflects the causal influence of the radiation they wanted to monitor, or a surge in ambient radiation. Chemists may worry about the purity of samples used to obtain data. Such questions are not, and are not well represented as, semantic questions to which semantic theory loading is relevant. Late 20 th century philosophers may have ignored such cases and exaggerated the influence of semantic theory loading because they thought of theory testing in terms of inferential relations between observation and theoretical sentences.

Nevertheless, some empirical results are reported as declarative sentences. Looking at a patient with red spots and a fever, an investigator might report having seen the spots, or measles symptoms, or a patient with measles. Watching an unknown liquid dripping into a litmus solution an observer might report seeing a change in color, a liquid with a PH of less than 7, or an acid. The appropriateness of a description of a test outcome depends on how the relevant concepts are operationalized. What justifies an observer to report having observed a case of measles according to one operationalization might require her to say no more than that she had observed measles symptoms, or just red spots according to another.

In keeping with Percy Bridgman’s view that

… in general, we mean by a concept nothing more than a set of operations; the concept is synonymous with the corresponding sets of operations (Bridgman 1927, 5)

one might suppose that operationalizations are definitions or meaning rules such that it is analytically true, e.g., that every liquid that turns litmus red in a properly conducted test is acidic. But it is more faithful to actual scientific practice to think of operationalizations as defeasible rules for the application of a concept such that both the rules and their applications are subject to revision on the basis of new empirical or theoretical developments. So understood, to operationalize is to adopt verbal and related practices for the purpose of enabling scientists to do their work. Operationalizations are thus sensitive and subject to change on the basis of findings that influence their usefulness (Feest 2005).

Definitional or not, investigators in different research traditions may be trained to report their observations in conformity with conflicting operationalizations. Thus instead of training observers to describe what they see in a bubble chamber as a whitish streak or a trail, one might train them to say they see a particle track or even a particle. This may reflect what Kuhn meant by suggesting that some observers might be justified or even required to describe themselves as having seen oxygen, transparent and colorless though it is, or atoms, invisible though they are (Kuhn 1962, 127ff). To the contrary, one might object that what one sees should not be confused with what one is trained to say when one sees it, and therefore that talking about seeing a colorless gas or an invisible particle may be nothing more than a picturesque way of talking about what certain operationalizations entitle observers to say. Strictly speaking, the objection concludes, the term ‘observation report’ should be reserved for descriptions that are neutral with respect to conflicting operationalizations.

If observational data are just those utterances that meet Feyerabend’s decidability and agreeability conditions, the import of semantic theory loading depends upon how quickly, and for which sentences reasonably sophisticated language users who stand in different paradigms can non-inferentially reach the same decisions about what to assert or deny. Some would expect enough agreement to secure the objectivity of observational data. Others would not. Still others would try to supply different standards for objectivity.

With regard to sentential observation reports, the significance of semantic theory loading is less ubiquitous than one might expect. The interpretation of verbal reports often depends on ideas about causal structure rather than the meanings of signs. Rather than worrying about the meaning of words used to describe their observations, scientists are more likely to wonder whether the observers made up or withheld information, whether one or more details were artifacts of observation conditions, whether the specimens were atypical, and so on.

Note that the worry about semantic theory loading extends beyond observation reports of the sort that occupied the logical empiricists and their close intellectual descendents. Combining results of diverse methods for making proxy measurements of paleoclimate temperatures in an epistemically responsible way requires careful attention to the variety of operationalizations at play. Even if no ‘observation reports’ are involved, the sticky question about how to usefully merge results obtained in different ways in order to satisfy one’s epistemic aims remains. Happily, the remedy for the worry about semantic loading in this broader sense is likely to be the same—investigating the provenance of those results and comparing the variety of factors that have contributed to their causal production.

Kuhn placed too much emphasis on the discontinuity between evidence generated in different paradigms. Even if we accept a broadly Kuhnian picture, according to which paradigms are heterogeneous collections of experimental practices, theoretical principles, problems selected for investigation, approaches to their solution, etc., connections between components are loose enough to allow investigators who disagree profoundly over one or more theoretical claims to nevertheless agree about how to design, execute, and record the results of their experiments. That is why neuroscientists who disagreed about whether nerve impulses consisted of electrical currents could measure the same electrical quantities, and agree on the linguistic meaning and the accuracy of observation reports including such terms as ‘potential’, ‘resistance’, ‘voltage’ and ‘current’. As we discussed above, the success that scientists have in repurposing results generated by others for different purposes speaks against the confinement of evidence to its native paradigm. Even when scientists working with radically different core theoretical commitments cannot make the same measurements themselves, with enough contextual information about how each conducts research, it can be possible to construct bridges that span the theoretical divides.

One could worry that the intertwining of the theoretical and empirical would open the floodgates to bias in science. Human cognizing, both historical and present day, is replete with disturbing commitments including intolerance and narrow mindedness of many sorts. If such commitments are integral to a theoretical framework, or endemic to the reasoning of a scientist or scientific community, then they threaten to corrupt the epistemic utility of empirical results generated using their resources. The core impetus of the ‘value-free ideal’ is to maintain a safe distance between the appraisal of scientific theories according to the evidence on one hand, and the swarm of moral, political, social, and economic values on the other. While proponents of the value-free ideal might admit that the motivation to pursue a theory or the legal protection of human subjects in permissible experimental methods involve non-epistemic values, they would contend that such values ought not ought not enter into the constitution of empirical results themselves, nor the adjudication or justification of scientific theorizing in light of the evidence (see Intemann 2021, 202).

As a matter of fact, values do enter into science at a variety of stages. Above we saw that ‘theory-ladenness’ could refer to the involvement of theory in perception, in semantics, and in a kind of circularity that some have worried begets unfalsifiability and thereby dogmatism. Like theory-ladenness, values can and sometimes do affect judgments about the salience of certain evidence and the conceptual framing of data. Indeed, on a permissive construal of the nature of theories, values can simply be understood as part of a theoretical framework. Intemann (2021) highlights a striking example from medical research where key conceptual resources include notions like ‘harm,’ ‘risk,’ ‘health benefit,’ and ‘safety.’ She refers to research on the comparative safety of giving birth at home and giving birth at a hospital for low-risk parents in the United States. Studies reporting that home births are less safe typically attend to infant and birthing parent mortality rates—which are low for these subjects whether at home or in hospital—but leave out of consideration rates of c-section and episiotomy, which are both relatively high in hospital settings. Thus, a value-laden decision about whether a possible outcome counts as a harm worth considering can influence the outcome of the study—in this case tipping the balance towards the conclusion that hospital births are more safe (ibid., 206).

Note that the birth safety case differs from the sort of cases at issue in the philosophical debate about risk and thresholds for acceptance and rejection of hypotheses. In accepting an hypothesis, a person makes a judgement that the risk of being mistaken is sufficiently low (Rudner 1953). When the consequences of being wrong are deemed grave, the threshold for acceptance may be correspondingly high. Thus, in evaluating the epistemic status of an hypothesis in light of the evidence, a person may have to make a value-based judgement. However, in the birth safety case, the judgement comes into play at an earlier stage, well before the decision to accept or reject the hypothesis is to be made. The judgement occurs already in deciding what is to count as a ‘harm’ worth considering for the purposes of this research.

The fact that values do sometimes enter into scientific reasoning does not by itself settle the question of whether it would be better if they did not. In order to assess the normative proposal, philosophers of science have attempted to disambiguate the various ways in which values might be thought to enter into science, and the various referents that get crammed under the single heading of ‘values.’ Anderson (2004) articulates eight stages of scientific research where values (‘evaluative presuppositions’) might be employed in epistemically fruitful ways. In paraphrase: 1) orientation in a field, 2) framing a research question, 3) conceptualizing the target, 4) identifying relevant data, 5) data generation, 6) data analysis, 7) deciding when to cease data analysis, and 8) drawing conclusions (Anderson 2004, 11). Similarly, Intemann (2021) lays out five ways “that values play a role in scientific reasoning” with which feminist philosophers of science have engaged in particular:

(1) the framing [of] research problems, (2) observing phenomena and describing data, (3) reasoning about value-laden concepts and assessing risks, (4) adopting particular models, and (5) collecting and interpreting evidence. (208)

Ward (2021) presents a streamlined and general taxonomy of four ways in which values relate to choices: as reasons motivating or justifying choices, as causal effectors of choices, or as goods affected by choices. By investigating the role of values in these particular stages or aspects of research, philosophers of science can offer higher resolution insights than just the observation that values are involved in science at all and untangle crosstalk.

Similarly, fine points can be made about the nature of values involved in these various contexts. Such clarification is likely important for determining whether the contribution of certain values in a given context is deleterious or salutary, and in what sense. Douglas (2013) argues that the ‘value’ of internal consistency of a theory and of the empirical adequacy of a theory with respect to the available evidence are minimal criteria for any viable scientific theory (799–800). She contrasts these with the sort of values that Kuhn called ‘virtues,’ i.e. scope, simplicity, and explanatory power that are properties of theories themselves, and unification, novel prediction and precision, which are properties a theory has in relation to a body of evidence (800–801). These are the sort of values that may be relevant to explaining and justifying choices that scientists make to pursue/abandon or accept/reject particular theories. Moreover, Douglas (2000) argues that what she calls “non-epistemic values” (in particular, ethical value judgements) also enter into decisions at various stages “internal” to scientific reasoning, such as data collection and interpretation (565). Consider a laboratory toxicology study in which animals exposed to dioxins are compared to unexposed controls. Douglas discusses researchers who want to determine the threshold for safe exposure. Admitting false positives can be expected to lead to overregulation of the chemical industry, while false negatives yield underregulation and thus pose greater risk to public health. The decision about where to set the unsafe exposure threshold, that is, set the threshold for a statistically significant difference between experimental and control animal populations, involves balancing the acceptability of these two types of errors. According to Douglas, this balancing act will depend on “whether we are more concerned about protecting public health from dioxin pollution or whether we are more concerned about protecting industries that produce dioxins from increased regulation” (ibid., 568). That scientists do as a matter of fact sometimes make such decisions is clear. They judge, for instance, a specimen slide of a rat liver to be tumorous or not, and whether borderline cases should count as benign or malignant (ibid., 569–572). Moreover, in such cases, it is not clear that the responsibility of making such decisions could be offloaded to non-scientists.

Many philosophers accept that values can contribute to the generation of empirical results without spoiling their epistemic utility. Anderson’s (2004) diagnosis is as follows:

Deep down, what the objectors find worrisome about allowing value judgments to guide scientific inquiry is not that they have evaluative content, but that these judgments might be held dogmatically, so as to preclude the recognition of evidence that might undermine them. We need to ensure that value judgements do not operate to drive inquiry to a predetermined conclusion. This is our fundamental criterion for distinguishing legitimate from illegitimate uses of values in science. (11)

Data production (including experimental design and execution) is heavily influenced by investigators’ background assumptions. Sometimes these include theoretical commitments that lead experimentalists to produce non-illuminating or misleading evidence. In other cases they may lead experimentalists to ignore, or even fail to produce useful evidence. For example, in order to obtain data on orgasms in female stumptail macaques, one researcher wired up females to produce radio records of orgasmic muscle contractions, heart rate increases, etc. But as Elisabeth Lloyd reports, “… the researcher … wired up the heart rate of the male macaques as the signal to start recording the female orgasms. When I pointed out that the vast majority of female stumptail orgasms occurred during sex among the females alone, he replied that yes he knew that, but he was only interested in important orgasms” (Lloyd 1993, 142). Although female stumptail orgasms occurring during sex with males are atypical, the experimental design was driven by the assumption that what makes features of female sexuality worth studying is their contribution to reproduction (ibid., 139). This assumption influenced experimental design in such a way as to preclude learning about the full range of female stumptail orgasms.

Anderson (2004) presents an influential analysis of the role of values in research on divorce. Researchers committed to an interpretive framework rooted in ‘traditional family values’ could conduct research on the assumption that divorce is mostly bad for spouses and any children that they have (ibid., 12). This background assumption, which is rooted in a normative appraisal of a certain model of good family life, could lead social science researchers to restrict the questions with which they survey their research subjects to ones about the negative impacts of divorce on their lives, thereby curtailing the possibility of discovering ways that divorce may have actually made the ex-spouses lives better (ibid., 13). This is an example of the influence that values can have on the nature of the results that research ultimately yields, which is epistemically detrimental. In this case, the values in play biased the research outcomes to preclude recognition of countervailing evidence. Anderson argues that the problematic influence of values comes when research “is rigged in advance” to confirm certain hypotheses—when the influence of values amounts to incorrigible dogmatism (ibid., 19). “Dogmatism” in her sense is unfalsifiability in practice, “their stubbornness in the face of any conceivable evidence”(ibid., 22).

Fortunately, such dogmatism is not ubiquitous and when it occurs it can often be corrected eventually. Above we noted that the mere involvement of the theory to be tested in the generation of an empirical result does not automatically yield vicious circularity—it depends on how the theory is involved. Furthermore, even if the assumptions initially made in the generation of empirical results are incorrect, future scientists will have opportunities to reassess those assumptions in light of new information and techniques. Thus, as long as scientists continue their work there need be no time at which the epistemic value of an empirical result can be established once and for all. This should come as no surprise to anyone who is aware that science is fallible, but it is no grounds for skepticism. It can be perfectly reasonable to trust the evidence available at present even though it is logically possible for epistemic troubles to arise in the future. A similar point can be made regarding values (although cf. Yap 2016).

Moreover, while the inclusion of values in the generation of an empirical result can sometimes be epistemically bad, values properly deployed can also be harmless, or even epistemically helpful. As in the cases of research on female stumptail macaque orgasms and the effects of divorce, certain values can sometimes serve to illuminate the way in which other epistemically problematic assumptions have hindered potential scientific insight. By valuing knowledge about female sexuality beyond its role in reproduction, scientists can recognize the narrowness of an approach that only conceives of female sexuality insofar as it relates to reproduction. By questioning the absolute value of one traditional ideal for flourishing families, researchers can garner evidence that might end up destabilizing the empirical foundation supporting that ideal.

Empirical results are most obviously put to epistemic work in their contexts of origin. Scientists conceive of empirical research, collect and analyze the relevant data, and then bring the results to bear on the theoretical issues that inspired the research in the first place. However, philosophers have also discussed ways in which empirical results are transferred out of their native contexts and applied in diverse and sometimes unexpected ways (see Leonelli and Tempini 2020). Cases of reuse, or repurposing of empirical results in different epistemic contexts raise several interesting issues for philosophers of science. For one, such cases challenge the assumption that theory (and value) ladenness confines the epistemic utility of empirical results to a particular conceptual framework. Ancient Babylonian eclipse records inscribed on cuneiform tablets have been used to generate constraints on contemporary geophysical theorizing about the causes of the lengthening of the day on Earth (Stephenson, Morrison, and Hohenkerk 2016). This is surprising since the ancient observations were originally recorded for the purpose of making astrological prognostications. Nevertheless, with enough background information, the records as inscribed can be translated, the layers of assumptions baked into their presentation peeled back, and the results repurposed using resources of the contemporary epistemic context, the likes of which the Babylonians could have hardly dreamed.

Furthermore, the potential for reuse and repurposing feeds back on the methodological norms of data production and handling. In light of the difficulty of reusing or repurposing data without sufficient background information about the original context, Goodman et al. (2014) note that “data reuse is most possible when: 1) data; 2) metadata (information describing the data); and 3) information about the process of generating those data, such as code, all all provided” (3). Indeed, they advocate for sharing data and code in addition to results customarily published in science. As we have seen, the loading of data with theory is usually necessary to putting that data to any serious epistemic use—theory-loading makes theory appraisal possible. Philosophers have begun to appreciate that this epistemic boon does not necessarily come at the cost of rendering data “tragically local” (Wylie 2020, 285, quoting Latour 1999). But it is important to note the useful travel of data between contexts is significantly aided by foresight, curation, and management for that aim.

In light of the mediated nature of empirical results, Boyd (2018) argues for an “enriched view of evidence,” in which the evidence that serves as the ‘tribunal of experience’ is understood to be “lines of evidence” composed of the products of data collection and all of the products of their transformation on the way to the generation of empirical results that are ultimately compared to theoretical predictions, considered together with metadata associated with their provenance. Such metadata includes information about theoretical assumptions that are made in data collection, processing, and the presentation of empirical results. Boyd argues that by appealing to metadata to ‘rewind’ the processing of assumption-imbued empirical results and then by re-processing them using new resources, the epistemic utility of empirical evidence can survive transitions to new contexts. Thus, the enriched view of evidence supports the idea that it is not despite the intertwining of the theoretical and empirical that scientists accomplish key epistemic aims, but often in virtue of it (ibid., 420). In addition, it makes the epistemic value of metadata encoding the various assumptions that have been made throughout the course of data collection and processing explicit.

The desirability of explicitly furnishing empirical data and results with auxiliary information that allow them to travel can be appreciated in light of the ‘objectivity’ norm, construed as accessibility to interpersonal scrutiny. When data are repurposed in novel contexts, they are not only shared between subjects, but can in some cases be shared across radically different paradigms with incompatible theoretical commitments.

4. The epistemic value of empirical evidence

One of the important applications of empirical evidence is its use in assessing the epistemic status of scientific theories. In this section we briefly discuss philosophical work on the role of empirical evidence in confirmation/falsification of scientific theories, ‘saving the phenomena,’ and in appraising the empirical adequacy of theories. However, further philosophical work ought to explore the variety of ways that empirical results bear on the epistemic status of theories and theorizing in scientific practice beyond these.

It is natural to think that computability, range of application, and other things being equal, true theories are better than false ones, good approximations are better than bad ones, and highly probable theoretical claims are better than less probable ones. One way to decide whether a theory or a theoretical claim is true, close to the truth, or acceptably probable is to derive predictions from it and use empirical data to evaluate them. Hypothetico-Deductive (HD) confirmation theorists proposed that empirical evidence argues for the truth of theories whose deductive consequences it verifies, and against those whose consequences it falsifies (Popper 1959, 32–34). But laws and theoretical generalization seldom if ever entail observational predictions unless they are conjoined with one or more auxiliary hypotheses taken from the theory they belong to. When the prediction turns out to be false, HD has trouble explaining which of the conjuncts is to blame. If a theory entails a true prediction, it will continue to do so in conjunction with arbitrarily selected irrelevant claims. HD has trouble explaining why the prediction does not confirm the irrelevancies along with the theory of interest.

Another approach to confirmation by empirical evidence is Inference to the Best Explanation (IBE). The idea is roughly that an explanation of the evidence that exhibits certain desirable characteristics with respect to a family of candidate explanations is likely to be the true on (Lipton 1991). On this approach, it is in virtue of their successful explanation of the empirical evidence that theoretical claims are supported. Naturally, IBE advocates face the challenges of defending a suitable characterization of what counts as the ‘best’ and of justifying the limited pool of candidate explanations considered (Stanford 2006).

Bayesian approaches to scientific confirmation have garnered significant attention and are now widespread in philosophy of science. Bayesians hold that the evidential bearing of empirical evidence on a theoretical claim is to be understood in terms of likelihood or conditional probability. For example, whether empirical evidence argues for a theoretical claim might be thought to depend upon whether it is more probable (and if so how much more probable) than its denial conditional on a description of the evidence together with background beliefs, including theoretical commitments. But by Bayes’ Theorem, the posterior probability of the claim of interest (that is, its probability given the evidence) is proportional to that claim’s prior probability. How to justify the choice of these prior probability assignments is one of the most notorious points of contention arising for Bayesians. If one makes the assignment of priors a subjective matter decided by epistemic agents, then it is not clear that they can be justified. Once again, one’s use of evidence to evaluate a theory depends in part upon one’s theoretical commitments (Earman 1992, 33–86; Roush 2005, 149–186). If one instead appeals to chains of successive updating using Bayes’ Theorem based on past evidence, one has to invoke assumptions that generally do not obtain in actual scientific reasoning. For instance, to ‘wash out’ the influence of priors a limit theorem is invoked wherein we consider very many updating iterations, but much scientific reasoning of interest does not happen in the limit, and so in practice priors hold unjustified sway (Norton 2021, 33).

Rather than attempting to cast all instances of confirmation based on empirical evidence as belonging to a universal schema, a better approach may be to ‘go local’. Norton’s material theory of induction argues that inductive support arises from background knowledge, that is, from material facts that are domain specific. Norton argues that, for instance, the induction from “Some samples of the element bismuth melt at 271°C” to “all samples of the element bismuth melt at 271°C” is admissible not in virtue of some universal schema that carries us from ‘some’ to ‘all’ but matters of fact (Norton 2003). In this particular case, the fact that licenses the induction is a fact about elements: “their samples are generally uniform in their physical properties” (ibid., 650). This is a fact pertinent to chemical elements, but not to samples of material like wax (ibid.). Thus Norton repeatedly emphasizes that “all induction is local”.

Still, there are those who may be skeptical about the very possibility of confirmation or of successful induction. Insofar as the bearing of evidence on theory is never totally decisive, insofar there is no single trusty universal schema that captures empirical support, perhaps the relationship between empirical evidence and scientific theory is not really about support after all. Giving up on empirical support would not automatically mean abandoning any epistemic value for empirical evidence. Rather than confirm theory, the epistemic role of evidence could be to constrain, for example by furnishing phenomena for theory to systematize or to adequately model.

