This article presents some equivalent forms of the common Kuder–Richardson Formula 21 and 20 estimators for nondichotomous data belonging to certain other exponential families, such as Poisson count data, exponential data, or geometric counts of trials until failure. Using the generalized framework of Foster (2020), an equation for the reliability for a subset of the natural exponential family have quadratic variance function is derived for known population parameters, and both formulas are shown to be different plug-in estimators of this quantity. The equivalent Kuder–Richardson Formulas 20 and 21 are given for six different natural exponential families, and these match earlier derivations in the case of binomial and Poisson data. Simulations show performance exceeding that of Cronbach’s alpha in terms of root mean square error when the formula matching the correct exponential family is used, and a discussion of Jensen’s inequality suggests explanations for peculiarities of the bias and standard error of the simulations across the different exponential families.

本文介绍了一些等效形式的常见 Kuder-Richardson 公式 21 和 20 估计量，用于属于某些其他指数族的非二分数据，例如泊松计数数据、指数数据或失败前试验的几何计数。使用 Foster (2020) 的广义框架，针对已知总体参数推导了具有二次方差函数的自然指数族子集的可靠性方程，并且两个公式都显示为该数量的不同插件估计量。为六个不同的自然指数系列给出了等效的 Kuder-Richardson 公式 20 和 21，并且这些公式与二项式和泊松数据情况下的早期推导相匹配。