The Role of Item Distributions on Reliability Estimation: The Case of Cronbach's Coefficient Alpha

Simulations concerning the distributional assumptions of coefficient alpha are contradictory. To provide a more principled theoretical framework, this article relies on the Fréchet-Hoeffding bounds, in order to showcase that the distribution of the items play a role on the estimation of correlations and covariances. More specifically, these bounds restrict the theoretical correlation range [-1, 1] such that certain correlation structures may be unfeasible. The direct implication of this result is that coefficient alpha is bounded above depending on the shape of the distributions. A general form of the Fréchet-Hoeffding bounds is derived for discrete random variables. R code and a user-friendly shiny web application are also provided so that researchers can calculate the bounds on their data.