Theories are said to ‘save’ observable phenomena if they satisfactorily predict, describe, or systematize them. How well a theory performs any of these tasks need not depend upon the truth or accuracy of its basic principles. Thus according to Osiander’s preface to Copernicus’ On the Revolutions , a locus classicus, astronomers “… cannot in any way attain to true causes” of the regularities among observable astronomical events, and must content themselves with saving the phenomena in the sense of using

… whatever suppositions enable … [them] to be computed correctly from the principles of geometry for the future as well as the past … (Osiander 1543, XX)

Theorists are to use those assumptions as calculating tools without committing themselves to their truth. In particular, the assumption that the planets revolve around the sun must be evaluated solely in terms of how useful it is in calculating their observable relative positions to a satisfactory approximation. Pierre Duhem’s Aim and Structure of Physical Theory articulates a related conception. For Duhem a physical theory

… is a system of mathematical propositions, deduced from a small number of principles, which aim to represent as simply and completely, and exactly as possible, a set of experimental laws. (Duhem 1906, 19)

‘Experimental laws’ are general, mathematical descriptions of observable experimental results. Investigators produce them by performing measuring and other experimental operations and assigning symbols to perceptible results according to pre-established operational definitions (Duhem 1906, 19). For Duhem, the main function of a physical theory is to help us store and retrieve information about observables we would not otherwise be able to keep track of. If that is what a theory is supposed to accomplish, its main virtue should be intellectual economy. Theorists are to replace reports of individual observations with experimental laws and devise higher level laws (the fewer, the better) from which experimental laws (the more, the better) can be mathematically derived (Duhem 1906, 21ff).

A theory’s experimental laws can be tested for accuracy and comprehensiveness by comparing them to observational data. Let EL be one or more experimental laws that perform acceptably well on such tests. Higher level laws can then be evaluated on the basis of how well they integrate EL into the rest of the theory. Some data that don’t fit integrated experimental laws won’t be interesting enough to worry about. Other data may need to be accommodated by replacing or modifying one or more experimental laws or adding new ones. If the required additions, modifications or replacements deliver experimental laws that are harder to integrate, the data count against the theory. If the required changes are conducive to improved systematization the data count in favor of it. If the required changes make no difference, the data don’t argue for or against the theory.

On van Fraassen’s (1980) semantic account, a theory is empirically adequate when the empirical structure of at least one model of that theory is isomorphic to what he calls the “appearances” (45). In other words, when the theory “has at least one model that all the actual phenomena fit inside” (12). Thus, for van Fraassen, we continually check the empirical adequacy of our theories by seeing if they have the structural resources to accommodate new observations. We’ll never know that a given theory is totally empirically adequate, since for van Fraassen, empirical adequacy obtains with respect to all that is observable in principle to creatures like us, not all that has already been observed (69).

The primary appeal of dealing in empirical adequacy rather than confirmation is its appropriate epistemic humility. Instead of claiming that confirming evidence justifies belief (or boosted confidence) that a theory is true, one is restricted to saying that the theory continues to be consistent with the evidence as far as we can tell so far. However, if the epistemic utility of empirical results in appraising the status of theories is just to judge their empirical adequacy, then it may be difficult to account for the difference between adequate but unrealistic theories, and those equally adequate theories that ought to be taken seriously as representations. Appealing to extra-empirical virtues like parsimony may be a way out, but one that will not appeal to philosophers skeptical of the connection thereby supposed between such virtues and representational fidelity.

On an earlier way of thinking, observation was to serve as the unmediated foundation of science—direct access to the facts upon which the edifice of scientific knowledge could be built. When conflict arose between factions with different ideological commitments, observations could furnish the material for neutral arbitration and settle the matter objectively, in virtue of being independent of non-empirical commitments. According to this view, scientists working in different paradigms could at least appeal to the same observations, and propagandists could be held accountable to the publicly accessible content of theory and value-free observations. Despite their different theories, Priestley and Lavoisier could find shared ground in the observations. Anti-Semites would be compelled to admit the success of a theory authored by a Jewish physicist, in virtue of the unassailable facts revealed by observation.

This version of empiricism with respect to science does not accord well with the fact that observation per se plays a relatively small role in many actual scientific methodologies, and the fact that even the most ‘raw’ data is often already theoretically imbued. The strict contrast between theory and observation in science is more fruitfully supplanted by inquiry into the relationship between theorizing and empirical results.

Contemporary philosophers of science tend to embrace the theory ladenness of empirical results. Instead of seeing the integration of the theoretical and the empirical as an impediment to furthering scientific knowledge, they see it as necessary. A ‘view from nowhere’ would not bear on our particular theories. That is, it is impossible to put empirical results to use without recruiting some theoretical resources. In order to use an empirical result to constrain or test a theory it has to be processed into a form that can be compared to that theory. To get stellar spectrograms to bear on Newtonian or relativistic cosmology, they need to be processed—into galactic rotation curves, say. The spectrograms by themselves are just artifacts, pieces of paper. Scientists need theoretical resources in order to even identify that such artifacts bear information relevant for their purposes, and certainly to put them to any epistemic use in assessing theories.

This outlook does not render contemporary philosophers of science all constructivists, however. Theory mediates the connection between the target of inquiry and the scientific worldview, it does not sever it. Moreover, vigilance is still required to ensure that the particular ways in which theory is ‘involved’ in the production of empirical results are not epistemically detrimental. Theory can be deployed in experiment design, data processing, and presentation of results in unproductive ways, for instance, in determining whether the results will speak for or against a particular theory regardless of what the world is like. Critical appraisal of the roles of theory is thus important for genuine learning about nature through science. Indeed, it seems that extra-empirical values can sometimes assist such critical appraisal. Instead of viewing observation as the theory-free and for that reason furnishing the content with which to appraise theories, we might attend to the choices and mistakes that can be made in collecting and generating empirical results with the help of theoretical resources, and endeavor to make choices conducive to learning and correct mistakes as we discover them.

Recognizing the involvement of theory and values in the constitution and generation of empirical results does not undermine the special epistemic value of empirical science in contrast to propaganda and pseudoscience. In cases where the influence of cultural, political, and religious values hinder scientific inquiry, it is often the case that they do so by limiting or determining the nature of the empirical results. Yet, by working to make the assumptions that shape results explicit we can examine their suitability for our purposes and attempt to restructure inquiry as necessary. When disagreements arise, scientists can attempt to settle them by appealing to the causal connections between the research target and the empirical data. The tribunal of experience speaks through empirical results, but it only does so through via careful fashioning with theoretical resources.

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• v.21(3); Fall 2022

Literature Reviews, Theoretical Frameworks, and Conceptual Frameworks: An Introduction for New Biology Education Researchers

Julie a. luft.

† Department of Mathematics, Social Studies, and Science Education, Mary Frances Early College of Education, University of Georgia, Athens, GA 30602-7124

Sophia Jeong

‡ Department of Teaching & Learning, College of Education & Human Ecology, Ohio State University, Columbus, OH 43210

Robert Idsardi

§ Department of Biology, Eastern Washington University, Cheney, WA 99004

Grant Gardner

∥ Department of Biology, Middle Tennessee State University, Murfreesboro, TN 37132

Associated Data

To frame their work, biology education researchers need to consider the role of literature reviews, theoretical frameworks, and conceptual frameworks as critical elements of the research and writing process. However, these elements can be confusing for scholars new to education research. This Research Methods article is designed to provide an overview of each of these elements and delineate the purpose of each in the educational research process. We describe what biology education researchers should consider as they conduct literature reviews, identify theoretical frameworks, and construct conceptual frameworks. Clarifying these different components of educational research studies can be helpful to new biology education researchers and the biology education research community at large in situating their work in the broader scholarly literature.

INTRODUCTION

Discipline-based education research (DBER) involves the purposeful and situated study of teaching and learning in specific disciplinary areas ( Singer et al. , 2012 ). Studies in DBER are guided by research questions that reflect disciplines’ priorities and worldviews. Researchers can use quantitative data, qualitative data, or both to answer these research questions through a variety of methodological traditions. Across all methodologies, there are different methods associated with planning and conducting educational research studies that include the use of surveys, interviews, observations, artifacts, or instruments. Ensuring the coherence of these elements to the discipline’s perspective also involves situating the work in the broader scholarly literature. The tools for doing this include literature reviews, theoretical frameworks, and conceptual frameworks. However, the purpose and function of each of these elements is often confusing to new education researchers. The goal of this article is to introduce new biology education researchers to these three important elements important in DBER scholarship and the broader educational literature.

The first element we discuss is a review of research (literature reviews), which highlights the need for a specific research question, study problem, or topic of investigation. Literature reviews situate the relevance of the study within a topic and a field. The process may seem familiar to science researchers entering DBER fields, but new researchers may still struggle in conducting the review. Booth et al. (2016b) highlight some of the challenges novice education researchers face when conducting a review of literature. They point out that novice researchers struggle in deciding how to focus the review, determining the scope of articles needed in the review, and knowing how to be critical of the articles in the review. Overcoming these challenges (and others) can help novice researchers construct a sound literature review that can inform the design of the study and help ensure the work makes a contribution to the field.

The second and third highlighted elements are theoretical and conceptual frameworks. These guide biology education research (BER) studies, and may be less familiar to science researchers. These elements are important in shaping the construction of new knowledge. Theoretical frameworks offer a way to explain and interpret the studied phenomenon, while conceptual frameworks clarify assumptions about the studied phenomenon. Despite the importance of these constructs in educational research, biology educational researchers have noted the limited use of theoretical or conceptual frameworks in published work ( DeHaan, 2011 ; Dirks, 2011 ; Lo et al. , 2019 ). In reviewing articles published in CBE—Life Sciences Education ( LSE ) between 2015 and 2019, we found that fewer than 25% of the research articles had a theoretical or conceptual framework (see the Supplemental Information), and at times there was an inconsistent use of theoretical and conceptual frameworks. Clearly, these frameworks are challenging for published biology education researchers, which suggests the importance of providing some initial guidance to new biology education researchers.

Fortunately, educational researchers have increased their explicit use of these frameworks over time, and this is influencing educational research in science, technology, engineering, and mathematics (STEM) fields. For instance, a quick search for theoretical or conceptual frameworks in the abstracts of articles in Educational Research Complete (a common database for educational research) in STEM fields demonstrates a dramatic change over the last 20 years: from only 778 articles published between 2000 and 2010 to 5703 articles published between 2010 and 2020, a more than sevenfold increase. Greater recognition of the importance of these frameworks is contributing to DBER authors being more explicit about such frameworks in their studies.

Collectively, literature reviews, theoretical frameworks, and conceptual frameworks work to guide methodological decisions and the elucidation of important findings. Each offers a different perspective on the problem of study and is an essential element in all forms of educational research. As new researchers seek to learn about these elements, they will find different resources, a variety of perspectives, and many suggestions about the construction and use of these elements. The wide range of available information can overwhelm the new researcher who just wants to learn the distinction between these elements or how to craft them adequately.

Our goal in writing this paper is not to offer specific advice about how to write these sections in scholarly work. Instead, we wanted to introduce these elements to those who are new to BER and who are interested in better distinguishing one from the other. In this paper, we share the purpose of each element in BER scholarship, along with important points on its construction. We also provide references for additional resources that may be beneficial to better understanding each element. Table 1 summarizes the key distinctions among these elements.

Comparison of literature reviews, theoretical frameworks, and conceptual reviews

This article is written for the new biology education researcher who is just learning about these different elements or for scientists looking to become more involved in BER. It is a result of our own work as science education and biology education researchers, whether as graduate students and postdoctoral scholars or newly hired and established faculty members. This is the article we wish had been available as we started to learn about these elements or discussed them with new educational researchers in biology.

LITERATURE REVIEWS

Purpose of a literature review.

A literature review is foundational to any research study in education or science. In education, a well-conceptualized and well-executed review provides a summary of the research that has already been done on a specific topic and identifies questions that remain to be answered, thus illustrating the current research project’s potential contribution to the field and the reasoning behind the methodological approach selected for the study ( Maxwell, 2012 ). BER is an evolving disciplinary area that is redefining areas of conceptual emphasis as well as orientations toward teaching and learning (e.g., Labov et al. , 2010 ; American Association for the Advancement of Science, 2011 ; Nehm, 2019 ). As a result, building comprehensive, critical, purposeful, and concise literature reviews can be a challenge for new biology education researchers.

Building Literature Reviews

There are different ways to approach and construct a literature review. Booth et al. (2016a) provide an overview that includes, for example, scoping reviews, which are focused only on notable studies and use a basic method of analysis, and integrative reviews, which are the result of exhaustive literature searches across different genres. Underlying each of these different review processes are attention to the s earch process, a ppraisa l of articles, s ynthesis of the literature, and a nalysis: SALSA ( Booth et al. , 2016a ). This useful acronym can help the researcher focus on the process while building a specific type of review.

However, new educational researchers often have questions about literature reviews that are foundational to SALSA or other approaches. Common questions concern determining which literature pertains to the topic of study or the role of the literature review in the design of the study. This section addresses such questions broadly while providing general guidance for writing a narrative literature review that evaluates the most pertinent studies.

The literature review process should begin before the research is conducted. As Boote and Beile (2005 , p. 3) suggested, researchers should be “scholars before researchers.” They point out that having a good working knowledge of the proposed topic helps illuminate avenues of study. Some subject areas have a deep body of work to read and reflect upon, providing a strong foundation for developing the research question(s). For instance, the teaching and learning of evolution is an area of long-standing interest in the BER community, generating many studies (e.g., Perry et al. , 2008 ; Barnes and Brownell, 2016 ) and reviews of research (e.g., Sickel and Friedrichsen, 2013 ; Ziadie and Andrews, 2018 ). Emerging areas of BER include the affective domain, issues of transfer, and metacognition ( Singer et al. , 2012 ). Many studies in these areas are transdisciplinary and not always specific to biology education (e.g., Rodrigo-Peiris et al. , 2018 ; Kolpikova et al. , 2019 ). These newer areas may require reading outside BER; fortunately, summaries of some of these topics can be found in the Current Insights section of the LSE website.

In focusing on a specific problem within a broader research strand, a new researcher will likely need to examine research outside BER. Depending upon the area of study, the expanded reading list might involve a mix of BER, DBER, and educational research studies. Determining the scope of the reading is not always straightforward. A simple way to focus one’s reading is to create a “summary phrase” or “research nugget,” which is a very brief descriptive statement about the study. It should focus on the essence of the study, for example, “first-year nonmajor students’ understanding of evolution,” “metacognitive prompts to enhance learning during biochemistry,” or “instructors’ inquiry-based instructional practices after professional development programming.” This type of phrase should help a new researcher identify two or more areas to review that pertain to the study. Focusing on recent research in the last 5 years is a good first step. Additional studies can be identified by reading relevant works referenced in those articles. It is also important to read seminal studies that are more than 5 years old. Reading a range of studies should give the researcher the necessary command of the subject in order to suggest a research question.

Given that the research question(s) arise from the literature review, the review should also substantiate the selected methodological approach. The review and research question(s) guide the researcher in determining how to collect and analyze data. Often the methodological approach used in a study is selected to contribute knowledge that expands upon what has been published previously about the topic (see Institute of Education Sciences and National Science Foundation, 2013 ). An emerging topic of study may need an exploratory approach that allows for a description of the phenomenon and development of a potential theory. This could, but not necessarily, require a methodological approach that uses interviews, observations, surveys, or other instruments. An extensively studied topic may call for the additional understanding of specific factors or variables; this type of study would be well suited to a verification or a causal research design. These could entail a methodological approach that uses valid and reliable instruments, observations, or interviews to determine an effect in the studied event. In either of these examples, the researcher(s) may use a qualitative, quantitative, or mixed methods methodological approach.

Even with a good research question, there is still more reading to be done. The complexity and focus of the research question dictates the depth and breadth of the literature to be examined. Questions that connect multiple topics can require broad literature reviews. For instance, a study that explores the impact of a biology faculty learning community on the inquiry instruction of faculty could have the following review areas: learning communities among biology faculty, inquiry instruction among biology faculty, and inquiry instruction among biology faculty as a result of professional learning. Biology education researchers need to consider whether their literature review requires studies from different disciplines within or outside DBER. For the example given, it would be fruitful to look at research focused on learning communities with faculty in STEM fields or in general education fields that result in instructional change. It is important not to be too narrow or too broad when reading. When the conclusions of articles start to sound similar or no new insights are gained, the researcher likely has a good foundation for a literature review. This level of reading should allow the researcher to demonstrate a mastery in understanding the researched topic, explain the suitability of the proposed research approach, and point to the need for the refined research question(s).

The literature review should include the researcher’s evaluation and critique of the selected studies. A researcher may have a large collection of studies, but not all of the studies will follow standards important in the reporting of empirical work in the social sciences. The American Educational Research Association ( Duran et al. , 2006 ), for example, offers a general discussion about standards for such work: an adequate review of research informing the study, the existence of sound and appropriate data collection and analysis methods, and appropriate conclusions that do not overstep or underexplore the analyzed data. The Institute of Education Sciences and National Science Foundation (2013) also offer Common Guidelines for Education Research and Development that can be used to evaluate collected studies.

Because not all journals adhere to such standards, it is important that a researcher review each study to determine the quality of published research, per the guidelines suggested earlier. In some instances, the research may be fatally flawed. Examples of such flaws include data that do not pertain to the question, a lack of discussion about the data collection, poorly constructed instruments, or an inadequate analysis. These types of errors result in studies that are incomplete, error-laden, or inaccurate and should be excluded from the review. Most studies have limitations, and the author(s) often make them explicit. For instance, there may be an instructor effect, recognized bias in the analysis, or issues with the sample population. Limitations are usually addressed by the research team in some way to ensure a sound and acceptable research process. Occasionally, the limitations associated with the study can be significant and not addressed adequately, which leaves a consequential decision in the hands of the researcher. Providing critiques of studies in the literature review process gives the reader confidence that the researcher has carefully examined relevant work in preparation for the study and, ultimately, the manuscript.

A solid literature review clearly anchors the proposed study in the field and connects the research question(s), the methodological approach, and the discussion. Reviewing extant research leads to research questions that will contribute to what is known in the field. By summarizing what is known, the literature review points to what needs to be known, which in turn guides decisions about methodology. Finally, notable findings of the new study are discussed in reference to those described in the literature review.

Within published BER studies, literature reviews can be placed in different locations in an article. When included in the introductory section of the study, the first few paragraphs of the manuscript set the stage, with the literature review following the opening paragraphs. Cooper et al. (2019) illustrate this approach in their study of course-based undergraduate research experiences (CUREs). An introduction discussing the potential of CURES is followed by an analysis of the existing literature relevant to the design of CUREs that allows for novel student discoveries. Within this review, the authors point out contradictory findings among research on novel student discoveries. This clarifies the need for their study, which is described and highlighted through specific research aims.

A literature reviews can also make up a separate section in a paper. For example, the introduction to Todd et al. (2019) illustrates the need for their research topic by highlighting the potential of learning progressions (LPs) and suggesting that LPs may help mitigate learning loss in genetics. At the end of the introduction, the authors state their specific research questions. The review of literature following this opening section comprises two subsections. One focuses on learning loss in general and examines a variety of studies and meta-analyses from the disciplines of medical education, mathematics, and reading. The second section focuses specifically on LPs in genetics and highlights student learning in the midst of LPs. These separate reviews provide insights into the stated research question.

Suggestions and Advice

A well-conceptualized, comprehensive, and critical literature review reveals the understanding of the topic that the researcher brings to the study. Literature reviews should not be so big that there is no clear area of focus; nor should they be so narrow that no real research question arises. The task for a researcher is to craft an efficient literature review that offers a critical analysis of published work, articulates the need for the study, guides the methodological approach to the topic of study, and provides an adequate foundation for the discussion of the findings.

In our own writing of literature reviews, there are often many drafts. An early draft may seem well suited to the study because the need for and approach to the study are well described. However, as the results of the study are analyzed and findings begin to emerge, the existing literature review may be inadequate and need revision. The need for an expanded discussion about the research area can result in the inclusion of new studies that support the explanation of a potential finding. The literature review may also prove to be too broad. Refocusing on a specific area allows for more contemplation of a finding.

It should be noted that there are different types of literature reviews, and many books and articles have been written about the different ways to embark on these types of reviews. Among these different resources, the following may be helpful in considering how to refine the review process for scholarly journals:

• Booth, A., Sutton, A., & Papaioannou, D. (2016a). Systemic approaches to a successful literature review (2nd ed.). Los Angeles, CA: Sage. This book addresses different types of literature reviews and offers important suggestions pertaining to defining the scope of the literature review and assessing extant studies.
• Booth, W. C., Colomb, G. G., Williams, J. M., Bizup, J., & Fitzgerald, W. T. (2016b). The craft of research (4th ed.). Chicago: University of Chicago Press. This book can help the novice consider how to make the case for an area of study. While this book is not specifically about literature reviews, it offers suggestions about making the case for your study.
• Galvan, J. L., & Galvan, M. C. (2017). Writing literature reviews: A guide for students of the social and behavioral sciences (7th ed.). Routledge. This book offers guidance on writing different types of literature reviews. For the novice researcher, there are useful suggestions for creating coherent literature reviews.

