The Theoretical Status of Latent Variables

Abstract

This article examines the theoretical status of latent variables as used in modern test theory models. First, it is argued that a consistent interpretation of such models requires a realist ontology for latent variables. Second, the relation between latent variables and their indicators is discussed. It is maintained that this relation can be interpreted as a causal one but that in measurement models for interindividual differences the relation does not apply to the level of the individual person. To substantiate intraindividual causal conclusions, one must explicitly represent individual level processes in the measurement model. Several research strategies that may be useful in this respect are discussed, and a typology of constructs is proposed on the basis of this analysis. The need to link individual processes to latent variable models for interindividual differences is emphasized.

Full text

The Theoretical Status of Latent Variables
Denny Borsboom, Gideon J. Mellenbergh, and Jaap van Heerden
University of Amsterdam
This article examines the theoretical status of latent variables as used in modern test theory models. First,
it is argued that a consistent interpretation of such models requires a realist ontology for latent variables.
Second, the relation between latent variables and their indicators is discussed. It is maintained that this
relation can be interpreted as a causal one but that in measurement models for interindividual differences
the relation does not apply to the level of the individual person. To substantiate intraindividual causal
conclusions, one must explicitly represent individual level processes in the measurement model. Several
research strategies that may be useful in this respect are discussed, and a typology of constructs is
proposed on the basis of this analysis. The need to link individual processes to latent variable models for
interindividual differences is emphasized.
Consider the following sentence: “Einstein would not have been
able to come up with his e ⫽mc
2
had he not possessed such an
extraordinary intelligence.” What does this sentence express? It
relates observable behavior (Einstein’s writing e ⫽mc
2
)toan
unobservable attribute (his extraordinary intelligence), and it does
so by assigning to the unobservable attribute a causal role in
bringing about Einstein’s behavior. In psychology, there are many
constructs that play this type of role in theories of human behavior;
examples are constructs like extraversion, spatial ability, self-
efficacy, and attitudes. Such variables are usually referred to as
latent variables. It is common to investigate the structure and
effect of unobservables like intelligence through the analysis of
interindividual differences data by statistically relating covariation
between observed variables to latent variables. This is done, for
example, in the widely used factor model. The idea is that although
the fit of a latent variable model to the data may not prove the
existence of causally operating latent variables, the model does
formulate this as a hypothesis; consequently, the fit of such models
can be adduced as evidence supporting this hypothesis. Finally, it
is often suggested that the type of causal relation tested in latent
variable modeling is similar to the relation between Einstein’s
intelligence and behavior in the above example; that is, the latent
variable exerts influence at the level of the individual.
Given the intuitive appeal of explaining a wide range of behav-
iors by invoking a limited number of latent variables, it is not
surprising that latent variables analysis has become a popular
technique in postbehaviorist psychology. The conceptual frame-
work of latent variables analysis, however, is older than cognitive
psychology and originates with the work of Spearman (1904), who
developed factor analytic models for continuous variables in the
context of intelligence testing. The basic statistical idea of latent
variables analysis is simple. If a latent variable underlies a number
of observed variables, then conditionalizing on that latent variable
will render the observed variables statistically independent. This is
known as the principle of local independence. The problem of
latent variables analysis is to find a set of latent variables that
satisfies this condition for a given set of observed variables.
With these insights, Spearman (1904) opened up a paradigm,
and the development of this paradigm in the 20th century has been
spectacular. The factor analytic tradition continued with the work
of Lawley (1943), Thurstone (1947), and Lawley and Maxwell
(1963), and it entered into the conceptual framework of confirma-
tory factor analysis (CFA) with Jo¨reskog (1971); Wiley, Schmidt,
and Bramble (1973); and So¨rbom (1974). In subsequent years,
CFA became a very popular technique, largely because of the
LISREL program by Jo¨reskog and So¨rbom (1993). In a research
program that developed mostly parallel to the factor analytic
tradition, the idea of latent variables analysis with continuous
latent variables was applied to dichotomous observed variables by
Guttman (1950), Lord (1952, 1980), Rasch (1960), Birnbaum
(1968), and Mokken (1971). These measurement models, primar-
ily used in educational testing, came to be known as Item Response
Theory (IRT) models. The IRT framework was extended to deal
with polytomous observed variables by Samejima (1969), Bock
(1972), and Thissen and Steinberg (1984). Meanwhile, in yet
another parallel research program, methods were developed to deal
with categorical latent variables. In this context, Lazarsfeld (1950),
Lazarsfeld and Henry (1968), and Goodman (1974) developed
latent structure analysis. Latent structure models may involve
categorical observed variables, in which case one speaks of latent
class analysis or metrical observed variables giving rise to latent
profile analysis (Bartholomew, 1987). After boundary-crossing
investigations by McDonald (1982), Thissen and Steinberg (1986),
Takane and de Leeuw (1987), and Goldstein and Wood (1989),
Denny Borsboom, Gideon J. Mellenbergh, and Jaap van Heerden, De-
partment of Psychology, University of Amsterdam, Amsterdam, the Neth-
erlands.
We thank Gitta Lubke, Sanneke Schouwstra, Maarten Speekenbrink,
and the members of the SEMNET electronic discussion group for many
useful discussions on latent variables analysis. Ingmar Visser, Helma van
den Berg, Conor Dolan, and Keith Markus have provided useful comments
on early versions of this article. Special thanks go out to Peter Molenaar,
whose work has stimulated many of the developments in this article.
Correspondence concerning this article should be addressed to Denny
Borsboom, Department of Psychology, Faculty of Social and Behavioral
Sciences, University of Amsterdam, Roetersstraat 15, 1018 WB Amster-
dam, the Netherlands. E-mail: dborsboom@fmg.uva.nl
Psychological Review Copyright 2003 by the American Psychological Association, Inc.
2003, Vol. 110, No. 2, 203–219 0033-295X/03/$12.00 DOI: 10.1037/0033-295X.110.2.203
203

Mellenbergh (1994) connected some of the parallel research pro-
grams by showing that most of the parametric measurement mod-
els could be formulated in a common framework.
At present, there are various developments that emphasize this
common framework for latent variables analysis, cases in point
being the work of Muthe´n and Muthe´n (1998), McDonald (1999),
and Moustaki and Knott (2000). Different terms are used to indi-
cate the general latent variable model. For example, Goldstein and
Wood (1989) use the term generalized linear item response model
(GLIRM), whereas Mellenbergh (1994) speaks of generalized
linear item response theory (GLIRT), and Moustaki and Knott
(2000) follow McCullagh and Nelder (1989) in using the term
generalized linear model (GLIM). We will adopt Mellenbergh’s
terminology and use the term GLIRT because it emphasizes the
connection with IRT and, in doing so, the fact that the model
contains at least one latent variable. Now, at the beginning of the
21st century, it would hardly be an overstatement to say that the
GLIRT model, at least among psychometricians and methodolo-
gists, has come to be the received view in the theory of psycho-
logical measurement.
The growing use of latent variables analysis in psychological
research means that explanations that make use of unobservable
theoretical entities are increasingly entertained in psychology. As
a consequence, the latent variable has come to play a substantial
role in the explanatory structure of psychological theories. Now,
concepts closely related to the latent variable have been discussed
extensively. These concepts include the meaning of the arrows in
diagrams of structural equation modeling (see, e.g., Edwards &
Bagozzi, 2000; Pearl, 1999; Sobel, 1994), the status of a strongly
related concept, namely the true score of classical test theory
(Klein & Cleary, 1967; Lord & Novick, 1968; Lumsden, 1976),
definitions of latent variables (Bentler, 1982; Bollen, 2002), spe-
cific instances of latent variables such as the Big Five Factors in
personality research (Lamiell, 1987; Pervin, 1994), and the trait
approach in general (Mischel, 1968, 1973). Also, the status of
unobservable entities is one of the major recurrent themes in the
philosophy of science of the past century, during which battles
were fought over the conceptual status of unobservable entities
such as electrons (for some contrasting views, see Cartwright,
1983; Devitt, 1991; Hacking, 1983; and Van Fraassen, 1980).
However, the theoretical status of the latent variable as it appears
in models of psychological measurement has not received a thor-
ough and general analysis as yet.
The following questions, for example, are relevant but seldom
addressed in detail. Should we assume that the latent variable
signifies a real entity or conceive of it as a useful fiction, con-
structed by the human mind? Should we say that we measure a
latent variable in the sense that it underlies and determines our
observations, or is it more appropriately considered to be con-
structed out of the observed scores? What exactly constitutes the
relation between latent variables and observed scores? Is this
relation of a causal nature? If so, in what sense? And, most
important, is latent variable theory neutral with respect to these
issues? In the course of discussing these questions, we will see that
latent variable theory is not philosophically neutral; specifically,
we will argue that, without a realist interpretation of latent vari-
ables, the use of latent variables analysis is hard to justify. At the
same time, however, the relation between latent variables and
individual processes proves to be too weak to defend causal
interpretations of latent variables at the level of the individual.
Further, we develop a distinction between several kinds of latent
variables on the basis of their relations with individual processes.
Before we start out on this investigation, some qualifications are
in order. Latent variable models for psychological measurement
are generally used in research in which a number of items, or tests,
are administered to a number of subjects at a single time point.
This type of model, which explains between-subjects covariation
by invoking latent variables on which subjects differ from each
other, is the primary topic of this paper. There are three reasons for
this. First, it is the most widely used model in psychology; second,
its formal theory is has been developed in great detail; and third,
it is the basis for some of the most influential latent variable
models around. These include those used in intelligence testing
(with the general intelligence model as a primary example) and
those used in personality research (with the five factor model as a
primary example). We denote this model as the standard measure-
ment model.
The structure of this article is as follows. First, it is argued that
the latent variable typically appears in two distinct ways: as a
formal–theoretical concept and as an operational–empirical con-
cept. In applications, these two concepts have to be connected. To
do this, however, we need a third—ontological—concept. We
distinguish three ontological frameworks that may be applied:
realism, constructivism, and operationalism. It is argued that a
realist account of the latent variable is required to maintain a
consistent connection between the formal and empirical concept of
a latent variable. The realist view requires an account of the
relation between the latent variable and its indicators, for which
causality is a natural candidate. We inquire whether such an
interpretation can be defended, and if so, how this causal relation
should be interpreted. Finally, we discuss the implications of our
analysis for research in psychology.
Three Ways of Looking at the Latent Variable
If one carefully examines the practice of testing, it appears that
there are at least two distinct ways in which the concept of a latent
variable is used. The first is as a formal, technical term, and the
second as an empirical term. The formal concept figures in math-
ematical treatments, whereas the empirical concept is a function of
the observed scores (often a weighted sumscore). For example, a
five factor model may be fitted to personality data. On the basis of
this model, factor scores can be constructed by summing appro-
priately weighted item (or subtest) scores. It is natural to connect
the formal and empirical concepts by conceiving of such a
weighted sumscore as an “estimate”of or as a “proxy”for the
latent variable of interest, as is customary in the literature; in the
example, the weighted sumscore of all items loading on the factor
extraversion would be considered an estimate of the level of
extraversion.
We will argue that this position is not without problems. Spe-
cifically, to make the connection, we need an ontology for the
latent variable. This requires an account from a third stance, which
we term the ontological stance. We will argue that the ontology
must be realist in nature. To clarify the problem situation, we will
discuss the formal and empirical connotations of the term latent
variable before establishing a connection between the two.
204 BORSBOOM, MELLENBERGH, AND VAN HEERDEN

