These models and their semantics now deserve a closer inspection. Distinct methodological traditions have arisen in various areas of research in the human sciences.
Included in these traditions are often strongly institutionalized preferences for particular sorts of statistical models. Although recent advances in generalized latent variable modeling e. By way of illustration, two popular classes of models will be described here, with particular attention to both their formal semantics and the manners in which they are commonly interpreted: MTMM CFA models and random-effects IRT testlet models.
The CTCM model is commonly presented as follows:. Under the assumption that T j , M j , and e j are mutually uncorrelated, the implied covariance structure is:.
A path diagram corresponding to this model is shown in Figure 1 here in the case in which there are three traits and three methods. This model, as a special case of linear confirmatory factor-analytic models more generally, models the conditional mean of the indicator variables, making it appropriate when said indicator variables are continuous. When estimated using maximum likelihood ML techniques as is common , it must be additionally assumed that these variables are normally distributed.
Since responses to individual test questions can rarely be scored continuously, item-level data are almost never modeled using CTCM models. This model, again like factor models more generally, also assumes that the latent variables are continuous. Formally, the CTCM model states that population-level variance 4 in each observed variable has three causes, each of which is modeled as a person-level random variable: the trait and method dimensions, and a specific factor. There are no formal semantic differences in the modeling of the trait and method causes other than their specific patterns of loadings the fact that T j and M j are separated out in Eq.
However, in practice, it is common for such dimensions to be interpreted primarily as nuisance dimensions, where only population-level parameters are of interest similarly to how unique factors are commonly interpreted.
Partly in reaction to this, modified forms of the CTCM model have been proposed, such as models with orthogonal methods factors correlated-trait uncorrelated-method CTUM. The method factor in the CTUM model could only be interpreted as denoting those attributes of persons that are truly specific to a particular method of measurement, and in many cases it may not be clear exactly what such attributes would be.
Another model proposed as an alternative to the CTCM model is the CU model Marsh and Grayson, , which drops the method factors entirely and allows the disturbances of observed variables that share a common method to correlate. There are a number of within-item multidimensional IRT models that have been developed that could be considered models for method effects. As mentioned previously, the concept of LID shares a conceptual relation with the concept of method effects: LID occurs when variation in some subset of item responses shares more in common than just their common cause represented by the primary latent variable, and methodological similarities among items are an obvious possible source of such shared variance.
Thus it could be said that method effects are one possible cause of LID, and, therefore, that models developed for LID may be used to model method effects. Various constrained versions of full information bi-factor models Gibbons and Hedeker, ; see also Holzinger and Swineford, have been proposed to model tests with testlet structure, one of the more famous of which is Bradlow et al.
Rijmen, testlet response model. The bi-factor model and Bradlow et al. Their model is as follows:. A separate equation for the covariance structure analogous to Eq.
In the case where slopes are estimated, there are various constraints that can be placed on the model for purposes of identification and interpretability: Bradlow et al. The logit link function makes these models appropriate for ordinal indicator variables.
The indicator variables are typically assumed to have Bernoulli distributions if dichotomous, and multinomial distributions if polytomous. As with all IRT models, it is assumed that the latent variables are continuous.
If the items within a testlet loaded onto more than one substantive dimension, this model would be equivalent to the CTUM model discussed earlier, albeit with a logit rather than identity link.
Wang and Wilson b also refer to Eq. In terms of its formal semantics, the model is even more general than that. A random testlet or facet effect is equivalent to a second dimension of individual differences that causes variation in a subset of item responses, and thus, induces stronger dependence amongst those items than would be expected due to their primary common cause s.
There is no reason why the logic of this cannot extend to other sources of variation in specific subsets of items on a test, such as the fact that different subsets of items represent different sources of information e. The previous two sections have illustrated how the formal semantics of models for method effects depend on the particulars of model specifications in terms of constraints, numbers of dimensions, choice of link functions, etc.
Informally, as discussed previously, interpretations of method effects depend largely on differences among research traditions in vocabulary, in the subject matter typically dealt with, and in the motivations commonly given for why method effects or LID are worthy of attention; these differences, combined with differences in the way the models are commonly presented illustrated by the different choices of symbols and the switch between vector and scalar notation between Eqs 1 and 3, as well as the differences in baseline assumptions concerning the link function and the number of substantive dimensions in the model , may give rise to the perception that these models and their associated semantics are entirely dissimilar.
This is, however, not the case: the models share a high degree of commonality at both the syntactic and semantic levels. It is easier to see the connections between the models if one starts with a more general model, and then derives the earlier models. Using the notation of Skrondal and Rabe-Hesketh , a generalized latent variable model can be formulated thusly:.