THEORETICAL FRAMEWORKS

Purpose of theoretical frameworks.

As new education researchers may be less familiar with theoretical frameworks than with literature reviews, this discussion begins with an analogy. Envision a biologist, chemist, and physicist examining together the dramatic effect of a fog tsunami over the ocean. A biologist gazing at this phenomenon may be concerned with the effect of fog on various species. A chemist may be interested in the chemical composition of the fog as water vapor condenses around bits of salt. A physicist may be focused on the refraction of light to make fog appear to be “sitting” above the ocean. While observing the same “objective event,” the scientists are operating under different theoretical frameworks that provide a particular perspective or “lens” for the interpretation of the phenomenon. Each of these scientists brings specialized knowledge, experiences, and values to this phenomenon, and these influence the interpretation of the phenomenon. The scientists’ theoretical frameworks influence how they design and carry out their studies and interpret their data.

Within an educational study, a theoretical framework helps to explain a phenomenon through a particular lens and challenges and extends existing knowledge within the limitations of that lens. Theoretical frameworks are explicitly stated by an educational researcher in the paper’s framework, theory, or relevant literature section. The framework shapes the types of questions asked, guides the method by which data are collected and analyzed, and informs the discussion of the results of the study. It also reveals the researcher’s subjectivities, for example, values, social experience, and viewpoint ( Allen, 2017 ). It is essential that a novice researcher learn to explicitly state a theoretical framework, because all research questions are being asked from the researcher’s implicit or explicit assumptions of a phenomenon of interest ( Schwandt, 2000 ).

Selecting Theoretical Frameworks

Theoretical frameworks are one of the most contemplated elements in our work in educational research. In this section, we share three important considerations for new scholars selecting a theoretical framework.

The first step in identifying a theoretical framework involves reflecting on the phenomenon within the study and the assumptions aligned with the phenomenon. The phenomenon involves the studied event. There are many possibilities, for example, student learning, instructional approach, or group organization. A researcher holds assumptions about how the phenomenon will be effected, influenced, changed, or portrayed. It is ultimately the researcher’s assumption(s) about the phenomenon that aligns with a theoretical framework. An example can help illustrate how a researcher’s reflection on the phenomenon and acknowledgment of assumptions can result in the identification of a theoretical framework.

In our example, a biology education researcher may be interested in exploring how students’ learning of difficult biological concepts can be supported by the interactions of group members. The phenomenon of interest is the interactions among the peers, and the researcher assumes that more knowledgeable students are important in supporting the learning of the group. As a result, the researcher may draw on Vygotsky’s (1978) sociocultural theory of learning and development that is focused on the phenomenon of student learning in a social setting. This theory posits the critical nature of interactions among students and between students and teachers in the process of building knowledge. A researcher drawing upon this framework holds the assumption that learning is a dynamic social process involving questions and explanations among students in the classroom and that more knowledgeable peers play an important part in the process of building conceptual knowledge.

It is important to state at this point that there are many different theoretical frameworks. Some frameworks focus on learning and knowing, while other theoretical frameworks focus on equity, empowerment, or discourse. Some frameworks are well articulated, and others are still being refined. For a new researcher, it can be challenging to find a theoretical framework. Two of the best ways to look for theoretical frameworks is through published works that highlight different frameworks.

When a theoretical framework is selected, it should clearly connect to all parts of the study. The framework should augment the study by adding a perspective that provides greater insights into the phenomenon. It should clearly align with the studies described in the literature review. For instance, a framework focused on learning would correspond to research that reported different learning outcomes for similar studies. The methods for data collection and analysis should also correspond to the framework. For instance, a study about instructional interventions could use a theoretical framework concerned with learning and could collect data about the effect of the intervention on what is learned. When the data are analyzed, the theoretical framework should provide added meaning to the findings, and the findings should align with the theoretical framework.

A study by Jensen and Lawson (2011) provides an example of how a theoretical framework connects different parts of the study. They compared undergraduate biology students in heterogeneous and homogeneous groups over the course of a semester. Jensen and Lawson (2011) assumed that learning involved collaboration and more knowledgeable peers, which made Vygotsky’s (1978) theory a good fit for their study. They predicted that students in heterogeneous groups would experience greater improvement in their reasoning abilities and science achievements with much of the learning guided by the more knowledgeable peers.

In the enactment of the study, they collected data about the instruction in traditional and inquiry-oriented classes, while the students worked in homogeneous or heterogeneous groups. To determine the effect of working in groups, the authors also measured students’ reasoning abilities and achievement. Each data-collection and analysis decision connected to understanding the influence of collaborative work.

Their findings highlighted aspects of Vygotsky’s (1978) theory of learning. One finding, for instance, posited that inquiry instruction, as a whole, resulted in reasoning and achievement gains. This links to Vygotsky (1978) , because inquiry instruction involves interactions among group members. A more nuanced finding was that group composition had a conditional effect. Heterogeneous groups performed better with more traditional and didactic instruction, regardless of the reasoning ability of the group members. Homogeneous groups worked better during interaction-rich activities for students with low reasoning ability. The authors attributed the variation to the different types of helping behaviors of students. High-performing students provided the answers, while students with low reasoning ability had to work collectively through the material. In terms of Vygotsky (1978) , this finding provided new insights into the learning context in which productive interactions can occur for students.

Another consideration in the selection and use of a theoretical framework pertains to its orientation to the study. This can result in the theoretical framework prioritizing individuals, institutions, and/or policies ( Anfara and Mertz, 2014 ). Frameworks that connect to individuals, for instance, could contribute to understanding their actions, learning, or knowledge. Institutional frameworks, on the other hand, offer insights into how institutions, organizations, or groups can influence individuals or materials. Policy theories provide ways to understand how national or local policies can dictate an emphasis on outcomes or instructional design. These different types of frameworks highlight different aspects in an educational setting, which influences the design of the study and the collection of data. In addition, these different frameworks offer a way to make sense of the data. Aligning the data collection and analysis with the framework ensures that a study is coherent and can contribute to the field.

New understandings emerge when different theoretical frameworks are used. For instance, Ebert-May et al. (2015) prioritized the individual level within conceptual change theory (see Posner et al. , 1982 ). In this theory, an individual’s knowledge changes when it no longer fits the phenomenon. Ebert-May et al. (2015) designed a professional development program challenging biology postdoctoral scholars’ existing conceptions of teaching. The authors reported that the biology postdoctoral scholars’ teaching practices became more student-centered as they were challenged to explain their instructional decision making. According to the theory, the biology postdoctoral scholars’ dissatisfaction in their descriptions of teaching and learning initiated change in their knowledge and instruction. These results reveal how conceptual change theory can explain the learning of participants and guide the design of professional development programming.

The communities of practice (CoP) theoretical framework ( Lave, 1988 ; Wenger, 1998 ) prioritizes the institutional level , suggesting that learning occurs when individuals learn from and contribute to the communities in which they reside. Grounded in the assumption of community learning, the literature on CoP suggests that, as individuals interact regularly with the other members of their group, they learn about the rules, roles, and goals of the community ( Allee, 2000 ). A study conducted by Gehrke and Kezar (2017) used the CoP framework to understand organizational change by examining the involvement of individual faculty engaged in a cross-institutional CoP focused on changing the instructional practice of faculty at each institution. In the CoP, faculty members were involved in enhancing instructional materials within their department, which aligned with an overarching goal of instituting instruction that embraced active learning. Not surprisingly, Gehrke and Kezar (2017) revealed that faculty who perceived the community culture as important in their work cultivated institutional change. Furthermore, they found that institutional change was sustained when key leaders served as mentors and provided support for faculty, and as faculty themselves developed into leaders. This study reveals the complexity of individual roles in a COP in order to support institutional instructional change.

It is important to explicitly state the theoretical framework used in a study, but elucidating a theoretical framework can be challenging for a new educational researcher. The literature review can help to identify an applicable theoretical framework. Focal areas of the review or central terms often connect to assumptions and assertions associated with the framework that pertain to the phenomenon of interest. Another way to identify a theoretical framework is self-reflection by the researcher on personal beliefs and understandings about the nature of knowledge the researcher brings to the study ( Lysaght, 2011 ). In stating one’s beliefs and understandings related to the study (e.g., students construct their knowledge, instructional materials support learning), an orientation becomes evident that will suggest a particular theoretical framework. Theoretical frameworks are not arbitrary , but purposefully selected.

With experience, a researcher may find expanded roles for theoretical frameworks. Researchers may revise an existing framework that has limited explanatory power, or they may decide there is a need to develop a new theoretical framework. These frameworks can emerge from a current study or the need to explain a phenomenon in a new way. Researchers may also find that multiple theoretical frameworks are necessary to frame and explore a problem, as different frameworks can provide different insights into a problem.

Finally, it is important to recognize that choosing “x” theoretical framework does not necessarily mean a researcher chooses “y” methodology and so on, nor is there a clear-cut, linear process in selecting a theoretical framework for one’s study. In part, the nonlinear process of identifying a theoretical framework is what makes understanding and using theoretical frameworks challenging. For the novice scholar, contemplating and understanding theoretical frameworks is essential. Fortunately, there are articles and books that can help:

• Creswell, J. W. (2018). Research design: Qualitative, quantitative, and mixed methods approaches (5th ed.). Los Angeles, CA: Sage. This book provides an overview of theoretical frameworks in general educational research.
• Ding, L. (2019). Theoretical perspectives of quantitative physics education research. Physical Review Physics Education Research , 15 (2), 020101-1–020101-13. This paper illustrates how a DBER field can use theoretical frameworks.
• Nehm, R. (2019). Biology education research: Building integrative frameworks for teaching and learning about living systems. Disciplinary and Interdisciplinary Science Education Research , 1 , ar15. https://doi.org/10.1186/s43031-019-0017-6 . This paper articulates the need for studies in BER to explicitly state theoretical frameworks and provides examples of potential studies.
• Patton, M. Q. (2015). Qualitative research & evaluation methods: Integrating theory and practice . Sage. This book also provides an overview of theoretical frameworks, but for both research and evaluation.

CONCEPTUAL FRAMEWORKS

Purpose of a conceptual framework.

A conceptual framework is a description of the way a researcher understands the factors and/or variables that are involved in the study and their relationships to one another. The purpose of a conceptual framework is to articulate the concepts under study using relevant literature ( Rocco and Plakhotnik, 2009 ) and to clarify the presumed relationships among those concepts ( Rocco and Plakhotnik, 2009 ; Anfara and Mertz, 2014 ). Conceptual frameworks are different from theoretical frameworks in both their breadth and grounding in established findings. Whereas a theoretical framework articulates the lens through which a researcher views the work, the conceptual framework is often more mechanistic and malleable.

Conceptual frameworks are broader, encompassing both established theories (i.e., theoretical frameworks) and the researchers’ own emergent ideas. Emergent ideas, for example, may be rooted in informal and/or unpublished observations from experience. These emergent ideas would not be considered a “theory” if they are not yet tested, supported by systematically collected evidence, and peer reviewed. However, they do still play an important role in the way researchers approach their studies. The conceptual framework allows authors to clearly describe their emergent ideas so that connections among ideas in the study and the significance of the study are apparent to readers.

Constructing Conceptual Frameworks

Including a conceptual framework in a research study is important, but researchers often opt to include either a conceptual or a theoretical framework. Either may be adequate, but both provide greater insight into the research approach. For instance, a research team plans to test a novel component of an existing theory. In their study, they describe the existing theoretical framework that informs their work and then present their own conceptual framework. Within this conceptual framework, specific topics portray emergent ideas that are related to the theory. Describing both frameworks allows readers to better understand the researchers’ assumptions, orientations, and understanding of concepts being investigated. For example, Connolly et al. (2018) included a conceptual framework that described how they applied a theoretical framework of social cognitive career theory (SCCT) to their study on teaching programs for doctoral students. In their conceptual framework, the authors described SCCT, explained how it applied to the investigation, and drew upon results from previous studies to justify the proposed connections between the theory and their emergent ideas.

In some cases, authors may be able to sufficiently describe their conceptualization of the phenomenon under study in an introduction alone, without a separate conceptual framework section. However, incomplete descriptions of how the researchers conceptualize the components of the study may limit the significance of the study by making the research less intelligible to readers. This is especially problematic when studying topics in which researchers use the same terms for different constructs or different terms for similar and overlapping constructs (e.g., inquiry, teacher beliefs, pedagogical content knowledge, or active learning). Authors must describe their conceptualization of a construct if the research is to be understandable and useful.

There are some key areas to consider regarding the inclusion of a conceptual framework in a study. To begin with, it is important to recognize that conceptual frameworks are constructed by the researchers conducting the study ( Rocco and Plakhotnik, 2009 ; Maxwell, 2012 ). This is different from theoretical frameworks that are often taken from established literature. Researchers should bring together ideas from the literature, but they may be influenced by their own experiences as a student and/or instructor, the shared experiences of others, or thought experiments as they construct a description, model, or representation of their understanding of the phenomenon under study. This is an exercise in intellectual organization and clarity that often considers what is learned, known, and experienced. The conceptual framework makes these constructs explicitly visible to readers, who may have different understandings of the phenomenon based on their prior knowledge and experience. There is no single method to go about this intellectual work.

Reeves et al. (2016) is an example of an article that proposed a conceptual framework about graduate teaching assistant professional development evaluation and research. The authors used existing literature to create a novel framework that filled a gap in current research and practice related to the training of graduate teaching assistants. This conceptual framework can guide the systematic collection of data by other researchers because the framework describes the relationships among various factors that influence teaching and learning. The Reeves et al. (2016) conceptual framework may be modified as additional data are collected and analyzed by other researchers. This is not uncommon, as conceptual frameworks can serve as catalysts for concerted research efforts that systematically explore a phenomenon (e.g., Reynolds et al. , 2012 ; Brownell and Kloser, 2015 ).

Sabel et al. (2017) used a conceptual framework in their exploration of how scaffolds, an external factor, interact with internal factors to support student learning. Their conceptual framework integrated principles from two theoretical frameworks, self-regulated learning and metacognition, to illustrate how the research team conceptualized students’ use of scaffolds in their learning ( Figure 1 ). Sabel et al. (2017) created this model using their interpretations of these two frameworks in the context of their teaching.

Conceptual framework from Sabel et al. (2017) .

A conceptual framework should describe the relationship among components of the investigation ( Anfara and Mertz, 2014 ). These relationships should guide the researcher’s methods of approaching the study ( Miles et al. , 2014 ) and inform both the data to be collected and how those data should be analyzed. Explicitly describing the connections among the ideas allows the researcher to justify the importance of the study and the rigor of the research design. Just as importantly, these frameworks help readers understand why certain components of a system were not explored in the study. This is a challenge in education research, which is rooted in complex environments with many variables that are difficult to control.

For example, Sabel et al. (2017) stated: “Scaffolds, such as enhanced answer keys and reflection questions, can help students and instructors bridge the external and internal factors and support learning” (p. 3). They connected the scaffolds in the study to the three dimensions of metacognition and the eventual transformation of existing ideas into new or revised ideas. Their framework provides a rationale for focusing on how students use two different scaffolds, and not on other factors that may influence a student’s success (self-efficacy, use of active learning, exam format, etc.).

In constructing conceptual frameworks, researchers should address needed areas of study and/or contradictions discovered in literature reviews. By attending to these areas, researchers can strengthen their arguments for the importance of a study. For instance, conceptual frameworks can address how the current study will fill gaps in the research, resolve contradictions in existing literature, or suggest a new area of study. While a literature review describes what is known and not known about the phenomenon, the conceptual framework leverages these gaps in describing the current study ( Maxwell, 2012 ). In the example of Sabel et al. (2017) , the authors indicated there was a gap in the literature regarding how scaffolds engage students in metacognition to promote learning in large classes. Their study helps fill that gap by describing how scaffolds can support students in the three dimensions of metacognition: intelligibility, plausibility, and wide applicability. In another example, Lane (2016) integrated research from science identity, the ethic of care, the sense of belonging, and an expertise model of student success to form a conceptual framework that addressed the critiques of other frameworks. In a more recent example, Sbeglia et al. (2021) illustrated how a conceptual framework influences the methodological choices and inferences in studies by educational researchers.

Sometimes researchers draw upon the conceptual frameworks of other researchers. When a researcher’s conceptual framework closely aligns with an existing framework, the discussion may be brief. For example, Ghee et al. (2016) referred to portions of SCCT as their conceptual framework to explain the significance of their work on students’ self-efficacy and career interests. Because the authors’ conceptualization of this phenomenon aligned with a previously described framework, they briefly mentioned the conceptual framework and provided additional citations that provided more detail for the readers.

Within both the BER and the broader DBER communities, conceptual frameworks have been used to describe different constructs. For example, some researchers have used the term “conceptual framework” to describe students’ conceptual understandings of a biological phenomenon. This is distinct from a researcher’s conceptual framework of the educational phenomenon under investigation, which may also need to be explicitly described in the article. Other studies have presented a research logic model or flowchart of the research design as a conceptual framework. These constructions can be quite valuable in helping readers understand the data-collection and analysis process. However, a model depicting the study design does not serve the same role as a conceptual framework. Researchers need to avoid conflating these constructs by differentiating the researchers’ conceptual framework that guides the study from the research design, when applicable.

Explicitly describing conceptual frameworks is essential in depicting the focus of the study. We have found that being explicit in a conceptual framework means using accepted terminology, referencing prior work, and clearly noting connections between terms. This description can also highlight gaps in the literature or suggest potential contributions to the field of study. A well-elucidated conceptual framework can suggest additional studies that may be warranted. This can also spur other researchers to consider how they would approach the examination of a phenomenon and could result in a revised conceptual framework.

It can be challenging to create conceptual frameworks, but they are important. Below are two resources that could be helpful in constructing and presenting conceptual frameworks in educational research:

• Maxwell, J. A. (2012). Qualitative research design: An interactive approach (3rd ed.). Los Angeles, CA: Sage. Chapter 3 in this book describes how to construct conceptual frameworks.
• Ravitch, S. M., & Riggan, M. (2016). Reason & rigor: How conceptual frameworks guide research . Los Angeles, CA: Sage. This book explains how conceptual frameworks guide the research questions, data collection, data analyses, and interpretation of results.

CONCLUDING THOUGHTS

Literature reviews, theoretical frameworks, and conceptual frameworks are all important in DBER and BER. Robust literature reviews reinforce the importance of a study. Theoretical frameworks connect the study to the base of knowledge in educational theory and specify the researcher’s assumptions. Conceptual frameworks allow researchers to explicitly describe their conceptualization of the relationships among the components of the phenomenon under study. Table 1 provides a general overview of these components in order to assist biology education researchers in thinking about these elements.

It is important to emphasize that these different elements are intertwined. When these elements are aligned and complement one another, the study is coherent, and the study findings contribute to knowledge in the field. When literature reviews, theoretical frameworks, and conceptual frameworks are disconnected from one another, the study suffers. The point of the study is lost, suggested findings are unsupported, or important conclusions are invisible to the researcher. In addition, this misalignment may be costly in terms of time and money.

Conducting a literature review, selecting a theoretical framework, and building a conceptual framework are some of the most difficult elements of a research study. It takes time to understand the relevant research, identify a theoretical framework that provides important insights into the study, and formulate a conceptual framework that organizes the finding. In the research process, there is often a constant back and forth among these elements as the study evolves. With an ongoing refinement of the review of literature, clarification of the theoretical framework, and articulation of a conceptual framework, a sound study can emerge that makes a contribution to the field. This is the goal of BER and education research.

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Theoretical Research: Definition, Methods + Examples

Research is the careful study of a particular research problem or concern using the scientific method. A theory is essential for any research project because it gives it direction and helps prove or disprove something. Theoretical basis helps us figure out how things work and why we do certain things.

Theoretical research lets you examine and discuss a research object using philosophical ideas and abstract theoretical structures.

In theoretical research, you can’t look at the research object directly. With the help of research literature, your research aims to define and sketch out the chosen topic’s conceptual models, explanations, and structures.

LEARN ABOUT: Research Process Steps

This blog will cover theoretical research and why it is essential. In addition to that, we are going to go over some examples.

What is the theoretical research?

Theoretical research is the systematic examination of a set of beliefs and assumptions.

It aims to learn more about a subject and help us understand it better. The information gathered in this way is not used for anything in particular because this kind of research aims to learn more.

All professionals, like biologists, chemists, engineers, architects, philosophers, writers, sociologists, historians, etc., can do theoretical research. No matter what field you work in, theoretical research is the foundation for new ideas.

It tries to answer basic questions about people, which is why this kind of research is used in every field of knowledge.

For example , a researcher starts with the idea that we need to understand the world around us. To do this, he begins with a hypothesis and tests it through experiments that will help him develop new ideas. 

What is the theoretical framework?

A theoretical framework is a critical component in research that provides a structured foundation for investigating a specific topic or problem. It encompasses a set of interconnected theories, existing theories, and concepts that guide the entire research process. 

The theoretical framework introduces a comprehensive understanding of the subject matter. Also, the theoretical framework strengthens the research’s validity and specifies the key elements that will be explored. Furthermore, it connects different ideas and theories, forming a cohesive structure that underpins the research endeavor.

A complete theoretical framework consists of a network of theories, existing theories, and concepts that collectively shape the direction of a research study. 