The Formal Stance: Syntax
In modern test theory models, such as the various IRT models or
confirmatory factor models, the relation between the latent vari-
able and the observed scores is mathematically explicit. In GLIRT,
the form for this relation is a generalized regression function of the
observed scores on the latent variable. This regression function
may differ in form (e.g., it is linear for the factor model but logistic
for the Rasch, 1960, model; see also Mellenbergh, 1994). For
instance, in a factor model for general intelligence, one would
specify that an increase of nunits in the latent variable leads to an
increase of ntimes the factor loading in the expected value of a
given item. So, formally, the model is just a regression model, but
the independent variable is latent rather than manifest. The inge-
nious idea in latent variable modeling is that although the model
cannot be tested directly for any given item because the indepen-
dent variable is latent, it can be tested indirectly through its
implications for the joint probability distribution of the item re-
sponses for a number of items.
Now there are two things one can do on the basis of the set of
formal assumptions underlying latent variables analysis. First, one
can determine how observed scores would behave if they were
“generated”under our model (this applies not only to mathemat-
ical derivations but also to simulation studies). Second, one can
develop plausible procedures to estimate parameters in the model
on the basis of manifest scores, given the assumption that these
scores were generated by our model. It is often implicitly sug-
gested that the formal derivations reveal something about reality,
but this is not the case. Each supposition inside the formal system
is a tautology, and tautologies in themselves cannot reveal any-
thing about the world. So this is all in the syntactic domain; that is,
it has no meaning outside the formal theory. We will denote the
latent variable as it appears in this “formal stance”(i.e., the
concept indicated by
␪
, in the IRT literature and by
␩
or
␰
in the
structural equation modeling [SEM] literature) as the formal latent
variable.
The Formal Stance: Semantics
The syntax of latent variable theory specifies a regression of the
observed scores on the latent variable. What are the semantics
associated with this relation? In other words, how do we interpret
this regression?
The syntax of latent variables analysis is taken from statistics,
and so are its semantics. Statistics is concerned with the behavior
of random variables, that is, with variables whose actual realiza-
tion is determined in a chance experiment. It is clear that the
interpretation of such variables as random, and the statistical
treatment that is based on that interpretation, is related to the
unpredictability of the processes that lead to the outcome of the
chance experiment. The justification for using statistical tech-
niques depends, in general, on the plausibility of such an interpre-
tation. This means that one has to show that the variable of interest
can, in some sense, be conceived of as a variable whose values are
determined by a chance experiment, so that the variable can be
considered a proper random variable.
In psychological measurement, the outcome variable that must
be conceived of as random is the item response. After all, it is the
expectation of the item response that goes into the regression
formulas. At first sight, however, it is not at all clear why a
response to an item in a psychological test should be considered a
random variable. It is therefore important to interpret the item
response in such a way as to justify this approach. This is rarely
stated explicitly in treatments of psychological measurement, but it
is crucial to the applicability of statistical models. This paragraph
is concerned with possible interpretations of the response to an
item, say, the item “2⫹2⫽...,”that may be used to justify
treating such a response as a random variable.
The main question is, how does one interpret the conditional
probability distribution of the observed variables, given the latent
variable? Although there may be many possible interpretations of
this distribution, we focus on two consistent interpretations that
were distinguished by Holland (1990). The first interpretation,
known as the stochastic subject interpretation, takes the probabil-
ity distribution as applying to the individual subject. This inter-
pretation implies a series of hypotheticals of the form, “Given that
Subject A has Value X on the latent variable, A has Probability
Distribution Y over the item responses.”Supposing that the imag-
inary subject John takes an intelligence test item, this would
become something like, “Given that John’s level of intelligence
is 2 standard deviations below the population mean, he has a
probability of .70 to answer the item ‘2⫹2⫽...’correctly.”For
subjects with different positions on the latent variable, different
parameters for the probability distribution in question are speci-
fied. So, for John’s brighter sister Jane we could get, “Given that
Jane’s level of intelligence is 1 standard deviation above the
population mean, Jane has a probability of .99 to answer the item
correctly.”The item response function (i.e., the regression of the
item response on the latent variable) then specifies how the prob-
ability of a correct answer changes with the position on the latent
variable.
The second interpretation we discuss is the repeated sampling
interpretation, which is more common in the literature on factor
analysis (see, e.g., Meredith, 1993) than in the literature on IRT.
This is a between-subjects formulation of latent variables analysis.
It focuses on characteristics of populations instead of characteris-
tics of individual subjects. The probability distribution of the item
responses, conditional on the latent variable, is conceived of as a
probability distribution that arises from repeated sampling from a
population of subjects with the same position on the latent vari-
able. In particular, parameters of these population distributions are
related to the latent variable in question.
Thus, the repeated sampling interpretation is in terms of a series
of sentences of the form, “The population of As with Value X on
the latent variable follows Distribution Y over the item responses.”
Now, the probability distribution over the item responses that
pertains to a specific Value X of the latent variable arises from
repeated sampling from the population of As having this value. In
this interpretation, the probability that John answers the item
correctly does not play a role. Rather, the focus is on the proba-
bility of drawing a person that answers the item correctly from a
population of people with John’s level of intelligence, and this
probability is .70. In other words, 70% of the population of people
with John’s level of intelligence (i.e., a level of intelligence that
is 2 standard deviations below the population mean) will answer
the item correctly with probability 1, and 30% of those people will
answer the item correctly with probability 0. There is no random
variation located within the person.
205
THEORETICAL STATUS OF LATENT VARIABLES

The difference between the stochastic subject and repeated
sampling interpretations is substantial, for it concerns the very
subject of the theory. The two interpretations entertain different
conceptions of what it is one is modeling: in the stochastic subject
formulation, one is modeling characteristics of individuals,
whereas in the repeated sampling interpretation, one is modeling
between-subjects variables. However, if one follows the stochastic
subject interpretation and assumes that everybody with John’s
level of intelligence has probability .70 of answering the item
correctly, then the expected proportion of subjects with this level
of intelligence who will answer the item correctly (repeated sam-
pling interpretation) is also .70. The assumption that the measure-
ment model has the same form within and between subjects has
been identified as the local homogeneity assumption (Ellis & Van
den Wollenberg, 1993). Via this assumption, the stochastic subject
formulation suggests a link between characteristics of the individ-
ual and between-subjects variables. Ellis and Van den Wollenberg
(1993) have shown, however, that the local homogeneity assump-
tion is an independent assumption that in no way follows from the
other assumptions of the latent variable model. Also, the assump-
tion is not testable, because it specifies what the probability of an
item response would be in a series of independent replications with
intermediate brainwashing in the Lord and Novick (1968, p. 29)
sense. Basically, this renders the connection between within-
subject processes and between-subjects variables speculative (in
the best case). In fact, we will argue later on that the connection is
little more than an article of faith; the standard measurement model
has virtually nothing to say about characteristics of individuals,
and even less about item response processes. This will prove
crucially important for the ontology of latent variables, to be
discussed later in this paper.
The Empirical Stance
Before we discuss the ontology of the latent variable, we make
an observation in the empirical domain. This observation is simple:
If observed variables behave in the right way, a latent variable
model will fit. By “in the right way,”we mean that the pattern of
scores behaves according to the model. For some models, this
requirement is more stringent than for others. In a standard CFA,
for example, only first-and second-order moments are involved in
the analysis, so that the requirement applies only to this part of the
data structure; for a Rasch (1960) model, additional requirements
concerning the pattern of scores are necessary. However, the
central point is both simple and instructive: The explanandum
(observed scores) can be discussed separately from the explanans
(the model).
The well known problem of underdetermination (any set of data
can be explained by an indefinite number of theories) illustrates
why the model cannot be considered identical with or implied by
the corresponding empirical structure and, as a matter of fact,
should be considered strongly distinct from that structure. In a
statistical context, the problem of underdetermination translates
into the idea that many data-generating mechanisms (i.e., models)
may lead to the same dataset. There is a connection here with the
issue of equivalent statistical models (see, e.g., Hershberger,
1994). In this context it has, for instance, been shown by Bar-
tholomew (1987; see also Molenaar & von Eye, 1994) that a latent
profile model with platent profiles generates the same first-and
second-order moments (means, variances, and covariances) for the
observed data as a factor model with p⫺1 continuous latent
variables. The models are conceptually different: The factor model
posits continuous latent variables (i.e., it specifies that subjects
vary in degree but not in kind), whereas the latent profile model
posits categorical latent variables at the nominal level (i.e., it
specifies that subjects vary in kind but not in degree). This sug-
gests, for example, that the five factor model in the personality
literature corresponds to a typology with six types. Moreover, on
the basis of the covariances used in factor analysis, the Big Five
factors would be indistinguishable from the Big Six types. That
such theoretically distinct models can be practically equivalent in
an empirical sense urges a strong distinction between the formal
and empirical structure of latent variables analysis.
We make this point because it emphasizes that the attachment of
theoretical content to a latent variable requires an inferential step
and is not in any way “given”in empirical data, just as it is not
given in the mathematical formulation of a model. The latent
variable as it is viewed from the empirical stance (i.e., the empir-
ical entity that is generally presented as an estimate of the latent
variable) will be denoted here as the operational latent variable
(after Sobel, 1994). Note that there is nothing latent about the
operational latent variable. It is simply a function of the observed
variables, usually a weighted sumscore (that the weights are de-
termined via the theory of the formal latent variable does not make
a difference in this respect). Note also that such a weighted
sumscore can always be obtained and will in general be judged
interpretable if the corresponding model fits the data adequately.
The foregoing discussion shows, however, that the fit of a model
does not entail the existence of a latent variable. A nice example
in this context is given by Wood (1978), who showed that letting
people toss a number of coins (interpreting the outcome of the
tosses as item responses) yields an item response pattern that is in
perfect agreement with the Rasch (1960) model. A more general
treatment is given in Suppes and Zanotti (1981) who show that for
three two-valued observed variables, a latent variable can be found
if and only if the observed scores have a joint distribution. The
developments in Bartholomew (1987) and Molenaar and von Eye
(1994) further show that model fit does not entail the form (e.g.,
categorical or continuous) of the latent variable, even if its exis-
tence is assumed a priori.
The above discussion shows that the connection between the
formal and operational latent variable is not self-evident. To make
that connection, we need an interpretation of the use of formal
theory in empirical applications. This, in turn, requires an ontology
for the latent variable.
The Ontological Stance
The formal latent variable is a mathematical entity. It figures in
mathematical formulas and statistical theories. Latent variable
theory tells us how parameters that relate the latent variable to the
data could be estimated, if the data were generated under the model
in question. The if in the preceding sentence is very important. It
points the way to the kind of ontology we require. The assumption
that it was this model, and not some other model, that generated
the data must precede the estimation process. In other words, if one
considers the weighted sumscore as an estimate of the position of
a given subject on a latent variable, one does so under the model
206 BORSBOOM, MELLENBERGH, AND VAN HEERDEN

