Internal Consistency

Estimators of Internal Consistency

Instead of comparing scores from replications of the whole test, internal consistency focuses on examining relationships among items within (i.e., internal to) the test. Each item can be treated as a separate test, or the items can be grouped into clusters or part tests. Depending on which estimator is used, the number of parts may be as few as two or as many as the number of individual items in the test. In any case, the part tests or items are treated as if they are interchangeable replicates of the test. The investigator then computes a statistic that summarizes the degree of consistency in the examinees’ responses to the various parts.

The estimators in the internal consistency family include, but are not limited to, the following:

• Spearman and Brown’s split-half coefficient
• Kuder and Richardson’s Formula 20 (KR-20)
• Cronbach’s coefficient α
• Hoyt’s coefficient
• Kristof’s coefficient
• Feldt’s coefficient
• Raju’s β coefficient
• McDonald’s ω coefficient
• Raykov’s ρ coefficient

The split-half coefficient proposed independently by Spearman and by Brown in 1910 was the first known estimator of internal consistency. Many of the estimators developed later in this field were attempts to improve on the split-half approach. One advantage of Cronbach’s coefficient α is that it is equivalent to the average split-half reliability of all possible half-length tests. The KR-20 estimator is a special case of Cronbach’s α that is appropriate when the items are scored dichotomously. More detailed descriptions of these three coefficients are provided under separate entries elsewhere in this volume.

Hoyt’s reliability coefficient is computed from the estimated mean squares obtained from the results of a two-way analysis of variance (persons crossed with items) with only one observation per cell. Hoyt’s formula is algebraically equivalent to Cronbach’s α, so it produces the same result within rounding error. One advantage of Hoyt’s approach is that it provides a conceptual link between classical test theory and generalizability theory, which also uses variance components obtained from [Page 854]analysis of variance. The simplest case of generalizability theory is a single-facet design in which persons are the object of measurement and test items are the only facet. That is the same design used by Hoyt. In this simple case, the value of the generalizability coefficient for relative decisions is the same as the value of both coefficient α and Hoyt’s coefficient obtained from the same data.

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