The theoretical framework is the fundamental principle that will be explored, strengthens the research’s credibility by aligning it with established knowledge, specifies the variables under investigation, and connects different aspects of the research to create a unified approach.

Theoretical frameworks are the intellectual scaffolding upon which the research is constructed. It is the lens through which researchers view their subject, guiding their choice of methodologies, data collection, analysis, and interpretation. By incorporating existing theory, and established concepts, a theoretical framework not only grounds the research but also provides a coherent roadmap for exploring the intricacies of the chosen topic.

Benefits of theoretical research

Theoretical research yields a wealth of benefits across various fields, from social sciences to human resource development and political science. Here’s a breakdown of these benefits while incorporating the requested topics:

Predictive power

Theoretical models are the cornerstone of theoretical research. They grant us predictive power, enabling us to forecast intricate behaviors within complex systems, like societal interactions. In political science, for instance, a theoretical model helps anticipate potential outcomes of policy changes.

Understanding human behavior

Drawing from key social science theories, it assists us in deciphering human behavior and societal dynamics. For instance, in the context of human resource development, theories related to motivation and psychology provide insights into how to effectively manage a diverse workforce.

Optimizing workforce

In the realm of human resource development, insights gleaned from theoretical research, along with the research methods knowledge base, help create targeted training programs. By understanding various learning methodologies and psychological factors, organizations can optimize workforce training for better results.

Building on foundations

It doesn’t exist in isolation; it builds upon existing theories. For instance, within the human resource development handbook, theoretical research expands established concepts, refining their applicability to contemporary organizational challenges.

Ethical policy formulation

Within political science, theoretical research isn’t confined to governance structures. It extends to ethical considerations, aiding policymakers in creating policies that balance the collective good with individual rights, ensuring just and fair governance. 

Rigorous investigations

Theoretical research underscores the importance of research methods knowledge base. This knowledge equips researchers in theory-building research methods and other fields to design robust research methodologies, yielding accurate data and credible insights.

Long-term impact

Theoretical research leaves a lasting impact. The theoretical models and insights from key social science theories provide enduring frameworks for subsequent research, contributing to the cumulative growth of knowledge in these fields.

Innovation and practical applications

It doesn’t merely remain theoretical. It inspires innovation and practical applications. By merging insights from diverse theories and fields, practitioners in human resource development devise innovative strategies to foster employee growth and well-being.

Theoretical research method

Researchers follow so many methods when doing research. There are two types of theoretical research methods.

• Scientific methods
• Social science method 

Let’s explore them below:

Scientific method

Scientific methods have some important points that you should know. Let’s figure them out below:

• Observation: Any part you want to explain can be found through observation. It helps define the area of research.
• Hypothesis: The hypothesis is the idea put into words, which helps us figure out what we see.
• Experimentation: Hypotheses are tested through experiments to see if they are true. These experiments are different for each research.
• Theory: When we create a theory, we do it because we believe it will explain hypotheses of higher probability.
• Conclusions: Conclusions are the learnings we derive from our investigation.

Social science methods

There are different methods for social science theoretical research. It consists of polls, documentation, and statistical analysis.

• Polls: It is a process whereby the researcher uses a topic-specific questionnaire to gather data. No changes are made to the environment or the phenomenon where the polls are conducted to get the most accurate results. QuestionPro live polls are a great way to get live audiences involved and engaged.
• Documentation: Documentation is a helpful and valuable technique that helps the researcher learn more about the subject. It means visiting libraries or other specialized places, like documentation centers, to look at the existing bibliography. With the documentation, you can find out what came before the investigated topic and what other investigations have found. This step is important because it shows whether or not similar investigations have been done before and what the results were.
• Statistic analysis : Statistics is a branch of math that looks at random events and differences. It follows the rules that are established by probability. It’s used a lot in sociology and language research. 

Examples of theoretical research

We talked about theoretical study methods in the previous part. We’ll give you some examples to help you understand it better.

Example 1: Theoretical research into the health benefits of hemp

The plant’s active principles are extracted and evaluated, and by studying their components, it is possible to determine what they contain and whether they can potentially serve as a medication.

Example 2: Linguistics research

Investigate to determine how many people in the Basque Country speak Basque. Surveys can be used to determine the number of native Basque speakers and those who speak Basque as a second language.

Example 3: Philosophical research

Research politics and ethics as they are presented in the writings of Hanna Arendt from a theoretical perspective.

LEARN ABOUT: 12 Best Tools for Researchers

From our above discussion, we learned about theoretical research and its methods and gave some examples. It explains things and leads to more knowledge for the sake of knowledge. This kind of research tries to find out more about a thing or an idea, but the results may take time to be helpful in the real world. 

This research is sometimes called basic research. Theoretical research is an important process that gives researchers valuable data with insight.

QuestionPro is a strong platform for managing your data. You can conduct simple surveys to more complex research using QuestionPro survey software.

At QuestionPro, we give researchers tools for collecting data, such as our survey software and a library of insights for any long-term study. Contact our expert team to find out more about it.

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Theories are formulated to explain, predict, and understand phenomena and, in many cases, to challenge and extend existing knowledge within the limits of critical bounded assumptions or predictions of behavior. The theoretical framework is the structure that can hold or support a theory of a research study. The theoretical framework encompasses not just the theory, but the narrative explanation about how the researcher engages in using the theory and its underlying assumptions to investigate the research problem. It is the structure of your paper that summarizes concepts, ideas, and theories derived from prior research studies and which was synthesized in order to form a conceptual basis for your analysis and interpretation of meaning found within your research.

Abend, Gabriel. "The Meaning of Theory." Sociological Theory 26 (June 2008): 173–199; Kivunja, Charles. "Distinguishing between Theory, Theoretical Framework, and Conceptual Framework: A Systematic Review of Lessons from the Field." International Journal of Higher Education 7 (December 2018): 44-53; Swanson, Richard A. Theory Building in Applied Disciplines . San Francisco, CA: Berrett-Koehler Publishers 2013; Varpio, Lara, Elise Paradis, Sebastian Uijtdehaage, and Meredith Young. "The Distinctions between Theory, Theoretical Framework, and Conceptual Framework." Academic Medicine 95 (July 2020): 989-994.

Importance of Theory and a Theoretical Framework

Theories can be unfamiliar to the beginning researcher because they are rarely applied in high school social studies curriculum and, as a result, can come across as unfamiliar and imprecise when first introduced as part of a writing assignment. However, in their most simplified form, a theory is simply a set of assumptions or predictions about something you think will happen based on existing evidence and that can be tested to see if those outcomes turn out to be true. Of course, it is slightly more deliberate than that, therefore, summarized from Kivunja (2018, p. 46), here are the essential characteristics of a theory.

• It is logical and coherent
• It has clear definitions of terms or variables, and has boundary conditions [i.e., it is not an open-ended statement]
• It has a domain where it applies
• It has clearly described relationships among variables
• It describes, explains, and makes specific predictions
• It comprises of concepts, themes, principles, and constructs
• It must have been based on empirical data [i.e., it is not a guess]
• It must have made claims that are subject to testing, been tested and verified
• It must be clear and concise
• Its assertions or predictions must be different and better than those in existing theories
• Its predictions must be general enough to be applicable to and understood within multiple contexts
• Its assertions or predictions are relevant, and if applied as predicted, will result in the predicted outcome
• The assertions and predictions are not immutable, but subject to revision and improvement as researchers use the theory to make sense of phenomena
• Its concepts and principles explain what is going on and why
• Its concepts and principles are substantive enough to enable us to predict a future

Given these characteristics, a theory can best be understood as the foundation from which you investigate assumptions or predictions derived from previous studies about the research problem, but in a way that leads to new knowledge and understanding as well as, in some cases, discovering how to improve the relevance of the theory itself or to argue that the theory is outdated and a new theory needs to be formulated based on new evidence.

A theoretical framework consists of concepts and, together with their definitions and reference to relevant scholarly literature, existing theory that is used for your particular study. The theoretical framework must demonstrate an understanding of theories and concepts that are relevant to the topic of your research paper and that relate to the broader areas of knowledge being considered.

The theoretical framework is most often not something readily found within the literature . You must review course readings and pertinent research studies for theories and analytic models that are relevant to the research problem you are investigating. The selection of a theory should depend on its appropriateness, ease of application, and explanatory power.

The theoretical framework strengthens the study in the following ways :

• An explicit statement of  theoretical assumptions permits the reader to evaluate them critically.
• The theoretical framework connects the researcher to existing knowledge. Guided by a relevant theory, you are given a basis for your hypotheses and choice of research methods.
• Articulating the theoretical assumptions of a research study forces you to address questions of why and how. It permits you to intellectually transition from simply describing a phenomenon you have observed to generalizing about various aspects of that phenomenon.
• Having a theory helps you identify the limits to those generalizations. A theoretical framework specifies which key variables influence a phenomenon of interest and highlights the need to examine how those key variables might differ and under what circumstances.
• The theoretical framework adds context around the theory itself based on how scholars had previously tested the theory in relation their overall research design [i.e., purpose of the study, methods of collecting data or information, methods of analysis, the time frame in which information is collected, study setting, and the methodological strategy used to conduct the research].

By virtue of its applicative nature, good theory in the social sciences is of value precisely because it fulfills one primary purpose: to explain the meaning, nature, and challenges associated with a phenomenon, often experienced but unexplained in the world in which we live, so that we may use that knowledge and understanding to act in more informed and effective ways.

The Conceptual Framework. College of Education. Alabama State University; Corvellec, Hervé, ed. What is Theory?: Answers from the Social and Cultural Sciences . Stockholm: Copenhagen Business School Press, 2013; Asher, Herbert B. Theory-Building and Data Analysis in the Social Sciences . Knoxville, TN: University of Tennessee Press, 1984; Drafting an Argument. Writing@CSU. Colorado State University; Kivunja, Charles. "Distinguishing between Theory, Theoretical Framework, and Conceptual Framework: A Systematic Review of Lessons from the Field." International Journal of Higher Education 7 (2018): 44-53; Omodan, Bunmi Isaiah. "A Model for Selecting Theoretical Framework through Epistemology of Research Paradigms." African Journal of Inter/Multidisciplinary Studies 4 (2022): 275-285; Ravitch, Sharon M. and Matthew Riggan. Reason and Rigor: How Conceptual Frameworks Guide Research . Second edition. Los Angeles, CA: SAGE, 2017; Trochim, William M.K. Philosophy of Research. Research Methods Knowledge Base. 2006; Jarvis, Peter. The Practitioner-Researcher. Developing Theory from Practice . San Francisco, CA: Jossey-Bass, 1999.

Strategies for Developing the Theoretical Framework

I.  Developing the Framework

Here are some strategies to develop of an effective theoretical framework:

• Examine your thesis title and research problem . The research problem anchors your entire study and forms the basis from which you construct your theoretical framework.
• Brainstorm about what you consider to be the key variables in your research . Answer the question, "What factors contribute to the presumed effect?"
• Review related literature to find how scholars have addressed your research problem. Identify the assumptions from which the author(s) addressed the problem.
• List  the constructs and variables that might be relevant to your study. Group these variables into independent and dependent categories.
• Review key social science theories that are introduced to you in your course readings and choose the theory that can best explain the relationships between the key variables in your study [note the Writing Tip on this page].
• Discuss the assumptions or propositions of this theory and point out their relevance to your research.

A theoretical framework is used to limit the scope of the relevant data by focusing on specific variables and defining the specific viewpoint [framework] that the researcher will take in analyzing and interpreting the data to be gathered. It also facilitates the understanding of concepts and variables according to given definitions and builds new knowledge by validating or challenging theoretical assumptions.

II.  Purpose

Think of theories as the conceptual basis for understanding, analyzing, and designing ways to investigate relationships within social systems. To that end, the following roles served by a theory can help guide the development of your framework.

• Means by which new research data can be interpreted and coded for future use,
• Response to new problems that have no previously identified solutions strategy,
• Means for identifying and defining research problems,
• Means for prescribing or evaluating solutions to research problems,
• Ways of discerning certain facts among the accumulated knowledge that are important and which facts are not,
• Means of giving old data new interpretations and new meaning,
• Means by which to identify important new issues and prescribe the most critical research questions that need to be answered to maximize understanding of the issue,
• Means of providing members of a professional discipline with a common language and a frame of reference for defining the boundaries of their profession, and
• Means to guide and inform research so that it can, in turn, guide research efforts and improve professional practice.

Adapted from: Torraco, R. J. “Theory-Building Research Methods.” In Swanson R. A. and E. F. Holton III , editors. Human Resource Development Handbook: Linking Research and Practice . (San Francisco, CA: Berrett-Koehler, 1997): pp. 114-137; Jacard, James and Jacob Jacoby. Theory Construction and Model-Building Skills: A Practical Guide for Social Scientists . New York: Guilford, 2010; Ravitch, Sharon M. and Matthew Riggan. Reason and Rigor: How Conceptual Frameworks Guide Research . Second edition. Los Angeles, CA: SAGE, 2017; Sutton, Robert I. and Barry M. Staw. “What Theory is Not.” Administrative Science Quarterly 40 (September 1995): 371-384.

Structure and Writing Style

The theoretical framework may be rooted in a specific theory , in which case, your work is expected to test the validity of that existing theory in relation to specific events, issues, or phenomena. Many social science research papers fit into this rubric. For example, Peripheral Realism Theory, which categorizes perceived differences among nation-states as those that give orders, those that obey, and those that rebel, could be used as a means for understanding conflicted relationships among countries in Africa. A test of this theory could be the following: Does Peripheral Realism Theory help explain intra-state actions, such as, the disputed split between southern and northern Sudan that led to the creation of two nations?

However, you may not always be asked by your professor to test a specific theory in your paper, but to develop your own framework from which your analysis of the research problem is derived . Based upon the above example, it is perhaps easiest to understand the nature and function of a theoretical framework if it is viewed as an answer to two basic questions:

• What is the research problem/question? [e.g., "How should the individual and the state relate during periods of conflict?"]
• Why is your approach a feasible solution? [i.e., justify the application of your choice of a particular theory and explain why alternative constructs were rejected. I could choose instead to test Instrumentalist or Circumstantialists models developed among ethnic conflict theorists that rely upon socio-economic-political factors to explain individual-state relations and to apply this theoretical model to periods of war between nations].

The answers to these questions come from a thorough review of the literature and your course readings [summarized and analyzed in the next section of your paper] and the gaps in the research that emerge from the review process. With this in mind, a complete theoretical framework will likely not emerge until after you have completed a thorough review of the literature .

Just as a research problem in your paper requires contextualization and background information, a theory requires a framework for understanding its application to the topic being investigated. When writing and revising this part of your research paper, keep in mind the following:

• Clearly describe the framework, concepts, models, or specific theories that underpin your study . This includes noting who the key theorists are in the field who have conducted research on the problem you are investigating and, when necessary, the historical context that supports the formulation of that theory. This latter element is particularly important if the theory is relatively unknown or it is borrowed from another discipline.
• Position your theoretical framework within a broader context of related frameworks, concepts, models, or theories . As noted in the example above, there will likely be several concepts, theories, or models that can be used to help develop a framework for understanding the research problem. Therefore, note why the theory you've chosen is the appropriate one.
• The present tense is used when writing about theory. Although the past tense can be used to describe the history of a theory or the role of key theorists, the construction of your theoretical framework is happening now.
• You should make your theoretical assumptions as explicit as possible . Later, your discussion of methodology should be linked back to this theoretical framework.
• Don’t just take what the theory says as a given! Reality is never accurately represented in such a simplistic way; if you imply that it can be, you fundamentally distort a reader's ability to understand the findings that emerge. Given this, always note the limitations of the theoretical framework you've chosen [i.e., what parts of the research problem require further investigation because the theory inadequately explains a certain phenomena].

The Conceptual Framework. College of Education. Alabama State University; Conceptual Framework: What Do You Think is Going On? College of Engineering. University of Michigan; Drafting an Argument. Writing@CSU. Colorado State University; Lynham, Susan A. “The General Method of Theory-Building Research in Applied Disciplines.” Advances in Developing Human Resources 4 (August 2002): 221-241; Tavallaei, Mehdi and Mansor Abu Talib. "A General Perspective on the Role of Theory in Qualitative Research." Journal of International Social Research 3 (Spring 2010); Ravitch, Sharon M. and Matthew Riggan. Reason and Rigor: How Conceptual Frameworks Guide Research . Second edition. Los Angeles, CA: SAGE, 2017; Reyes, Victoria. Demystifying the Journal Article. Inside Higher Education; Trochim, William M.K. Philosophy of Research. Research Methods Knowledge Base. 2006; Weick, Karl E. “The Work of Theorizing.” In Theorizing in Social Science: The Context of Discovery . Richard Swedberg, editor. (Stanford, CA: Stanford University Press, 2014), pp. 177-194.

Writing Tip

Borrowing Theoretical Constructs from Other Disciplines

An increasingly important trend in the social and behavioral sciences is to think about and attempt to understand research problems from an interdisciplinary perspective. One way to do this is to not rely exclusively on the theories developed within your particular discipline, but to think about how an issue might be informed by theories developed in other disciplines. For example, if you are a political science student studying the rhetorical strategies used by female incumbents in state legislature campaigns, theories about the use of language could be derived, not only from political science, but linguistics, communication studies, philosophy, psychology, and, in this particular case, feminist studies. Building theoretical frameworks based on the postulates and hypotheses developed in other disciplinary contexts can be both enlightening and an effective way to be more engaged in the research topic.

CohenMiller, A. S. and P. Elizabeth Pate. "A Model for Developing Interdisciplinary Research Theoretical Frameworks." The Qualitative Researcher 24 (2019): 1211-1226; Frodeman, Robert. The Oxford Handbook of Interdisciplinarity . New York: Oxford University Press, 2010.

Another Writing Tip

Don't Undertheorize!

Do not leave the theory hanging out there in the introduction never to be mentioned again. Undertheorizing weakens your paper. The theoretical framework you describe should guide your study throughout the paper. Be sure to always connect theory to the review of pertinent literature and to explain in the discussion part of your paper how the theoretical framework you chose supports analysis of the research problem or, if appropriate, how the theoretical framework was found to be inadequate in explaining the phenomenon you were investigating. In that case, don't be afraid to propose your own theory based on your findings.

Yet Another Writing Tip

What's a Theory? What's a Hypothesis?

The terms theory and hypothesis are often used interchangeably in newspapers and popular magazines and in non-academic settings. However, the difference between theory and hypothesis in scholarly research is important, particularly when using an experimental design. A theory is a well-established principle that has been developed to explain some aspect of the natural world. Theories arise from repeated observation and testing and incorporates facts, laws, predictions, and tested assumptions that are widely accepted [e.g., rational choice theory; grounded theory; critical race theory].

A hypothesis is a specific, testable prediction about what you expect to happen in your study. For example, an experiment designed to look at the relationship between study habits and test anxiety might have a hypothesis that states, "We predict that students with better study habits will suffer less test anxiety." Unless your study is exploratory in nature, your hypothesis should always explain what you expect to happen during the course of your research.

The key distinctions are:

• A theory predicts events in a broad, general context;  a hypothesis makes a specific prediction about a specified set of circumstances.
• A theory has been extensively tested and is generally accepted among a set of scholars; a hypothesis is a speculative guess that has yet to be tested.

Cherry, Kendra. Introduction to Research Methods: Theory and Hypothesis. About.com Psychology; Gezae, Michael et al. Welcome Presentation on Hypothesis. Slideshare presentation.

Still Yet Another Writing Tip

Be Prepared to Challenge the Validity of an Existing Theory

Theories are meant to be tested and their underlying assumptions challenged; they are not rigid or intransigent, but are meant to set forth general principles for explaining phenomena or predicting outcomes. Given this, testing theoretical assumptions is an important way that knowledge in any discipline develops and grows. If you're asked to apply an existing theory to a research problem, the analysis will likely include the expectation by your professor that you should offer modifications to the theory based on your research findings.

Indications that theoretical assumptions may need to be modified can include the following:

• Your findings suggest that the theory does not explain or account for current conditions or circumstances or the passage of time,
• The study reveals a finding that is incompatible with what the theory attempts to explain or predict, or
• Your analysis reveals that the theory overly generalizes behaviors or actions without taking into consideration specific factors revealed from your analysis [e.g., factors related to culture, nationality, history, gender, ethnicity, age, geographic location, legal norms or customs , religion, social class, socioeconomic status, etc.].

Philipsen, Kristian. "Theory Building: Using Abductive Search Strategies." In Collaborative Research Design: Working with Business for Meaningful Findings . Per Vagn Freytag and Louise Young, editors. (Singapore: Springer Nature, 2018), pp. 45-71; Shepherd, Dean A. and Roy Suddaby. "Theory Building: A Review and Integration." Journal of Management 43 (2017): 59-86.

