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## 1. Introduction

One of the most striking and challenging phenomena in the Social Sciences is the unreliability of its measurements: Measuring the same attribute twice often yields two diﬀerent results. If the same measurement instrument is applied twice, such a diﬀerence may sometimes be due to a change in the measured attribute itself. Sometimes these changes in the measured attribute are due to the mere fact of measuring. For example, people learn when solving tasks and they change their attitude when they reﬂect on statements in an attitude questionnaire. In other cases the change of the measured attribute is due to developmental phenomena, or it might be due to learning between occasions of measurement. However, if change of the attribute can be excluded two diﬀerent results in measuring the same attribute can be explained only by ‘measurement error.’

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Classical (Psychometric) Test Theory (CTT) aims at studying the reliability of a (real-valued) test score variable (measurement, test) that maps a crucial aspect of qualitative or quantitative observations into the set of real numbers. Aside from determining the reliability of a test score variable itself, CTT allows answering questions such as:

(a) How do two random variables correlate once the measurement error is ﬁltered out (correction for attenuation)?

(b) How dependable is a measurement in characterizing an attribute of an individual unit, i.e., which is the conﬁdence interval for the true score of that individual with respect to the measurement considered?

(c) How reliable is an aggregated measurement consisting of the average (or sum) of several measurements of the same unit or object (Spearman–Brown formula for test length)?

(d) How reliable is a diﬀerence, e.g., between a pretest and post-test?

## 2. Basic Concepts Of Classical Test Theory

### 2.1 Primitives

In the framework of CTT, each measurement (test score) is considered being a value of a random variable Y consisting of two components: a ‘true score’ and an ‘error score.’ Two levels, or more precisely, two random experiments may be distinguished: (a) sampling an observational unit (e.g., a person) and (b) sampling a score within a given unit. Within a given unit, the true score is a parameter, i.e., a given but unknown number characterizing the attribute of the unit, whereas the error is a random variable with an unknown distribution. The true score of the unit is deﬁned to be the expectation of this intraindividual distribution.

Taking the across units perspective, i.e., joining the two random experiments, the true score is itself considered to be a value of a random variable (the ‘true score variable’). The ‘error variable’ is again a random variable, the distribution of which is a mixture of the individual units’ error distributions. Most theorems of CTT (e.g., Lord and Novick, 1968) are formulated from this across units’ perspective allowing talking about the correlation of true scores with other variables, for instance.

More formally, CTT refers to a ( joint) random experiment of (a) sampling an observational unit u (such as a person) from a set Ω_{U} of units (called the population), and (b) registering one or more observations out of a set Ω_{O} of possible observations. The set of possible outcomes of the random experiment is the set product:Ω=Ω_{U} ×Ω_{O}. The elements of Ω_{O}, the observations, might be qualitative (such as ‘answering in category a of item 1 and in category b of item 2’), quantitative (such as reaction time and alcohol concentration in the blood), or consisting of both qualitative and quantitative components. In Psychology, the measurements are often deﬁned by test scoring rules prescribing how the observations are transformed into test scores. (Hence, these measurement are also often called ‘tests’ or ‘test score variables.’) These scoring rules may just consist of summing initial scores of items (deﬁning a psychological scale) or might be more sophisticated representations of observable attributes of the units. CTT does not prescribe the deﬁnition of the test score variables. It just additively decomposes them into true score variables and error variables. Substantive theory and empirical validation studies are necessary in order to decide whether or not a given test score variable is meaningful. CTT only helps disentangling the variances of its true score and error components.

Referring to the joint random experiment described above the mapping U: Ω→ Ω_{U}, U(ω)= u, (the unit or person projection) may be considered a qualitative random variable having a joint distribution with the test scores variables Yi. Most theorems of CTT deal with two or more test score variables (tests) Y_{i} and the relationship between their true score and error components. (The index i refer to one of several tests considered.)

