Marginal Rate Ratios For Count Outcomes

Why a Count Rate Ratio Can Depend on the Scale

In a count GLMM with a log link, a fixed-effect coefficient often has a simple conditional interpretation: exponentiating it gives a rate ratio for units with the same random effects. A population-averaged comparison asks a different question. It compares expected rates after averaging over the random-effects distribution. These two ratios need not agree, particularly when the model contains random slopes.

The distinction matters when the scientific comparison concerns rates in a population of clusters rather than a comparison at a specified cluster effect. The calculations below hold other covariates fixed and average over the same random-effects distribution at each covariate setting. They do not average over a distribution of covariates.

Random-Intercept Model: The Averaging Multiplier Cancels

Suppose cluster \(j\) has a count \(Y_{ij}\) and exposure \(E_{ij}>0\). A Poisson random-intercept model with a log link can be written

Here \(\log(E_{ij})\) is an offset, so the conditional rate per exposure unit is \(\mu_{ij}/E_{ij}\). Assume the random effect is independent of the covariate settings being compared and has the same distribution at both settings. The marginal rate at \(x\) is therefore the conditional rate integrated over \(b_j\):

The final equality follows from the moment-generating function of a normal random variable: if \(b\sim N(0,\sigma^2)\), then \(E[\exp(b)]=\exp(\sigma^2/2)\). For two covariate settings \(x_a\) and \(x_b\), the marginal rate ratio is

Thus, in this model, the random-intercept multiplier raises both marginal rates by the same factor and cancels from their ratio. This equality does not mean that the conditional and marginal rates are equal: each marginal rate includes the factor \(\exp(\sigma_b^2/2)\). It means that their ratio is unchanged when the random-intercept distribution is common across settings.

Random Slopes: The Averaging Multiplier Can Differ

Now allow the covariate effect to vary across clusters. For a single covariate \(x\), write the random intercept and slope as \(b_{0j}\) and \(b_{1j}\), with \((b_{0j},b_{1j})^\mathsf{T}\sim N(\mathbf{0},\Sigma)\). The conditional mean is

Define \(\mathbf{z}(x)=(1,x)^\mathsf{T}\). The random contribution to the linear predictor is \(\mathbf{z}(x)^\mathsf{T}\mathbf{b}_j\), which is normally distributed with mean zero and variance \(v(x)=\mathbf{z}(x)^\mathsf{T}\Sigma\mathbf{z}(x)\). Integrating its exponential gives

For \(\Sigma\) with intercept variance \(\sigma_{00}\), slope variance \(\sigma_{11}\), and intercept-slope covariance \(\sigma_{01}\), this variance is \(v(x)=\sigma_{00}+2x\sigma_{01}+x^2\sigma_{11}\). Consequently,

This equation separates the fixed-effect contrast from the change in the random-effect variance across settings. By contrast, at a particular cluster with random slope \(b_{1j}\), the conditional rate ratio is \(\exp\{(\beta_1+b_{1j})(x_b-x_a)\}\). At zero random effects, the conditional ratio is \(\exp\{\beta_1(x_b-x_a)\}\). The marginal ratio generally differs from that zero-random-effect comparison when \(v(x_b)\ne v(x_a)\). Intercept-slope covariance contributes to that difference as well as slope variance.

Offsets, Rates, and Expected Counts

A rate ratio and an expected-count ratio answer different questions when exposures differ. With the same random-intercept distribution at both settings, the marginal rate ratio remains \(\exp\{\beta_1(x_b-x_a)\}\). But the expected count is exposure multiplied by the marginal rate. If exposure differs between settings, the expected-count ratio includes the exposure ratio too.

Assumptions, Diagnostics, and Limits

These integrations require a correctly specified mean model and a random-effects distribution for which the exponential moment exists. In the formulas above, random effects are multivariate normal with mean zero and finite covariance matrix, and their distribution is shared across the covariate settings being compared. The exposure enters as a known offset with coefficient fixed at one. Conditional Poisson sampling is a conventional count-model assumption; the marginal mean formulas follow by integrating the conditional mean over the stated random-effects distribution.

For a random-intercept model, examine whether the common random-intercept distribution is plausible and whether residual variation suggests that the conditional count model is inadequate. For random slopes, also check whether the fitted covariance matrix is admissible, whether the data support estimating slope variation, and whether predicted rates are sensitive to the assumed random-effects distribution. Sparse data or limited within-cluster variation can make a random-slope variance and covariance unstable. Numerical integration or simulation may be useful for models whose random-effects structure is more complicated than the normal random-intercept or slope cases derived here.

A common mistake is to exponentiate the fixed slope and call it the marginal rate ratio in every log-link GLMM. That is correct for a random-intercept model under the shared-distribution assumptions, but not generally for a random-slope model: the marginal log ratio also contains \(\{v(x_b)-v(x_a)\}/2\). Another mistake is to compare expected counts without accounting for exposure. State whether the target is a rate per exposure unit or a count over a specified exposure.

The conditional comparison is useful when the target is a cluster with specified random effects; the marginal comparison is useful when the target is a rate averaged over the model’s random-effects distribution. Neither is automatically preferable: the estimand should match the scientific question. These model-based marginal rates also do not, by themselves, average over differing covariate distributions or establish a causal effect.