Centering Predictors Within Nested Groups

Why Separate Within-Group and Between-Group Variation?

Suppose a clinical service records how many hours each patient spends in a self-management program, with patients nested within clinics. A patient who attends more hours than other patients in the same clinic differs in a within-clinic way. A clinic whose patients attend more hours on average differs in a between-clinic way. Those are distinct comparisons, and their associations with an outcome need not be equal.

A model using only the raw predictor can combine these sources into one coefficient. Group-mean centering makes the within-group deviation explicit; including the group mean as a second predictor represents between-group differences separately. In a GLMM, this distinction applies to the conditional model, given the random effects. It does not by itself make an association causal or remove all assumptions about how predictors relate to random effects.

Here \(\bar{x}\) may be the observation-weighted grand mean, \(\sum_j n_j\bar{x}_j/\sum_j n_j\), or a group-weighted mean, \(\sum_j\bar{x}_j/J\). The choice changes the reference point and intercept, not the within-group deviations. State the choice, especially with unequal group sizes. A group-mean centered predictor alone is not the full within-between decomposition: including \(x_{B,j}\) allows a separate between-group association.

A Conditional GLMM for the Decomposition

For a nested random-intercept and random-within-slope model, write

The conditional response distribution is from the chosen GLM family, with its variance and link specified for the outcome. Responses are conditionally independent given the random effects and predictors. The group effects satisfy \((b_{0j},b_{1j})^\mathsf{T}\sim N_2(0,\Sigma_b)\), independently across groups, with an estimated covariance matrix \(\Sigma_b\). The model also assumes that this random-effects distribution is appropriate and that the predictors are suitably independent of the random effects under the model’s conditioning assumptions. Inference requires enough groups and enough informative within-group data for the parameters to be estimated; centering does not guarantee that.

The fixed within-group coefficient \(\beta_W\) describes the average change in the conditional linear predictor for a one-unit increase in the predictor relative to the group mean. In this random-slope model, group \(j\)’s conditional within-group slope is \(\beta_W+b_{1j}\). The fixed between-group coefficient \(\beta_B\) describes the change in the conditional linear predictor associated with a one-unit increase in the centered group mean, holding the within deviation and random effects fixed. These are link-scale interpretations; conditional means generally require applying the inverse link.

Deriving the Within-Between Representation

The decomposition follows directly by adding and subtracting the group mean. Its value is that it makes explicit what a raw-predictor model assumes.

This algebra holds for the linear predictor in a GLMM just as it does for a linear model. It does not assert that the marginal mean, after averaging over random effects, has the same coefficient interpretation. The link-scale coefficients are conditional on the random effects.

Worked Conditional Interpretation

Which Variation Informs Which Slope?

Within-group effects are learned from differences among observations in the same group. Between-group effects are learned from differences among group means. A random slope for \(x_{W,ij}\) allows the within-group association to vary across groups, so its variance is informed by the pattern of within-group contrasts across groups. Groups with no within-group variation in \(x\) have \(x_{W,ij}=0\) for every observation and provide no direct within-group slope contrast. Repeated observations and adequate within-group spread are important; a large total sample does not compensate automatically for too few groups or little within-group variation.

The following design calculation illustrates the separation. For the clinic data above, the within-group sum of squares is

The weighted sum of squares of group means around the grand mean is

These are predictor-design summaries, not estimates of outcome variance or a direct comparison of coefficient precision. They show that the data contain both within-clinic and between-clinic predictor variation. If the within sum of squares were zero, the data could not estimate a within-clinic slope from observed within-clinic contrasts. If all clinic means were identical, there would be no between-clinic variation to estimate \(\beta_B\).

Alternatives, Diagnostics, and Limits

Useful checks include tabulating group sizes and within-group ranges, plotting \(x_{W,ij}\) by group, and checking the distribution of group means. Inspect fitted random-slope variation, its uncertainty, and convergence; a variance estimate near zero or a poorly determined intercept-slope correlation can indicate weak information or a model that is too ambitious for the design. Check whether the assumed conditional response distribution and link are reasonable, and examine residual or predictive diagnostics appropriate to the GLMM. A converged fit alone does not establish that the decomposition or random-effects assumptions are appropriate.

Centering is a change of predictor representation, not a remedy for confounding or a substitute for design. The model still relies on its assumptions about the relationship between covariates and random effects. Measurement error in group means, very small groups, missing observations, and informative group sizes can complicate interpretation. The random slope specified here concerns within-group deviations; adding a random slope for a group-level predictor is a different model choice, requiring enough groups and support for that variation.