specified. Now this weighted sumscore is not an estimate of the
formal latent variable; one does not use an IQ score to estimate the
general concept usually indicated by the Greek letter
␪
, but to
estimate intelligence. Thus, one uses the formal side of the model
to acquire knowledge about some part of the world; then it follows
that one estimates something that is also in that part of the world.
What is that something?
It will be clear that in answering this question, one must con-
sider the ontology of the latent variable, which is, in quite a crucial
way, connected to its theoretical status. An ontological view is
needed to connect the operational latent variable to its formal
counterpart, but at first sight there seems to be a considerable
freedom of choice regarding this ontology. We will argue that this
is not the case.
There are basically three positions one can take with respect to
this issue. The first position adheres to a form of entity realism in
that it ascribes an ontological status to the latent variable in the
sense that it is assumed to exist independent of measurement. The
second position could be coined constructivist in that it regards the
latent variable as a construction of the human mind, which need
not be ascribed existence independent of measurement. The third
position maintains that the latent variable is nothing more than the
empirical content it carries—a“numerical trick”used to simplify
our observations: This position holds that there is nothing beyond
the operational latent variable and could be called operationalist.
Strictly taken, operationalism is a kind of constructivism, but we
intend the latter term to indicate a broader class of views (e.g., the
more sophisticated empiricist view of Van Fraassen, 1980). In fact,
we think that only the first of these views can be consistently
attached to the formal content of latent variable theory.
Note that our discussion of these views is not meant to constitute
an exhaustive categorization of the possible positions one may
take. For present purposes, however, the gap between realism and
constructivism is more important than the fine line separating
various forms of each position. For this reason, we limit our
attention to these views.
Operationalism and the Numerical Trick
We will first discuss the last view—that the latent variable is
nothing but the result of a numerical trick to simplify our obser-
vations. In this view, the latent variable is a (possibly weighted)
sumscore and nothing more. There are several objections that can
be raised against this view. A simple way to see that it is flawed
is to take any standard textbook on latent variable theory and to
replace the term latent variable with weighted sumscore. This will
immediately render the text incomprehensible. It is, for example,
absurd to assert that there is a sumscore underlying the item
responses. The obvious response to this argument is that one
should not take such texts literally or, worse, that one should
maintain an operationalist point of view. Such a move, however,
raises more serious objections.
If the latent variable is to be conceived of in an operationalist
sense, then it follows that there is a distinct latent variable for
every single test one constructs. This is a direct consequence of the
operationalist view (Bridgman, 1927), which holds that the mean-
ing of a concept is synonymous with the set of operations used to
measure it. Therefore, distinct sets of operations define distinct
concepts (Suppe, 1974). In the present context, this implies that
different sets of items must necessarily measure different latent
variables. This is inconsistent with the basic idea of latent variable
theory. To see this, consider a simple test consisting of three items
a,b, and c. In the operationalist view, the latent variable that
accounts for the item responses on the subtest consisting of items
aand bis different from the latent variable that accounts for the
item response pattern on the subtest consisting of items band c.
So, the test consisting of items a,b, and cdoes not measure the
same latent variable and therefore cannot be unidimensional. In
fact, in the operationalist view, it is impossible even to formulate
the requirement of unidimensionality; consequently, an operation-
alist would have a very hard time making sense of procedures
commonly used in latent variable theory, such as adaptive testing,
in which different tests are administered to different subjects with
the objective to measure a single latent variable. We conclude that
operationalism and latent variable theory are fundamentally
incompatible.
A related view holds that the use of latent variable theory is
merely instrumental, a means to an end. This is the instrumentalist
point of view (Toulmin, 1953), which is akin to operationalism. In
this view, the latent variable is a pragmatic concept, a “tool,”that
is merely useful for its purpose (the purpose being prediction or
data reduction, for example). No doubt, methods such as explor-
atory factor analysis may be used as data reduction techniques, and
although principal components analysis seems more suited as a
reduction technique, they are often used in this spirit. Also, such
models can be used for prediction, although it has been forcefully
argued by several authors (e.g., Maxwell, 1962) that the instru-
mentalist view leaves us entirely in the dark when confronted with
the question of why our predictive machinery (i.e., the model)
works. We do not have to address such issues in detail, however,
because the instrumentalist view simply fails to provide us with a
structural connection between the formal and operational latent
variable. In fact, the instrumental interpretation begs the question.
Suppose that we interpret latent variable models as data reduction
devices. Why, then, are the factor loadings determined via formal
latent variable theory in the first place? Obviously, in this view, no
weighting of the sumscore can be structurally defended over any
other. Any defense of this position must therefore be as ad hoc as
the use of latent variables analysis for data reduction itself.
1
Realism and Constructivism
So, if there is more to the latent variable than just a calculation
used to simplify our observations, what is it? We are left with a
choice between realism, maintaining that latent variable theory
should be taken literally—the latent variable signifying a real
entity—and constructivism, stating that it is a fiction, constructed
by the human mind.
The difference between realism and constructivism resides
mainly in the constructivists’denial of one or more of the realists’
claims. Realism exists in a number of forms, but in general a realist
will maintain one or several of the following theses (Devitt, 1991;
1
This should not be read as a value judgment. We think data reduction
techniques are very important, especially in the exploratory phases of
research. That these techniques are important, however, does not entail that
they are not ad hoc.
207
THEORETICAL STATUS OF LATENT VARIABLES

