A comparison of objective priors for Cronbach’s coefficient alpha using a three-component hierarchical model

In this article, the reference and probability matching priors for Cronbach’s alpha will be derived, and the performance of these two priors will be compared to that of the well-known Jeffreys prior and a divergence prior. Cronbach’s alpha is a measure used to assess the reliability of a set of test items. A simulation study will be considered to compare the performance of the priors, where the coverage rates, average interval lengths, and average 95% credibility intervals will be computed. The performance of the Bayesian intervals will be compared to an interval using the maximum likelihood estimate (MLE) of alpha. It will be shown that the method of simulation is computationally fast and efficient. Three illustrative examples will also be considered. Throughout the article, the three-component hierarchical model will be used. The three-component hierarchical model should be used when the data is nested or hierarchical. Ignoring this structure by averaging across levels or fitting a single variance biases estimates and underestimates uncertainty.