Predicting a Random Effect for a New Group

Why a New Group Is Different

For a group already represented in the data, its outcomes can update what the model says about that group’s latent deviation from the population pattern. A group with no observed outcomes offers no such update. This does not mean the model cannot make predictions for the group: it means those predictions come from the fitted distribution of group effects, together with any observed covariates and the response model, rather than from group-specific outcome evidence.

That distinction matters because “predict the random effect” and “predict a response from the new group” are different tasks. The first concerns the latent group deviation itself. The second also includes the conditional response variability specified by the GLMM. Both can be addressed using the fitted model, but neither should be mistaken for a data-informed estimate of a particular new group’s deviation.

For example, write a GLMM’s conditional linear predictor for observation \(i\) in group \(j\) as \(\eta_{ij}=x_{ij}^{T}\beta+z_{ij}^{T}b_j\). Here \(b_j\) is the group’s random-effect vector, and its distribution is specified by the model. For a new group, use a new draw \(b_*\) from that distribution. If the model says group effects are independent and identically distributed, then \(b_*\) is independent of the effects for previously observed groups, conditional on the model parameters.

Why There Is No Data-Informed Group Estimate

The key result follows directly from Bayes’ rule. For an observed group with outcomes \(y_j\), its conditional distribution for the random effect is proportional to the group’s likelihood times the population distribution for the effect. For a group with no outcomes, the likelihood contribution is the constant 1. Multiplying the population distribution by 1 leaves it unchanged.

The proportionality constant is therefore already one, so the conditional distribution is exactly \(F(\,\cdot\,;\hat\psi)\). In particular, a posterior mean or mode for this group is just the corresponding summary of the fitted population distribution. If the random effect is modeled to have mean zero, the posterior mean is zero; that is a population-centered summary, not evidence that the particular group has no deviation. If the fitted distribution is asymmetric, its mean and mode need not coincide.

The result assumes the group truly contributes no outcome information and that its random effect follows the fitted distribution appropriate to that group. If the model specifies a distribution conditional on known group-level covariates, those covariates can affect the distribution even when the group has no outcomes. They still do not supply outcome-based evidence about that group’s realized deviation.

From the Random-Effect Distribution to Response Predictions

A future response depends on both the new group’s latent effect and the conditional response model. Let \(Y_*\) denote a response in the new group at specified covariates \(x_*,z_*\). The plug-in predictive distribution averages the conditional response distribution over possible group effects. This averaging is often called integration over the random effects; for a discrete distribution it is a weighted sum.

Here \(A\) is any set of response values. The conditional probability inside the integral comes from the GLMM’s response distribution and link function. The integral accounts for uncertainty about which random effect the new group will have. In general, evaluating the response model at the mean random effect is not equivalent to averaging the conditional response probabilities.

Assumptions, Diagnostics, and Alternatives

These predictions rely on the fitted model being a reasonable description of the population of groups to which the new group belongs. In particular, the model must specify a suitable random-effect distribution, the new group must be exchangeable with the groups used to fit it (or have its own correctly specified conditional distribution), and the response distribution and link must be suitable. The prediction also conditions on estimated parameters in the examples above; treating estimates as fixed omits parameter-estimation uncertainty.

For a new-group response, the predictive distribution includes variation from both the random effect and the conditional response distribution. A predictive interval or tail probability should therefore be calculated from the mixture distribution, not just from the uncertainty in a group-effect summary. If uncertainty in \(\hat\beta\) and \(\hat\psi\) is material, a fuller prediction should additionally account for uncertainty in those fitted parameters.

Useful checks include comparing the fitted random-effect distribution with the information supported by observed groups, checking whether predictions are sensitive to its assumed shape or estimated variability, and assessing conditional response fit. For example, systematic lack of fit in predicted binary probabilities or count distributions can undermine new-group predictions even if the random-effect distribution appears plausible. A variance estimate near zero also makes the fitted distribution concentrate near its center, so predictions may approach those from a model without group variation; this is not proof that all future groups are identical.

The main alternatives answer different questions. A prediction at \(b_*=0\) describes a group at the population center, not a randomly selected group averaged over the fitted effects. A conditional prediction at a specified \(b_*\) assumes that effect value is known. A response prediction that ignores random effects can omit between-group variation. The new-group mixture instead averages over the fitted effect distribution. None of these choices can recover an outcome-informed estimate for a group that has supplied no outcomes.

Practical Workflow

The central limitation is that the distribution is learned from the observed groups and the model assumptions. It may be poorly estimated when few groups are available, misspecified if the new group comes from a different population, or insufficient when relevant group-level predictors have been omitted. New-group prediction is therefore a model-based population prediction, not a substitute for data from the particular group.