Why Model an Exposure-by-Covariate Interaction?
An exposure-response relationship may differ across patients. For example, the change in clinical response associated with higher exposure might be smaller among patients with impaired renal function, or might vary with baseline disease severity. A model can represent this possibility by allowing the exposure-response slope to depend on a covariate. That is an exposure-by-covariate interaction.
An interaction is not the same as a difference in exposure between covariate groups. If one group tends to have higher exposure, its average outcome can differ even when the exposure-response slope is identical in both groups. The interaction concerns the relationship between exposure and outcome at different covariate values, conditional on the model and its assumptions.
Model and Interpretation
Let \(E\) denote an individual’s exposure and \(Z\) a covariate measured before the relevant outcome period. Choose reference values \(E_{\mathrm{ref}}\) and \(Z_{\mathrm{ref}}\), and define scaled variables \(x=(E-E_{\mathrm{ref}})/s_E\) and \(z=(Z-Z_{\mathrm{ref}})/s_Z\), where \(s_E\) and \(s_Z\) are specified scaling units. For example, \(x\) could count increments of 10 exposure units and \(z\) increments of 20 mL/min in renal function. With additional covariates \(C\), a general interaction model is
Here \(g\) is the chosen link function, \(Y\) is the outcome, and \(E(Y\mid x,z,C)\) denotes its conditional mean. The model includes both main effects as well as the product term. If \(x=0\) and \(z=0\) are the reference values, \(\beta_E\) describes the exposure effect on the link scale at \(z=0\); \(\beta_Z\) describes the covariate effect at \(x=0\). The coefficient \(\beta_{EZ}\) describes how the exposure coefficient changes as \(z\) changes. Centering makes these reference-point interpretations explicit; it does not, by itself, remove confounding or change whether the interaction is present.
For an identity link, the modelled mean at fixed \(z\) and \(C\) has exposure slope \(\beta_E+\beta_{EZ}z\). Thus, comparing two covariate values \(z_1\) and \(z_0\), the difference in exposure slopes is \(\beta_{EZ}(z_1-z_0)\). This follows directly by subtracting the two slopes:
If \(z\) is an indicator equal to 0 or 1, \(\beta_E\) is the exposure slope in the reference category and \(\beta_E+\beta_{EZ}\) is the slope in the other category. For a logit link, these are slopes for log odds, not probabilities: exponentiating them gives the conditional odds ratio for a one-unit increase in \(x\) at the specified \(z\). The interaction coefficient exponentiated is the ratio of those exposure odds ratios between \(z=1\) and \(z=0\).
Worked Examples
Worked Example: Renal Function Modifies a Continuous Exposure Slope
Suppose a hypothetical study models change in a clinical efficacy score, with larger values representing greater improvement. Let \(x\) be exposure in increments of 10 units above the reference exposure, and let \(z\) be renal function in increments of 20 mL/min above 60 mL/min. The fitted identity-link model, with no additional covariates for this illustration, is
At the reference renal function, \(z=0\), the exposure slope is \(\beta_E=3\): each 10-unit increase in exposure is associated with 3 additional score points in mean improvement. At \(z=1\), corresponding to renal function 80 mL/min, the slope is \(3-1.2=1.8\) points per 10 exposure units. At \(z=-1\), corresponding to 40 mL/min, the slope is \(3+1.2=4.2\) points per 10 units. The interaction coefficient is therefore a difference of \(-1.2\) score points per exposure increment for each 20 mL/min increase in renal function.
For a direct prediction check, at \(x=1,z=1\), the predicted mean is \(12+3+1.5-1.2=15.3\) points. At the same renal function but reference exposure, \(x=0,z=1\), it is \(12+1.5=13.5\). Their difference is 1.8 points, matching the slope at \(z=1\). These are model-based conditional predictions; whether they support a causal claim also depends on confounding control and exposure overlap.
Worked Example: A Logistic Interaction and Its Uncertainty
In a hypothetical binary efficacy analysis, \(Y=1\) denotes clinical success. Let \(x\) be exposure in 10-unit increments above a reference, and let \(z=1\) indicate renal impairment while \(z=0\) indicates the reference renal-function category. Suppose the fitted logit model is
For patients in the reference category, a one-unit increase in \(x\) changes the log odds by 0.60, giving an odds ratio of \(\exp(0.60)=1.82\). For patients with renal impairment, the log-odds slope is \(0.60-0.35=0.25\), giving an odds ratio of \(\exp(0.25)=1.28\). The interaction odds ratio is \(\exp(-0.35)=0.705\): the exposure odds ratio in the impaired group is estimated to be 0.705 times the corresponding odds ratio in the reference group, per 10-unit exposure increment. This is a comparison of odds ratios, not a probability difference.
Suppose the estimated interaction coefficient has standard error \(SE=0.16\). A Wald statistic for \(H_0:\beta_{EZ}=0\) is \(z=-0.35/0.16=-2.19\), giving a two-sided \(p\)-value of approximately 0.0287 under the large-sample normal approximation. The coefficient-scale 95% Wald interval is calculated using the unrounded endpoints:
On the model assumptions, this interval estimates the ratio of exposure odds ratios; it excludes 1. The \(p\)-value is the probability, under the null model and the test’s assumptions, of a statistic at least as extreme as the observed one. It is not the probability that the null hypothesis is true. Wald inference can be unreliable with sparse data or poorly estimated coefficients, so the estimate should be considered alongside its precision and the data supporting each covariate-exposure region.
