Classical Test Theory

Assumptions of CTT

Classical test theory assumes linearity—that is, the regression of the observed score on the true score is linear. This linearity assumption underlies the practice of creating tests from the linear combination of items or subtests. In addition, the following assumptions are often made by classical test theory:

• The expected value of measurement error within a person is zero.
• The expected value of measurement error across persons in the population is zero.
• True score is uncorrelated with measurement error in the population of persons.
• The variance of observed scores across persons is equal to the sum of the variances of true score and measurement error.
• Measurement errors of different tests are not correlated.

The first four assumptions can be readily derived from the definitions of true score and measurement error. Thus, they are commonly shared by all the models of CTT. The fifth assumption is also suggested by most of the models because it is needed to estimate reliability. All of these assumptions are generally considered “weak assumptions,” that is, assumptions that are likely to hold true in most data. Some models of CTT make further stronger assumptions that, although they are not needed for deriving most formulas central to the theory, provide estimation convenience:

• Measurement error is normally distributed within a person and across persons in the population.
• Distributions of measurement error have the same variance across all levels of true score.

Important Concepts in CTT