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A Bayesian deviance is minus twice the log-likelihood, with any chosen additive data-only constant held consistent across the models being compared. Its Bayesian posterior expectation measures average fit. In the deviance information criterion, evaluating it at a parameter's posterior mean also enters the effective complexity penalty. This likelihood-based convention differs by a data-only constant from a saturated-model exponential-family deviance when a common saturated model exists.

Smaller mean Bayesian deviance indicates better average likelihood fit. Model B has the smallest $\overline{D}$, improving on A by $3.9$ and on C by $3.1$. Model C has the smallest effective parameter count in DIC, reflecting its shared timing and growth coefficients. Model B's hierarchical covariance parameters and residual flexibility contribute to a larger effective count than C.

Adding fit and complexity gives the reported deviance information criterion: A has $260.1$, B $258.1$, and C $255.6$. Thus DIC favors Model C, whose simpler shape more than compensates for its slightly worse mean fit than B. The differences are modest, and the criteria must use the same observation likelihood and deviance convention. These numbers are not posterior model probabilities; predictive assessment remains useful.

Using the same Bayesian deviance convention, the deviance information criterion and Akaike information criterion areBoth combine a lack-of-fit term with a complexity penalty; smaller values are preferred among comparable models. Under the locally flat prior distribution and the regular multivariate normal distribution approximation, $D(\overline{\theta })\approx D(\hat{\theta })$ and $p_D\approx p$, soAll fitted parameters counted in the likelihood, including unknown dispersion parameters, belong in $p$. The equivalence need not hold for informative priors, nonregular models, or different choices of the likelihood's latent-variable representation.

The reported deviance information criterion differs byModel 3 has a mean Bayesian deviance worse by only $0.2$, but its effective parameter count in DIC is lower by $2.1$. The small fit cost is more than offset by the reduced complexity penalty, so the criterion gives a mild preference to Model 3. A difference this small should not be presented as decisive evidence, especially without the Monte Carlo error of the criterion difference.

The counts $143.6$ and $141.5$ are effective rather than literal parameter numbers. The likelihood includes child-specific intercepts and uncertain true baselines as well as global coefficients. With 106 of each local quantity and seven global parameters, Model 2 has 219 stochastic unknowns in a natural full representation, but partial pooling and other prior information reduce their effective contribution. Fixing two slopes lowers the effective count by about two, with a further small change from the fitted posterior. There is no reason for $p_D$ to be an integer, or to equal just the number of population-level regression coefficients.

For illustration, the deviances evaluated at posterior means implied by the table are $\overline{D}-p_D=984.5$ for Model 2 and $986.8$ for Model 3. These differ from the average deviances because averaging a nonlinear deviance is not the same as evaluating it at an averaged parameter. This interpretation presumes the same observed-data likelihood convention in both fits; integrating local variables out before forming the deviance would define a different comparison. The DIC difference is not a Bayes factor.

For the common Bayesian deviance convention $D=-2\log L$ used in all three models,Here Dhat is $D(\overline{\theta })$, an at-posterior-mean fit measure, and $p_D$ is an effective parameter count. The independent model's Dhat of 53.1 is almost identical to the exchangeable model's 53.2; both improve on the common model's 57.8. The common model's $p_D=7$ corresponds to six intercepts plus one shared effect. Independence uses roughly twelve effective parameters. Partial pooling reduces the exchangeable model's effective complexity to about 8.7 while retaining nearly the same fitted likelihood function as independence.

The exchangeable model has the lowest reported DIC, but the common model is competitive. Their difference is only about 1.3, whereas independence is worse by about 6.3. The deviance information criterion measures penalized fit for a predictive comparison, not model posterior probabilities, and these numbers do not establish overwhelming evidence for heterogeneity. The displayed exchangeable $\overline{D}+p_D$ is $61.9+8.7=70.6$, rather than the printed 70.5; rounding of the underlying values can account for a tenth and does not change this interpretation.

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.