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Research Process Guide

• Step 1 - Identifying and Developing a Topic
• Step 2 - Narrowing Your Topic
• Step 3 - Developing Research Questions
• Step 4 - Conducting a Literature Review
• Step 5 - Choosing a Conceptual or Theoretical Framework
• Step 6 - Determining Research Methodology
• Step 6a - Determining Research Methodology - Quantitative Research Methods
• Step 6b - Determining Research Methodology - Qualitative Design
• Step 7 - Considering Ethical Issues in Research with Human Subjects - Institutional Review Board (IRB)
• Step 8 - Collecting Data
• Step 9 - Analyzing Data
• Step 10 - Interpreting Results
• Step 11 - Writing Up Results

Step 5: Choosing a Conceptual or Theoretical Framework

For all empirical research, you must choose a conceptual or theoretical framework to “frame” or “ground” your study. Theoretical and/or conceptual frameworks are often difficult to understand and challenging to choose which is the right one (s) for your research objective (Hatch, 2002). Truthfully, it is difficult to get a real understanding of what these frameworks are and how you are supposed to find what works for your study. The discussion of your framework is addressed in your Chapter 1, the introduction and then is further explored through in-depth discussion in your Chapter 2 literature review.

“Theory is supposed to help researchers of any persuasion clarify what they are up to and to help them to explain to others what they are up to” (Walcott, 1995, p. 189, as cited in Fallon, 2016). It is important to discuss in the beginning to help researchers “clarify what they are up to” and important at the writing stage to “help explain to others what they are up to” (Fallon, 2016).  

What is the difference between the conceptual and the theoretical framework?

Often, the terms theoretical framework and conceptual framework are used interchangeably, which, in this author’s opinion, makes an already difficult to understand idea even more confusing. According to Imenda (2014) and Mensah et al. (2020), there is a very distinct difference between conceptual and theoretical frameworks, not only how they are defined but also, how and when they are used in empirical research.

Imenda (2014) contends that the framework “is the soul of every research project” (p.185). Essentially, it determines how the researcher formulates the research problem, goes about investigating the problem, and what meaning or significance the research lends to the data collected and analyzed investigating the problem.  

Very generally, you would use a theoretical framework if you were conducting deductive research as you test a theory or theories. “A theoretical framework comprises the theories expressed by experts in the field into which you plan to research, which you draw upon to provide a theoretical coat hanger for your data analysis and interpretation of results” (Kivunja, 2018, p.45 ).  Often this framework is based on established theories like, the Set Theory, evolution, the theory of matter or similar pre-existing generalizations like Newton’s law of motion (Imenda, 2014). A good theoretical framework should be linked to, and possibly emerge from your literature review.

Using a theoretical framework allows you to (Kivunja, 2018):

• Increase the credibility and validity of your research
• Interpret meaning found in data collection
• Evaluate solutions for solving your research problem

According to Mensah et al.(2020) the theoretical framework for your research is not a summary of your own thoughts about your research. Rather, it is a compilation of the thoughts of giants in your field, as they relate to your proposed research, as you understand those theories, and how you will use those theories to understand the data collected.

Additionally, Jabareen (2009) defines a conceptual framework as interlinked concepts that together provide a comprehensive  understanding of a phenomenon. “A conceptual framework is the total, logical orientation and associations of anything and everything that forms the underlying thinking, structures, plans and practices and implementation of your entire research project” (Kivunja, 2018, p. 45). You would largely use a conceptual framework when conducting inductive research, as it helps the researcher answer questions that are core to qualitative research, such as the nature of reality, the way things are and how things really work in a real world (Guba & Lincoln, 1994).

Some consideration of the following questions can help define your conceptual framework (Kinvunja, 2018):

• What do you want to do in your research? And why do you want to do it?
• How do you plan to do it?
• What meaning will you make of the data?
• Which worldview will you situate your study in? (i.e. Positivist? Interpretist? Constructivist?)

Examples of conceptual frameworks include the definitions a sociologist uses to describe a culture and the types of data an economist considers when evaluating a country’s industry. The conceptual framework consists of the ideas that are used to define research and evaluate data. Conceptual frameworks are often laid out at the beginning of a paper or an experiment description for a reader to understand the methods used (Mensah et al., 2020).

Writing it up

After choosing your framework is to articulate the theory or concept that grounds your study by defining it and demonstrating the rationale for this particular set of theories or concepts guiding your inquiry.  Write up your theoretical perspective sections for your research plan following your choice of worldview/ research paradigm. For a quantitative study you are particularly interested in theory using the procedures for a causal analysis. For qualitative research, you should locate qualitative journal articles that use a priori theory (knowledge that is acquired not through experience) that is modified during the process of research (Creswell & Creswell, 2018). Also, you should generate or develop a theory at the end of your study. For a mixed methods study which uses a transformative (critical theoretical lens) identify how the lens specifically shapes the research process.                                   

Creswell, J. W., & Creswell, J. D. (2 018). Research design: Qualitative, quantitative, and mixed methods approaches. Sage.

Fallon, M. (2016). Writing up quantitative research in the social and behavioral sciences. Sense. https://kean.idm.oclc.org/login?url=https://search.ebscohost.com/login.aspx?direct=true&AuthType=cookie,ip,url,cpid&custid=keaninf&db=nlebk&AN=1288374&site=ehost-live&scope=site&ebv=EB&ppid=pp_C1

Guba, E. G., & Lincoln, Y. S. (1994). Competing paradigms in qualitative research. Handbook of Qualitative Research, 2 (163-194), 105.

Hatch, J. A. ( 2002). Doing qualitative research in education settings. SUNY Press.

Imenda, S. (2014). Is there a conceptual difference between theoretical and conceptual frameworks?  Journal of Social Sciences, 38 (2), 185-195.

Jabareen, Y. (2009). Building a conceptual framework: Philosophy, definitions, and procedure. International Journal of Qualitative Methods, 8 (4), 49-62.

Kivunja, C. ( 2018, December 3). Distinguishing between theory, theoretical framework, and conceptual framework. The International Journal of Higher Education, 7 (6), 44-53. https://files.eric.ed.gov/fulltext/EJ1198682.pdf  

Mensah, R. O., Agyemang, F., Acquah, A., Babah, P. A., & Dontoh, J. (2020). Discourses on conceptual and theoretical frameworks in research: Meaning and implications for researchers. Journal of African Interdisciplinary Studies, 4 (5), 53-64.

• Last Updated: Jun 29, 2023 1:35 PM
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How to... Conduct empirical research

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Empirical research is research that is based on observation and measurement of phenomena, as directly experienced by the researcher. The data thus gathered may be compared against a theory or hypothesis, but the results are still based on real life experience. The data gathered is all primary data, although secondary data from a literature review may form the theoretical background.

On this page

What is empirical research, the research question, the theoretical framework, sampling techniques, design of the research.

• Methods of empirical research
• Techniques of data collection & analysis
• Reporting the findings of empirical research
• Further information

Typically, empirical research embodies the following elements:

• A  research question , which will determine research objectives.
• A particular and planned  design  for the research, which will depend on the question and which will find ways of answering it with appropriate use of resources.
• The gathering of  primary data , which is then analysed.
• A particular  methodology  for collecting and analysing the data, such as an experiment or survey.
• The limitation of the data to a particular group, area or time scale, known as a sample: for example, a specific number of employees of a particular company type, or all users of a library over a given time scale. The sample should be somehow representative of a wider population.
• The ability to  recreate  the study and test the results. This is known as  reliability .
• The ability to  generalise  from the findings to a larger sample and to other situations.

The starting point for your research should be your research question. This should be a formulation of the issue which is at the heart of the area which you are researching, which has the right degree of breadth and depth to make the research feasible within your resources. The following points are useful to remember when coming up with your research question, or RQ:

• your doctoral thesis;
• reading the relevant literature in journals, especially literature reviews which are good at giving an overview, and spotting interesting conceptual developments;
• looking at research priorities of funding bodies, professional institutes etc.;
• going to conferences;
• looking out for calls for papers;
• developing a dialogue with other researchers in your area.
• To narrow down your research topic, brainstorm ideas around it, possibly with your colleagues if you have decided to collaborate, noting all the questions down.
• Come up with a "general focus" question; then develop some other more specific ones.
• they are not too broad;
• they are not so narrow as to yield uninteresting results;
• will the research entailed be covered by your resources, i.e. will you have sufficient time and money;
• there is sufficient background literature on the topic;
• you can carry out appropriate field research;
• you have stated your question in the simplest possible way.

Let's look at some examples:

Bisking et al. examine whether or not gender has an influence on disciplinary action in their article  Does the sex of the leader and subordinate influence a leader's disciplinary decisions?  ( Management Decision , Volume 41 Number 10) and come up with the following series of inter-related questions:

• Given the same infraction, would a male leader impose the same disciplinary action on male and female subordinates?
• Given the same infraction, would a female leader impose the same disciplinary action on male and female subordinates?
• Given the same infraction, would a female leader impose the same disciplinary action on female subordinates as a male leader would on male subordinates?
• Given the same infraction, would a female leader impose the same disciplinary action on male subordinates as a male leader would on female subordinates?
• Given the same infraction, would a male and female leader impose the same disciplinary action on male subordinates?
• Given the same infraction, would a male and female leader impose the same disciplinary action on female subordinates?
• Do female and male leaders impose the same discipline on subordinates regardless of the type of infraction?
• Is it possible to predict how female and male leaders will impose disciplinary actions based on their respective BSRI femininity and masculinity scores?

Motion et al. examined co-branding in  Equity in Corporate Co-branding  ( European Journal of Marketing , Volume 37 Number 7/8) and came up with the following RQs:

RQ1:  What objectives underpinned the corporate brand?

RQ2:  How were brand values deployed to establish the corporate co-brand within particular discourse contexts?

RQ3:  How was the desired rearticulation promoted to shareholders?

RQ4:  What are the sources of corporate co-brand equity?

Note, the above two examples state the RQs very explicitly; sometimes the RQ is implicit:

Qun G. Jiao, Anthony J. Onwuegbuzie are library researchers who examined the question:  "What is the relationship between library anxiety and social interdependence?"  in a number of articles, see  Dimensions of library anxiety and social interdependence: implications for library services   ( Library Review , Volume 51 Number 2).

Or sometimes the RQ is stated as a general objective:

Ying Fan describes outsourcing in British companies in  Strategic outsourcing: evidence from British companies  ( Marketing Intelligence & Planning , Volume 18 Number 4) and states his research question as an objective:

The main objective of the research was to explore the two key areas in the outsourcing process, namely:

• pre-outsourcing decision process; and
• post-outsourcing supplier management.

or as a proposition:

Karin Klenke explores issues of gender in management decisions in  Gender influences in decision-making processes in top management teams   ( Management Decision , Volume 41 Number 10).

Given the exploratory nature of this research, no specific hypotheses were formulated. Instead, the following general propositions are postulated:

P1.  Female and male members of TMTs exercise different types of power in the strategic decision making process.

P2.  Female and male members of TMTs differ in the extent in which they employ political savvy in the strategic decision making process.

P3.  Male and female members of TMTs manage conflict in strategic decision making situations differently.

P4.  Female and male members of TMTs utilise different types of trust in the decision making process.

Sometimes, the theoretical underpinning (see next section) of the research leads you to formulate a hypothesis rather than a question:

Martin et al. explored the effect of fast-forwarding of ads (called zipping) in  Remote control marketing: how ad fast-forwarding and ad repetition affect consumers  ( Marketing Intelligence & Planning , Volume 20 Number 1) and his research explores the following hypotheses:

The influence of zipping H1. Individuals viewing advertisements played at normal speed will exhibit higher ad recall and recognition than those who view zipped advertisements.

Ad repetition effects H2. Individuals viewing a repeated advertisement will exhibit higher ad recall and recognition than those who see an advertisement once.

Zipping and ad repetition H3. Individuals viewing zipped, repeated advertisements will exhibit higher ad recall and recognition than those who see a normal speed advertisement that is played once.

Empirical research is not divorced from theoretical considerations; and a consideration of theory should form one of the starting points of your research. This applies particularly in the case of management research which by its very nature is practical and applied to the real world. The link between research and theory is symbiotic: theory should inform research, and the findings of research should inform theory.

There are a number of different theoretical perspectives; if you are unfamiliar with them, we suggest that you look at any good research methods textbook for a full account (see Further information), but this page will contain notes on the following:

This is the approach of the natural sciences, emphasising total objectivity and independence on the part of the researcher, a highly scientific methodology, with data being collected in a value-free manner and using quantitative techniques with some statistical measures of analysis. Assumes that there are 'independent facts' in the social world as in the natural world. The object is to generalise from what has been observed and hence add to the body of theory.

Very similar to positivism in that it has a strong reliance on objectivity and quantitative methods of data collection, but with less of a reliance on theory. There is emphasis on data and facts in their own right; they do not need to be linked to theory.

Interpretivism

This view criticises positivism as being inappropriate for the social world of business and management which is dominated by people rather than the laws of nature and hence has an inevitable subjective element as people will have different interpretations of situations and events. The business world can only be understood through people's interpretation. This view is more likely to emphasise qualitative methods such as participant observation, focus groups and semi-structured interviewing.

While reality exists independently of human experience, people are not like objects in the natural world but are subject to social influences and processes. Like  empiricism  and  positivism , this emphasises the importance of explanation, but is also concerned with the social world and with its underlying structures.

Inductive and deductive approaches

At what point in your research you bring in a theoretical perspective will depend on whether you choose an:

• Inductive approach  – collect the data, then develop the theory.
• Deductive approach  – assume a theoretical position then test it against the data.

It should be emphasised that none of the above approaches are mutually exclusive and can be used in combination.

Sampling may be done either:

• On a  random  basis – a given number is selected completely at random.
• On a  systematic  basis – every  n th element  of the population is selected.
• On a  stratified random  basis – the population is divided into segments, for example, in a University, you could divide the population into academic, administrators, and academic related. A random number of each group is then selected.
• On a  cluster  basis – a particular subgroup is chosen at random.
• Convenience  – being present at a particular time e.g. at lunch in the canteen.
• Purposive  – people can be selected deliberately because their views are relevant to the issue concerned.
• Quota  – the assumption is made that there are subgroups in the population, and a quota of respondents is chosen to reflect this diversity.

Useful articles

Richard Laughlin in  Empirical research in accounting: alternative approaches and a case for "middle-range" thinking  provides an interesting general overview of the different perspectives on theory and methodology as applied to accounting. ( Accounting, Auditing & Accountability Journal,  Volume 8 Number 1).

D. Tranfield and K. Starkey in  The Nature, Social Organization and Promotion of Management Research: Towards Policy  look at the relationship between theory and practice in management research, and develop a number of analytical frameworks, including looking at Becher's conceptual schema for disciplines and Gibbons et al.'s taxonomy of knowledge production systems. ( British Journal of Management , vol. 9, no. 4 – abstract only).

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Qualitative and Quantitative Research

What is "empirical research".

• empirical research
• Locating Articles in Cinahl and PsycInfo
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Empirical research  is based on observed and measured phenomena and derives knowledge from actual experience rather than from theory or belief. 

How do you know if a study is empirical? Read the subheadings within the article, book, or report and look for a description of the research "methodology."  Ask yourself: Could I recreate this study and test these results?

Key characteristics to look for:

• Specific research questions  to be answered
• Definition of the  population, behavior, or   phenomena  being studied
• Description of the  process  used to study this population or phenomena, including selection criteria, controls, and testing instruments (such as surveys)

Another hint: some scholarly journals use a specific layout, called the "IMRaD" format, to communicate empirical research findings. Such articles typically have 4 components:

• Introduction : sometimes called "literature review" -- what is currently known about the topic -- usually includes a theoretical framework and/or discussion of previous studies
• Methodology:  sometimes called "research design" --  how to recreate the study -- usually describes the population, research process, and analytical tools
• Results : sometimes called "findings"  --  what was learned through the study -- usually appears as statistical data or as substantial quotations from research participants
• Discussion : sometimes called "conclusion" or "implications" -- why the study is important -- usually describes how the research results influence professional practices or future studies
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Empirical or Theoretical?

Empirical: Based on data gathered by original experiments or observations.

Theoretical: Analyzes and makes connections between empirical studies to define or advance a theoretical position.

Videos on Finding Empirical Articles

Where Can I Find Empirically-Based Education Articles?

The most direct route is to search PsycInfo, linked above.

This will take you to the Advanced Search, where you can type in your key words at the top. Then scroll down through all the limiting options to the Methodology menu. Select Empirical Study. 

In other databases without the Methodology limiter, such as Education Source , try keywords like empirical , study , and research .

How Can I Tell if an Article is Empirical?

Check for these components:

• Peer-reviewed
• Charts, graphs, tables, and/or statistical analyses
• More than 5 pages
• Sections with names like: Abstract, Introduction, Literature Review, Method, Data, Analysis, Results, Discussion, References

Look for visual cues of data collection and analysis:

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Empirical & Non-Empirical Research

What is non-empirical research.

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Examples of Non-Empirical Research

Empirical vs. non-empirical infographic, what distinguishes non-empirical research from empirical research.

The collection and analysis of evidence to test a proposition is one of the hallmarks of empirical research .

Non-empirical research articles focus more on theories, methods, well-supported opinions, and their implications for research. Non-empirical research includes comprehensive review or analysis of journal literature or articles that focus on methodology. It may rely on or analyze empirical research literature but does not need to be essentially data- or evidence-driven as empirical research is.

Are Empirical Articles also Peer-Reviewed?

The question of whether empirical research is always peer-reviewed confuses apples and oranges, namely, peer-reviewed articles publications and common research methodologies.

• Peer-reviewed, scholarly, or academic articles are written by scholars for publication in academic journals.
• Peer-reviewed journals are edited by teams of scholars who approve and endorse what is published in their journals.
• Empirical and non-empirical research are methodologies used in research. An academic journal may publish articles reflecting both types of methodologies

Common Types of Articles That Are Not Empirical

Knowing how to quickly identify some common types of non-empirical research articles in peer-reviewed journals can help speed up your search. Here are some brief definitions and examples.

• Peer-reviewed articles that systematically discuss and propose abstract concepts and methods for a field without primary data collection.
• Example: Grosser, K. & Moon, J. (2019). CSR and feminist organization studies: Towards an integrated theorization for the analysis of gender issues .
• Peer-reviewed articles that systematically describe, summarize, and often categorize and evaluate previous research on a topic without collecting new data.
• Example: Heuer, S. & Willer, R. (2020). How is quality of life assessed in people with dementia? A systematic literature review and a primer for speech-language pathologists .
• Note: empirical research articles and doctoral dissertations will have a literature review section, which exists to give context to the empirical research proposed or conducted. In a literature review article, the literature review is the focus.
• While literature reviews or meta-analyses are not empirical, they are often a great source of information on previous empirical research on a topic, with citations to find that research.
• Non-peer-reviewed articles where the authors discuss their thoughts on a particular topic without data collection and a systematic method. There are a few differences between these types of articles.
• Written by the editors or guest editors of the journal.
• Example: Naples, N. A., Mauldin, L., & Dillaway, H. (2018). From the guest editors: Gender, disability, and intersectionality .
• Written by guest authors. The journal may have a non-peer-reviewed process for authors to submit these articles, and the editors of the journal may invite authors to write opinion articles.
• Example: García, J. J.-L., & Sharif, M. Z. (2015). Black lives matter: A commentary on racism and public health .
• Written by a journal's readers, often in response to an article previously published in the journal.
• Example: Nathan, M. (2013). Letters: Perceived discrimination and racial/ethnic disparities in youth problem behaviors .
• Non-peer-reviewed articles that describe and evaluate books, products, services, and other things the audience of the journal would be interested in.
• Example: Robinson, R. & Green, J. M. (2020). Book review: Microaggressions and traumatic stress: Theory, research, and clinical treatment .

Based on an original created by the University of Memphis libraries.

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What Is A Theoretical Framework? A Practical Answer

• Published: 30 November 2015
• Volume 26 , pages 593–597, ( 2015 )

Cite this article

• Norman G. Lederman 1 &
• Judith S. Lederman 1  

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Avoid common mistakes on your manuscript.

Other than the poor or non-existent validity and/or reliability of data collection measures, the lack of a theoretical framework is the most frequently cited reason for our editorial decision not to publish a manuscript in the Journal of Science Teacher Education . A poor or missing theoretical framework is similarly a critical problem for manuscripts submitted to other journals for which Norman or Judith have either served as Editor or been on the Editorial Board. Often the problem is that an author fails to justify his/her research effort with a theoretical framework. However, there is another level to the problem. Many individuals have a rather narrow conception of what constitutes a theoretical framework or that it is somehow distinct from a conceptual framework. The distinction on lack thereof is a story for another day. The following story may remind you of an experience you or one of your classmates have had.

Doctoral students live in fear of hearing these now famous words from their thesis advisor: “This sounds like a promising study, but what is your theoretical framework?” These words instantly send the harried doctoral student to the library (giving away our ages) in search of a theory to support the proposed research and to satisfy his/her advisor. The search is often unsuccessful because of the student’s misconception of what constitutes a “theoretical framework.” The framework may actually be a theory, but not necessarily. This is especially true for theory driven research (typically quantitative) that is attempting to test the validity of existing theory. However, this narrow definition of a theoretical framework is commonly not aligned with qualitative research paradigms that are attempting to develop theory, for example, grounded theory, or research falling into the categories of description and interpretation research (Peshkin, 1993 ). Additionally, a large proportion of doctoral theses do not fit the narrow definition described. The argument here is not that various research paradigms have no overarching philosophies or theories about knowing. Clearly quantitative research paradigms are couched in a realist perspective and qualitative research paradigms are couched in an idealist perspective (Bogdan & Biklen, 1982 ). The discussion here is focused on theoretical frameworks at a much more specific and localized perspective with respect to the justification and conceptualization of a single research investigation. So, what is a theoretical framework?