Hacking, 1983). First, there is realism about theories; the core
thesis of this view is that theories are either true or false. Second,
one can be a realist about the entities that figure in scientific
theories; the core thesis of this view is that at least some theoretical
entities exist. Third, realism is typically associated with causality;
theoretical entities are causally responsible for observed phenom-
ena. These three ingredients of realism offer a simple explanation
for the success of science; we learn about entities in the world
through a causal interaction with them, the effect of this being that
our theories get closer to the truth. The constructivist, however,
typically denies both realism about theories and about entities. The
question is whether a realist commitment is implied in latent
variables analysis. We will argue that this is the case; latent
variable theory maintains both theses in the set of assumptions
underlying the theory.
Entity realism is weaker than theory realism. For example, one
may be a realist about electrons, in which case one would maintain
that the theoretical entities that we call electrons correspond to
particles in reality. This does not imply realism about theories; for
example, one may view theories about electrons as abstractions,
describing the behavior of such particles in idealized terms (so that
these theories are, literally taken, false). Cartwright (1983) takes
such a position. Theory realism without entity realism is much
harder to defend, for a true theory that refers to nonexistent entities
is difficult to conceive of. We will first discuss entity realism
before turning to the subject of theory realism.
Entity Realism
Latent variable theory adheres to entity realism, because this
form of realism is needed to motivate the choice of model in
psychological measurement. The model that is customary in psy-
chological measurement is the model depicted in the left panel of
Figure 1. (We borrow the symbolic language from the structural
equation modeling literature, but the structure of the model gen-
eralizes to IRT and other latent variable models.) The model
specifies that the pattern of covariation between the indicators can
be fully explained by a regression of the indicators on the latent
variable, which implies that the indicators are independent after
conditioning on the latent variable (this is the assumption of local
independence). An example of the model in the left panel of the
figure would be a measurement model for, say, dominance, in
which the indicators are item responses on items like, “I would like
a job where I have power over others,”“I would make a good
military leader,”and “I try to control others.”Such a model is
called a reflective model (Edwards & Bagozzi, 2000), and it is the
standard conceptualization of measurement in psychology. An
alternative model that is more customary in sociological and
economical modeling is the model in the right panel of Figure 1.
In this model, called a formative model, the latent variable is
regressed on its indicators. An example of a formative model is the
measurement model for socioeconomic status (SES). In such a
model a researcher would, for example, record the variables in-
come, educational level, and neighborhood as indicators of SES.
The models in Figure 1 are psychometrically and conceptually
different (Bollen & Lennox, 1991). There is, however, no a priori
reason why, in psychological measurement, one should prefer one
type of measurement model to the other.
2
The measurement mod-
els that psychologists use are typically of the reflective kind. Why
is this?
The obvious answer is that the choice of model depends on the
ontology of the latent variables that it invokes. A realist point of
view motivates the reflective model because the response on the
questionnaire items is thought to vary as a function of the latent
variable. In this case, variation in the latent variable precedes
variation in the indicators. In ordinary language, dominant people
will be more inclined to answer the questions affirmatively than
submissive people. In this interpretation, dominance comes first
and “leads to”the item responses. This position implies a regres-
sion of the indicators on the latent variable and thus motivates the
choice of model. In the SES example, however, the relationship
between indicators and latent variable is reversed. Variation in the
indicators now precedes variation in the latent variable; SES
changes as a result of a raise in salary and not the other way
around.
Latent variables of the formative kind are not conceptualized as
determining our measurements but as a summary of these mea-
surements. These measurements may very well be thought to be
determined by a number of underlying latent variables (which
would give rise to the spurious model with multiple common
causes of Edwards & Bagozzi, 2000), but one is not forced in any
way to make such an assumption. Now, if one wanted to know
how to weigh the relative importance of each of the measurements
comprising SES in predicting, say, health, one could use a forma-
tive model like the one depicted in the right panel of Figure 1. In
such a model, one could also test whether SES acts as a single
variable in predicting health. In fact, this predictive value would be
the main motivation for conceptualizing SES as a single latent
2
It is in itself an interesting (and neglected) question as to where to draw
the line separating these classes of models at the substantive level. For
example, which of the formal models should be applied to the relation
between diagnostic criteria and mental disorders in the Diagnostic and
Statistical Manual of Mental Disorders (4th ed.; American Psychiatric
Association, 1994)?
Figure 1. Two models for measurement. The left panel is the reflective
measurement model. The Xs are observed variables,
␰
is the latent variable,
␭
s are factor loadings, and the
␦
s are error terms. The right panel shows the
formative model. The latent variable is denoted
␩
, the
␥
s are the weights
of the indicators, and
␨
is a residual term.
208 BORSBOOM, MELLENBERGH, AND VAN HEERDEN

variable. However, nowhere in this development has it been shown
that SES exists independent of the measurements.
The formative model thus does not necessarily require a realist
interpretation of the latent variable that it invokes. In fact, if a
realist interpretation were to be given, it would be natural to
conceptualize this as a spurious model with multiple common
causes in the sense of Edwards and Bagozzi (2000). This would
again introduce a reflective part in the model, which would cor-
respond to that part of the model that has a realist interpretation.
Thus, the realist interpretation of a latent variable implies a reflec-
tive model, whereas constructivist, operationalist, or instrumental-
ist interpretations are more compatible with a formative model.
In conclusion, the standard model in psychological measure-
ment is a reflective model that specifies that the latent variable is
more fundamental than the item responses. This implies entity
realism about the latent variable, at least on the hypothetical side
of the argument (the assumptions of the model). Maybe more
important than this is that psychologists use the model in this spirit.
In this context, Hacking’s (1983) remark that “the final arbitrator
in philosophy is not how we think but what we do”(p. 31) is
relevant; the choice for the reflective measurement model in psy-
chology expresses realism with respect to the latent variable.
Theory Realism
Theory realism is different from entity realism in that it con-
cerns the status of the theory, over and above the status of the
entities that figure in the theory. It is therefore a stronger philo-
sophical position. The realist interpretation of theories is naturally
tied to a correspondence view of truth (O’Connor, 1975). This
theory constructs truth as a “match”between the state of affairs as
posed by the theory and the state of affairs in reality and is the
theory generally endorsed by realists (Devitt, 1991). The reason
why such a view is connected to realism is that to have a match
between theoretical relations and relations in reality, these rela-
tions in reality have to exist quite independently of what we say
about them. For the constructivist, of course, this option is not
open. Therefore, the constructivist will either deny the correspon-
dence theory of truth and claim that truth is coherence between
sentences (this is the so-called coherence theory of truth) or deny
the relevance of the notion of truth altogether, for example by
positing that not truth, but empirical adequacy (consistency of
observations with predictions) is to be taken as the central aim of
science (Van Fraassen, 1980).
The formal side of latent variable theory, of course, does not
claim correspondence truth; it is a system of tautologies and has no
empirical content. The question, however, is whether a correspon-
dence type of assumption is formulated in the application of latent
variable theory. There are three points in the application where this
may occur: first, in the evaluation of the position of a subject on
the latent variable; second, in the estimation of parameters; and
third, in conditional reasoning based on the assumption that a
model is true.
In the evaluation of the position of a subject on the latent
variable, correspondence-truth sentences are natural. The simple
reason for this is that the formal theory implies that one could be
wrong about the position of a given subject on the latent variable,
which is possible only with the assumption that there is a “true”
position. To see this, consider the following. Suppose you have
administered an intelligence test, and you successfully fit a unidi-
mensional latent variable model to the data. Suppose that the single
latent variable in the model represents general intelligence. Now
you determine the position on the latent variable for 2 subjects, say
John and Jane. You find that the weighted sumscore (i.e., the
operational latent variable) is greater for John than for Jane, and
you tentatively conclude that John occupies a higher position on
the trait in question than Jane (i.e., you conclude that John is more
intelligent). Now could it be that you have made a mistake, in that
John actually has a lower score on the trait than Jane? The formal
theory certainly implies that this is possible (in fact, this is what
much of the theory is about; the theory will even be able to specify
the probability of such a mistake, given the positions of John and
Jane on the latent variable), so that the answer to this question must
be affirmative. This forces commitment to a realist position be-
cause there must be something to be wrong about. That is, there
must be something like a true (relative) position of the subjects on
the latent trait in order for your assessment to be false. You can, as
a matter of fact, never be wrong about a position on the latent
variable if there is no true position on that variable. Messick (1989)
concisely expressed this point when he wrote, “One must be an
ontological realist in order to be an epistemological fallibilist”
(p. 26).
This argument is related to the second point in the application
where one finds a realist commitment, namely in the estimation of
parameters. Here, we find essentially the same situation, but in a
more general sense. Estimation is a realist concept: Roughly
speaking, one could say that the idea of estimation is meaningful
only if there is something to be estimated. Again, this requires the
existence of a true value; in a seriously constructivist view of latent
variable analysis, the term parameter estimation should be re-
placed by the term parameter determination, for it is impossible to
be wrong about something if it is not possible to be right about it.
And estimation theory is largely concerned with being wrong: It is
a theory about the errors one makes in the estimation process. At
this point, one may object that this is a problem only within a
frequentist framework, because the idea of a “true”parameter
value is typically associated with frequentism (Fisher, 1925; Hack-
ing, 1965; Neyman & Pearson, 1967). It may further be argued that
using Bayesian statistics (Lee, 1997; Novick & Jackson, 1974)
could evade the problem. Within a Bayesian framework, however,
the realist commitment becomes even more articulated. A Bayes-
ian conception of parameter estimation requires one to specify a
prior probability distribution over a set of parameter values. This
probability distribution reflects one’s degree of belief over that set
of parameter values. Because it is a probability distribution, how-
ever, the total probability over the set of parameter values must be
equal to 1. This means that, in specifying a prior distribution, one
explicitly acknowledges that the probability (i.e., one’s degree of
belief) that the parameter actually has a value in the particular set
is equal to 1. In other words, one states that one is certain about
that. The statement that one is certain that the parameter has a
value in the set implies that one can be wrong about that value.
And now we are back in the original situation: It is very difficult
to be wrong about something if one cannot be right about it. In
parameter estimation, this requires the existence of a true value.
The third point in the application of latent variables analysis in
which one encounters correspondence truth is in conditionals that
are based on the assumption that a model is true. In the evaluation
209
THEORETICAL STATUS OF LATENT VARIABLES