This response model can be combined with a structural model:. Although this account leaves out many details, it summarizes the essence of a generalized latent variable model.
On the other hand, the random-effects Rasch testlet model Eq. Returning to a visual examination of Figures 1 and 2 , it should be readily apparent that the two models share many features.
Path diagrams such as these are traditionally silent as to the nature of the link function represented by the arrows — in classical CFA models, the arrows represent linear effects i. These two omissions aside, however, it can be seen that in both cases variance in each indicator is influenced by two primary dimensions of individual differences, one of which is typically interpreted as denoting an attribute that the test was designed to measure, and the other of which denotes sources of variation associated with a particular method.
Once one is aware that a there is no upper bound on the number of indicator variables that load onto each dimension and b there can be multiple substantive dimensions in IRT models just as in linear CFA models, it becomes clear that both models can easily be represented by the same path diagrams, with the exception of the absence of indicator-specific unique factors or error terms in the model with a non-linear link function.
Thus, perhaps despite appearances to the contrary, latent variable models employed in different research traditions share deep syntactic connections, and, accordingly, share much of their formal semantics as well. In addition to the norms of statistical and interpretive practice associated with particular research traditions, thinking about method effects is also affected by beliefs many of which may not be explicitly recognized by researchers regarding the meaning of measurement itself.
There is not a single consensus definition of measurement accepted by all human scientists, or indeed by all physical scientists, and debates over the meaning of measurement will likely not see resolution any time soon. Obviously, unclear semantics about measurement can propagate to unclear semantics about any measurement-related concept, including but not limited to method effects.
It is worth reviewing some of the most influential lines of thought concerning measurement, and exploring how each of them has contributed to discourse on method effects, sometimes in contradictory ways. The various ways of thinking about measurement can broadly be categorized as either empiricist or realist. The term empiricism can refer to a broad range of philosophical positions; they share in common a commitment to direct observation as the basis for knowledge though what counts as observation is a perennially unsettled issue.
Empiricism has been a major force in shaping Western thinking about science and natural philosophy since at least as far back as Aristotle, and standard accounts of the history of Western science emphasize how, over the centuries, empiricist lines of thinking have dovetailed with other views in epistemology particularly those based on rationalism. In the early twentieth century, the movement known as logical positivism synthesized many ideas from classical empiricism along with then-current advances in the philosophy of language and mathematics.
Positivism was associated with a strong emphasis on direct observation as the basis for knowledge and a category rejection of metaphysics; statements regarding unobservable theoretical entities or forces were only regarded as meaningful if such statements could be linked to observations in a clear and consistent manner.
There are two major strands of thought on measurement that are consistent with much of positivist thinking. The first is representational measurement theory RMT , which is characterized by the stance that measurement is the construction of morphisms between numerical relations and empirical relations e. The second is operationalism , which is characterized by the stance that the meaning of any theoretical concept is exhausted by the operations undertaken to measure instances of the concept Bridgman, Representationalism is regularly described as the mainstream position on measurement in the general literature the philosophy of science.
It has also had a significant influence on thinking about measurement in the human sciences; however, with the exception of the relatively small body of literature in mathematical psychology from which the theory originated, most of this influence has been indirect.
Representational measurement theory holds that to measure is to construct a representation of an empirical relational system via a numerical relational system. On this view, the starting point for measurement is the determination of empirical relations amongst objects e. Michell, ; Borsboom, Once empirical relations are determined, numbers are assigned to empirical entities in such a way as to preserve the qualities of their empirical relations.
Relational systems can possess different sorts of structures, and the particular sort of mapping of empirical onto numerical relations determines the scale properties. This is an example of the aforementioned indirect influence of RMT on thinking about measurement in the human sciences. One of the principal reasons that RMT has not been more widely influential in the human sciences is that standard accounts of the theory have difficulty accounting for the role of measurement error.
RMT holds that relations must be directly observable; in contrast, statistical models employed in the human sciences such as those discussed in the previous section take observations to be error-prone reflections of latent variables with idealized structures. One could formulate this hypothesis in at least two ways. In the first case, the method acts as a perfect conduit running from true relations in the world to sensed relations. In this case, one could either hold that the world does not exist apart from our perceptions of it, or that its existence is simply irrelevant.
Where, then, do methods play a role in RMT? Without an account for how observations can contain error, it seems the only answer can be that either a the method of measurement plays a trivial role in being a perfect conduit from the real to the sensed world, or b the very concept of a method of measurement is unnecessary, as measurement is simply the mapping of directly experienced relations onto numerical relations. In either case, if two different methods of measurement yield two different relational systems, they cannot be said to be measuring the same attribute.