It is, perhaps, easier to understand the nature and function of a theoretical framework if it is viewed as the answer to two basic questions:

What is the problem or question?

Why is your approach to solving the problem or answering the question feasible?

Indeed, the answers to these questions are the substance and culmination of Chapters I and II of the proposal and completed dissertation, or the initial sections preceding the Methods section of a research article. The answers to these questions can come from only one source, a thorough review of the literature (i.e., a review that includes both the theoretical and empirical literature as well as apparent gaps in the literature). Perhaps, a hypothetical situation can best illustrate the development and role of the theoretical framework in the formalization of a dissertation topic or research investigation. Let us continue with the doctoral student example, keeping in mind that a parallel situation also presents itself to any researcher planning research that he/she intends to publish.

As an interested reader of educational literature, a doctoral student becomes intrigued by the importance of questioning in the secondary classroom. The student immediately begins a manual and computer search of the literature on questioning in the classroom. The student notices that the research findings on the effectiveness of questioning strategies are rather equivocal. In particular, much of the research focuses on the cognitive levels of the questions asked by the teacher and how these questions influence student achievement. It appears that the research findings exhibit no clear pattern. That is, in some studies, frequent questioning at higher cognitive levels has led to more achievement than frequent questioning at the lower cognitive levels. However, an equal number of investigations have shown no differences between the achievement of students who are exposed to questions at distinctly different cognitive levels, but rather the simple frequency of questions.

The doctoral student becomes intrigued by these equivocal findings and begins to speculate about some possible explanations. In a blinding flash of insight, the student remembers hearing somewhere that an eccentric Frenchman named Piaget said something about students being categorized into levels of cognitive development. Could it be that a student’s cognitive level has something to do with how much and what he/she learns? The student heads back to the library and methodically searches through the literature on cognitive development and its relationship to achievement.

At this point, the doctoral student has become quite familiar with two distinct lines of educational research. The research on the effectiveness of questioning has established that there is a problem. That is, does the cognitive level of questioning have any effect on student achievement? In effect, this answers the first question identified previously with respect to identification of a theoretical framework. The research on the cognitive development of students has provided an intriguing perspective. That is, could it be possible that students of different cognitive levels are affected differently by questions at different cognitive levels? If so, an answer to the problem concerning the effectiveness questioning may be at hand. This latter question, in effect, has addressed the second question previously posed about the identification of a theoretical framework. At this point, the student has narrowed his/her interests as a result of reviewing the literature. Note that the doctoral student is now ready to write down a specific research question and that this is only possible after having conducted a thorough review of the literature.

The student writes down the following research hypotheses:

Both high and low cognitive level pupils will benefit from both high and low cognitive levels of questions as opposed to no questions at all.

Pupils categorized at high cognitive levels will benefit more from high cognitive level questions than from low level questions.

Pupils categorized at lower cognitive levels will benefit more from low cognitive level questions than from high level questions.

These research questions still need to be transformed into testable statistical hypotheses, but they are ready to be presented to the dissertation advisor. The advisor looks at the questions and says: “This looks like a promising study, but what is your theoretical framework?” There is no need, however for a sprint to the library. The doctoral student has a theoretical framework. The literature on questioning has established that there is a problem and the literature on cognitive development has provided the rationale for performing the specific investigation that is being proposed. ALL IS WELL!

If some of the initial research completed by Norman concerning what classroom variables contributed to students’ understandings of nature of science (Lederman, 1986a , 1986b ; Lederman & Druger, 1985 ) had to align with the overly restricted definition of a theoretical framework, which necessitates the presence of theory, it never would have been published. In these initial studies, various classroom variables were identified that were related to students’ improved understandings of nature of science. The studies were descriptive and correlational and were not driven by any theory about how students learn nature of science. Indeed, the design of the studies was derived from the fact that there were no existing theories, general or specific, to explain how students might learn nature of science more effectively. Similarly, the seminal study of effective teaching, the Beginning Teacher Evaluation Study (Tikunoff, Berliner, & Rist, 1975 ), was an ethnographic study that was not guided by the findings of previous research on effective teaching. Rather, their inductive study simply compared 40 teachers “known” to be effective and ineffective of mathematics and reading to derive differences in classroom practice. Their study had no theoretical framework if one were to use the restrictive conception that a theory needed to provide a guiding framework for the investigation. There are plenty of other examples that have guided lines of research that could be provided, but there is no need to beat a dead horse by detailing more examples. The simple, but important, point is that research following qualitative research paradigms or traditions (Jacob, 1987 ; Smith, 1987 ) are particularly vulnerable to how ‘theoretical framework’ is defined. Indeed, it could be argued that the necessity of a theory is a remnant from the times in which qualitative research was not as well accepted as it is today. In general, any research design that is inductive in nature and attempts to develop theory would be at a loss. We certainly would not want to eliminate multiple traditions of research from the Journal of Science Teacher Education .

Harry Wolcott’s discussion about validity in qualitative research (Wolcott, 1990 ) is quite explicit about the lack of theory or necessity of theory in driving qualitative ethnography. Interestingly, he even rejects the idea of validity as being a necessary criterion in qualitative research. Additionally, Bogdan and Biklen ( 1982 ) emphasize the importance of qualitative researchers “bracketing” (i.e., masking or trying to forget) their a priori theories so that it does not influence the collection of data or any meanings assigned to data during an investigation. Similar discussions about how qualitative research differs from quantitative research with respect to the necessity of theory guiding the research have been advanced by many others (e.g., Becker, 1970 ; Bogdan & Biklen, 1982 ; Erickson, 1986 ; Krathwohl, 2009 ; Rist, 1977 ; among others). Perhaps, Peshkin ( 1993 , p. 23) put it best when he expressed his concern that “Research that is not theory driven, hypothesis testing, or generalization producing may be dismissed as deficient or worse.” Again, the key point is that qualitative research is as valuable and can contribute as much to our knowledge of teaching and learning as quantitative research.

There is little doubt that qualitative researchers often invoke theory when analyzing the data they have collected or try to place their findings within the context of the existing literature. And, as stated at the beginning of this editorial, different research paradigms have large overarching theories about how one comes to know about the world. However, this is not the same thing has using a theory as a framework for the design of an investigation from the stating of research questions to developing a design to answer the research questions.

It is quite possible that you may be thinking that this editorial about the meaning of a theoretical framework is too theoretical. Trust us in believing that there is a very practical reason for us addressing this issue. At the beginning of the editorial we talked about the lack of a theoretical framework being the second most common reason for manuscripts being rejected for publication in the Journal of Science Teacher Education . Additionally, we mentioned that this is a common reason for manuscripts being rejected by other prominent journals in science education, and education in general. Consequently, it is of critical importance that we, as a community, are clear about the meaning of a theoretical framework and its use. It is especially important that our authors, reviewers, associate editors, and we as Editors of the journal are clear on this matter. Let us not fail to mention that most of us are advising Ph.D. students in the conceptualization of their dissertations. This issue is not new. In 1992, the editorial board of the Journal of Research in Science Teaching was considering the claim, by some, that qualitative research was not being evaluated fairly for publication relative to quantitative research. In their analysis of the relative success of publication for quantitative and qualitative research, Wandersee and Demastes ( 1992 , p. 1005) noted that reviewers often noted, “The manuscript had a weak theoretical basis” when reviewing qualitative research.

Theoretical frameworks are critically important to all of our work, quantitative, qualitative, or mixed methods. All research articles should have a valid theoretical framework to justify the importance and significance of the work. However, we should not live in fear, as the doctoral student, of not having a theoretical framework, when we actually have such, because an Editor, reviewer, or Major Professor is using any unduly restrictive and outdated meaning for what constitutes a theoretical framework.

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Lederman, N.G., Lederman, J.S. What Is A Theoretical Framework? A Practical Answer. J Sci Teacher Educ 26 , 593–597 (2015). https://doi.org/10.1007/s10972-015-9443-2

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Influence of regional air pollution pressure on the green transformation of higher education: an empirical study based on pm2.5 in chinese cities, 1. introduction, 2. theoretical analysis and hypothesis, 2.1. regional air pollution pressure and green transformation of higher education, 2.2. the moderating effect of policy support intensity, 2.3. the moderating effect of public environmental awareness, 3. empirical test and result analysis, 3.1. data source and sample determination, 3.2. research design, 3.3. measurement of green transformation in higher education, 3.3.1. methodology basis, 3.3.2. qualitative analysis.

• How do you view the impact of current air quality conditions on the campus environment?
• What measures have your institution taken to address air pollution?
• What challenges did you encounter while implementing these measures?
• What factors do you believe motivated your institution to adopt green initiatives?
• What are your thoughts on the environmental activities organized by your school?
• Have you participated in any environmental activities at your school? If so, which ones?
• What do you think your school could do to promote green transformation?
• What environmental actions have you taken in your personal life?

3.3.3. Construction of Higher Education Green Transformation System

3.3.4. measurement of other variables, 3.4. empirical analysis, 3.4.1. descriptive analysis, 3.4.2. correlation analysis, 3.4.3. regression analysis, 4. robustness checks, 4.1. analysis regarding variations in pm2.5, 4.2. examination of sample selection bias, 5. further analysis, 5.1. analysis of time lag in policy response, 5.2. analysis of regional variations, 5.3. dynamic association between air pollution pressure and higher education green transformation, 6. conclusions, 6.1. summary of findings.

• The Role of Regional Air Pollution Pressure
• The Critical Nature of Government Policy Support
• Influence of Social Cultural Factors
• Robustness Testing
• Impact at the Policy Level
• Geographical Differences

6.2. Research Recommendations

6.2.1. at the higher education institution level, 6.2.2. at the government level, 6.2.3. at the societal and public level, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Ying, R.; Wang, X. Influence of Regional Air Pollution Pressure on the Green Transformation of Higher Education: An Empirical Study Based on PM2.5 in Chinese Cities. Sustainability 2024 , 16 , 7153. https://doi.org/10.3390/su16167153

Ying R, Wang X. Influence of Regional Air Pollution Pressure on the Green Transformation of Higher Education: An Empirical Study Based on PM2.5 in Chinese Cities. Sustainability . 2024; 16(16):7153. https://doi.org/10.3390/su16167153

Ying, Rui, and Xiuli Wang. 2024. "Influence of Regional Air Pollution Pressure on the Green Transformation of Higher Education: An Empirical Study Based on PM2.5 in Chinese Cities" Sustainability 16, no. 16: 7153. https://doi.org/10.3390/su16167153

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Five years with the GDPR: an empirical study emphasising information privacy and the consumer

• Wanda Presthus Kristiania University College
• Hanne Sørum Kristiania University College

Consumers’ privacy rights have been enshrined in law, long before information systems and the Internet was brought to life. In 2018, stricter regulations relating to information privacy came into force, named the General Data Protection Regulation (GDPR). Using elements of Roger’s diffusion of innovations theory, we investigated the research question: How has five years of the GDPR influenced consumer’s knowledge, attitude, and practice of their enhanced rights? We draw on empirical data collected in Norway through four online survey questionnaires over five years (N=1293). Quantitative (descriptive statistics) and qualitative analyses (manual cluster text mining) were performed to obtain a state-of-the-art mapping of insights on consumers and their information privacy. Our findings show that the respondents’ answers remained similar over the years, and that the GDPR has not had a significant influence on the consumer. The respondents demonstrated a high degree of knowledge regarding both the regulation and technology, such as cookies. Their attitude was sceptical, as they valued their enhanced rights but questioned the feasibility. Regarding their practice, our findings reveal diversity. Some respondents took careful actions to protect their privacy, while most did not. The present paper should be interesting to both the industry (practitioners) and academia (researchers).

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• Wanda Presthus, Kaja Felix Sønslien, An analysis of violations and sanctions following the GDPR , International Journal of Information Systems and Project Management: Vol. 9 No. 1 (2021)

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Unified momentum model for rotor aerodynamics across operating regimes

• Jaime Liew   ORCID: orcid.org/0000-0002-5858-4614 1 ,
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• Electrical and electronic engineering
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• Fluid dynamics
• Mechanical engineering

Despite substantial growth in wind energy technology in recent decades, aerodynamic modeling of wind turbines relies on momentum models derived in the late 19th and early 20th centuries, which are well-known to break down under flow regimes in which wind turbines often operate. This gap in theoretical modeling for rotors that are misaligned with the inflow and also for high-thrust rotors has resulted in the development of numerous empirical corrections which are widely applied in textbooks, research articles, and open-source and industry design codes. This work reports a Unified Momentum Model to efficiently predict power production, thrust force, and wake dynamics of rotors under arbitrary inflow angles and thrust coefficients without empirical corrections. The Unified Momentum Model is additionally coupled with a blade element model to enable blade element momentum modeling predictions of wind turbines in high thrust coefficient and yaw misaligned states without using corrections for these states. This Unified Momentum Model can form a new basis for wind turbine modeling, design, and control tools from first principles and may enable further development of innovations necessary for increased wind production and reliability to respond to 21st century climate change challenges.

Introduction

To meet mid-century global net-zero carbon emissions targets, wind energy capacity is estimated to require between a 9- and 11-fold increase 1 , 2 . Within the U.S. alone, as much as a 28-fold increase in wind power capacity is required to achieve net-zero by 2050 3 . This unprecedented scale-up must be guided by modeling tools that are sufficiently accurate for the design and control of wind turbines and wind farms 4 , 5 . Despite substantial growth in wind energy technology in recent decades 6 , aerodynamic modeling of wind turbine rotors relies heavily on models originally derived in the late 19th and early 20th centuries 7 , 8 , 9 , which are well-known to break down in flow regimes that modern wind turbines often operate within 10 , 11 , 12 , 13 . To overcome discrepancies associated with this fundamental breakdown, the predictions of wind turbine forces and power output that drive contemporary design and control protocols are based on empirical formulas 14 , 15 . Besides numerically intensive computational fluid dynamics (CFD) simulations that have limited utility in high-throughput optimization applications, there is no existing first-principles theory that can accurately predict rotor aerodynamics across the range of thrust forces and misalignment angles commonly encountered by wind turbines. This work develops a Unified Momentum Model for rotor aerodynamics that is valid across operating regimes, from low to high thrust coefficient magnitudes, including positive and negative thrust, and at arbitrary misalignment angles between inflow and rotor. The Unified Momentum Model overcomes limitations of classical one-dimensional momentum theory by accounting for both misalignment between the rotor and inflow and the pressure deficit in the rotor wake, as predicted by a solution to the differential Euler equations. The resulting aerodynamic model predicts rotor thrust, power, and wake velocities at arbitrary misalignments and thrust coefficients without empirical corrections. This Unified Momentum Model represents a departure from traditional one-dimensional momentum theory aided with empirical corrections, offering a first-principles foundation that can serve as a new basis for modeling, designing, and controlling wind turbines.

One-dimensional momentum modeling, originally derived in the late 19th century by Rankine (1865) 7 , W. Froude (1878) 8 , and R.E. Froude (1889) 9 , is the predominant model used in engineering analysis and design of rotors including wind turbines, propellers, helicopters, drones, and hydrokinetic turbines 11 , 16 , 17 , 18 , 19 , 20 , 21 , 22 . The theory, which is the starting point for any textbook on rotor aerodynamics across engineering applications 23 , represents the rotor as a porous actuator disk that imparts a thrust force on the surrounding flow. The induced velocities generated by the rotor thrust are related to the upstream and downstream velocities via the conservation of mass and momentum in one dimension normal to the disk. One-dimensional momentum modeling provides reasonably accurate predictions of rotor performance at low to moderate thrust coefficients, depicted in Fig.  1 as the windmill state . However, the theory is well known to break down at higher thrust coefficients as well as in situations of misalignment between the inflow and the rotor, which is commonly encountered in practice 11 , 24 . In these regimes, the one-dimensional momentum theory exhibits high error for critical quantities including rotor thrust, power, wake velocities, and outlet pressure, demonstrating quantitative and qualitative deviation from measurements. Given the one-dimensionality of the classical model, effects caused by misalignment between the rotor and the inflow are not captured. Experiments and CFD simulations show that thrust continues to increase as induction increases 10 , 24 , 25 , whereas classical momentum modeling is unable to capture this behavior, predicting the opposite trend (see Fig.  1) . The discrepancies in the classical momentum theory arise from two main limiting assumptions: (1) one-dimensional flow perpendicular to the rotor and (2) recovery of wake pressure to the freestream value.

(Left) Schematic illustrating the rotor thrust coefficient variations with rotor-normal induction across various operational scenarios (propeller state, windmill state, and turbulent wake state 24 ) for a yaw-aligned actuator disk. Model predictions are shown using classical one-dimensional momentum modeling 7 , 8 , 9 , Glauert’s empirical relation 10 , and the Unified Momentum Model introduced in this study. (Right) Schematic representation of the control volume enclosing the porous actuator disk that is used to derived the Unified Momentum Model developed in this study.

The assumption of one-dimensional flow neglects all lateral velocities induced by the rotor misalignment 26 . Wind turbines continuously operate in some degree of yaw misalignment with respect to the incident wind direction due to a slowly reacting yaw controller and error or bias in wind direction measurements 27 . These misalignment errors are predicted to be larger for floating offshore turbines 28 , which are anticipated to account for a large fraction of future U.S. offshore wind energy generation 29 . One-dimensional momentum modeling is typically adjusted using empirical skewed wake corrections to represent the influence of rotor misalignment 11 , 14 , 17 . Textbooks state that the power production of a rotor yaw misaligned at angle γ scales with \({\cos }^{3}(\gamma )\) 11 , while empirical observations and CFD output do not exhibit this relationship, instead showing sub-cubic behavior which varies with rotor operating conditions 30 , 31 , 32 . In tandem, given the proliferation of wind energy and the densification of wind farms that leads to unfavorable aerodynamic wake interactions between neighboring turbines 33 , 34 , research has focused on methods to collectively operate turbines within a farm by controlling the wind flow to maximize farm power production 5 . The primary wind farm flow control methodology, termed wake steering, entails intentionally yaw misaligning wind turbines with respect to the freestream wind 35 , 36 , a misalignment that results in an explicit breakdown of the one-dimensional momentum modeling used to predict the power, loads, and wake velocities associated with the yawed turbine.

The assumption that the wake pressure recovers to the freestream value is violated at higher thrust coefficients. At higher thrust coefficients, the static pressure in the wake downstream of the rotor fails to return to the freestream pressure. This persistent pressure drop behind the rotor, known as base suction 37 , corresponds to an additional thrust force contribution that is not captured by current theoretical models. Specifically, classical momentum modeling is widely understood and accepted to break down at a value of the induction factor ( a  = 1 −  u d / u ∞ , where u d is the velocity at the rotor disk and u ∞ is the freestream wind speed) that is only 10% higher than the optimal value of a  =  1/3 23 , 38 predicted by Betz (1920) 11 , 25 , 38 , based on classical one-dimensional momentum modeling. There is no comprehensive theoretical or analytical model rooted in first-principles that can adequately capture the effects caused by rotor misalignment and wake pressure. This gap in theoretical modeling has prompted the development of numerous empirical corrections, which have found widespread application in textbooks 11 , 39 , research articles 15 , 21 , 40 , and open-source, as well as industry, design codes 14 , 41 , 42 . Examples include the empirical Glauert correction to model high thrust operation 10 , 11 , 12 , 13 , 14 , 15 , 21 , 40 , skewed wake corrections 11 , 14 , 17 , and an empirically tuned power-yaw formula to model rotor misalignment 31 , 35 , 43 .

This work extends first-principles understanding and develops a new analytical aerodynamic model for rotors operating in arbitrary thrust and misalignment conditions. To address long-standing discrepancies between classical momentum modeling and empirical observations, the actuator disk model is re-examined from first principles. Using conservation of mass, momentum, and energy, the limiting assumptions of the classical theory are eliminated by modeling the pressure deficit in the rotor wake, using a solution to the differential Euler equations, and by accounting for arbitrary rotor misalignment with a lifting line model. This results in a set of five coupled equations that simultaneously govern rotor induction, thrust, wake velocities, wake pressure, and power production. The unified equations predict the empirically observed monotonic increase in thrust coefficient with increasing induction factor, offering a first-principles solution to a persistent limitation of classical one-dimensional momentum modeling that predicts a trend in the opposite direction. The equations further predict the quantitative influence of rotor misalignment on wake velocities, thrust, and power at arbitrary, joint misalignments and thrust coefficients. As in classical momentum theory, the model is derived under uniform inflow. But relative to the classical theory, the Unified Model generalizes to predict the effects of both misalignment and high thrust operation without empirical corrections. The model therefore provides a new basis to account for additional effects such as turbulence and wind shear in the atmospheric boundary layer, rather than starting from classical theory coupled with empiricism. The Unified Momentum Model is coupled with blade element modeling to result in a new blade element momentum (BEM) modeling approach that can predict the effects of rotor misalignment and high thrust operation without empirical corrections in the momentum model. This study provides new theoretical insight and a simple, computationally-efficient model for complex turbine aerodynamics that have previously relied on empirical corrections or necessitated expensive CFD to resolve 12 , 13 .