of model fit, statistical formulations use the term true model; for
example, the pvalue resulting from a goodness-of-fit chi-square
test is computed under the null hypothesis that the model is true.
Psychometricians are, of course, aware that this is a very stringent
condition for psychological measurement models to fulfill. So, in
discussions on this topic, one often hears that there is no such thing
as a true model (Browne & Cudeck, 1992; Cudeck & Browne,
1983). For example, McDonald and Marsh (1990) stated, “It is
commonly recognized, although perhaps not explicitly stated, that
in real applications no restrictive model fits the population, and all
fitted restrictive models are approximations and not hypotheses
that are possibly true”(p. 247). It would seem that such a suppo-
sition, which is in itself not unreasonable, expresses a move away
from realism. This is not necessarily the case. The supposition that
there is no true model actually leaves two options: Either all
models are false, or truth is not relevant at all. The realist who
adheres to a correspondence view of truth must take the first
option. The constructivist will take the second and replace the
requirement of truth with one of empirical adequacy.
If the first option is taken, the natural question to ask is, in what
sense is the model false? Is it false, for example, because it
assumes that the latent variable follows a normal distribution
although this is not the case? So interpreted, one is still a realist;
there is a true model, but it is a different model from the one we
specified, that is, one in which the latent variable is not normally
distributed. The fact that the model is false is, in this sense,
contingent on the state of affairs in reality. The model is false, but
not necessarily false (i.e., it might be correct in some cases, but it
is false in the present application). One could, in this view,
reformulate the statement that there is no such thing as a true
model as the statement that all models are misspecified. That this
interpretation of the sentence “all models are false”is not contrary
to, but in fact parasitic on realism, can be seen because the whole
notion of misspecification requires the existence of a true model,
for how can we misspecify if there is no true model? Now, one
may say that one judges the (misspecified) model close enough to
reality to warrant the estimation procedures. One then interprets
the model as “approximately true.”So, with this interpretation, one
is firmly in the realist camp, even though one acknowledges that
one has not succeeded in formulating the true model. This is as far
as realists could go in the acknowledgement that their models are
usually wrong. Popper (1963) was a realist who held such a view
concerning theories.
The constructivist must take the second option and move away
from the truth concept. The constructivist will argue that one
should not interpret the statement that the model is true literally,
but weaken the requirement to one of empirical adequacy. The
whole concept of truth is thus judged irrelevant. The assumption
that the model is true could then be restated as the assumption that
the model fits the observable item response patterns perfectly at
the population level. This renders the statistical assumption that a
model is true (now interpreted as “empirically adequate”) mean-
ingful, because it allows for disturbances in the observed fit due to
random sampling, without assuming a realist view of truth. How-
ever, so interpreted, underdetermination rears its ugly head.
For example, take a simple case of statistically equivalent co-
variance structure models such as the ones graphically represented
in Figure 2 (based on Hershberger, 1994). These models are
empirically equivalent. This means that if one of them fits the data,
the other will fit the data equally well. If the assumption that
Model A is true is restated as the assumption that it is empirically
adequate (i.e., it fits the item responses perfectly at the population
level), the assumption that Model A is true is fully equivalent to
the assumption that Model B is true.
Now try to reconstruct the estimation procedure. The estimation
of the correlation between the latent variables
␰
1
and
␰
2
takes place
under the assumption that Model B is true. Under the empirical
adequacy interpretation, however, this assumption is equivalent to
the assumption that Model A is true, for the adjective true as it is
used in statistical theory now merely refers to empirical adequacy
at the population level. This implies that the assumption that
Model B is true may be replaced by the assumption that Model A
is true, for these assumptions are the same. However, this would
Figure 2. Two equivalent models. The structural equation models in the
figure predict the same variance–covariance matrix and are thus empiri-
cally equivalent. Xs indicate observed variables,
␰
s are latent variables,
␭
s
are factor loadings,
␦
s are error terms, and
␾
is the correlation between
latent variables.
210 BORSBOOM, MELLENBERGH, AND VAN HEERDEN

mean that the correlation between the latent variables
␰
1
and
␰
2
can
be estimated under the assumption that Model A is true. In Model
A, however, there is only one latent variable. It follows that in the
empirical adequacy view, the correlation between two latent vari-
ables can be estimated under the assumption that there is only one
latent variable underlying the measurements. In our view, this is
not particularly enlightening. But it must be said that the situation
need not necessarily bother the constructivist, because the con-
structivist did not entertain a realist interpretation of these latent
variables in the first place. However, it would take some ingenious
arguments to defend this interpretation.
In summary, the evaluation of the position of a subject on the
latent variable, the process of estimating parameters, and the
conditional reasoning based on the assumption that a model is true
are characterized by realist commitments. It would be difficult to
interpret these procedures without an appeal to some sort of
correspondence truth. However, what we have shown is only that
the natural interpretation of what one is doing in latent variables
analysis is a realist one, not that it is the only interpretation. It may
be that the constructivist could make sense of these procedures
without recourse to truth. For now, however, we leave this task to
the constructivist and contend that theory realism is required to
make sense of latent variables analysis.
Causality
The connection between the formal and the operational latent
variable requires a realist ontology. The question then becomes,
what constitutes the relation between the latent variable and its
indicators? Note that this question is not pressing for the opera-
tionalist who argues that the latent variable does not signify
anything beyond the data, which implies that the relation between
the latent variable and its indicators is purely logical. Nor need it
bother the constructivist who argues that people construct this
relation themselves; it is not an actual but a mental relation,
revealing the structure of the theories rather than a structure in
reality. The realist will have to come up with something different,
for the realist cannot maintain either of these interpretations.
The natural candidate, of course, is causality. That a causal
interpretation may be formulated for the relation between latent
variables and their indicators has been argued by several authors
(e.g., Edwards & Bagozzi, 2000; Glymour, 2001; Pearl, 1999,
2000), and we will not repeat these arguments. The structure of the
causal relation is known as a common cause relation (the latent
variable is the common cause of its indicators) and has been
formulated by Reichenbach (1956). Here, we will concentrate on
the form of the relation in a standard measurement model. Specif-
ically, we will argue that a causal connection can be defended in
a between-subjects sense, but not in a within-subject sense.
For this purpose, we must distinguish between two types of
causal statements that one can make about latent variable models.
First, one can say that population differences in position on the
latent variable cause population differences in the expectation of
the item responses. In accordance with the repeated sampling
interpretation, this interpretation posits no stochastic aspects
within persons; the expectation of the item response is defined
purely in terms of repeated sampling from a population of subjects
with a particular position on the latent variable. Second, one can
say that a particular subject’s position on the latent variable causes
his or her item response probabilities. This interpretation corre-
sponds to the stochastic subject interpretation and does pose prob-
abilities at the level of the individual. The first of these views can
be defended, but the second is very problematic.
To start with the least problematic, consider the statement that
differences in the latent variable positions (between populations of
subjects) causes the difference in expected item responses (be-
tween populations of subjects). This posits the causal relation at a
between-subjects level. The statement would fit most accounts of
causality, for example the three criteria of Mill (1843). These hold
that X can be considered a cause of Y if (a) X and Y covary; (b)
X precedes Y; and (c) ceteris paribus, Y does not occur if X does
not occur. In the present situation, we have (a) covariation between
the difference in position on the latent variable and the difference
in expected item responses; (b) in the realist viewpoint, the dif-
ference in position on the latent variable precedes the difference in
expected item responses; and (c) if there is no difference in
position on the latent variable, there is no difference in expected
item responses. The between-subjects causal statement can also be
framed in a way consistent with other accounts of causality, for
example the counterfactual account of Lewis (1973) or the related
graph–theoretical account of Pearl (1999, 2000). We conclude that
a causal relation can be maintained in a between-subjects form. Of
course, many problems remain. For example, most latent variables
cannot be identified independently of their indicators. As a result,
the causal account violates the criterion of separate identifiability
of effects and causes, so that circularity looms. However, this is a
problem for any causal account of measurement (Trout, 1999), and
the main point is that the relation between the latent variable and
its indicators can at least be formulated as a causal one.
The individual account of causality, however, is problematic.
Consider the statement that Subject A’s position on the latent
variable causes Subject A’s item response. The main problem here
is the following. One of the essential ingredients of causality is
covariation. All theories of causality use this concept, be it in a real
or in a counterfactual manner. If X is to cause Y, X and Y should
covary. If there is no covariation, there cannot be causation (the
reverse is, of course, not the case). One can say, for example, that
striking a match caused the house to burn down. One of the reasons
that this is possible is that a change in X (the condition of the
match) precedes a change in Y (the condition of the house). One
cannot say, however, that Subject A’s latent variable value caused
his item responses, because there is no covariation between his
position on the latent variable and his item responses. An individ-
ual’s position on the latent variable is, in a standard measurement
model, conceptualized as a constant, and a constant cannot be a
cause. The same point is made in a more general context by
Holland (1986) when he says that an attribute cannot be a cause.
The obvious way out of this issue is to invoke a counterfactual
account of causation (see, e.g., Lewis, 1973; Sobel, 1994). With
this account, one analyzes causality using counterfactual alterna-
tives. This is done by constructing arguments such as, “X caused
Y, because if X had not happened, ceteris paribus, Y would not
have happened.”This is called a counterfactual account because X
did in fact happen. For the previous example, one would have to
say, “The striking of the match caused the house to burn down,
because the house would not have burned down if the match had
not been struck.”For our problem, however, this account of
causality does not really help. Of course, we could construct
211
THEORETICAL STATUS OF LATENT VARIABLES

