Cronbach's alpha reliability: Interval estimation, hypothesis testing, and sample size planning

Abstract

Cronbach's alpha is one of the most widely used measures of reliability in the social and organizational sciences. Current practice is to report the sample value of Cronbach's alpha reliability, but a confidence interval for the population reliability value also should be reported. The traditional confidence interval for the population value of Cronbach's alpha makes an unnecessarily restrictive assumption that the multiple measurements have equal variances and equal covariances. We propose a confidence interval that does not require equal variances or equal covariances. The results of a simulation study demonstrated that the proposed method performed better than alternative methods. We also present some sample size formulas that approximate the sample size requirements for desired power or desired confidence interval precision. R functions are provided that can be used to implement the proposed confidence interval and sample size methods.Cronbach's alpha reliability (Cronbach, 1951) is one of the most widely used measures of reliability in the social and organizational sciences. Cronbach's alpha reliability describes the reliability of a sum (or average) of q measurements where the q measurements may represent q raters, occasions, alternative forms, or questionnaire/test items. When the measurements represent multiple questionnaire/test items, which is the most common application, Cronbach's alpha is referred to as a measure of "internal consistency" reliability. If the measurements are "parallel" (see, e.g., McDonald, 1999), they will have equal variances and equal covariances. If the measurements are "essentially tau-equivalent" (see, e.g., McDonald, 1999), they will have equal covariances but will not necessarily have equal variances. It can be shown (see, e.g., McDonald, 1999) that Cronbach's alpha correctly describes the reliability of the sum or average of q measurements that satisfy the parallel assumption or the less restrictive essentially tau-equivalent assumption. If the measurements are "congeneric" measurements (Joreskog, 1971), they can be represented by a one-factor model with uncorrelated measurement errors. The congeneric assumption is more realistic than the tau-equivalence assumption because the measurements are not required to have equal variances and the covariances among the measurements are not required to be equal.As an example, suppose we have a five-item questionnaire (q = 5) to measure employee work ethic with each item scored on a 1-7 Likert scale. In this example, the parallel measurement assumption requires all five items to have equal variances and all 5(4)/2 = 10 covariances among the five items are required to be equal. A tau-equivalent assumption does not require the five items to have equal variances but does require the 10 covariances to be equal. With a congeneric assumption, the variances of the five items can be unequal and the 10 covariances can be unequal.Sijtsma (2009) has criticized the widespread use of Cronbach's alpha because it understates the reliability of the sum...

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