Classical test theory

Classical test theory is a body of related psychometric theory that predict outcomes of psychological testing such as the difficulty of items or the ability of test-takers. Generally speaking, the aim of classical test theory is to understand and improve the reliability of psychological tests.

Classical test theory may be regarded as roughly synonymous with true score theory. The term "classical" refers not only to the chronology of these models but also contrasts with the more recent psychometric theories, generally referred to collectively as item response theory, which sometimes bear the appellation "modern" as in "modern latent trait theory".

Classical test theory as we know it today was codified by Novick (1966) and described in classic texts such as Lord & Novick 1968) and Allen & Yen (1979/2002). The description of classical test theory below follows these seminal publications.

Definitions

Classical test theory assumes that each person has a true score,T, that would be obtained if there were no errors in measurement. A person's true score is defined as the expected number-correct score over an infinite number of independent administrations of the test. Unfortunately, test users never observe a person's true score, only an observed score, X. It is assumed that observed score = true score plus some error:

                X         =       T      +    E
          observed score     true score     error

Classical test theory is concerned with the relations between the three variables $X$ , $T$ , and $E$ in the population. These relations are used to say something about the quality of test scores. In this regard, the most important concept is that of reliability. The reliability of the observed test scores $X$ , which is denoted as ${\rho _{XT}^{2}}$ , is defined as the ratio of true score variance ${\sigma _{T}^{2}}$ to the observed score variance ${\sigma _{X}^{2}}$ :

${\rho _{XT}^{2}}={\frac {\sigma _{T}^{2}}{\sigma _{X}^{2}}}$

Because the variance of the observed scores can be shown to equal the sum of the variance of true scores and the variance of error scores, this is equivalent to

${\rho _{XT}^{2}}={\frac {\sigma _{T}^{2}}{\sigma _{X}^{2}}}={\frac {\sigma _{T}^{2}}{{\sigma _{T}^{2}}+{\sigma _{E}^{2}}}}$

This equation, which formulates a signal-to-noise ratio, has intuitive appeal: The reliability of test scores becomes higher as the proportion of error variance in the test scores becomes lower and vice versa. The reliability is equal to the proportion of the variance in the test scores that we could explain if we knew the true scores. The square root of the reliability is the correlation between true and observed scores.

Reliability

Note that reliability is not, as is often assumed by test users, a fixed property of tests, but a property of the use of a test with a particular population. Reliability should be estimated in each population where the test is to be used, because test scores will not be equally reliable in every population or even every sample. For instance, as is the case for any correlation, the reliability of test scores will be lowered by restriction of range. Thus, IQ-test scores that are highly reliable in the general population will be less reliable in a population of college students and even less reliable in a sample of sophomores. Also note that test scores are perfectly unreliable for any given individual $i$ , because, as has been noted above, the true score is a constant at the level of the individual, which implies it has zero variance, so that the ratio of true score variance to observed score variance, and hence reliability, is zero.

Reliability cannot be estimated directly since that would require one to know the true scores, which according to classical test theory is impossible. However, estimates of reliability can be obtained by various means. One way of estimating reliability is by constructing a so-called parallel test. The fundamental property of a parallel test is that it yields the same true score and the same observed score variance as the original test for every individual. If we have parallel tests x and x', then this means that

${\varepsilon }(X_{i})={\varepsilon }(X'_{i})$

and

${\sigma }_{E_{i}}^{2}={\sigma }_{E'_{i}}^{2}$

Under these assumptions, it follows that the correlation between parallel test scores is equal to reliability (see Lord & Novick, 1968, Ch. 2, for a proof).

${\rho }_{XX'}={\frac {{\sigma }_{XX'}}{{\sigma }_{X}{\sigma }_{X'}}}={\frac {{\sigma }_{T}^{2}}{{\sigma }_{X}^{2}}}={\rho }_{XT}^{2}$

Using parallel tests to estimate reliability is cumbersome because parallel tests are very hard to come by. In practice the method is rarely used. Instead, researchers use a measure of internal consistency known as Cronbach's ${\alpha }$ . Consider a test consisting of $k$ items $u_{j}$ , $j=1,\ldots ,k$ . The total test score is defined as the sum of the individual item scores, so that for individual $i$

$X_{i}=\sum _{j=1}^{k}{U_{ij}}$

Then Cronbach's alpha equals

$\alpha ={\frac {k}{k-1}}\left(1-{\frac {\sum _{j=1}^{k}{\sigma _{U_{j}}^{2}}}{\sigma _{X}^{2}}}\right)$

Cronbach's ${\alpha }$ can be shown to provide a lower bound for reliability under rather mild assumptions. Thus, the reliability of test scores in a population is always higher than the value of Cronbach's ${\alpha }$ in that population. Thus, this method is empirically feasible and, as a result, it is very popular among researchers. Calculatiion of Cronbach's ${\alpha }$ is included in many standard statistical packages such as SPSS and SAS.^[1]

As has been noted above, the entire exercise of classical test theory is done to arrive at a suitable definition of reliability. Reliability is supposed to say something about the general quality of the test scores in question. The general idea is that, the higher reliability is, the better. Classical test theory does not say how high reliability is supposed to be. Too high a value for ${\alpha }$ , say over .9, indicates redundancy of items. Around .8 is recommended for personality research, while .9+ is desirable for individual high-stakes testing.^[2] It must be noted that these 'criteria' are not based on reasonable arguments but the result of convention. Whether they make any sense or not is unclear.

Alternatives

Classical test theory is by far the most influential theory of test scores in the social sciences. In psychometrics, the theory has been superseded by the more sophisticated models in Item Response Theory (IRT) and Generalizability theory (G-theory). IRT models, however, are catching on slowly in mainstream research. One of the main problems causing this is the lack of widely available, user-friendly software; also, IRT is not included in standard statistical packages like SPSS and SAS, whereas these packages routinely provide estimates of Cronbach's ${\alpha }$ . Until this problem is solved, classical test theory will probably remain the theory of choice for many researchers.

Notes

^ Pui-Wa Lei and Qiong Wu (2007). "CTTITEM: SAS macro and SPSS syntax for classical item analysis" (PDF). Behavior Research Methods. 39 (3): 527–530. PMID 17958163.
^ Streiner, D. L. (2003). Starting at the Beginning: An Introduction to Coefficient Alpha and Internal Consistency. Journal of Personality Assessment, 80, 99-103

References

Allen, M.J., & Yen, W. M. (2002). Introduction to Measurement Theory. Long Grove, IL: Waveland Press.
Novick, M.R. (1966) The axioms and principal results of classical test theory Journal of Mathematical Psychology Volume 3, Issue 1, February 1966, Pages 1-18
Lord, F. M. & Novick, M. R. (1968). Statistical theories of mental test scores. Reading MA: Addison-Welsley Publishing Company

External links

International Test Commission article on Classical Test Theory