sentences like, “If Subject A had had a different position on the
latent variable, Subject A would have produced different item
responses,”but this raises some difficult problems.
Suppose, for example, that one has administered Einstein a
number of IQ items. Consider the counterfactual statement, “If
Einstein had been less intelligent, he would have scored lower on
the IQ items.”This seems like a plausible formulation of the
hypothesis tested in a between-subjects model, and it also seems as
if it adequately expresses the causal efficacy of Einstein’s intelli-
gence, but there are strong reasons for doubting whether this is the
case. For example, we may reformulate the above counterfactual
statement as follows: “If Einstein had had John’s level of intelli-
gence, he would have scored lower on the IQ items.”But does this
counterfactual statement express the causal efficacy of intelligence
within Einstein? It seems to us that what we express here is not a
within-subject causal statement at all, but a between-subjects con-
clusion in disguise, namely, the conclusion that Einstein scored
higher than John because he is more intelligent than John. Simi-
larly, “If Einstein had had the intelligence of a fruit fly, he would
not have been able to answer the IQ items correctly”does not
express the causal efficacy of Einstein’s intelligence, but the
difference between the population of humans and the population of
fruit flies. We know that fruit flies act rather stupidly, and so are
inclined to agree that Einstein would act equally stupidly if he had
the intelligence of a fruit fly. And it seems as if this line of
reasoning conveys the idea that Einstein’s intelligence has some
kind of causal efficacy. However, the counterfactual statement is
completely unintelligible except when interpreted as expressing
knowledge concerning the difference between human beings (a
population) and fruit flies (another population). It does not contain
information on the structure of Einstein’s intellect and much less
on the alleged causal power of Einstein’s intelligence. It contains
only the information that Einstein will score higher on an IQ test
than a fruit fly because he is more intelligent than a fruit fly—but
this is exactly the between-subjects formulation of the causal
account. Clearly, the individual causal account transfers knowl-
edge of between-subjects differences to the individual and posits a
variable that is defined between subjects as a causal force within
subjects.
In other words, the within-subjects causal interpretation of
between-subjects latent variables rests on a logical fallacy (the
fallacy of division; Rorer, 1990). Once you think about it, this is
not surprising. What between-subjects latent variables models do
is specify sources of between-subjects differences, but because
they are silent with respect to the question of how individual scores
are produced, they cannot be interpreted as posing intelligence as
a causal force within Einstein. Thus, the right counterfactual
statement (which is actually the one implied by the repeated
sampling formulation of the measurement model) is between sub-
jects: the IQ score we obtained from the nth subject (who hap-
pened to be Einstein) would have been lower had we drawn
another subject with a lower position on the latent variable from
the population. Note, however, that our argument does not estab-
lish that it is impossible that some other conceptualization of
intelligence may be given a causal within-subject interpretation. It
establishes that such an interpretation is not formulated in a
between-subjects model and therefore cannot be extracted from
such a model; a thousand clean replications of the general intelli-
gence model on between-subjects data would not establish that
general intelligence plays a causal role in producing Einstein’s
item responses.
But what about variables like height? Is it not unreasonable to
say, “If Einstein had been taller, he would have been able to reach
the upper shelves in the library”? No, this is not unreasonable, but
it is unreasonable to assume a priori that intelligence, as a between-
subjects latent variable, applies in the same way as height does.
The concept of height is not defined in terms of between-subjects
differences, but in terms of an empirical concatenation operation
(Krantz, Luce, Suppes, & Tversky, 1971; Michell, 1999). Roughly,
this means that we know how to move Einstein around in the
height dimension (for example by giving him platform shoes) and
that the effect of doing this is tractable (namely, wearing platform
shoes will enable Einstein to reach the upper shelves). Moreover,
it can be assumed that the height dimension applies to within-
subject differences in the same way that it applies to between-
subjects differences. This is to say that the statements, “If Einstein
had been taller, he would have been able to reach the upper shelves
in the library”and “If we had replaced Einstein with a taller
person, this person would have been able to reach the upper
shelves in the library”are equivalent with respect to the dimension
under consideration. They are equivalent in this sense, exactly
because the dimensions pertaining to within- and between-subjects
variability are qualitatively the same: If we give Einstein platform
shoes that make him taller, he is, in all relevant respects, exchange-
able with the taller person in the example. We do not object to
introducing height in a causal account of this kind, because vari-
ations in height have demonstrably the same effect within and
between subjects. But it remains to be shown that the same holds
true for psychological variables like intelligence.
The analogy does, however, provide an opening: The individual
causal account could be defended on the assumption that intelli-
gence is like height, in that the within-subjects and between-
subjects dimensions are equivalent. However, the between-
subjects model does not contain this equivalence as an assumption.
Therefore, such an argument would have to rest on the idea that, by
necessity, there has to be a strong relation between models for
within-subjects variability and models for between-subjects vari-
ability. It turns out that this idea is untenable because there is a
surprising lack of relation between within-subjects models and
between-subjects models. To discuss within-subject models, we
now need to extend our discussion to the time domain. This is
necessary because to model within-subjects variability, there has to
be variability, and variability requires replications of some kind;
moreover, if variability cannot result from sampling across sub-
jects, it has to come from sampling within subjects. In this para-
digm, one could, for example, administer Einstein a number of IQ
items repeatedly over time, and analyze the within-subject covaria-
tion between item responses. The first technique of this kind was
Cattell’s so-called P-technique (Cattell & Cross, 1952), and the
factor analysis of repeated measurements of an individual subject
have been refined, for example, by Molenaar (1985). The exact
details of such models need not concern us here; what is important
is that in this kind of analysis, systematic covariation over time is
explained on the basis of within-subject latent variables. So, in-
stead of between-subjects dimensions that explain between-
subjects covariation, we now have within-subject dimensions that
explain within-subject covariation. One could imagine that if the
within-subject model for Einstein had the same structure as the
212 BORSBOOM, MELLENBERGH, AND VAN HEERDEN

between-subjects model, then the individual causal account would
make sense despite all the difficulties we encountered above.
In essence, such a situation would imply that the way in which
Einstein differs from himself over time is qualitatively the same as
the way in which he differs from other subjects at one single time
point. This way, the clause “If Einstein were less intelligent”
would refer to a possible state of Einstein at a different time point,
however hypothetical. More important, this state would, in all
relevant respects, be identical to the state of a different subject, say
John, who is less intelligent at this time point. In such a state of
affairs, Einstein and John would be exchangeable, like a child and
a dwarf are exchangeable with respect to the variable height. It
would be advantageous, if not truly magnificent, if a between-
subjects model would imply or even test such exchangeability.
This would mean, for example, that the between-subjects five
factor model of personality would imply a five factor model for
each individual subject. If this were to be shown, our case against
the individual causal account would be reduced from a substantial
objection to philosophical hairsplitting. However, the required
equivalence has not been shown, and the following reasons lead us
to expect that it will not, in general, be a tenable assumption.
The link connecting between-subjects variables to characteris-
tics of individuals is similar to the link we have been discussing in
the stochastic subject formulation of latent variable models, in
which the model for the individual is counterfactually defined in
terms of repeated measurements with intermediate brainwashing.
We have already mentioned that Ellis and Van den Wollenberg
(1993) have shown that the assumption that the measurement
model holds for each individual subject (local homogeneity) has to
be added to and is in no way implied by the model. One may,
however, suppose that although finding a particular structure in
between-subjects data may not imply that the model holds for each
subject, it would at least render this likely. Even this is not the
case. It is known that if a model fits in a given population, this does
not entail the fit of the same model for any given element from a
population, or even for the majority of elements from that popu-
lation (Molenaar, 1999; Molenaar, Huizenga, & Nesselroade, in
press).
So, the five factors in personality research are between subjects,
but if a within-subjects time series analysis would be performed on
each of these subjects, we could get a different model for each of
the subjects. In fact, Molenaar et al. (in press) have performed
simulations in which they had different models for each individual
(so, one individual followed a one-factor model, another a two-
factor model, etc.). It turned out that when a between-subjects
model was fitted to between-subjects data at any specific time
point, a factor model with low dimensionality (i.e., a model with
one or two latent variables) provided an excellent fit to the data,
even if the majority of subjects had a different latent variable
structure.
With regard to the five factor model in personality, substantial
discrepancies between intraindividual and interindividual struc-
tures have been empirically demonstrated in Borkenau and Osten-
dorf (1998). Mischel and Shoda (1998), Feldman (1995), and
Cervone (1997) have illustrated similar discrepancies between
intraindividual and interindividual structures. This shows that
between-subjects models and within-subject models bear no obvi-
ous relation to each other, at least not in the simple sense discussed
above. This is problematic for the individual causal account of
between-subjects models, because it shows that the premise “if
Einstein were less intelligent. . .”cannot be supplemented with the
conclusion “. . . then his expected item response pattern would be
identical to John’s expected item response pattern.”It cannot be
assumed that Einstein and John (or any other subject, for that
matter) are exchangeable in this respect, because at the individual
level, Einstein’s intelligence structure may differ from John’sin
such a way that the premise of the argument cannot be fulfilled
without changing essential components of Einstein’s intellect.
Thus, the data-generating mechanisms at the level of the individual
are not captured, not implied, and not tested by between-subjects
analyses without heavy theoretical background assumptions that,
in psychology, are simply not available.
The individual causal account is not merely implausible for
philosophical or mathematical reasons; for most psychological
variables, there is also no good theoretical reason for supposing
that between-subjects variables do causal work at the level of the
individual. For example, what causal work could the between-
subjects latent variable we call general intelligence do in the
process leading to Einstein’s answer to an IQ item? Let us recon-
struct the procedure. Einstein enters the testing situation, sits
down, and takes a look at the test. He then perceives the item. This
means that the bottom-up and top-down processes in his visual
system generate a conscious perception of the task to be fulfilled;
it happens to be a number series problem. Einstein has to complete
the series 1, 1, 2, 3, 5, 8, ...?Now he starts working on the
problem; this takes place in working memory, but he also draws
information from long-term memory (e.g., he probably applies the
concept of addition, although he may also be trying to remember
the name of a famous Italian mathematician of whom this series
reminds him). Einstein goes through some hypotheses concerning
the rules that may account for the pattern in the number series.
Suddenly he has the insight that each number is the sum of the
previous two (and simultaneously remembers that it was Fi-
bonacci). Now he applies that rule and concludes that the next
number must be 13. Einstein then goes through various motoric
processes that result in the appearance of the number 13 on the
piece of paper, which is coded as 1by the person hired to do the
typing. Einstein now hasa1inhisresponse pattern, indicating that
he gave a correct response to the item. This account has used
various psychological concepts, such as working memory, long-
term memory, perception, consciousness, and insight. But where in
this account of the processes leading to Einstein’s item response
did intelligence enter? The answer is nowhere. Intelligence is a
concept that is intended to account for individual differences, and
the model that we apply is to be interpreted as such. Again, this
implies that the causal statement drawn from such a measurement
model retains this between-subjects form.
The last resort for anyone willing to endorse the individual
causal account of between-subjects models is to view the causal
statement as an elliptical (i.e., a shorthand) explanation. The ex-
planation for which it is a shorthand would, in this case, be one in
terms of processes taking place at the individual level. This re-
quires stepping down from the macro level of repeated testing (as
conceptualized in the within-subjects modeling approach) to the
micro level of the processes leading up to the item response in this
particular situation. We will argue in the next paragraph that there
is merit to this approach in several respects, but it does not really
help in the individual causal account as discussed in this section.
213
THEORETICAL STATUS OF LATENT VARIABLES

