In modern scientific research, small-scale studies with limited participants are increasingly common. However, interpreting individual outcomes can be challenging, making it standard practice to combine data across studies using random effects to draw broader scientific conclusions. In this work, we introduce an optimal methodology for assessing the goodness of fit of a reference distribution for the random effects arising from binomial counts. For meta-analyses, we also derive optimal tests to evaluate whether multiple studies are in agreement before pooling the data. In all cases, we prove that the proposed tests optimally distinguish null and alternative hypotheses separated in the 1-Wasserstein distance.