Operationalism or operationism; Bridgman, shares with RMT a focus on observables as the basis of knowledge and a rejection of metaphysics. Operationalism was proposed as a semantic doctrine about the meaning of theoretical terms rather than a theory of measurement per se : operationalism holds that the meaning of theoretical terms is exhausted by the particular operations undertaken to observe them, which means that the results of a particular set of operations or measurement procedure are interpreted as measurements simply by fiat.
Operationalism was originally proposed as a form of extreme epistemic humility in reaction to the upending of seemingly basic concepts such as length by the special theory of relativity: Bridgman felt that one of the reasons that it had been so difficult to see that the Newtonian notion of absolute time and space was flawed was that our theoretical terms came with too much baggage.
Thus, asking why the lengths of objects seemed to be different depending on the speed with which they were traveling was already an ill-formed question, in that hidden within it was a false assumption about the nature of space.
Operationalism has since been almost uniformly rejected as irreconcilable with general scientific practice and vocabulary. Following the collapse of logical positivism and an associated general retreat from extreme forms of empiricism, many scholars became increasingly willing to accept that the interpretation of concepts like temperature and intelligence outrun their associated measurement procedures — and, in fact, it is very difficult to make sense of both scientific and lay discourse about such concepts without this belief.
Operationalism had a strong influence on psychology and in particular, behaviorism , especially through Boring e. More generally, and again like RMT, the concept of measurement error is ill-fitting with operationalism: if the results of applying a procedure are by definition a measurement of the theoretical term, what is there to be in error about? If one were willing to accept that repeated applications of the same measurement procedure under the same conditions could yield different results, and one were willing to accept a definition of the theoretical term in terms of the average of a series of replications of a procedure rather than the outcome of a single application of that procedure, one could define measurement error as random deviations from a true long-run average; in fact, this is exactly how measurement error is defined in Classical Test Theory, a point argued by Borsboom Moreover, given our lack of access to the true counterfactual of running the same procedure under the same conditions, it is unclear why results should actually be expected to differ over identical applications.
If results differ because the conditions are themselves different, then, according to the doctrine of operationalism, one does not have measurement error — one has distinct theoretical concepts. Thus, at least in their original, strict forms, the two major lines of empiricist thought on measurement have little room for the concept of a method effect, as it is commonly interpreted in human science measurement.
As soon as one has formulated the idea that an attribute of an object or person can be observed in more than one way, it seems one has also assigned an independent identity to the attribute, and embraced at least some version of a realist stance on measurement. The term realism also refers to a broad range of positions; what they share in common is the belief that a natural world exists independently of observation.
Scientific realism further proposes that at least one aim of science is to promote the acquisition of knowledge about this natural world. In the context of measurement e. It should be noted that while the strict forms of empiricism discussed in the previous section are either antirealist or simply arealist, there is nothing inherently contradictory about a commitment to observation as the basis of knowledge and the belief that a natural world exists independently of observation; thus, realist philosophies are often compatible with more moderated forms of empiricism.
There are various possible ways to conceive of the relationship between a measured attribute and the outcomes of a measurement procedure. Borsboom et al. For example, a mercury thermometer measures temperature because variation in temperature the attribute causally produces variation in the expansion of mercury in precisely calibrated glass tubes the observations.
The link of causality from the attribute to the outcomes of the procedure justifies the inference from those observed outcomes back to the unobserved attribute. The validity of such a procedure is clearly threatened to the extent to which anything besides the targeted attribute can causally produce variation in the outcomes of the procedure. That is, if there is some other attribute of objects e. Another source of such variance would be actual variance in methods, insofar each the outcomes of different methods applied to the same objects may have different expectations.
Decoupling method-specific variance from attribute variance under a realist framework thus requires nothing more than knowing what attribute is the target of measurement, and how variance in this attribute is transmitted to variance in the outcomes of the measurement procedure. If it is possible to give a complete account of the causal processes leading from variation in the attribute to variation in observations, any additional causes of variation in observations can be clearly identified as attribute-irrelevant, and threats to the validity of the measurement procedure.
To the extent to which such additional sources of variation are associated with the particular method of observation used, they could be termed method effects. This account raises an important conceptual point about method effects: inherent in the idea of a method effect is that, at least in principle, more than one measurement procedure i. This hypothesis is broadly consistent with the semantics of the MTMM and testlet models discussed earlier and displayed in Figures 1 and 2.
However, it may not always be clear to what extent an attribute is conceptually independent of the methods of measurement, especially in human science applications.
