Predicting New Combinations in Crossed Models

Why a new combination is not necessarily a new group

In a crossed-effects model, an observation belongs to levels of more than one factor: for example, a clinician and a case type, or a school and a test item. A future observation may involve a pairing that has never appeared in the data even though both factor levels have appeared many times. That is not the same prediction problem as one involving a genuinely new clinician or a new item. The model can use information about each observed level separately, but it has no directly estimated effect for the unobserved pairing unless such an interaction was included in the model.

The key is to classify each factor level, not just the combination. A level is known here if it was represented in the fitted data. The factor levels may have been observed in different pairings; that still makes their individual random effects estimable to some degree. Conversely, a level that was never represented has no data-based level-specific estimate. How much information a known level contributes depends on its data and the model, not merely on its label.

Model and prediction target

For clarity, consider a GLMM with two crossed random-intercept factors, A and B. The response distribution conditional on the random effects belongs to an exponential-family model, and its conditional mean is related to a linear predictor by a link function. A simple form is:

Here \(i\) indexes levels of A, \(j\) indexes levels of B, and \(\mathbf{x}_{ij}\) contains the fixed-effect predictors for the observation. The usual model assumptions include the specified conditional response distribution and link, normally distributed mean-zero random effects, and independence between the two random-effect families in this simple model. Observations are conditionally independent given the random effects and fixed predictors, unless the model specifies further dependence. The model may use a more general covariance structure, but predictions must follow the structure actually fitted.

Write \(\hat{\boldsymbol{\beta}}\) for the fitted fixed effects and \(\hat u_i,\hat v_j\) for the fitted conditional estimates of known-level random effects. These are generally shrunken estimates, not the unknown true effects. For an observation at known levels \(i,j\), the plug-in conditional linear predictor is \(\hat\eta_{ij}=\mathbf{x}_{ij}^{\mathsf T}\hat{\boldsymbol{\beta}}+\hat u_i+\hat v_j\), and its conditional mean prediction is \(g^{-1}(\hat\eta_{ij})\).

This rule applies to the random effects the model contains. If the model includes a random A-by-B interaction, an unobserved pairing has no pairing-specific estimate; it must instead be treated as new under that interaction’s distribution. If there is no such interaction, do not invent an additional effect for the pair. With random slopes, use the estimated random-effect vector for each known level multiplied by that observation’s random-effect design vector; a level contributes only through the random-effect terms present in the fitted model.

Known effects, unseen effects, and response-scale means

For a new level, its random effect has mean zero on the linear-predictor scale under the model. Setting that effect to zero gives a useful link-scale plug-in prediction. It does not generally give the average response for a randomly sampled new level, because applying a nonlinear inverse link after averaging differs from averaging after applying the inverse link.

For example, with a log link and independent normal random intercepts, suppose a prediction has fixed part \(\eta_0\) and includes unseen effects \(U_1,\ldots,U_k\), where \(U_r\sim N(0,\tau_r^2)\). The plug-in value at zero unseen effects is \(\exp(\eta_0)\). The mean response averaged over those new effects is:

This follows because the sum of independent normal effects is normal with mean zero and variance \(\sum_r\tau_r^2\), and a normal variable \(U\sim N(0,\tau^2)\) satisfies \(E\{\exp(U)\}=\exp(\tau^2/2)\). Thus, averaging over unseen levels raises the mean above the zero-effect plug-in value for a log link. For a known level, a plug-in conditional prediction instead uses its fitted effect; averaging over uncertainty about that effect is a further prediction-inference task. Under a logit link, the corresponding response-scale average generally has no such simple closed form and can be evaluated by numerical integration or simulation.

A practical construction and its checks

Before computing a prediction, define what is being predicted: a conditional mean for specified known levels, a mean averaged over new levels, or a future observation including its response variation. Then verify the factor membership against the fitted data. The distinction between a new pairing and a new level is essential; a pairing can be new while both fitted effects are available. A compact construction is:

Several checks help identify a prediction that may be unreliable. Confirm that the new observation uses the same factor definitions and coding as the fitted model. Inspect how much information supports each known level: a fitted random effect based on very few observations is typically strongly shrunk and uncertain. In crossed designs with sparse or disconnected membership patterns, estimates and predictions can be weakly supported; the existence of a numerical fitted effect does not guarantee precise prediction. Also check whether the prediction is outside the covariate range used for fitting. These checks concern the design and model support, not just arithmetic.

Alternatives and limitations

A conditional prediction uses fitted random effects for known levels and is useful when the target specifically concerns those realized levels. A population-level prediction averages over random effects for newly sampled levels; setting new effects to zero is a link-scale reference value, not always that response-scale average. Treating every factor as fixed is another possible modeling choice, but it changes the model and the intended inference rather than providing a shortcut for the fitted random-effects model. Earlier material on predicting responses for new nested groups uses the same distinction between observed and unseen levels, but crossed models require making that decision separately for each factor.

A predicted mean is not an observed outcome. For a future count, for instance, the count varies around its conditional mean even if all model parameters and random effects were known. A prediction interval for a future observation must include this response variation and may also need to reflect uncertainty in fitted parameters and known-level random effects. A confidence interval for a mean and a prediction interval for an outcome therefore have different targets.

The calculation also relies on the fitted structure being appropriate. If the scientific process contains pair-specific variation but the model has no interaction term, the prediction cannot represent that source of variation. If an interaction is included, a new pairing has no observed interaction estimate and must be handled as a new random effect. This tutorial’s simple variance corrections assume independent normal random intercepts and a log link; other links, correlated effects, and random slopes require averaging under their own fitted distributions.

Common mistake

A frequent error is to see a pairing absent from the data and set both factor effects to zero. That discards information whenever the two levels have appeared in other pairings. The correction is to check level membership independently: use the fitted random effect for each known level, handle only unseen levels as new, and introduce a pair-specific term only if the fitted model includes one. A second error is to call \(\exp(\eta_0)\) the average response for new levels under a nonlinear link; for a log link, the normal random-effect variance correction shows why that is generally too small.