### 2.2 The Core Concepts: True Score And Error Variables

Using the primitives introduced above, the true score variable τ_{i}:= E(Y_{i} |U ) is deﬁned by the conditional expectation of the test Y_{i} given the variable U. The values of the ‘true score variable’ τ_{i }are the conditional expected values E(Y_{i}|U= u) of Y_{i} given the unit u. They are also called the ‘true scores’ of the unit u with respect to i. Hence, these true scores are the expected values of the intraindividual distributions of the Yi. The ‘measurement error variables’ ε_{i} are simply deﬁned by the diﬀerence ε_{i}: -Y_{i} -τ_{i}. Table 1 summarizes the primitives and deﬁnitions of the basic concepts of CTT.

## 3. Properties Of True Score And Error Variables

Once the true score variables and error variables are deﬁned a number of properties (see Table 2) can be derived, some of which are known as the ‘axioms of CTT.’ However, since the work done by Novick (1966) and Zimmerman (1975, 1976) it is well known that all these properties already follow from the deﬁnition of true score and error variables. They are not new and independent assumptions as has been originally proposed (e.g., Gulliksen 1950). All equations in Table 2 are no assumptions. They are inherent properties of true scores and errors. Hence, trying to test or falsify these properties empirically would be meaningless in just the same way, as it is meaningless to test whether or not a bachelor is really unmarried. The property of being unmarried is an inherent part or logical consequence of the concept of a bachelor.

Only one of the ‘axioms of CTT’ does not follow from the deﬁnition of true score and error variables: ‘uncorrelatedness of errors variables’ among each other. Hence, uncorrelatedness of errors has another epistemological status as the properties displayed in Table 2 (the other ‘axioms’). Uncorrelatedness of errors is certainly a desirable and useful property; but it might be wrong in speciﬁc empirical applications (e.g., Zimmerman and Williams 1977). In fact it is an assumption and it plays a crucial rule in deﬁning models of CTT.

Equation 1 of Table 2 is a simple rearrangement of the deﬁnition of the error variable. Equation. 2 shows that the variance of a test score variable, too, has two additive components: the ‘variance of the true score variable’ and the ‘variance of the error variable.’ This second property follows from Eqn. 3 according to which a true score variable is uncorrelated with a measurement error variable, even if they pertain to diﬀerent test score variables Y_{i} and Y_{j}. Equation 4 states that the expected value of an error variable is zero, whereas Eqn. 5 implies that the expected value of an error variable is zero within each individual observational unit u. Finally, according to Eqn. 6 the conditional expectation of an error variable is also zero for each mapping of U. This basically means that the expected value of an error variable is zero in each subpopulation of observational units.

### 3.1 Additional Concepts: Reliability, Unconditional And Conditional Error Variances

Although the true score and error variables deﬁned above are the core concepts of CTT, in empirical applications, the true scores can only be estimated. What is also possible, is to estimate the ‘variances’ of the true score and error variables in a random sample (consisting of repeating many times the random experiment described earlier). The variance Var (ε_{i}) of the measurement error may be considered a gross parameter representing the degree of unreliability. A normed parameter of unreliability is Var (ε_{i}) Var (Y_{i}), the proportion of the variance of Y_{i }due to measurement error. Its counterpart is 1-Var (ε_{i})/Var (Y_{i}), i.e.,

the ‘reliability’ of Yi. This coeﬃcient varies between zero and one. In fact, most theorems and most empirical research deal with this ‘coeﬃcient of reliability.’ The reliability coeﬃcient is a convenient information about the dependability of the measurement ‘in one single number.’

In early papers on CTT, reliability of a test has been deﬁned by its correlation with itself (e.g., Thurstone 1931, p. 3). However, this deﬁnition is only metaphoric, because a variable always correlates perfectly with itself. What is meant is to deﬁne reliability by the correlation of ‘parallel tests’ (see below). The assumptions deﬁning parallel tests in fact imply that the correlation between two test score variables is the reliability. Note that the deﬁnition of ‘reliability’ via Eqn. (1) does not rest on any assumption other than 0<Var (Y_{i})<∞.

