## Reliabilities of Item Score, Person’s Score, Attribute Mastery Probability, and Their Relationship to the Classical Test Theory

In the classical test theory, the observed score X is defined as the sum of the true score T and the error of measurement E, where the covariance of T and E is 0, and the variances are not zero. The reliability of score X is defined by the ratio of the variances of T and X, ρ = σ2(T)/σ2(X). When X is the sum of n item scores, X = X1 + X2 + … + Xn, then the lower bound of the reliability, Cronbach’s α, is given by the following equation, which is expressed by the observed item score variances of X and item scores Xi:

α σ

σ ≥

- -

=n n

X X

1 1 1

S ( ) ( )

(9.1)

Because it is not possible to measure the true scores directly, the reliability of a test must be inferred from the observed scores. Cronbach derived a lower bound of the test reliability using total scores and item scores. Since then, Cronbach’s α has been used as the lower bound of the reliability of scores, but the relationship (Equation [9.0.1]) will not provide the reliability of a single item because the denominator n – 1 becomes 0 for n = 1; however, the information from attributes has never been utilized in any psychometric theory. Use of such information enables us to obtain the reliability of a single item. In this chapter, we introduce the reliability of attributes and an item, and then relate them to Cronbach’s α (Lord & Novick, 1968; Tatsuoka, 1975). Then, we introduce attribute characteristic curves and relate them to item response curves and observed item difficulties.