Bayesian process control for Cronbach’s alpha

Abstract

In this paper, Bayesian statistical process control limits are derived for Cronbach's coefficient alpha  in the case of the balanced one-way random effects model. Cronbach's alpha is one of the most commonly used measures for assessing a set of items' internal consistency or reliability, thereby assessing the assumption that they measure the same latent construct. By using the available data and the Jeffreys independence prior, the posterior distribution of  and the predictive density of a future (unknown) Cronbach's alpha can be derived. Given a stable Phase I process, the predictive density function and the conditional predictive density functions are used to calculate central values, variances, control limits, run-lengths and the average run-length. The predictive density of a future run-length is the average of a large number of geometrical distributions, each with its own parameter value. Three applications of interest are included in this paper. From the results, it can be seen that the average and median run-lengths are usually larger than the theoretical values. An advantage of the Bayesian procedure, however, is that the control limits can be adjusted in such a way that the average or median run-length has a specific value.