‘Reliability’ is useful to compare diﬀerent instruments to each other if they are applied in the same population. Used in this way, reliability in fact helps evaluating the quality of measurement instruments. However, it may not be useful under all circumstances to infer the dependability of measures of an individual unit. For the latter purpose one might rather look at the ‘conditional error variance’ Var (ε_{i }|U= u) given a speciﬁc observational unit u or at the ‘conditional error variances’ Var (ε_{i}|τ_{i}=t ) given the subpopulation with true score τ_{i} = t.

## 4. Models of Classical Test Theory

The deﬁnitions of true score and error variables have to be supplemented by assumptions deﬁning a model if the theoretical parameters such as the reliability are to be computed by empirically estimable parameters such as the means, variances, covariance, or correlation of the test score variables. Table 3 displays the most important of these assumptions and the most important models deﬁned by combining some of these assumptions.

The assumption (a ) to (a ) specify in diﬀerent ways the assumption that two tests Yi and Yj measure the same attribute. Such an assumption is crucial for inferring the degree of reliability from the discrepancy between two measurements of the same attribute of the same person. Perfect identity or ‘τ-equivalence’ of the two true score variables is assumed with (a ). With (a ) this assumption is relaxed: the two true score variables may diﬀer by an additive constant. Two balances, for instance, will follow this assumption if one of them yields a weight that is always one pound larger than the weight indicated by the other balance, irrespective of the object to be weighed. According to Assumption (a ), the two tests measure the same attribute in the sense that their true score variables are linear functions of each other.

The other two assumptions deal with properties of the measurement errors. With (b) one assumes measurement errors pertaining to diﬀerent test score variables to be uncorrelated. In (c) ‘equal error variances are assumed,’ i.e., these tests are assumed to measure equally well.

### 4.1 Parallel Tests

4.1.1 Deﬁnition. The most simple and convenient set of assumptions is the model of ‘parallel tests.’ Two tests Y_{i} and Y_{j} are deﬁned to be parallel if they are τ-equivalent, if their error variables are uncorrelated, and if they have identical error variances. Note that Assumption (a ) implies that there is a uniquely deﬁned latent variable being identical to each of the true score variables. Hence, one may drop the index i and denote this latent variable by η. The assumption of t-equivalence may equivalently be written Y_{i}=η+ε_{i}, where ε_{i} : =Y_{i} E(Y_{i|}U ).

4.1.2 Identiﬁcation. For parallel tests the theoretical parameters may be computed from the para- meters characterizing the distribution of at least two test score variables, i.e., the theoretical parameters are identiﬁed in this model if m ≥2. According to Table 4 the expected value of η is equal to the expected value of each of the tests, whereas the variance of η can be computed from the covariance of two diﬀerent tests. The variance Var (ε_{i}) of the measurement error variables may be computed by the difference Var (Y_{i})-Co (Y_{i}, Y_{j}), i≠j. Finally, the reliability Rel(Y_{i}) is equal to the correlation Corr(Y_{i}, Y_{j}) of two diﬀerent test score variables.

4.1.3 Testability. The model of parallel tests implies several consequences that may be tested empirically. First, all parallel tests Y_{i }have equal expectations E(Y_{i}), equal variances Var (Y_{i}), and equal covariances Co (Y_{i}, Y_{j}) in the total population. Second, parallel tests also have equal expectations within each subpopulation (see Table 4).

Note that these hypotheses may be tested separately and or simultaneously as a single multidimensional hypothesis in the framework of ‘simultaneous equation models’ via AMOS (Arbuckle 1997), EQS (Bentler 1995), LISREL 8 (Joereskog and Soerbom 1998), MPLUS (Muthen and Muthen 1998), MX (Neale 1997), RAMONA (Browne and Mels 1998), SEPATH (Steiger 1995), and others. Such a simultaneous test may even include the hypotheses about the parameters in several subpopulations (see Table 4). What is not implied by the assumptions of parallel tests is the equality of the variances and the co- variances of the test score variables in subpopulations.

