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Conditional on $\mu ,\psi >0$, let $\lambda \sim {\chi }_4^2$ and independently $Z\sim N(0,1)$. The normal distribution with the stated precision parameter can be generated asThereforeby the defining normal distribution and chi-squared distribution representation of Student's t-distribution. Its density isThus $\mu$ is the location and $\psi$ is the scale, not the standard deviation: $\mathrm{Var}({\beta }_j\mid \mu ,\psi )=2{\psi }^2$. Each study gets its own independent chi-squared draw in this Student t random-effect model.

The Student t random-effect model is useful when most studies are comparable but occasional genuine departures are more frequent than a normal distribution hierarchy allows. Its heavier tails permit a study effect far from $\mu$ without forcing a large common between-study heterogeneity scale on every study. In the Gaussian scale mixture representation, a small study-specific ${\lambda }_j$ lowers its precision parameter and weakens its shrinkage.

Use this as robust partial pooling when occasional atypical effects are plausible. Known systematic population or design differences should still be modeled explicitly; a heavy tail cannot identify or correct within-study bias by itself.

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 $\sqrt{2}\psi$.

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.

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.

A Student t random-effect model assigns Student's t-distributions to exchangeable study effects, allowing heavier tails than a normal distribution hierarchy. An equivalent Gaussian scale mixture is ${\lambda }_j\sim {\chi }_{\nu }^2$, ${\beta }_j\mid {\lambda }_j,\mu ,\psi \sim N(\mu ,\nu {\psi }^2/{\lambda }_j)$, with independent latent draws. For $\nu >2$ the variance is $\nu {\psi }^2/(\nu -2)$, so $\psi$ is a scale rather than a standard deviation. Small latent precisions weaken shrinkage for atypical studies while retaining partial pooling for the rest.