Subgroup Exposure-Response Comparisons

What a Subgroup Comparison Must Establish

A treatment may have a different exposure-response relationship in different patient groups. For example, the change in an efficacy score associated with a given increase in exposure might differ by renal function category. The relevant question is not simply whether an exposure-response association is detectable in each subgroup. It is whether the estimated relationships differ from one another, by how much, and with what uncertainty.

The comparison should be defined before interpreting results: specify the subgroup, the exposure contrast, the outcome scale, and the model scale on which differences will be assessed. An interaction term in a pooled model gives a direct estimate of subgroup difference on the specified model scale. Its interpretation depends on that scale: a difference in linear slopes is not the same quantity as a ratio of odds ratios or a difference in predicted probabilities.

Subgroups should be prespecified when possible, with a clinically interpretable definition and a clear reference group. This helps distinguish a planned comparison from a pattern noticed after inspecting many possible partitions. Prespecification does not, by itself, remove uncertainty, confounding, or the need to consider multiplicity when many subgroup claims are made.

A Pooled Interaction Model

For a continuous outcome measured once per participant, a simple linear model is

Here \(Y_i\) is the outcome for participant \(i\), \(x_i\) is exposure on a specified scale, and \(G_i=0\) or \(1\) identifies subgroup membership. The error \(\varepsilon_i\) has mean zero conditional on the modeled predictors. For the usual model-based standard errors and tests, assume independent participants, a correctly specified conditional mean, and constant error variance; normal errors support exact small-sample inference, while large-sample inference is often approximately normal. If these assumptions are not suitable, the uncertainty calculation must be adapted rather than treating the usual standard error as automatically valid.

When \(G_i=0\), the exposure slope is \(\beta_E\). When \(G_i=1\), it is \(\beta_E+\beta_{EG}\). Therefore the slope difference, subgroup 1 minus subgroup 0, is \(\beta_{EG}\). The interaction estimate directly addresses the comparison; the subgroup-specific slopes describe its components.

If \(x\) is measured in units of 10 mg/L, a slope is a change in outcome units per 10 mg/L. Centering exposure at a meaningful value can make the intercept and subgroup main effect easier to interpret; it does not change the linear slope difference. The subgroup main effect \(\beta_G\) is the difference between groups at \(x=0\), not a general measure of subgroup difference at every exposure.

Estimating Uncertainty in the Contrast

The interaction estimate is a linear contrast of model coefficients. If \(\hat{\boldsymbol{\beta}}\) is the vector of fitted coefficients, \(c\) is a vector selecting the desired contrast, and \(\widehat{\mathrm{Cov}}(\hat{\boldsymbol{\beta}})\) is their estimated covariance matrix, then the estimated contrast is \(c^\mathsf{T}\hat{\boldsymbol{\beta}}\) and its standard error is the square root of \(c^\mathsf{T}\widehat{\mathrm{Cov}}(\hat{\boldsymbol{\beta}})c\). A large-sample 95% confidence interval is the estimate plus or minus 1.96 standard errors. For smaller samples, use an appropriate \(t\) critical value and degrees of freedom when justified.

This calculation matters because coefficients in the same model are generally correlated. For the subgroup 1 slope, \(\hat{\beta}_E+\hat{\beta}_{EG}\), its variance includes the covariance between the two estimates:

Interactions in Binary-Outcome Models

For a binary outcome, a logistic model can express the exposure relationship on the log-odds scale:

This model assumes a correctly specified conditional log-odds relationship and, for standard likelihood-based inference, independent participant outcomes. Here \(\beta_E\) is the exposure effect on log odds in subgroup 0. In subgroup 1, it is \(\beta_E+\beta_{EG}\). Consequently, \(\exp(\beta_{EG})\) is the ratio of the exposure odds ratios for the same exposure increment in subgroups 1 and 0. It is not a difference in event probabilities. To compare probabilities, calculate predicted probabilities at specified exposure values and use an uncertainty calculation for that probability contrast.

Why Separate Significance Tests Do Not Compare Subgroups

A common error is to conclude that exposure-response relationships differ because the estimate is statistically significant in one subgroup but not in another. A test in each subgroup asks whether that subgroup's effect differs from zero. It does not test whether the two subgroup effects differ from each other. The difference depends on both estimates and the uncertainty of their contrast.

Assumptions, Diagnostics, and Alternative Comparisons

An interaction estimate is only as meaningful as the model and data supporting it. Consider the following before interpreting a subgroup contrast:

• Comparable exposure scale and range. State the exposure unit and compare effects over a range represented in both groups. If exposure values barely overlap, a model may compare slopes largely through extrapolation rather than information from both subgroups.
• Appropriate outcome model. Check that the selected outcome distribution, link, and exposure-response form are reasonable for the question. A linear interaction compares linear slopes; it cannot capture a true difference in curvature if curvature is absent from the model.
• Subgroup and covariate specification. Define subgroup membership clearly. If adjustment covariates are needed, state what is adjusted for and ensure the subgroup contrast remains interpretable. A single exposure interaction does not automatically account for other subgroup-varying relationships.
• Dependence and uncertainty. Repeated outcomes, clustering, or other dependence require an uncertainty method that accounts for that structure. Sparse observations or events in a subgroup can produce imprecise estimates or unstable fitting.
• Exposure-response interpretation. Differences in observed exposure relationships do not by themselves establish causal effect modification. As emphasized in the earlier material on causal assumptions in individual-level exposure-response, confounding and the way exposure arises must be considered before making causal claims.

A pooled model with an interaction is often the clearest primary approach because it estimates the contrast directly and can share specified model components across groups. Separate subgroup fits can be useful when models genuinely need different structures, but the difference and its uncertainty must still be calculated; separate \(p\)-values are not a substitute. A third option is to report subgroup-specific predicted outcomes at clinically meaningful exposure values. This can aid interpretation, especially for nonlinear links, but the chosen exposure values and probability-scale contrast should be explicit, and uncertainty must be propagated from the fitted model.

The scale of comparison is a substantive choice, not just a technical detail. A linear model interaction compares outcome-unit slopes. A logistic interaction compares log-odds slopes, equivalently the ratio of odds ratios for a fixed exposure increment. Two groups can differ on one scale and appear similar on another. When the clinical question concerns absolute risks or outcome changes, report relevant predictions or contrasts on that scale as well as the model interaction, with uncertainty.

Common Pitfalls and Reporting

• “Significant here, not significant there” means different. It does not. Estimate the between-group contrast and its uncertainty directly.
• An interaction \(p\)-value says how large the difference is. It does not. Report the estimated difference, its units or scale, and an uncertainty interval.
• A non-significant interaction proves the relationships are the same. It does not. The interval may include both negligible and clinically important differences. A claim of similarity requires a defined range of differences considered acceptably small and an analysis designed to assess that claim.
• An interaction has one scale-independent meaning. It does not. State whether it is a slope difference, log-odds interaction, ratio of odds ratios, or another defined contrast.
• A precise-looking estimate is necessarily reliable. It may be sensitive to weak exposure overlap, sparse subgroup data, model misspecification, or subgroup selection. Prespecification and a reasonable uncertainty calculation do not eliminate these limitations.

A concise report should name the prespecified groups and reference group, identify the exposure increment and outcome scale, give the subgroup-specific estimates, and report the direct interaction contrast with its interval. For example: “The estimated difference in linear slopes (group 1 minus group 0) was \(-1.10\) score points per 10 mg/L (95% interval \(-1.982\) to \(-0.218\)).” Add whether the result is an association or a causal estimate, as warranted by the analysis assumptions.