Group-Mean Centering and Conditional Effects

Why Separate Within-Cluster and Between-Cluster Associations?

Suppose a health service records a predictor such as weekly exercise for patients treated at several clinics. Two questions are distinct: within a clinic, are patients who exercise more than their clinic’s average different in outcome from their peers? And across clinics, do clinics with higher average exercise have different outcomes? A single predictor coefficient can blur these associations, even when the predictor varies both between patients and between clinics.

Group-mean centering makes the distinction explicit. It does not, by itself, establish a causal effect: patient and clinic characteristics may explain some or all of an observed association. In a GLMM, the two associations are expressed on the model’s link scale; their translation to the response scale depends on the link and the random effects.

For each observation, the decomposition \(x_{ij}=x_{ij}^{W}+\bar{x}_j\) is exact. A positive \(x_{ij}^{W}\) means the observation is above its cluster’s mean; a negative value means it is below. A value of zero means the observation is at the cluster mean. The centered value and the cluster mean answer different questions and should not be treated as interchangeable versions of the same predictor.

A Random-Slope Model With Both Components

Let \(Y_{ij}\) be the response for observation \(i\) in cluster \(j\), and let \(b_{0j}\) and \(b_{1j}\) be that cluster’s random intercept and random slope. A model that separates the two associations is

Here \(g\) is the link function. The fixed coefficient \(\beta_W\) describes the average within-cluster association: the change in the conditional linear predictor for a one-unit increase in \(x_{ij}\) relative to the cluster mean, holding the cluster mean fixed. The fixed coefficient \(\beta_B\) describes the association with a one-unit increase in the cluster mean when the individual’s centered value is zero. The random slope \(b_{1j}\) allows the within-cluster association to vary by cluster. Thus, the cluster-specific within-cluster slope on the link scale is \(\beta_W+b_{1j}\), not \(\beta_W\) alone.

The coefficient \(\beta_B\) is not the random slope. The random slope applies to within-cluster deviations; the cluster mean appears only with its fixed coefficient in this model. The random intercept shifts the conditional linear predictor for a cluster. When comparing the within- and between-cluster coefficients, \(\beta_B-\beta_W\) is sometimes called a contextual contrast: it quantifies how the between-cluster association differs from the within-cluster association on the link scale. It is not automatically a causal contextual effect.

Why the Decomposition Separates the Associations

The separation follows directly from substituting \(x_{ij}=x_{ij}^{W}+\bar{x}_j\) into a model that allows distinct within- and between-cluster coefficients. The fixed part of the linear predictor is \(\beta_W x_{ij}^{W}+\beta_B\bar{x}_j\). A change in the deviation changes the first term, while a change in the cluster mean changes the second. If instead the model uses one fixed coefficient \(\beta\) for the raw predictor, then

That raw-predictor specification constrains the within- and between-cluster coefficients to be equal. It therefore cannot represent distinct associations unless the equality is reasonable. Including both components relaxes that constraint and makes the intended comparisons visible in the model.

Interpretation, Diagnostics, and Alternatives

Interpret each coefficient at the level it represents. The within-cluster association concerns differences among observations in the same cluster, conditional on the model terms. The between-cluster association concerns differences in cluster means. For a log link, exponentiating a slope gives a conditional rate or mean ratio per unit, where appropriate. For a logit link, exponentiating gives a conditional odds ratio. For other links, interpret the coefficient on the link scale or transform the fitted linear predictor using the response function. A coefficient alone does not give a response-scale change that is constant across all baseline values for a nonlinear link.

Several checks help determine whether the decomposition is informative and adequately fitted:

• Inspect the amount of within-cluster predictor variation. If most clusters have little or no variation, the within-cluster slope and its random variation may be weakly informed.
• Check the calculation and definition of each cluster mean. Means based on different observation sets or missing-data patterns can change what the between-cluster term represents.
• Examine convergence, estimated random-effect variances, and the intercept-slope covariance. Convergence problems or a random-slope variance estimated near zero can signal that the requested random-effects structure is not supported well by the data.
• Assess whether the link-scale mean structure is plausible, including whether a linear within-cluster association is adequate over the observed predictor range. A decomposition does not repair nonlinearity or an inappropriate response distribution.
• Consider whether cluster means are confounded with unmeasured cluster characteristics. A between-cluster association can reflect those characteristics as well as the predictor mean.

The main alternative is to use the raw predictor with one fixed coefficient, which imposes equality of the within- and between-cluster associations. That simpler constraint may be useful when substantively justified, but it should not be mistaken for an estimated separation. A cluster fixed-effects approach can control for all cluster-constant differences when estimating within-cluster associations, but it does not estimate effects of cluster-constant predictors in the same way. The group-mean-centered model instead includes both the within-cluster deviation and cluster mean, while relying on its random-effects assumptions.

The decomposition is limited by the observed data and model specification. Cluster means can be measured imprecisely when clusters are small; unequal cluster sizes and missing observations affect their precision and interpretation. Neither centering nor a random slope removes confounding, guarantees a causal interpretation, or ensures that the chosen link-scale relationship is correct.

Common Mistake