Comparing Adjusted Rates Across Groups

Why Compare Rates at Shared Conditions?

A group with more observed events does not necessarily have a higher event rate. It may simply have more people at risk, longer follow-up, or a different distribution of risk-related covariates. A model-based adjusted comparison asks a more controlled question: what rates does the fitted model assign to the groups under the same covariate and exposure conditions?

In a Poisson rate model, the expected count depends on both the exposure and the rate. Write \(Y_i\) for the count in record \(i\), \(E_i>0\) for its exposure, and \(\mu_i=E[Y_i]\). A log-link model with an exposure offset has the form \(\log(\mu_i)=\log(E_i)+\eta_i\). The quantity \(\exp(\eta_i)\) is the modeled rate per unit of exposure. Comparing groups means evaluating this rate at shared covariate values, or averaging their predicted rates over the same specified covariate distribution.

The comparison must specify its target conditions. If the model contains group-by-covariate interactions, the adjusted contrast can change with the selected profile. If the goal is a single comparison over a covariate distribution rather than at one profile, both groups must be predicted over the same target distribution and combined using the same weights.

Defining the Comparison from the Model

For illustration, let \(G_i\) indicate membership in group B, with group A as the reference, and let \(z_i\) be a covariate. A model that allows the group difference to depend on \(z_i\) is

The exposure \(E_i\) enters as an offset: its coefficient is fixed at one. The group A rate at \(z\) is \(\exp(\beta_0+\beta_z z)\); the group B rate is \(\exp(\beta_0+\beta_G+(\beta_z+\beta_{Gz})z)\). Dividing the two rates cancels the shared terms, leaving a rate ratio that may depend on \(z\).

Without the interaction, \(\beta_{Gz}=0\), the model implies a constant rate ratio across values of \(z\). With the interaction, \(\beta_G\) alone is the log rate ratio only at \(z=0\). For a contrast at another value, the interaction contribution must be included.

Shared Exposure and the Difference Between Rates and Counts

For a fixed covariate profile, a model-based rate prediction is \(\exp\{\eta(g,z)\}\). At a chosen exposure \(E^\ast\), the corresponding expected count is \(E^\ast\exp\{\eta(g,z)\}\). If the same positive \(E^\ast\) is used for both groups, the ratio of predicted counts equals the ratio of predicted rates: the common exposure cancels. This does not make the observed counts interchangeable with rates; observed counts can reflect unequal exposure and different covariate profiles.

To verify the result, let the predicted means at the shared profile be \(\mu_A^\ast=E^\ast r_A\) and \(\mu_B^\ast=E^\ast r_B\). Because \(E^\ast>0\),

If the exposures instead are \(E_A\) and \(E_B\), then \(\mu_B/\mu_A=(E_B/E_A)(r_B/r_A)\). A ratio of counts is then not, by itself, a rate ratio.

Adjusted Comparisons Over a Shared Covariate Distribution

A single covariate profile may not be the most relevant target. For a finite set of target profiles \(z_1,\ldots,z_K\), choose weights \(w_k\geq 0\) with \(\sum_{k=1}^K w_k=1\). The adjusted rate for group \(g\), averaged over this common target, is

Both groups use the same profiles and weights. One may report \(R_B/R_A\) or \(R_B-R_A\). When the model has interactions, these averaged contrasts generally cannot be obtained by exponentiating the group coefficient: each profile’s predicted rate must first be calculated, then averaged. The target profiles and weights determine what population the comparison describes.

Uncertainty for a Model-Based Rate Ratio

At a fixed profile in a log-link model, a log rate ratio is often a linear contrast of the model coefficients. Write the coefficient vector as \(\beta\), its estimator as \(\hat{\beta}\), and let \(c\) be a known contrast vector. Then \(\log(r_B/r_A)=c^\mathsf{T}\beta\), estimated by \(c^\mathsf{T}\hat{\beta}\). For the interaction model above at covariate value \(z_0\), the contrast is \(\beta_G+z_0\beta_{Gz}\).

The variance expression follows from the variance rule for linear combinations: for a random vector \(U\) with covariance matrix \(\Sigma\), \(\operatorname{Var}(c^\mathsf{T}U)=c^\mathsf{T}\Sigma c\). Applying this to the asymptotic distribution of \(\hat{\beta}\) gives the stated estimated standard error. Under the stated asymptotic-normality and consistency conditions, dividing the centered contrast by that standard error converges in distribution to \(N(0,1)\), by Slutsky’s theorem. The Wald interval is therefore a large-sample interval, not an exact finite-sample guarantee.

Common Mistakes and Interpretation

• Comparing event totals instead of rates. A larger total can arise from greater exposure. Calculate or model rates using the exposure information before interpreting a group contrast.
• Calling a coefficient an adjusted comparison without checking interactions. In an interaction model, the group coefficient is the log rate ratio only at the reference covariate value. Include the interaction terms for the chosen profile.
• Comparing predictions at different covariate values. An adjusted group contrast should use the same profile, or the same target profiles and weights, for both groups.
• Exponentiating a rate difference. Exponentiating a log rate ratio yields a rate ratio. A rate difference is obtained by subtracting predicted rates on their original scale.
• Ignoring the exposure units. A rate stated per person-year differs numerically from a rate stated per 1,000 person-years by a factor of 1,000. State the unit when reporting rates and rate differences.

Adjusted comparisons are useful because they make the conditions of a model-based group contrast explicit. Their meaning depends on the fitted model, the covariates included, and the selected profile or target distribution. They do not automatically establish a causal effect: that interpretation requires additional design and causal assumptions beyond the rate-model calculation.