For parallel tests Y_{n}+…+Y_{m} as deﬁned in Table 4, the reliability of the sum score S : =Y_{1}+…+ Y_{m} may be computed by the ‘Spearman-Brown formula for lengthened tests:’

Using this formula the reliability of an aggregated measurement consisting of the sum (or average) of m parallel measurements of the same unit can be computed. For m =2, each with Rel(Y_{i})= 0.80, for instance

The Spearman–Brown formula may also be used to answer the opposite question. Suppose there is a test being the sum of m parallel tests and this test has reliability Rel(S). What would be the reliability Rel(Yi) of the m parallel tests? For example, if m =2, what would be the reliability of a test half?

### 4.2 Essentially τ-Equivalent Tests

4.2.1 Deﬁnition. The model of essentially τ-equivalent tests is less restrictive than the model of parallel tests. Two tests Y_{i }and Y_{j} are deﬁned to be ‘essentially τ-equivalent’ if their true score variables diﬀer only by an additive constant (Assumption a_{2} in Table 3) and if their error variables are uncorrelated (Assumption b in Table 3). Assumption (a_{2}) implies that there is a latent variable η that is a translation of each of the true score variables, i.e.,

Also note that the latent variable η is uniquely deﬁned up to a translation. Hence, it is necessary to ﬁx the scale of the latent variable η. This can be done by ﬁxing one of the coeﬃcients λ_{i} (e.g., λ_{1}= 0) or by ﬁxing the expected value of η [e.g., E(η)=0].

Table 5 summarizes the assumptions deﬁning the model and the consequences for identiﬁcation and testability. In this model, the reliability cannot be identiﬁed any more by the correlation between two tests. Instead the reliability is identiﬁed by

Furthermore, the expected values of diﬀerent tests are not identical any more within each subpopulation. Instead, the diﬀerences between the expected values E^{(s)(}Y_{i}) E^{(s)}(Y_{j}) of two essentially τ-equivalent tests Y_{i} and Y_{j }are the same in each and every subpopulation. All other properties are the same as in the model of parallel tests. Again, all these hypotheses may be tested via structural equation modeling.

For essentially τ-equivalent tests Y_{1}, …, Y_{m}, the reliability of the sum score S: =Y_{1}+…+Y_{m} may be computed by the ‘Cronbach’s coeﬃcient α:’

This coeﬃcient is a lower bound for the reliability of S if only uncorrelated errors are assumed.

### 4.3 τ-Congeneric Tests

4.3.1 Deﬁnition. The model of τ-congeneric tests is deﬁned by the Assumptions (a_{3}) and (b) in Table 3. Hence, two tests Y_{i} and Y_{j} are called τ-congeneric if their true score variables are positive linear functions of each other and if their error variables are uncorrelated. Assumption a_{3} implies that there is a latent variable η such that each true score variable is a positive linear function of it, i.e.,

or equivalently:

where ε_{i}:=Y_{i}-E(Y_{i }|U).

The latent variable η is uniquely deﬁned up to positive linear functions. Hence, in this model, too, it is necessary to ﬁx the scale of η. This can be done by ﬁxing a pair of the coeﬃcients (e.g., λ_{i0}= 0 and λ_{i1}=1) or by ﬁxing the expected value and the variance of η [e.g., E(η) = 0 and Var (η) = 1].

