 Solution

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.