Classical test theory

Classical test theory is a body of related psychometric theory that predicts outcomes of psychological testing such as the difficulty of items or the ability of test-takers. It is a theory of testing based on the idea that a person’s observed or obtained score on a test is the sum of a true score (error-free score) and an error score. 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.

Classical Test Theory was born only after the following three achievements or ideas were conceptualized: one, a recognition of the presence of errors in measurements, two, a conception of that error as a random variable, and third, a conception of correlation and how to index it. In 1904, Charles Spearman was responsible for figuring out how to correct a correlation coefficient for attenuation due to measurement error and how to obtain the index of reliability needed in making the correction. Spearman's finding is thought to be the beginning of Classical Test Theory by some (Traub, 1997). Others who had an influence in the Classical Test Theory's framework include: George Udny Yule, Truman Lee Kelley, those involved in making the Kuder-Richardson Formulas, Louis Guttman, and, most recently, Melvin Novick, not to mention others over the next quarter century after Spearman's initial findings.

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:

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