Standard analysis of variance assumes observations are normally distributed within groups. This paper develops some analysis of variance tests for data which are Bernoulli, Poisson, exponential, or geometric distributed within groups. The tests are shown in Table 1. For natural exponential family data with conjugate priors for the distribution of means, reliability estimators directly estimate the posterior shrinkage. Using the linear posterior expectation induced by conjugate prior, a method is developed to construct an analysis of variance test by determining an appropriate transformation of a reliability estimator. The sampling distribution of the transformed reliability estimator under the assumption of group mean equality is derived to construct an appropriate test statistic. This method is used to invert the generalized KR21 estimators of Foster (2021) for some non-normal data, and it is also shown that the standard analysis of variance F-test statistic can be transformed into a consistent reliability estimator under the same assumptions. A limited simulation study shows that the inverted KR21 test has, in some scenarios, higher power than a standard analysis of variance or a generalized linear model analysis of variance.