The main reason for this is that the between-subjects latent vari-
able will not indicate the same process in each subject. Therefore,
the causal agent (i.e., the position on the latent variable) that is
posited within subjects on the basis of a between-subjects model
does not refer to the same process in all subjects. This contrasts
sharply with measures of, say, temperature, in which the same
process is responsible for different readings on a thermometer. In
such a case, the position on the latent variable could be taken as a
proxy for a process, and the causal explanation of observed scores
in terms of a latent variable could be viewed as an elliptical
explanation.
In psychological measurement, however, such an elliptical ex-
planation would refer to a qualitatively different process for dif-
ferent positions on the latent variable, probably even to different
processes for different people with the same position on the latent
variable. Jane, high on the between-subjects dimension general
intelligence, will in all likelihood approach many IQ items using a
strategy that is qualitatively different from her brother John’s. John
and his nephew Peter, equally intelligent, may both fail to answer
an item correctly, but for different reasons (e.g., John has difficul-
ties remembering series of patterns in the Raven task, whereas
Peter has difficulties in imagining spatial rotations). It is obvious
that this problem is even more serious in personality testing, in
which one generally does not even have the faintest idea of what
happens between item administration and item response. For this
reason, it would be difficult to conceive of a meaningful interpre-
tation of such an elliptical causal statement without rendering it
completely vacuous, in the sense that the position on the latent
variable is shorthand for whatever process leads to person’s re-
sponse. In such an interpretation, the within-subject causal account
would be trivially true, but uninformative.
On the basis of this analysis, we must conclude that the within-
subjects causal statement, that Subject A’s position on the latent
variable causes his item responses, does not sit well with existing
accounts of causality. A between-subjects causal relation can be
defended, although it is certainly not without problems. Such an
interpretation conceives of latent variables as sources of individual
differences but explicitly abstracts away from the processes taking
place at the level of the individual. The main reason for the failure
of the within-subjects causal account seems to be that it rests on
the misinterpretation of a measurement model as a process model,
that is, as a mechanism that operates at the level of the individual.
This fallacy is quite pervasive in the behavioral sciences. For
instance, part of the nature–nurture controversy, as well as con-
troversies surrounding the heritability coefficients used in genetics,
may also be due to this misconception. The fallacious idea that a
heritability coefficient of .50 for IQ scores means that 50% of an
individual’s intelligence is genetically determined remains one of
the more pervasive misunderstandings in the nature–nurture dis-
cussion. Ninety percent of variations in height may be due to
genetic factors, but this does not imply that my height is 90%
genetically determined. Similarly, a linear model for interindi-
vidual variations in height does not imply that individual growth
curves are linear; that 30% of the interindividual variation in
success in college may be predicted from the grade point average
in high school does not mean that 30% of the exams you passed
were predictable from your high school grades; and that there is a
sex difference in verbal ability does not mean that your verbal
ability will increase if you undergo a sex change operation. It is
clear to all that these interpretations are fallacious. Still, for some
reason, such misinterpretations are very common in the interpre-
tation of results obtained in latent variables analysis. However,
they can all be considered to be specific violations of the general
statistical maxim that between-subjects conclusions should not be
interpreted in a within-subjects sense.
Implications for Psychology
It is clear that between-subjects models do not imply, test, or
support causal accounts that are valid at the individual level. In
turn, the causal accounts that can be formulated and supported in
a between-subjects model do not address individuals. However,
connecting psychological processes to the latent variables that are
so prominent in psychology is of obvious importance. It is essen-
tial that such efforts be made, because the between-subjects ac-
count in itself does not correspond to the kind of hypotheses that
many psychological theories would imply, as these theories are
often formulated at the level of individual processes. The relation
(or relations) that may exist between latent variables and individ-
ual processes should therefore be studied in greater detail, and
preferably within a formalized framework, than has so far been
done. In this section, we provide an outline of the different ways
in which the relation between individual processes and between-
subject latent variables can be conceptualized. These different
conceptualizations correspond to different kinds of psychological
constructs. They also generate different kinds of research questions
and require different research strategies to substantiate conclusions
concerning these constructs.
First, theoretical considerations may suggest that a latent vari-
able is at the appropriate level of explanation for both between-
subjects and within-subjects differences. Examples of psycholog-
ical constructs that could be conceptualized in this manner are
various types of state variables such as mood, arousal, or anxiety,
and perhaps some attitudes. That is, it may be hypothesized for
differences in the state variable arousal, that the dimension on
which I differ from myself over time and the dimension on which
I differ from other people at a given time point are the same. If this
is the case, the latent variable model that explains within-subjects
differences over time must be the same model as the model that
explains between-subjects differences. Fitting latent variable mod-
els to time series data for a single subject is possible (Molenaar,
1985), and such techniques suggest exploring statistical analyses
of case studies to see whether the structure of the within-subject
latent variable model matches between-subjects latent variables
models. If this is the case, there is support for the idea that we are
talking about a dimension that pertains to both variability within a
subject and between-subjects variability. Possible states of a given
individual would then match possible states of different individu-
als, which means that in relevant respects, the exchangeability
condition discussed in the previous section holds. Thus, in this
situation we may say that a latent variable does explanatory work
both at the within-subject and the between-subjects level, and a
causal account may be set up at both of these. Following the
terminology introduced by Ellis and Van den Wollenberg (1993)
we call this type of construct locally homogeneous, in which
locally indicates that the latent variable structure pertains to the
level of the individual, and homogeneous refers to the fact that this
structure is the same for each individual.
214 BORSBOOM, MELLENBERGH, AND VAN HEERDEN

Locally homogeneous constructs will not often be encountered
in psychology in which myriads of individual differences can be
expected to be the rule rather than the exception. We would not be
surprised if for the majority of constructs, time series analyses on
individual subjects would indicate that different people exhibit
different patterns of change over time, which are governed by
different latent variable structures. So, for some people, psycho-
logical distress may be unidimensional, whereas for others it may
be multidimensional. If this is the case, it would seem that we
cannot lump these people together in between-subjects models to
test hypotheses concerning psychological processes, for they
would constitute a heterogeneous population in a theoretically
important sense. At present, however, we do not know how often
and to what degree such a situation occurs, which makes this one
of the big unknowns in psychology. This is because there is an
almost universal—but surprisingly silent—reliance on what may
be called a uniformity-of-nature assumption in doing between-
subjects analyses; the relation between mechanisms that operate at
the level of the individual and models that explain variation
between individuals is often taken for granted, rather than
investigated.
For example, in the attitude literature (Cacioppo & Berntson,
1999; Russell & Carroll, 1999), there is currently a debate on
whether the affective component of attitudes is produced by a
singular mechanism, which would produce a bipolar attitude struc-
ture (with positive and negative affect as two ends of a single
continuum), or whether it should be conceptualized as consisting
of two relatively independent mechanisms (one for positive and
one for negative affect). This debate is characterized by a strong
uniformity assumption: It either is a singular dimension (for ev-
eryone), or we have two relatively independent subsystems (for
everyone). It is, however, not obvious that the affect system should
be the same for all individuals, for it may turn out that the affective
component in attitudes is unidimensional for some people but not
for others. We emphasize that such a finding would not render the
concept of attitude obsolete; but clearly, a construct governed by
different latent variable models within different individuals will
have to play a different role in psychological theories than a locally
homogeneous construct. We call such constructs locally heteroge-
neous. Locally heterogeneous constructs may have a clear dimen-
sional structure between subjects, but they pertain to different
structures at the level of individuals. Thus, we now have a dis-
tinction between two types of constructs: locally homogeneous
constructs, for which the latent dimension is the same within and
between subjects, and locally heterogeneous constructs, for which
this is not the case. Locally homogeneous constructs allow for
testing hypotheses concerning individual processes, modules, and
subsystems through the analysis of between-subjects variability,
whereas locally heterogeneous constructs do not. In applications, it
is imperative to find out which of the two are being discussed,
especially when we are testing hypotheses concerning processes at
the individual level with between-subjects models.
It will be immediately obvious that constructs that are hypoth-
esized as stable traits, such as the factors in the five factor model,
are not expected to exhibit either of these structures. If a trait is
highly stable, covariation of repeated measurements will not obey
a latent variable model at all. Most variance of the observed
variables will be error variance, so that this implies that these
observed variables will be almost independent over time. This
hypothesis could and should be tested using time series analysis
(for the five factor model, the data of Borkenau & Ostendorf, 1998,
actually seem to reject it). If it holds, the latent variable in question
would be one that produces between-subjects variability but does
no work at the individual level. We call this type of construct a
locally irrelevant construct. This terminology should not be taken
to imply a value judgment, as locally irrelevant constructs have
played, and will probably continue to play, an important role in
psychology. However, the terminology should be read unambigu-
ously as indicating the enormous degree to which such constructs
abstract from the level of the individual. They should, for this
reason, not be conceptualized as explaining behavior at the level of
the individual. In the personality literature, this has been argued on
independent grounds by authors such as Lamiell (1987), Pervin
(1994), and Epstein (1994).
It is disturbing and slightly embarrassing for psychology that
one cannot say with sufficient certainty in which of these classes
particular psychological constructs (e.g., personality traits, intelli-
gence, attitudes) fall. This is the result of a century of operating on
silent uniformity-of-nature assumptions by focusing almost exclu-
sively on between-subjects models. It seems that psychological
research has adapted to the limitations of common statistical
procedures (e.g., by abandoning case studies because analysis of
variance requires sample sizes larger than 1) instead of inventing
new procedures that allow for the testing of theories at the proper
level, which is often the level of the individual, or at the very least
exploiting time series techniques that have been around in other
disciplines (e.g., econometrics) for a very long time (Durbin &
Koopman, 2001). Clearly, extending measurements into the time
domain is essential, and fortunately the statistical tools for doing
this are rapidly becoming available. Models that are suited for this
task have seen substantial developments over the last two decades
(Collins & Sayer, 2001, provide an informative overview; for
further information, see, e.g., Fischer & Parzer, 1991; McArdle,
1987; Molenaar, 1985; and Wilson, 1989). Powerful software for
estimating and testing these models has been developed (Jo¨reskog
&So¨rbom, 1993; Muthe´n & Muthe´n, 1998; Neale, 1999), which
makes this type of analysis relatively accessible to nonstatisticians.
It would be especially worthwhile to try latent variable analyses at
the level of the individual, which would bring the all but aban-
doned case study back into scientific psychology—be it, perhaps,
from an unexpected angle.
There remains an open question pertaining to the ontological
status of latent variables, and especially those that fall into the
class of locally irrelevant constructs. We have shown that latent
variables, at least those of the reflective kind, imply a realist
ontology. How should we conceptualize the existence of such
latent variables if they cannot be found at the level of the individ-
ual? It seems that the proper conceptualization of the latent vari-
able (if its reality is maintained) is as an emergent property, in the
sense that it is a characteristic of an aggregate (the population) that
is absent at the level of the constituents of this aggregate (individ-
uals). Of course, this does not mean that there is no relation
between the processes taking place at the level of the individual
and between-subjects latent variables. In fact, the between-subjects
latent variable must be parasitic on individual processes, because
these must be the source of between-subjects variability. If it is
shown that a given set of cognitive processes leads to a particular
215
THEORETICAL STATUS OF LATENT VARIABLES