Table 6 summarizes the assumptions deﬁning the model and the consequences for identiﬁcation and testability assuming E(η) =0 and Var (η) =1. Other ways of ﬁxing the scale of η would imply diﬀerent formula. As can be seen from the formula in Table 6, the model of τ-congeneric variables and all its parameters are identiﬁed if there are at least three diﬀerent tests for which Assumptions a and b hold. The covariance structure in the total population implied by the model may be tested empirically if there are at least four test score variables. Only in this case the model has fewer theoretical parameters determining the covariance matrix of the test score variables than there elements in this covariance matrix. The implications for the mean structure are testable already for three test score variables provided the means of the test score variables are available in at least four subpopulations.

### 4.4 Other Models Of CTT

The models treated previously are not the only ones that can be used to determine the theoretical parameters of CTT such as reliability, true score variance, and error variance. In fact, the models dealt with are limited to unidimensional models. However, true score variables may also be decomposed into several latent variables. ‘Conﬁrmatory factor analysis’ provides a powerful methodology to construct, estimate, and test models with multidimensional decompositions of true score variables. Note, however, that not each factor model is based on CTT. For instance, there are one-factor models that are not models of τ-congeneric variables in terms of CTT. A model with one common factor and several speciﬁc but uncorrelated factors is a counter example. The common factor is not necessarily a linear function of the true score variables and the speciﬁc factors are not necessarily the measurement error variables as deﬁned in CTT.

### 4.5 Some Practical Issues

Once a measurement or test score has been obtained for a speciﬁc individual, one might want to know how dependable that individual measurement is. If the reliability of the measurement is known and if one assumes a normal distribution of the measurement errors which is homogeneous for all individuals, the 95 percent-conﬁdence interval for the true score of that individual with respect to the measurement Y_{i} can be computed by:

Another result deals with the correlation between two true score variables. If the reliabilities for two test score variables Y_{1} and Y_{2} are known, and assuming uncorrelated measurement errors, one may compute

This eqn. is known as the ‘correction for attenuation.’

Another important issue deals with the ‘reliability of a diﬀerence variable,’ for example, a diﬀerence between a pretest Y and a posttest Y . Assuming equal true score and error variances between pre and posttest implies identical reliabilities, i.e., Rel(Y_{1})=Rel(Y_{2})= Rel(Y ). If additionally uncorrelated measurement errors are assumed, the reliability Rel(Y_{1}-Y_{2}) := Var (E(Y_{1}-Y_{2|} U)/Var (Y_{1}– Y_{2}) of the diﬀerence Y_{1}-Y_{2} may be computed by:

According to this formula, the reliability of a diﬀerence between pre- and posttest is always smaller than the reliability of the pre- and posttest, provided the assumptions mentioned above hold. In the extreme case in which there is no diﬀerential change, i.e., τ_{2}=τ_{1}+ constant, the reliability coeﬃcient Rel(Y_{1}-Y_{2}) will be zero. Obviously, this does not mean that the change is not dependable. It only means that there is no variance in the change, since each individual changes by the same amount. This phenomenon has lead to much confusion about the usefulness of measuring change (e.g., Cronbach and Furby 1970, Harris 1963, Rogosa 1995). Most of these problems are now solved by structural equation modeling (allowing the include latent change variables such as in growth curve models (e.g., McArdle and Epstein 1987, Willet and Sayer 1996) or, more directly, in true change models (Steyer, Eid, and Schwenkmezger 1997, Steyer, Partchev, and Shanahan, 2000). Models of this kind are no longer hampered by reliability problems and allow the explanation of inter-individual diﬀerences in intraindividual change.

## 5. Discussion

It should be noted that CTT refers to the population level, i.e., to the random experiment of sampling a single observational unit and assessing some of its behavior (see Table 1). CTT does not refer to sampling models that consist of repeating this random experiment many times. Hence, no questions of statistical estimation and hypothesis testing are dealt with. Of course, the population models of CTT have to be supplemented by sampling models when it comes to applying statistical analyses, e.g., via structural equation modeling.

Aside from this more technical aspect, what are the limitations of CTT? First, CTT and its models are not really adequate for modeling answers to individual items in a questionnaire. This purpose is more adequately met by models of item response theory (IRT) which specify how the probability of answering in a speciﬁc category of an item depends on the attribute to be measured, i.e., on the value of a latent variable.