The definition of temperature as an attribute of objects or systems is now very precise, and thermodynamic theory can specify the causal mechanisms that lead from variation in temperature to variation in the outcomes of the application of a range of specific measurement procedures including but not limited to the aforementioned mercury thermometer in a great amount of detail. Arguably, there are no cognitive theories so precisely developed, and the causal mechanisms that link attributes to observations are rarely if ever specified in such detail.
More generally, it is not always clear to what extent the method of observation is truly attribute-irrelevant, and to what extent the methods of observation help inform or even construct the meaning of the attribute. Though such interpretive difficulties have been acknowledged by a number of scholars, including Cronbach e. In part, this may be because researchers are intuitively working from a metaphysical position that might be termed constructive-realism rather than a stricter form of realism that holds that attributes exist fully independently of human-designed measurement procedures.
The concept of realism applied to psychological attributes is often taken to imply that the attributes in question are hypothesized to exist independently of human intentionality.
That is, stating that an attribute exists or is real is taken to imply that it exists in observer-independent ontologically objective fashion, just like supposedly physical attributes such as temperature and mass. This, in turn, is often interpreted as implying physical i. However, it is not necessary for psychological attributes to be ontologically objective for them to be real components of the natural world. Briefly, psychological attributes can be a to some extent ontologically subjective, in that they involve conscious phenomena with subjective first-person ontology, and b to some extent be composites delineated by contextually and pragmatically driven linguistic frames of reference, rather than being natural kinds or natural attributes, as the case may be in the classic sense e.
From this perspective, what constitutes a method effect is a contextualized and pragmatic issue, and methodological features of the very same procedure may be considered method effects or not relative to the conception of the attribute s being measured by the test. A contemporary example comes from the renewed interest on the part of the U. On performance tasks, students may be asked to for example perform short experiments or produce specified products. Suppose that there is some degree of disparity between the results of performance events and multiple-choice items concerning the relative levels of knowledge of the students.
It could be said that the two testing modalities each require a different set of method-specific ancillary skills in addition to the attribute intended to be measured e. Interestingly, much of the past and current rhetoric around the use of performance events in educational assessments is consistent with both possibilities cf. Stevens appears to have determined—though his exact conclusion is somewhat unclear—that the phrase keep and bear arms was a unitary term of art. Such single linguistic units, called binomials or multinomials, are common in legal writing.
Think of cease and desist or lock, stock, and barrel. As a result, Stevens concluded, there was no need to consider whether keep arms had a different meaning from bear arms.
Therefore, he had no reason to determine whether keep arms , by itself, could refer to an individual right. Joshua Feinzig and Joshua Zoffer: A constitutional case for gun control. The phrase keep and bear arms was a novel term. In short, keep and bear arms was not a term of art with a fixed meaning.
Indeed, the meaning of this phrase was quite unsettled then, as it had barely been used in other governmental documents. Ultimately, a careful study of the Second Amendment would have to treat keep arms and bear arms as two separate linguistic units, and thus two separate rights.
We performed another search in COFEA, about the meaning of keep arms , looking for documents in which keep and arms and their variants appear within six words of each other. The results here were somewhat inconclusive. The remainder of the hits did not support either reading. We could not find a dominant usage for what keep arms meant at the founding.
Thus, even if Scalia was wrong about the most common meaning of bear arms , he may still have been right about keep arms. Based on our findings, an average citizen of the founding era would likely have understood the phrase keep arms to refer to possessing arms for both military and personal uses. Finally, it is not enough to consider keep and bear arms in a vacuum. About 40 percent of the results had a militia sense, about 25 percent used an individual sense, and about 30 percent referred to both militia and individual senses.
The remainder were ambiguous. Political economists and political scientists have offered a plethora of explanations for why competition in laxity remains rare in environmental and consumer policy.
It remains to be seen why 43 loses its validity for long times. The obvious question, however, is why they are not used more in teaching design. I would like to know why they took place. Finding an answer to the question of why a transformation took place in prehistory, then, would ideally mean taking all factors involved into account.
It also demonstrates why that process took almost twenty-five years to begin. But why exactly is this supposed to be a problem?
Now there is no reason why they could not at the same time derive symbolic or evidential value from something else than these outcomes. Even so, an explanation why just these symptoms co-occur would be welcome. So why has such a synthesis been so slow to develop? Why then were those other thirty also quashed?
However, if something is too complex, or is perceived to be too complex, then why should biologists bother? The diversity of themes covered in the papers is the reason why this seemed to be such an eclectic mix. See all examples of why. These examples are from corpora and from sources on the web.
Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors. Translations of why in Chinese Traditional. See more. Need a translator? Translator tool. What is the pronunciation of why? Browse whorl. Test your vocabulary with our fun image quizzes. Image credits. Word of the Day have a heart of gold. Blog Outsets and onsets! Read More. November 08, To top.
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