Unified Momentum Model

In this study, we derive and validate an analytical relationship between the induction, streamwise and spanwise wake velocities, and the wake pressure through integral analysis of a control volume enclosing the actuator disk. This Unified Momentum Model addresses the two limiting assumptions of one-dimensionality and pressure recovery to freestream from classical momentum modeling. As in classical momentum theory, the analytical model is derived under uniform, inviscid inflow and neglects turbulence. As a result, the model is applicable at the rotor, predicting induction, thrust force, and power, and in the near-wake, which is the portion of the wake region before turbulent mixing contributes to the dynamics 44 , 45 , 46 , 47 . The induction is modeled using the Bernoulli equation, with the pressure drop at the actuator disk predicted by the thrust force. The radial flow is continuous over the porous actuator disk and therefore cancels in the Bernoulli equation, as further discussed in the  Supplementary Information . The streamwise wake velocity is modeled using momentum conservation in the streamwise direction, along with mass conservation in the control volume and the streamtube enclosing the actuator disk. Both the induction and the streamwise wake velocity depend on each other and on the pressure deficit in the wake. Given an arbitrary misalignment angle between the actuator disk and the incident flow, the flow acquires a lateral component, deviating from the one-dimensional assumption of classical momentum theory. We model the lateral wake velocity using a lifting line model 26 , 48 , 49 , that relates the lateral wake velocity to the lift component of the rotor-normal thrust force.

To close the system of equations, the remaining unknown is the pressure in the wake, which classical one-dimensional momentum modeling assumes to be equal to the freestream pressure 11 . As the induced velocity is increased, flow separation and a large pressure drop can occur 37 , 50 , which prevents the wake pressure from recovering to its freestream value, and correspondingly increases the thrust 11 . To model this behavior, the wake pressure in this study is modeled using a solution to the steady, two-dimensional Euler equations with an actuator disk turbine body forcing, recently proposed by Madsen (2023) 51 , based on the analytical method of von Kármán and Burgers (1935) 52 . The objective is to model the inviscid rotor and near-wake regions, therefore the inviscid Euler equations are used rather than the Reynolds-Averaged Navier–Stokes equations with a turbulence model, consistent with the assumption to neglect the effects of turbulence. In the far-wake, turbulence 46 and three-dimensional effects such as wake curling 53 become important. The two-dimensional Euler equations are used in this study to result in an analytical model form, rather than requiring a CFD solver to predict a full three-dimensional pressure field. The quantitative fidelity of this modeling approximation is validated in the subsequent results.

The solution to the pressure Poisson equation is decomposed into contributions from the linear actuator disk body forces and the nonlinear advective terms. In the absence of the nonlinear advective pressure component, the Unified Momentum Model is entirely analytical. Model predictions with and without the nonlinear advective term are shown in the  Supplementary Information , with the results demonstrating that including the nonlinear portion of the wake pressure lowers prediction error in the region of induction approximately between 0.7 and 0.9, and therefore, we determine that it is necessary to include. The nonlinear advective component of the pressure exhibits instability at very high thrust coefficients and induction factors (induction  ≳  0.7), in both the Euler equation solution and in large eddy simulations. We investigated mitigation solutions including convergence enhancement techniques and dynamic mode decomposition analysis, not shown for brevity. The final methodology selected records the minimum pressure at the centerline location throughout iterations of a range of relaxations, yielding an approximate upper bound for the magnitude of the nonlinear pressure drop. The full methodology is described in the Supplementary Information . The quantitative accuracy of this pressure modeling approach is evaluated in the results section, compared to large eddy simulations.

To complete this set of equations, a model for the near-wake length is needed to determine the pressure in the wake at the outlet of the inviscid near-wake region. The outlet of the inviscid near-wake region is coincident with the onset of the turbulent far-wake region, modeled using so-called wake models. Therefore, near-wake predictions from momentum theory are used to provide initial conditions to wake models 44 , 45 , 46 , 47 . Bastankhah & Porté-Agel (2016) 46 used the shear layer model of Lee & Chu (2003) 54 and one-dimensional momentum modeling to predict the near-wake length for low thrust and induction rotors. To predict the near-wake length, we adapt this approach to remain consistent with both high-thrust and yaw misaligned regimes simultaneously. The shear layer model introduces a single parameter, β , that relates the shear layer growth to the characteristic velocity shear between the wake and freestream velocity 46 . This parameter has a well-accepted range in the literature 54 . Here, the parameter is estimated using data from CFD of an actuator disk and the obtained value falls within the typical range documented in prior studies. Further, the quantitative influence of the parameter uncertainty is assessed in the results below, and in  Supplementary Information . The wake expansion is also predicted by the model, consistent with a decrease in streamwise velocity and the lateral velocity, as connected through the conservation of mass and momentum.

The final system of equations, given in “Methods”, provides an analytical relationship between the induction, streamwise velocity, lateral velocity, near-wake length, and wake pressure for a rotor with arbitrary yaw misalignments and thrust coefficients. The corresponding power production for a wind turbine or hydrokinetic rotor is computed using the predicted induction factor.

Unified Momentum Model predictions in uniform flow

Predictions from the model proposed here are assessed through a priori and a posteriori analysis, compared to high-fidelity large eddy simulations (LES). To first focus on model assessment and validation, simulations are performed in the same dynamical context as the model development, namely, uniform inflow and an actuator disk modeled rotor. The numerical implementation of the LES is described in “Methods”. To demonstrate the robustness of the numerical method, LES results are shown using two standard approaches for modeling the thrust force (see Calaf et al. 55 ), depending on input of either the thrust coefficient C T and a thrust force dependent on the freestream velocity \({\vec{{u}}}_{\infty }\) , or input of a modified thrust coefficient \({C}_{T}^{{\prime} }\) and a thrust force dependent on the velocity at the rotor \({\vec{u}}_{d}\) . Figure  2 a illustrates the thrust coefficient C T as a function of the rotor-normal induction factor a n ( \({a}_{n}=1-{\vec{u}}_{d}\cdot \hat{n}/{\vec{u}}_{\infty }\cdot \hat{n}\) ), which is the generalization of the classical one-dimensional induction factor. In addition to the present LES results, LES results from Martínez-Tossas et al. (2022, NREL) 13 are shown for reference. To demonstrate the validity of the model form, a priori model predictions are shown, where the wake pressure in LES is measured and provided to the model. The fully predictive a posteriori model output, where the pressure is modeled based on the methodology described previously, is also shown. The a priori model output exhibits remarkable agreement with LES in terms of predicting induction, wake velocities and power production, validating the model-form proposed here. The fully-predictive model, which requires no pressure input from LES, exhibits low error across the full range of rotor-normal induction factors realized in LES, showing substantial qualitative and quantitative predictive accuracy improvements compared to classical one-dimensional momentum modeling. The clear and notable deficiency of classical momentum modeling to predict the qualitative and quantitative response of the thrust coefficient C T for induction factors of a n   ≳  1/3 has resulted in numerous empirical formulas to be proposed in the literature 14 , 15 , 21 , 40 . In this study, through first-principles modeling of the wake pressure, substantial increases in predictive accuracy are achieved without empirical formulas. Similar observations can be made through analysis of the coefficient of power C P depending on the rotor-normal induction factor a n as shown in Fig.  2 c.

Coefficient of thrust, C T , as a function of ( a ) rotor-normal induction factor, a n , and ( b ) the local thrust coefficient \({C}_{T}^{{\prime} }\) . Coefficient of power, C P , as a function of ( c ) a n , and ( d ) \({C}_{T}^{{\prime} }\) . The variables estimated by the presented Unified Momentum Model (Methods), are shown. The shaded region corresponds to ± 10% uncertainty in parameter β . Note that this uncertainty range is visually negligible for the plotted quantities and therefore may not be visible. Results from large eddy simulation (LES) from MIT and NREL and classical one-dimensional momentum modeling are shown as a reference. A priori model results are also shown for reference, where the LES-measured pressure deficit ( p 1  −  p 4 )/ \((\rho {u}_{\infty }^{2})\) is provided as an input to the model to facilitate prediction of C T and C P . The Betz limit of a n  = 1/3 ( \({C}_{T}^{{\prime} }=2\) ) and C P  = 16/27 is also shown by the dashed green line.

Interestingly, the proposed model predicts the maximum value of C P as 0.5984, occurring at an induction factor of 0.345, which is ~1.0% and 3.5% higher than the classical Betz limit 38 (also known as the Lanchester-Betz-Joukowsky limit 23 , 56 , 57 ) for the coefficient of power (16/27) and power-maximizing induction factor (1/3), respectively. Since the Betz limit is derived using classical one-dimensional momentum modeling with the assumption that the wake pressure recovers to freestream conditions, it inherently dictates that the turbine cannot extract energy from the static pressure. In contrast, the model presented here does account for the persistent pressure drop in the far wake, resulting in a marginal increase in energy extraction. While these are modest departures from the classical Betz limit, this result elucidates that neglecting the energetic contributions of the wake pressure deficit will incorrectly model energy conservation. This result is further highlighted by the consistent underprediction of the coefficient of power C P of one-dimensional momentum modeling for induction factors of a n   ≳  1/3. The consistent one-dimensional momentum modeling error in the coefficient of power for a n   ≳  1/3 is alleviated by the pressure model proposed here. In Fig. 2 b, and d, the coefficients of thrust and power are shown as a function of the modified thrust coefficient \({C}_{T}^{{\prime} }\) 55 for \(0 \, < \, {C}_{T}^{{\prime} } \, < \, 12\) , which is a common alternative method to model actuator disks in CFD. As is evident in Fig.  2 a, the marginal increase in C T is reduced for higher values of \({C}_{T}^{{\prime} }\) , such that \({C}_{T}^{{\prime} }=12\) results in an induction factor of only a n  ≈ 0.7. This results from the definition of the thrust described previously, where the thrust force depends on the local thrust coefficient \({C}_{T}^{{\prime} }\) and the disk velocity \({\vec{u}}_{d}\) . As the local thrust coefficient \({C}_{T}^{{\prime} }\) is increased, the induction also increases. Correspondingly, the disk velocity \({\vec{u}}_{d}\) is lowered, which in turn lowers the thrust force. The counteracting effects of increasing \({C}_{T}^{{\prime} }\) and decreasing disk velocity limit the growth in the thrust coefficient C T , which depends on these variables jointly. Since thrust force and power produced by rotors are based on the velocity at the rotor ( \({\vec{u}}_{d}\) ), the local thrust coefficient model offers a more physically consistent representation of a wind turbine rotor 49 , 55 .

Since the modeling framework proposed here represents the relationship between thrust, power, and wake variables from first-principles, the predicted wake velocities conserve mass, momentum, and energy across arbitrary thrust coefficients and misalignment angles. The streamwise wake velocity and the density-normalized wake pressure deficit are shown in Fig.  3 a and c, respectively, as a function of the rotor-normal induction a n , and in Fig.  3 b,d as a function of the modified thrust coefficient \({C}_{T}^{{\prime} }\) . The methodology for computing wake quantities u 4 ,  v 4 ,  x 0 , and wake pressure from LES is described in the “Streamtube analysis numerical setup” in “Methods”. Above a n  = 0.5 and \({C}_{T}^{{\prime} }=4\) , classical momentum modeling predicts that the wake velocity is negative, which differs from the LES output. Instead, both the LES and the model predictions (a priori and a posteriori ) asymptote to zero wake velocity for a n  = 1 ( \({C}_{T}^{{\prime} }\to \infty\) ).

Wake streamwise velocity u 4 as a function of ( a ) rotor-normal induction factor, a n , and ( b ) local thrust coefficient \({C}_{T}^{{\prime} }\) . Density-normalized wake pressure deficit ( p 1  −  p 4 )/ \((\rho {u}_{\infty }^{2})\) as a function of ( c ) a n and ( d ) \({C}_{T}^{{\prime} }\) . The variables estimated by the presented Unified Momentum Model (Methods) are shown. The shaded region corresponds to  ± 10% uncertainty in β . Note that this uncertainty range is visually negligible for some plotted quantities and therefore may not be visible. Results from LES and classical one-dimensional momentum modeling are shown as a reference. Two additional a priori model results are shown. For the wake streamwise velocity u 4 ( a , b ), a priori model results are shown where the LES measured pressure deficit \(({p}_{1}-{p}_{4})/(\rho {u}_{\infty }^{2})\) is provided as an input to the model to facilitate prediction of wake velocity u 4 . For the wake pressure deficit \(({p}_{1}-{p}_{4})/(\rho {u}_{\infty }^{2})\) ( c , d ), the pressure deficits that are implied from energy conservation, with the input of LES measured a n and u 4 , are shown. The Betz limit of a n  = 1/3 and \({C}_{T}^{{\prime} }=2\) is also shown by the dashed green line.

The density-normalized wake pressure deficit (Fig.  3c ) from LES is first compared to the wake pressure that is implied by energy conservation in the control volume (further described in  Supplementary Information ). To estimate the wake pressure implied by energy conservation, we solve for the wake pressure deficit using the model equation for the induction factor a n , and we input the LES measured values for a n and the wake velocities. As shown in Fig.  3c , there is excellent agreement between the LES measured wake pressure and the wake pressure implied by energy conservation for a n  < 0.7, further validating the model form. As a n approaches unity, the accuracy of the Bernoulli equation in the wake is reduced, which lowers the quantitative accuracy of the pressure prediction in this region, but this has a limited quantitative impact on the primary variables of interest ( C T ,  C P , and wake velocities shown in Fig.  2 and Fig.  3 a). The reduction in the accuracy of the Bernoulli equation is related to the increasing degree of turbulence in the region immediately downwind of the turbine. The length of the near-wake reduces as the thrust coefficient increases (further detailed in  Supplementary Information ). This trend is captured in the model. However, even with a shrinking near-wake length, neglecting turbulent mixing incurs increasing error with increasing thrust. Still, the Bernoulli equation is used in the model form to preserve the computational efficiency of the analytical formulation. Finally, we show the fully predictive wake pressure results from the differential Euler equations and near-wake length closure proposed here. The predictive pressure model exhibits qualitative and quantitative agreement, with some increasing predictive error at values of induction above a n   > 0.7 from the Bernoulli equation as described previously. As the induction a n approaches unity, there is a nonlinear increase in the wake pressure deficit, with the limiting state of a n  = 1 resulting in flow separation 37 , 50 . At and above an induction of unity ( a n  ≥ 1), the wake flow is separated, resulting in the bluff body wake of a flat circular plate. In such cases, the presented model is no longer valid. While the regime of unity induction ( a n  ≥ 1) is likely not often relevant to wind power applications, it is useful to connect this Unified Momentum Model with research focused on bluff body wake dynamics 58 , 59 .

A parallel modeling approach to the integral analysis presented here represents the porous disk with a distribution of equal magnitude sources in potential flow 60 , 61 , 62 . However, these approaches have also in large part neglected the wake pressure deficit, similar to classical one-dimensional momentum modeling. Steiros & Hultmark (2018) 50 extended the source modeling approach by providing a more detailed representation of the wake, using momentum conservation and the Bernoulli equation, and by including a wake pressure term. To close the derived system of equations analytically without an additional pressure model equation, a simple wake factor was introduced to model the streamwise wake velocity. The one-dimensional model proposed by Steiros & Hultmark (2018) 50 exhibited excellent agreement compared to experimental measurements of the coefficient of drag (thrust) of porous plates immersed in a water tank. While the thrust predictions of this model exhibit substantial improvements for high values of induction compared to classical one-dimensional momentum modeling 50 , it is notable that the wake factor-based wake velocity model differs from classical one-dimensional momentum model predictions and CFD data, even for low induction values (below the heavily loaded limit of a n   ≈  0.37 25 ). Detailed comparisons between the model proposed here and the Steiros & Hultmark (2018) 50 model are made in the  Supplementary Information .

We further evaluate the model predictions for yaw misaligned rotors. Accurately predicting the power output of a yaw-misaligned wind turbine is crucial, given that turbines typically operate in yaw 27 , and for the effective flow control in wind farms through wake steering 5 , 36 . Recently, Heck et al. (2023) 49 demonstrated that the deviation of yawed rotor power production from \({\cos }^{3}(\gamma )\) results from the impact of the yaw misalignment on the induction factor a n , but the proposed model had increasing error with increasing thrust coefficients because it assumed that the wake pressure recovers to freestream pressure. The presented Unified Momentum Model extends the framework introduced by Heck et al. (2023), ensuring its validity at high thrust coefficients by relaxing the pressure recovery assumption. As previously discussed, the one unknown parameter β unspecified in the Unified Momentum Model was calibrated using LES data of yaw-aligned turbines to find that it was well within the range accepted in literature. Here, we use that fixed value of β calibrated to yaw-aligned LES data to predict out-of-sample conditions for a yaw misaligned turbine. In the  Supplementary Information , we show predictions for the near-wake length as a function of the yaw misalignment without re-calibrating β , demonstrating confidence in both the value of the parameter and its ability to accurately generalize to unseen conditions. In Fig.  4 , the coefficient of power C P is shown for yaw misalignments between 0  <  γ  <  50° and local thrust coefficients between \(0 < {C}_{T}^{{\prime} } < 4\) for the proposed model and for LES, along with a baseline model from classical one-dimensional momentum modeling. The model proposed here exhibits high levels of qualitative and quantitative accuracy compared to LES. Specifically, the model demonstrates an 84% and a 21% reduction in the mean absolute error of C P across the thrust coefficient and yaw misalignment values considered here compared to classical one-dimensional momentum modeling (with \(P(\gamma )=P(\gamma=0)\cdot {\cos }^{3}(\gamma )\) ) and to the model that neglects the wake pressure deficit 49 , respectively.

Coefficient of power C P as a function of the yaw misalignment γ and the local thrust coefficient \({C}_{T}^{{\prime} }\) for ( a ) classical one-dimensional momentum modeling and empirically modeling the effect of yaw misalignment as \(P(\gamma )=P(\gamma=0)\cdot {\cos }^{3}(\gamma )\) which is a common model 11 given the previous lack of a first-principles approach, ( b ) the presented Unified Momentum Model (Methods), and ( c ) LES. The power-maximizing \({C}_{T}^{{\prime} }\) set points as a function of γ are indicated in red.

A Blade Element Momentum (BEM) model based on the Unified Momentum Model

The Unified Momentum Model generalizes and directly replaces classical one-dimensional momentum theory in its many uses. We note that in the limit of neglecting the wake pressure and yaw misalignment, the Unified Momentum Model yields identical predictions to the limited classical theory. Two important applications for inviscid momentum modeling of rotors are to provide initial conditions to far-wake models that account for turbulence 44 , 45 , 46 , 47 and to provide the necessary closure for blade element momentum (BEM) modeling for rotors 14 , 15 , 63 . In state-of-the-art and widely-used BEM modeling, classical momentum theory deficiencies in high thrust and yaw misalignment are universally handled with empirical corrections 14 , 15 .

We develop a first-principles blade element momentum model based on the Unified Momentum Model that predicts high thrust and yawed operation without empirical corrections for these states for the first time. The classical momentum theory conventionally employed alongside numerous empirical corrections for high-thrust and yaw-misaligned scenarios is directly replaced in the BEM model with the Unified Momentum Model. There is no need to modify the blade element portion of BEM. The advantage of BEM modeling over momentum modeling alone is that BEM models realistic turbine controller parameters, such as blade pitch angle and rotor angular velocity (typically nondimensionalized as a tip-speed ratio) and their effect on rotor aerodynamics and forces (loads). Using the unified BEM model, the coefficient of power over a range of blade pitch angles and rotor tip-speed ratios is predicted and shown in Fig.  5 . We compare the unified BEM model developed here to standard approaches currently used. Using a BEM model dependent on classical one-dimensional momentum theory without a high-thrust correction (Fig.  5 a) fails to converge in regions where the thrust coefficient exceeds unity, rendering its solution undefined. Consequently, the global optimal operating point using this classical model lies on the convergence boundary, preventing the accurate determination of the optimal controller set point. By incorporating an empirical high-thrust correction (Fig.  5 b), the upper-left quadrant of the coefficient of power surface can be realized, allowing a globally optimal set point to be retrieved, but it requires empiricism to achieve which yields uncertainty. Replacing the classical momentum theory and empirical high-thrust correction with the Unified Momentum Model (Fig.  5 c) enables the prediction of the coefficient of power across all of the relevant pitch and tip-speed ratio regimes, and therefore the identification of the optimal control strategy without any empirical corrections.

Contour plots showing the variation of power coefficient ( C P ) with blade pitch angle ( θ p ) and blade tip-speed ratio ( λ ) using different thrust-induction momentum modeling closures in a blade element momentum (BEM) model implementation. Fully aligned ( a – c ) and yaw misaligned ( d – f ) conditions at γ  = 30° are considered. The momentum modeling closure used in the BEM includes ( a and d ) classical momentum theory, ( b and e ) classical momentum with high-thrust correction, and ( c and f ) the Unified Momentum Model, all incorporating Prandtl tip and root correction 83 in the blade element model.  C P -maximizing set-points are indicated with a marker.