Worked Example: Different Exposure Distributions Without an Interaction
Consider a hypothetical identity-link model for an efficacy score, with \(z=0\) for one covariate group and \(z=1\) for another:
There is no \(xz\) term, so the modelled exposure slope is 2 score points per exposure unit in both groups. Suppose the mean exposure is \(x=0\) in the first group and \(x=1.5\) in the second. The model-predicted group means at those exposure means are \(10+2(0)+0=10\) and \(10+2(1.5)+1=14\). The four-point difference decomposes into a three-point contribution associated with the 1.5-unit exposure difference, \(2(1.5)=3\), and a one-point covariate contribution. It does not demonstrate effect modification: the slopes remain equal.
A slope comparison requires information about the exposure-outcome relationship at different covariate values, rather than a comparison of group averages alone. In practice, that comparison also requires adequate overlap: if almost all observations in one covariate range have low exposure and almost all in another have high exposure, the fitted interaction may depend heavily on extrapolation.
Estimation, Diagnostics, and Alternatives
For a specified model, the interaction can be estimated by including the product \(xz\) alongside the main effects. A confidence interval for \(\beta_{EZ}\), or a test of \(H_0:\beta_{EZ}=0\), assesses the interaction on the model’s link scale. With several prespecified interaction terms, a joint test can assess whether their coefficients are all zero. Standard errors must respect the data structure; for example, repeated observations from one patient cannot automatically be treated as independent.
Interpretation should report meaningful conditional exposure effects, not only the product-term coefficient. For an identity-link model, calculate slopes at clinically relevant covariate values and express them in outcome units per exposure increment. For a nonlinear link, calculate the corresponding link-scale effects or model-predicted outcomes at stated covariate and exposure values. Because interactions are scale-dependent, an absent interaction on the log-odds scale does not imply identical probability differences at every covariate value.
Several checks are particularly relevant to interactions. Examine whether each covariate range has observations across the exposure range used to estimate the slopes; sparse combinations can produce unstable estimates. Check whether the assumed functional forms are plausible within the available data and whether a few observations dominate the product term. Consider uncertainty in estimated individual exposures, since measurement error can distort exposure slopes and their apparent variation. A large interaction estimate with a very wide interval, changing markedly under reasonable model specifications, or supported only by extrapolation is weak evidence for a stable difference in exposure response.
The main alternatives answer related but distinct questions. Fitting separate models by covariate group gives group-specific slopes, but makes direct estimation and testing of their difference less straightforward and can yield inconsistent adjustment choices. A pooled model with an interaction estimates that difference directly and uses shared model structure. A model without an interaction is more parsimonious, but imposes a common link-scale exposure slope. Mechanistic nonlinear models may be appropriate when the scientific question concerns a specific nonlinear exposure-response mechanism; an interaction should not be used as a substitute for a misspecified exposure-response shape. In each case, the chosen form should be justified by the scientific question, the observed data, and the intended interpretation.
Common Pitfalls and Limitations
- Confusing different exposure levels with effect modification. Group-average outcomes can differ because exposure distributions differ. Compare conditional exposure slopes or modelled exposure contrasts instead.
- Omitting main effects. Including \(xz\) without \(x\) and \(z\) generally imposes unintended restrictions. Include both main effects when fitting the product term.
- Ignoring units and reference values. The numerical interaction coefficient changes with the scaling of exposure or covariate. State the increment and reference point so its meaning is clear.
- Over-interpreting a small \(p\)-value. Statistical evidence for a nonzero coefficient does not establish a clinically important difference, a causal interaction, or reliable extrapolation beyond observed data.
- Assuming a null interaction means identical effects on every scale. The interaction is defined on the model’s chosen scale; probability-scale contrasts can vary even when a logit-scale interaction is zero.
Interaction estimates are often less precise than main-effect estimates because they depend on variation across combinations of exposure and covariate values. Their interpretation can also be sensitive to model form, covariate measurement, and the causal assumptions used for individual-level exposure-response analyses. Report the scale, units, conditional effects, uncertainty, and relevant overlap rather than treating one coefficient as a complete summary.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- In an identity-link model with exposure slope \(\beta_E=2.4\) and interaction coefficient \(\beta_{EZ}=-0.5\), what is the exposure slope at \(z=2\)?
- In a logistic model, an interaction coefficient for a binary covariate is 0.20. What does \(\exp(0.20)\) compare?
- Two covariate groups have different mean exposures and different mean outcomes. Does this alone establish an exposure-by-covariate interaction? Explain.
- Why should the exposure and covariate main effects ordinarily remain in a model that includes their product?
Solutions. (1) The slope is \(\beta_E+\beta_{EZ}z=2.4-0.5(2)=1.4\) outcome units per exposure unit. (2) \(\exp(0.20)\approx1.22\) is the ratio of the exposure odds ratio in the covariate category \(z=1\) to that in the reference category \(z=0\), for the model’s specified exposure increment. (3) No. Different exposure distributions can produce different group means even with a common exposure slope; estimate or compare conditional slopes, with adequate exposure overlap. (4) The main effects allow the outcome to depend on exposure at the covariate reference value and on the covariate at the exposure reference value. Omitting them generally imposes restrictions that are not part of the usual interaction question.