latent variable structure, we could therefore say that this set of
processes realizes the latent variables in question.
The relevant research question for scientists should then be,
which processes generate which latent variable structures? What
types of individual processes, for example in intelligence testing,
are compatible with the general intelligence model? Obviously,
time series analyses will not provide an answer to this question in
the case of constructs that are hypothesized to be temporally stable,
such as general intelligence. In this case, we need to connect
between-subjects models to models of processes taking place at the
level of the individual. This may involve a detailed analysis of
cognitive processes that are involved in solving IQ test items, for
example. Such inquiries have already been carried out by those at
the forefront of quantitative psychology. Embretson (1994), for
example, has shown how to build latent variable models based on
theories of cognitive processes, and one of the interesting features
of such inquiries is that they show clearly how a single latent
variable can originate or emerge out of a substantial number of
distinct cognitive processes. This kind of research is promising and
may lead to important results in psychology. We would not be
surprised, for example, if it turned out that Sternberg’s (1985)
triarchic theory of intelligence, which is largely a theory about
cognitive processes and modules at the level of the individual, is
not necessarily in conflict with the between-subjects conceptual-
ization of general intelligence. Finally, we note that the connection
of cognitive processes and between-subjects latent variables re-
quires the use of results from both experimental and correlational
psychological research traditions, which Cronbach (1957) has
called the two disciplines of scientific psychology. This paragraph
may therefore be read as a restatement of his call for integration of
these schools.
Discussion
In this article, we have inquired what philosophical position is
implied by latent variable theory. One may reframe this question as
the question of whether latent variable models are philosophically
neutral. It has been argued that this is not the case. The mathe-
matical and empirical connotations of the latent variable may be
considered neutral. In a sense, neither requires the word latent; the
formal latent variable is a mathematical concept, and the opera-
tional latent variable is a weighted sumscore. It is in the connection
between these two concepts when we use the syntax of latent
variable theory to estimate something with the weighted sumscore
that the theory takes side with realism. Entity realism about latent
variables is needed to motivate the choice for the reflective model
over the formative model. Theory realism follows from the obser-
vation that the formal side of the theory implies that it is possible
to be wrong about the position of a subject on the latent variable,
and that weaker formulations—using empirical adequacy instead
of truth—are difficult to interpret. Finally, in a standard measure-
ment model, the causal ingredient of realism can be defended in a
between-subjects sense but not in a within-subject sense. The
within-subjects causal interpretation may be viewed as a fallacious
application of between-subjects results to individuals. To substan-
tiate causal conclusions at the level of the individual, one must
investigate patterns of covariation at the individual level, that is,
one must fit within-subject latent variable models to repeated
measurements in the sense of Cattell and Cross (1952) and Mo-
lenaar (1985).
On the basis of this line of thinking, the possible relations
between within-subjects models and between-subjects models
were used as the foundation for a classification of psychological
constructs as locally homogeneous, locally heterogeneous, and
locally irrelevant. The main implication of this analysis for psy-
chological research is as simple as it is instructive: If one wants to
know what happens in a person, one must study that person. This
requires representing individual processes where they belong,
namely at the level of the individual. On the other hand, if the
study of the individual is dismissed as too difficult, too labor
intensive, or simply as irrelevant, one cannot expect between-
subjects analyses to miraculously yield information at this level.
Before we discuss some implications of these results, there are
two important asides to make concerning what we are not saying.
First, it is not suggested here that one cannot use a standard
measurement model and still think of the latent variable as con-
structed out of the observed variables or as a fiction. But we do
insist that this is an inconsistent position, in that it cannot be used
to connect the operational latent variable to its formal counterpart
in a consistent way. Whether one should or should not allow such
an inconsistency in one’s reasoning is a different question that is
beyond the scope of this article. Second, if one succeeds in fitting
a latent variable model in a given situation, the present discussion
does not imply that one is forced to believe in the reality of the
latent variable. In fact, this would require a logical strategy known
as “inference to the best explanation”or “abduction,”which is
especially problematic in the light of underdetermination. So we
are not saying that, for example, the fit of a factor model with one
higher order factor to a set of IQ measurements implies the
existence of a general intelligence factor; what we are saying is
that the consistent connection between the empirical and formal
side of a factor model requires a realist position. Whether realism
about specific instances of latent variables, such as general intel-
ligence, can be defended is an epistemological issue that is the
topic of heated discussion in the philosophy of science (see, e.g.,
Cartwright, 1983; Devitt, 1991; Hacking, 1983; Van Fraassen,
1980). On the epistemological side of the problem, there are
probably few latent entities in psychology that fulfill the episte-
mological demands of realists such as Hacking (1983).
The realism implicit in latent variables analysis resides in the
hypothetical side of the argument. Here, the theory cannot do
without theory realism. The assumption that a model is true must
be taken literally, more literally, perhaps, than many latent vari-
ables theorists would be comfortable with. However, to do science
means one has to immerse oneself in the scientific world pic-
ture—a fact that is admitted even by such antirealists as Van
Fraassen (1980)—and that world picture is thoroughly realist. It
does not mean that—in a rather trivial way—latent variables exist
by fiat, as they would in a constructivist account. On the contrary,
from the realist viewpoint, the existence of latent entities is an
assumption that may or may not be fulfilled, and assuming their
existence could be regarded as an “as if”approach to the data. This
may be considered analogous to, for example, the treatment of data
as if they were the result of random sampling; random sampling is
extremely rare (if it exists at all), but the bulk of statistical analyses
assume it. As a result, a researcher will approach the data as if they
were the result of a random sampling procedure.
216 BORSBOOM, MELLENBERGH, AND VAN HEERDEN

It will be felt that there are certain tensions in this article. We
have not tried to cover these up, because we think they are
indicative of some fundamental problems in psychological
measurement and require a clear articulation. The realist inter-
pretation of latent variable theory seems to lead to conclusions
that we are not willing to draw. Psychology has a strong empiri-
cist tradition, and we do not want to go beyond the obser-
vations—at least, no further than strictly necessary. As a result,
there is a feeling that realism about latent variables takes us too
far into metaphysical speculations. At the same time, we would
probably like latent variables models to yield conclusions of a
causal nature (the model should at the very least allow for the
formulation of such relations). But we cannot defend any sort of
causal structure invoking latent variables if we are not realists
about these latent variables, in the sense that they exist indepen-
dent of our measurements: One cannot claim that A causes B, and
at the same time maintain that A is constructed out of B. If we then
reluctantly accept realism, invoking perhaps more metaphysics
than we would like, it appears that the type of causal conclusions
available are not the ones we desired. Namely, the causality in our
measurement models is consistently formulated only at the
between-subjects level. And although the boxes, circles, and ar-
rows in the graphical representation of the model suggest that the
model is dynamic and applies to the individual, on closer scrutiny
no such dynamics are to be found. Indeed, this has been pinpointed
as one of the major problems of mathematical psychology by Luce
(1997): Our theories are formulated in a within-subjects sense, but
the models we apply are often based solely on between-subjects
comparisons.
The need to extend the conceptual framework of psychology
by linking individual processes to between-subjects compari-
sons has been emphasized by a number of psychologists, for
example by Sternberg (1985) in the context of intelligence
research and by Eysenck and Eysenck (1985) in the field of
personality theories. The need for models that can incorporate
individual processes has also been acknowledged by psycho-
metricians, such as Goldstein and Wood (1989). Modeling
individual processes and linking them to between-subjects la-
tent variables is possible and has become a growing field in
psychometrics (Collins & Sayer, 2001; Embretson, 1994; Fi-
scher & Parzer, 1991; McArdle, 1987; Molenaar, 1985; Wilson,
1989). These developments are promising, and we have indi-
cated a number of ways in which research into latent variables
structures could benefit from making the connection between
individual processes and between-subjects latent variables. It is
clear that such research will often have to involve the analysis
of repeated measurements of individuals, because it is impera-
tive to ascertain whether our constructs are locally homoge-
neous, locally heterogeneous, or locally irrelevant. Theory for-
mation could also benefit greatly from an analysis along these
lines, for in many fields, it is unclear what role psychological
constructs play at the level of the individual. So, there is a
substantial amount of work to do, both in theoretical analysis
and in empirical research. For now, we have to acknowledge
that individual processes are not represented in our standard
measurement models, but we hope that, with respect to this
issue, this article will soon be outdated.
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Received December 8, 2000
Revision received June 14, 2002
Accepted June 17, 2002 䡲
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