The BEM models are used to identify the pitch and tip-speed ratio that maximizes the coefficient of power for the turbine. When the turbine is yaw misaligned, classical momentum theory without or with corrections (Fig.  5 d, e, respectively) predicts a decrease in power maximizing tip-speed ratio. This tip-speed ratio reduction causes a reduction in thrust force. In contrast, the BEM implementation using the Unified Momentum Model (Fig.  5 f) predicts that the power maximizing control strategy entails a decrease in pitch angle, which increases the thrust level, relative to fixing control at the yaw-aligned power maximizing set-point. This result agrees with recent literature which indicates that optimal control of wind turbines with yaw misalignment requires an increase in the thrust coefficient 32 , 64 . The novel BEM model based on the Unified Momentum Model is compared to blade-resolved wind turbine simulations 65 , demonstrating a threefold reduction in error for yaw of 30° compared to classical momentum theory. Further results comparing the BEM based on the Unified Momentum Model to standard empirical approaches are provided in the  Supplementary Information . In summary, the Unified Momentum Model yields a new approach to blade element momentum modeling that predicts yawed and high thrust operation from first-principles without empirical corrections in the momentum model. The new first-principles model also enables new engineering insights, for example, identifying the control strategy for a yaw misaligned turbine to maximize power production. This recommended control strategy qualitatively differs from the control recommendations predicted by classical momentum theory.

Unified Momentum Model predictions in atmospheric boundary layer flow

As in one-dimensional momentum theory, the Unified Momentum Model developed here is derived in the context of turbulence-free, uniform inflow across the disk area. However, one-dimensional momentum theory is ubiquitously used for predictions in realistic environments with turbulence and wind shear in BEM and wake modeling applications. Similarly, we investigate the fidelity of the Unified Momentum Model in turbulent atmospheric boundary layer (ABL) conditions typical for these wind power applications. First, in Fig.  6 a, c, e, and g, comparisons between the Unified Momentum Model and uniform inflow LES are shown for rotor quantities a n and C T , as well as near wake velocities in the streamwise δ u 0  =  u ∞   −   u 4 and spanwise δ v 0   =   v ∞   −  v 4 directions, where the ∞ subscript denotes the inflow velocities. Over a wide range of yaw angles γ ∈ [0, 40°] and local thrust coefficients \({C}_{T}^{{\prime} }\in [0.4,9.6]\) , the Unified Model quantitatively predicts relevant near-wake properties with high accuracy. Second, we investigate the predictions of these same quantities for an actuator disk immersed in ABL conditions using a comprehensive suite of large eddy simulations over a range of thrust coefficients and yaw angles. The numerical setup used for the seventy independent LES runs of the ABL is provided in “Methods”. We show a comparison between near-wake quantities extracted from LES with turbulent ABL inflow and the Unified Momentum Model in Fig.  6 b, d, f, and h for a range of yaw misalignment angles γ 1 ∈ [0, 45°] and thrust coefficients \({C}_{T}^{{\prime} }\in [0.4,4]\) . In general, model errors are greater in ABL inflow than for uniform inflow, particularly at the coincidence of high yaw-misalignment angles and thrust coefficients, as expected based on the dynamical regime considered in the model derivation. Still, the Unified Momentum Model lowers prediction error in turbulent ABL inflow across these yaw and thrust coefficient regimes by 60%, 83%, and 78% for the induction, streamwise wake velocity, and spanwise wake velocity, respectively, compared to classical one-dimensional momentum theory.

Model comparison for rotor and near-wake properties, measured as streamtube-averaged quantities, in uniform and atmospheric boundary layer (ABL) inflow as a function of local thrust coefficient \({C}_{T}^{{\prime} }\) and yaw γ . Subfigures show ( a , b ) rotor-normal induction factor, ( c , d ) thrust coefficient C T , ( e , f ) initial streamwise velocity deficit δ u 0  =  u ∞  −  u 4 , and ( g , h ) initial lateral velocity deficit δ v 0  =  v ∞  −  v 4 . Model predictions are compared with LES in ( a , c , e , g ) uniform inflow and ( b , d , f , h ) conventionally neutral ABL conditions.

The simple, first-principles analytical relationship between the induction, thrust, power, wake velocities, near-wake length, and wake pressure proposed here is ideally suited for implementation in a wide range of applications, such as blade-element momentum (BEM) modeling frameworks for wind and hydrokinetic turbines and propellers, such as OpenFAST 42 and HAWC2 66 , as well as in wake models used for wind farm design and control, such as the FLORIS model 41 and PyWake 67 . The predictive Unified Momentum Model developed here, which has a runtime on the order of microseconds on a standard desktop computer, has lower predictive error than classical one-dimensional momentum modeling for all thrust coefficients and misalignment angles considered, enabling rotor modeling for arbitrary thrust and yaw conditions without empirical corrections for the first time. The first-principles BEM model developed here based on the Unified Momentum Model accurately predicts the power and forces of a turbine without empirical corrections tailored to the high thrust and yaw-misaligned states, yielding new insights into the design and optimization of wind turbine control. The Unified Momentum Model improves our physical understanding of rotors operating across all thrust regimes, including a modification to the Betz limit 23 , 38 that is enabled by momentum and kinetic energy extraction from the pressure. Future work is encouraged to enhance the understanding of the shear layer growth parameter, β . Although this parameter exhibits a weak sensitivity on rotor quantities such as thrust and power, it exerts a more significant impact on wake variables, particularly governing the balance between wake pressure and wake velocity (see  Supplementary Information) . Further investigation is recommended to explore the universality of this parameter in comparison to high Reynolds number experimental measurements.

The growing trends in wind turbine design, such as larger rotor diameters, taller hub heights, and increasing complexity in design and control strategies, are pushing the limits of the applicability of existing modeling tools. Consequently, the scientific community is motivated to address grand scientific challenges 4 , focused on the design 68 and control 5 of modern utility-scale wind turbines and wind farms. The proposed Unified Momentum Model is a first-principles-based approach that unifies rotor thrust, yaw, and induction with the outlet velocity and pressure of flow through an actuator disk. It addresses major theoretical limitations in modeling rotor performance under high thrust and yaw misalignment. These complex operating conditions have historically relied on empirical models due to gaps in underlying theory. Such empirical approaches have been deployed widely across wind power applications, affecting BEM modeling employed for wind turbine design and control, as well as the wake models utilized for wind farm design and operation. Derived under assumptions of uniform, zero-turbulence inflow, the Unified Momentum Model provides an ideal starting point for extensions to more complex operational regimes such as turbulent, non-uniform inflow and rotational effects, rather than starting from a basis that already includes empirical models that are unlikely to extrapolate out-of-sample. The Unified Momentum Model yields equal or lower predictive error compared to LES than classical one-dimensional momentum theory for the yaw angles and thrust coefficients investigated here. Low predictive error is exhibited by the presented results across uniform inflow regimes and most atmospheric cases, yet opportunities persist to enhance modeling accuracy for high thrust coefficients and yaw angles when accounting for ABL flows. Future work should extend the analysis and model to consider unsteadiness, such as from floating motion for an offshore turbine, wind speed and direction shear, and turbulence.

This study employs integral analysis of the control volume enclosing the actuator disk (Fig.  1) to derive a system of analytical equations describing the relationships between rotor-normal induction, thrust force, power production, wake pressure, near-wake length, and wake outlet velocities. The derivation is described in detail in the model derivation section of the  Supplementary Information . The analytical equations consider both yaw-aligned and yaw-misaligned (porous) actuator disks. The inputs are the modified thrust coefficient \({C}_{T}^{{\prime} }\) and yaw misalignment angle γ . The modified thrust coefficient controls the thrust force as \({\vec{F}}_{\!T}=-\frac{1}{2}\rho {C}_{T}^{{\prime} }{A}_{d}{({\vec{u}}_{d}\cdot \hat{n})}^{2}\hat{n},\) where \({\vec{u}}_{d}\) is the velocity at the actuator disk, ρ is the fluid density, A d is the disk area, and \(\hat{n}\) is the unit vector normal to the disk. The thrust force can be written as a function of the rotor-normal induction factor a n and the yaw angle  \(\gamma\) ,  \({\vec{F}}_{T}=-\frac{1}{2}\rho {C}_{T}^{{\prime} }{A}_{d}{(1-{a}_{n})}^{2}{\cos }^{2}(\gamma ){u}_{\infty }^{2}\left[\cos (\gamma )\hat{\imath }+\sin (\gamma )\hat{\jmath}\right].\) The equations then solve for the rotor-normal induction a n , streamwise wake velocity u 4 , lateral wake velocity v 4 , near-wake length x 0 , and the pressure difference ( p 4  −  p 1 ) as outputs. The final form of the equations is:

where the freestream incident wind speed is u ∞ , the actuator disk diameter is D , and the unknown shear layer growth rate parameter is β  = 0.1403 as outlined in the  Supplementary Information . The pressure equation (Eq. ( 5 )) contains two terms. The first term is the pressure contribution from the actuator disk forcing, and the second term ( p N L ) is a nonlinear term that results from the advection. The variable p N L is the nonlinear pressure contribution to the outlet wake pressure, which is described in detail in the model derivation in  Supplementary Information .

In some applications, utilizing the thrust coefficient C T as the input parameter proves to be convenient compared to using the local thrust coefficient \({C}_{T}^{{\prime} }\) , for instance, when performing blade-element modeling of a rotor where the thrust coefficient is output by the blade-element model to be input into the momentum model. Another circumstance is when a utility-scale wind turbine’s thrust curve is available as a function of freestream wind speed u ∞ , adopting C T as the input variable can offer convenience. Because C T is a derived quantity that depends on both the induction and the yaw misalignment angle, to use C T as an input variable requires a sixth equation to be included in the set of equations above:

where the thrust coefficient is defined as \({C}_{T}=2\parallel \! {\vec{F}}_{T}\! \! \parallel /(\rho {A}_{d}{u}_{\infty }^{2}),\) where \(\parallel {\vec{F}}_{T}\parallel\) is the magnitude of the thrust force, which is further defined in the following section. Although it is possible to algebraically reformulate Eq. ( 1 )–( 5 ) in different ways to utilize the thrust coefficient C T as an input parameter, the presented set of six equations offers numerically robust solutions applicable to a broad range of input values and initial conditions when solved using fixed-point iteration as described in the  Supplementary Information . We emphasize that the two forms of the model, whether \({C}_{T}^{{\prime} }\) or C T is input, are mathematically and physically exactly equivalent. The only difference is how one chooses to represent the thrust force \({\vec{F}}_{T}\) in notation, but it is important that the thrust coefficient is defined to be consistent with the model form and geometry presented in this study.

Large eddy simulation numerical setup

Large eddy simulations are performed using PadéOps 69 , 70 , an open-source incompressible flow code 71 . The horizontal directions use Fourier collocation and a sixth-order staggered compact finite difference scheme is used in the vertical direction 72 . A fourth-order strong stability preserving (SSP) variant of the Runge-Kutta scheme is used for time advancement 73 and the sigma subfilter scale model is used 74 . Simulations are performed with uniform inflow with zero freestream turbulence, consistent with the derivation of classical momentum modeling. The boundary conditions are periodic in the x and y directions with a fringe region 75 used in the x -direction to remove the wake from recirculating. The simulations are performed with a domain size of L x  = 25 D in length and cross-sectional size L y  = 20 D ,  L z   = 10 D with 256  ×  512  ×  256 grid points and with the turbine 5 D downwind of the inlet.

Additional simulations with atmospheric boundary layer (ABL) inflow conditions are run using the concurrent-precursor method 76 . An empty, horizontally homogeneous domain without turbines is used to spin up turbulence using the initialization methodology from Liu et al. 77 with a surface roughness z 0  = 1 mm, a driving geostrophic wind speed of G   =  8 m s −1 , and a free atmosphere lapse rate of Γ   = 1 K km −1 . The Coriolis parameter f c is set to 10 −4  rad s −1 . The ABL LES domain is 3.84 km × 1.28 km × 1.28 km in the streamwise, lateral, and vertical directions, with a grid spacing of Δ x  = 10 m,  Δ y  = 5 m, and Δ z   =  5 m, respectively. Periodic boundary conditions are used in the lateral direction, and the bottom wall uses a wall model based on Monin-Obukhov similarity theory 70 , 78 . A single actuator disk with a diameter of 100 m is placed 5 D from the inlet at a hub height of 100 m in the ABL LES simulations.

The porous disk is modeled using an actuator disk model (ADM), that imparts a thrust force that depends on the modified thrust coefficient \({C}_{T}^{{\prime} }\) and the disk velocity \({\vec{u}}_{d}\cdot \hat{n}\) 55

where ρ is the fluid density, A d  =  π D 2 /4 is the area of the disk where D is the diameter, and \(\hat{n}\) is the unit normal vector perpendicular to the disk. Note that this differs from the simplified model of thrust force from an actuator disk that depends on the freestream rotor-normal wind speed  \({\vec{u}}_{\infty}\) , \({\vec{F}}_{T,{\rm{ideal}}}=-\frac{1}{2}\rho {C}_{T}{A}_{d}\parallel {\vec{u}}_{\infty }{\parallel }^{2}\hat{n}\) , where \({\vec{u}}_{\infty }={u}_{\infty }\hat{\imath }+0\hat{\jmath}\) is the freestream wind velocity vector and \({C}_{T}=2 \parallel {\vec{F}}_{T,{\rm{ideal}}} \parallel / (\rho {A}_{d}\parallel {\vec{u}}_{\infty }{\parallel }^{2} )\) . Porous disks produce thrust based on the wind velocity at the rotor, which has been modified by induction. The coefficient of thrust C T is an empirical quantity that depends on the magnitude of the induction, and it needs to be measured or predicted using a model. It is preferable for both analytical and numerical modeling, and more physically intuitive, to model the thrust force based on the velocity that is accessible to the porous disk, which is the disk velocity \({\vec{u}}_{d}\) , and thus \({C}_{T}^{{\prime} }\) is the input thrust coefficient. For a yaw-aligned turbine, \({C}_{T}^{{\prime} }={C}_{T}/{(1-a(\gamma=0))}^{2}\) 55 .

The rotor-normal, rotor-averaged induction factor a n for a disk with yaw misalignment angle γ is defined as

In the yaw-aligned case, the rotor-normal induction factor reduces to the standard axial induction factor a  = 1 −  u d / u ∞ . Combining Eqs. ( 7 ) and ( 8 ), the thrust force written in terms of the rotor-normal induction factor is then

The power for an actuator disk-modeled wind turbine is computed as \(P=-{\vec{F}}_{T}\cdot {\vec{u}}_{d}\) .

The numerical ADM implementation follows the regularization methodology introduced by Calaf et al. 55 and further developed by Shapiro et al. 79 . The porous disk thrust force \(\vec{f}(\vec{x})={\vec{F}}_{T}{\mathcal{R}}(\vec{x})\) is implemented in the domain ( \(\vec{x}\) ) through an indicator function \({\mathcal{R}}(\vec{x})\) . The indicator function \({\mathcal{R}}(\vec{x})\) is \({\mathcal{R}}(\vec{x})={{\mathcal{R}}}_{1}(x){{\mathcal{R}}}_{2}(y,z)\) where

where H ( x ) is the Heaviside function, \({\rm{erf}}(x)\) is the error function, s  = 3 Δ x /2 is the ADM disk thickness, and Δ is the ADM filter width. The disk velocity \({\vec{u}}_{d}\) , used in the thrust force calculation Eq. ( 7 ), is calculated using the indicator function, \({\vec{u}}_{d}=M\iiint {\mathcal{R}}(\vec{x})\vec{u}(\vec{x})\,{{\rm{d}}}^{3}\vec{x}\) , where \(\vec{u}(\vec{x})\) is the filtered velocity in the LES domain and M is a correction factor that depends on the filter width Δ 79 . Small values of filter width Δ tend towards the theoretical actuator disk representation (a true actuator disk represents the disk as infinitesimally thin with a discontinuity in forcing at radial position r   =   R ), but lead to numerical oscillations in LES. Given the nature of spatially distributing the thrust force with a larger value of Δ , numerical implementations of the ADM typically underestimate the induction and therefore overestimate power production 79 , 80 . The correction factor \(M=\left.\right(1+{({C}_{T}^{{\prime} }\Delta /(2\sqrt{3\pi }D))}^{-1}\) derived by Shapiro et al. 79 is used to correct this error by ensuring that the filtered ADM in Eq. ( 10 ) sheds the same amount of vorticity as the infinitesimally thin disk, depending on \({C}_{T}^{{\prime} }\) and the ADM filter width. Here, we use Δ / D   =  3 h /(2 D ) where \(h={(\Delta {x}^{2}+\Delta {y}^{2}+\Delta {z}^{2})}^{1/2}\) is the effective grid spacing. The qualitative and quantitative conclusions of this paper are not affected by this choice, as shown in “Comparison between different actuator disk model regularization methods in LES” in the  Supplementary Information , provided that the correction factor M derived by Shapiro et al. 79 is used for larger Δ . For small Δ values (e.g. Δ / D  = 0.032), numerical (grid-to-grid) oscillations contaminate the wake pressure measurements. Therefore, we have selected Δ / D   =  3 h /(2 D ) with M given above. In summary, sensitivity experiments are performed for different numerical ADM implementations in  Supplementary Information , and the results demonstrate that the qualitative conclusions of this study do not depend on the numerical implementation of the ADM.

Streamtube analysis numerical setup

We consider a three-dimensional streamtube analysis. While a yaw-aligned actuator disk in uniform inflow presents an axisymmetric streamtube, yaw misalignment results in wake curling 53 and three-dimensional variations that motivate a three dimensional streamtube analysis. A theoretical actuator disk model has uniform thrust force for all positions r  <  R within the rotor, where r is a radial position defined relative to the disk center and R is the radius of the disk. Numerical implementations of actuator disk models for computational fluid dynamics use regularization methodologies to avoid sharp discontinuities in the wind turbine body force 49 , 55 , 79 , as discussed in “Large eddy simulation numerical setup”, which results in some variation in the thrust force towards the outer extent of the disk radius. The degree to which there is variation in the thrust force for r  <   R depends on the numerical implementation of the regularization (i.e. filter length Δ in the present implementation). Following previous studies 26 , 49 , to focus the streamtube analysis on the portion of the wake over which the thrust force is constant, the streamtube seed points are defined at the actuator disk with an initial radius of R s  <  R , where R s / R  = 0.7. Numerical tests (not shown) demonstrate a small quantitative sensitivity between 0.5  <  R s / R   < 0.9, but the qualitative results are insensitive to R s .

While flow quantities within the streamtube depend on x , a core assumption in near-wake models is that there is a particular x location (or a range of x locations), that is characteristic of the inviscid near-wake (potential core), such that near-wake flow quantities can be described with a single value per variable, rather than a one-dimensional field variable depending on x . The fidelity of this assumption is investigated in “Streamtube analysis and budgets” in  Supplementary Information . Here, we describe the methodology to define these individual flow quantities from LES data. The near-wake streamwise velocity u 4 is taken as the minimum value of the streamtube-averaged streamwise velocity on the interval 0 <  x / D  < 5. This is equivalent to picking u 4 at the location of maximum wake strength. The near-wake length x 0 is also taken to be the x position where u 4 is sampled because this inflection point marks the onset of the turbulent wake where wake recovery begins. The wake pressure p 4 is also calculated at the near-wake length x  =  x 0 . Following Shapiro et al. 26 , the lateral velocity v 4 is taken as its maximum, which is closer to the actuator disk, approximately at x / D  = 0.5 (see discussion by Shapiro et al. 26 ). The same procedure for extracting the wake properties u 4 ,  v 4 ,  x 0 , and p 4 is also used for wind turbine wakes in ABL inflows.

Reporting summary

Further information on research design is available in the  Nature Portfolio Reporting Summary linked to this article.

Data availability

All data and figure-generating code is available as an open-source Python library at https://doi.org/10.5281/zenodo.10524066 81 .  Source data are provided with this paper.

Code availability

A reference implementation of the Unified Momentum Model is available as an open-source Python library at https://doi.org/10.5281/zenodo.10524066 81 . A blade element momentum implementation using the Unified Momentum Model is available at 10.5281/zenodo.11175618 82 . The large eddy simulation software source code, PadéOps 71 , is available at https://github.com/Howland-Lab/PadeOps . The Unified Momentum Model is also available open-source on Github ( https://github.com/Howland-Lab/Unified-Momentum-Model ). The blade element momentum model based on the Unified Momentum Model is open-source on Github ( https://github.com/Howland-Lab/MITRotor ).

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Acknowledgements

K.S.H. and M.F.H. acknowledge funding from the National Science Foundation (Fluid Dynamics program, grant number FD-2226053, Program Manager: Dr. Ronald D. Joslin). J.L. acknowledges support from Siemens Gamesa Renewable Energy. K.S.H. additionally acknowledges funding through a National Science Foundation Graduate Research Fellowship under grant no. DGE-2141064. Simulations were performed on the Stampede2 and Stampede3 supercomputers under the NSF ACCESS project ATM170028.

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M.F.H. conceived the research. M.F.H., J.L., and K.S.H. developed the Unified Momentum Model. J.L. developed the Unified Momentum Model code. J.L. and M.F.H. analysed the data. M.F.H. and K.S.H. performed large eddy simulations. All authors contributed to manuscript writing and edits.

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Liew, J., Heck, K.S. & Howland, M.F. Unified momentum model for rotor aerodynamics across operating regimes. Nat Commun 15 , 6658 (2024). https://doi.org/10.1038/s41467-024-50756-5

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DOI : https://doi.org/10.1038/s41467-024-50756-5

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