= Solution
Fit both the normal and <Student t random-effect model> with comparable proper <prior distributions>. Compare priors on the same spread measure: a <normal distribution> scale $\tau$ is a <standard deviation>, whereas the $t_4$ <standard deviation> is $\sqrt2\psi$.
Use a <posterior predictive check>: draw study effects and binomial counts from each fitted hierarchy and compare replicated dispersion and extreme study contrasts with the observations. For predicting a new study, generate a new effect from the hierarchy rather than reusing an existing fitted effect. A <Leave-one-out cross-validation> with entire studies held out can compare integrated predictive probabilities for both arms of each omitted trial, averaging over <hyperparameters> and its unobserved study effect. <Leave-one-study-out influence analysis> also reveals whether the difference is driven by a single trial.
\b[Prefer the heavier-tailed hierarchy if it improves the relevant predictive checks and held-out study predictions robustly to reasonable prior choices.] The <deviance information criterion> can supplement the comparison, but its effective parameter count can depend on the latent-variable representation, and six studies give limited information about tail shape. A small numerical criterion difference alone is insufficient evidence.
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