Classical Test Theory

Assumptions of CTT

CTT views a test score as having two components, the true score and a random error. The true score is considered the average of identical values taken from repeated measurements without limit. The error is regarded as unrelated either to the true score or to the error that can occur in another measurement of the same attribute. Although the theory is an oversimplification and does not represent the results, it brings out relationships that are informative and useful in test design and construction as well as in evaluation of test scores. The first basic assumption of CTT is that the obtained score is the sum of true score plus error; that is, the true score and the error score are inextricably mixed. This concept can be expressed as a simple equation:

X is the obtained score, T is the true score, and

E represents errors of measurement.

It must be pointed out that the true score is never known. It remains within a certain interval, however, and a best estimate of it can be obtained.

Measurement error is all things except the true score. Errors of measurement can arise from numerous sources, such as item selection, test administration, test scoring, and systematic errors of measurement. The first three sources of error are jointly called unsystematic measurement error, meaning that their impact is unexpected and inconsistent. A systematic measurement error occurs when a test consistently measures something different from the trait it was designed to measure.

Measurement error reduces reliability or repeatability of test results. The assumption that the obtained score is made up of the true score and the error score reveals several additional assumptions. An assumption derived from true scores is that unsystematic measurement error affects test scores randomly. The randomness of measurement error is a fundamental assumption of CTT. Since there are random events, unsystematic measurement errors have some probability of being positive or negative, and consequently they amount to an average of zero across a large group of subjects. It follows that the mean error of measurement is zero. Another assumption of CTT is that measurement errors are not correlated with true scores. A final assumption is that measurement errors are not correlated with errors on other tests. All these assumptions can be summarized as follows: (a) Measurement errors are random, (b) the mean error of measurement is zero, (c) true scores and error are uncorrelated, and (d) errors on different tests are uncorrelated.

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