Contrasting Marginal Means And Marginal Variances

Why Marginal Means and Variances Need Separate Attention

A GLMM describes an outcome conditional on covariates and random effects. A population-averaged response-scale mean instead averages the conditional mean over the random-effects distribution. That integration can change the mean because the inverse link is nonlinear. It also contributes to the outcome's marginal variance, but the way it contributes depends on the outcome family.

The distinction is especially instructive for binary and count outcomes. A binary mixture has a variance determined entirely by its marginal success probability. A Poisson mixture, by contrast, can have a variance greater than its marginal mean because its conditional rate varies across random effects. In both cases, marginal variance is a property of the outcome distribution, not the uncertainty in an estimated mean.

General Moment Decomposition

Fix a covariate pattern \(x\), and let \(B\) denote the random effect or random-effects vector for a cluster. Assume the GLMM specifies a conditional outcome distribution given \(B\) and \(x\), and that the stated random-effects distribution is the one over which the population average is defined. Write the conditional mean as \(m(B,x)=E(Y\mid B,x)\) and the conditional variance as \(v(B,x)=\operatorname{Var}(Y\mid B,x)\). Assume these moments exist.

The governing identities are the laws of total expectation and total variance:

This decomposition requires no special link function. It follows directly by adding and subtracting the conditional mean in the squared deviation from the marginal mean.

Binary Outcomes: Variation in Probabilities, Not Extra Bernoulli Variance

For a binary outcome, suppose \(Y\mid B,x\sim\operatorname{Bernoulli}\{p(B,x)\}\), where \(p(B,x)=P(Y=1\mid B,x)\). Then \(m(B,x)=p(B,x)\) and \(v(B,x)=p(B,x)\{1-p(B,x)\}\). Consequently,

and

Expanding the right-hand side gives \(M(x)-M(x)^2\), so \(V(x)=M(x)\{1-M(x)\}\). A single marginal binary outcome is still Bernoulli, with success probability \(M(x)\). The random effects divide this variance into average conditional Bernoulli variation and variation in success probabilities across clusters; they do not create variance beyond that of a Bernoulli variable with the same marginal mean.

Count Outcomes: Conditional Poisson Variation Plus Rate Heterogeneity

For a Poisson GLMM, suppose \(Y\mid B,x\sim\operatorname{Poisson}\{\lambda(B,x)\}\). Its conditional mean and variance both equal \(\lambda(B,x)\). The general decomposition becomes

The first variance component is the average conditional Poisson variance. The second is the variance of the conditional rates across random effects. Unless the rates are constant across the random-effects distribution, the marginal count variance exceeds its mean, so the marginal outcome is not Poisson with mean \(M(x)\).

To justify the result, use the normal moment-generating identity \(E\{\exp(tB)\}=\exp(t^2\tau^2/2)\). It gives \(E(\lambda)=\exp\{\eta+\tau^2/2\}=M\), and \(E(\lambda^2)=\exp(2\eta+2\tau^2)\). Thus \(\operatorname{Var}(\lambda)=E(\lambda^2)-M^2=M^2\{\exp(\tau^2)-1\}\). Substitution into the Poisson variance decomposition yields the stated \(V(x)\).

Interpretation, Diagnostics, and Limitations

The formulas rely on the conditional outcome family and random-effects distribution being appropriate for the estimand and data. For the binary case, the conditional response must be Bernoulli; for the count results, it must be conditionally Poisson and the stated random effects must have the required moments. A different conditional distribution changes the \(E_B\{v(B,x)\}\) term. A different random-effects distribution can also change both the marginal mean and the variance.

Useful checks include comparing observed cluster-level outcome patterns with model-based predictions, examining whether conditional residual variation remains systematic, and checking whether the fitted model reproduces the degree of within-cluster heterogeneity relevant to the analysis. For counts, persistent variance beyond the fitted Poisson-lognormal model, excess zeros, or influential clusters can signal misspecification; a large variance by itself does not identify which mechanism caused it. For binary outcomes, the marginal variance at a fixed covariate pattern is fixed by the marginal probability, so it is not an independent diagnostic of extra dispersion.

The main alternatives answer different questions. A conditional GLMM prediction at a specified random effect describes a cluster conditional on that effect; an integrated marginal prediction averages over the specified random-effects distribution. A model directly specified for population-averaged responses can target marginal quantities without interpreting the same conditional random-effects structure. These approaches need not yield the same coefficients or variance assumptions. The formulas here concern response-scale moments at a fixed covariate pattern; they do not by themselves define a target covariate population or account for uncertainty in estimated model parameters.

• Pitfall: treating the random effect's mean as the marginal prediction. A centered random effect has mean zero, but generally \(E_B\{g^{-1}(\eta+B)\}\ne g^{-1}(\eta)\). Integrate the response-scale conditional mean, rather than simply setting \(B=0\).
• Pitfall: calling every random-effect contribution “extra variance.” For binary data, the between-probability component is part of the decomposition, but the total remains \(M(1-M)\). For Poisson mixtures, rate heterogeneity adds to the mean-scale variance and can produce marginal overdispersion.
• Pitfall: confusing outcome variance with standard error. \(V(x)\) describes variation among outcomes under the model. It is not the standard error of an estimated marginal mean or contrast.