A second limitation of CTT is the exclusive focus on measurement errors. ‘Generalizability theory’ presented by Cronbach et al. (1972) (see also Shavelson and Webb 1991) generalized CTT to include other factors determining test scores.

Inspired by Generalizability Theory Tack (1980), Steyer et al. (1989) presented a generalization of CTT, called ‘Latent State-Trait Theory,’ which explicitly takes into account the situation factor, introduced formal deﬁnitions of states and traits, and presented models allowing to disentangle person, as well as situation and or interaction eﬀects and from measurement error. More recent presentations are Steyer et al. (1992) as well as Steyer, et al. (1999). Eid (1995, 1996) extended this approach to the normal ogive model for analyses on the item level.

The parameters of CTT are often said to be ‘population dependent,’ i.e., meaningful only with respect to a given population. This is true for the variance of the true score variable and the reliability coeﬃcient. The reliability (coeﬃcient) of an intelligence test is diﬀerent in the population of students than the general population. This is a simple consequence of the restriction of the (true score) variance of intelligence in the population of the students. However, such a restriction neither exists for the true score estimates neither of individual persons nor for the item parameters λ_{i} of the model of ‘essentially τ-equivalent tests,’ for instance. Proponents of IRT models have often forwarded the population dependence critique. They contrast it with the ‘population independence’ of the person and the item parameters of IRT models. However, ‘population independence’ also holds for the person and the item parameters of the model of essentially τ-equivalent tests, for instance.

In applications of CTT it is often assumed that the error variances are the same for each individual, irrespective of the true score of that individual. This assumption may indeed be wrong in many applications. In IRT models no such assumption is made. However, it is possible to assume diﬀerent error variances for diﬀerent (categories of ) persons in CTT models as well. In this case, the unconditional error variance and the reliability coeﬃcient are not the best available information for inferring the dependability of individual true score estimates. In this case one should seek to obtain estimates of conditional measurement error variances for speciﬁc classes of persons. It is to be expected that persons with high true scores have a higher measurement error variance than those with medium true scores and that those with low true scores have a higher error variance again. (This would be due to ‘ﬂoor and ceiling eﬀects.’) Other patterns of the error variance depending on the size of the true score may occur as well.

Such phenomena do not mean that ‘true scores’ and ‘error scores’ would be correlated; only the ‘error variances’ would depend on the true scores. None of the properties listed in Table 2 would be violated. As mentioned before, the properties listed in Table 2 cannot be wrong in empirical applications. What could be wrong, however, is the true score interpretation of the latent variable in a concrete structural equation model. Misinterpretations of this sort can be most eﬀectively prevented by empirical tests of the hypotheses listed in the testability sections of Tables 4 to 6.

The most challenging critique of many applications of CTT is that they are based on rather arbitrarily deﬁned test score variables. If these test score variables are not well chosen any model based on them is also not well founded. Are there really good reasons to base models on sum scores across items in questionnaires? Why take the sum of the items as test score variables Yi and not another way of aggregation such as a weighted sum, or a product, or a sum of logarithms? And why aggregate and not look at the items themselves?

There is no doubt that IRT models are more informative than CTT models if samples are big enough to allow their application, if the items obey the laws deﬁning the models, and if detailed information about the items (and even about the categories of ‘polytomous items,’ such as in ‘ratings scales’) is sought. In most applications, the decision how to deﬁne the test score variables Yi on which models of CTT are built is arbitrary, to some degree. It should be noted, however, that arbitrariness in the choice of the test score variables cannot be avoided altogether. Even if models are based on the item level, such as in IRT models, one may ask ‘Why these items and not other ones?’ Whether or not a good choice has been made will only prove in model tests and in validation studies. This is true for models of CTT as well as for models of alternative theories of psychometric tests.

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