Risk-Scale Interaction Under Case-Control Sampling

Why Case-Control Sampling Changes the Risk Scale

A case-control study deliberately selects people according to their disease status. It can therefore provide useful information about how genotype and exposure relate to disease odds, but the fraction of sampled participants who are cases is set by the sampling design. That fraction is not, in general, the disease prevalence in the source population. This distinction matters for gene–environment interaction on the risk scale: an additive interaction contrast is built from population risks, not from the case fraction in the study.

Let \(D\) indicate disease, with \(D=1\) for a case and \(D=0\) for a control. Let \(G\) and \(E\) be binary genotype and exposure indicators, respectively, where 0 denotes the reference category. Write \(r_{ge}=P(D=1\mid G=g,E=e)\) for the source-population disease risk in cell \((g,e)\). The additive interaction contrast compares how the risk difference associated with one factor changes across levels of the other.

The contrast is an absolute difference in probabilities, so its units are risk points (or percentage points). A relative version is the relative excess risk due to interaction, or RERI:

When \(r_{00}>0\), the additive contrast equals \(r_{00}\times\mathrm{RERI}\). These measures describe interaction on the risk scale. Interpreting them causally requires additional assumptions about confounding, selection, consistency, and the meaning of the exposure and genotype contrasts; a case-control design alone does not establish those assumptions.

What the Case-Control Sample Identifies

Suppose the probability of being sampled is \(\pi_1\) for a case and \(\pi_0\) for a control, with sampling otherwise independent of \(G\) and \(E\) within disease status. For a particular genotype-exposure cell, the odds of disease in the sample are:

Here \(S=1\) means the person was sampled. The sampling factor \(\pi_1/\pi_0\) multiplies the population disease odds. Consequently, the case fraction and fitted intercept from an unweighted case-control sample generally do not represent population prevalence or population baseline risk. But the sampling factor is the same across cells under the stated design, so odds ratios comparing cells can still be identified. This is the basis for standard logistic regression analysis of case-control data when selection depends on disease status alone.

An odds ratio is not generally a risk ratio. Even if the cell-specific odds ratios are known, risks cannot be recovered from them without a baseline risk or other population information. If a compatible baseline risk is known, the conversion is:

This formula is exact for the odds ratio comparing cell \((g,e)\) with the reference cell, provided both risks refer to the same target population and conditioning or standardization scheme. Replacing an odds ratio by a risk ratio is only an approximation when disease is sufficiently rare; it is not an automatic consequence of using logistic regression.

Recovering Risks With a Baseline Risk

If a reliable source supplies the population risk \(r_{00}\) for the same reference cell and target population, the cell odds ratios can be converted to risks one cell at a time. The resulting additive contrast is interpretable in risk points. The baseline risk must match the target setting: a risk from a different population, time period, disease definition, or covariate stratum may produce a misleading conversion.

Calibrating a Risk Model With External Prevalence

Often the external information is overall disease prevalence rather than a reference-cell risk. Prevalence can calibrate the intercept of a risk model if the model's relative effects are estimated from the case-control data and the target-population distribution of predictors is available. For example, let \(X_i\) denote the predictors for a person in the target population, and let \(\eta_i\) be the model's linear predictor excluding its intercept. A logistic risk model has \(P(D_i=1\mid X_i)=\operatorname{expit}(\alpha+\eta_i)\), where \(\operatorname{expit}(u)=1/(1+\exp(-u))\). Choose \(\alpha\) so that the model's average risk over the target distribution equals the external prevalence \(K\):

The sum can represent a target-population dataset or be replaced by a weighted average over covariate patterns. This equation usually needs to be solved numerically. It requires the external prevalence and a suitable target distribution of predictors; prevalence by itself does not reveal how risk is distributed across genotype and exposure cells. Once calibrated, risks for specified cells can be predicted and, if appropriate, averaged over other covariates to define standardized cell risks.

Recovering Risks With Sampling Weights

A second route is to use known sampling probabilities. If each person in the source population has known inclusion probability \(\pi_i\), an inverse-probability weight is \(w_i=1/\pi_i\). Weighting sampled individuals by \(w_i\) makes their contribution represent the number of source-population individuals they stand for. For a conventional case-control design, the weights may be \(1/\pi_1\) for cases and \(1/\pi_0\) for controls. A weighted risk model can then target population risks, provided the sampling probabilities are correct, selection is independent of predictors within the strata used to define those probabilities, and the model and target population are appropriately specified. The design must also give positive inclusion probability to the relevant groups.

If inclusion probabilities are unknown, weights cannot be inferred just from the case and control counts without further information. Known population sampling fractions, external prevalence, or other valid design information may supply what is needed. Standard errors must reflect the sampling design and weight estimation; treating weighted pseudo-counts as an ordinary unweighted sample can understate uncertainty.

Interpretation and Common Pitfalls

Risk-scale interaction is useful because it expresses departure from additivity in absolute disease probability, a scale that can help communicate excess risk. In a genetic epidemiology study, however, the numerical result depends on the population and its disease prevalence. The same odds ratios can imply different risk contrasts in populations with different baseline risks or genotype-exposure distributions. A report should therefore state the target population, how risks were obtained, and whether the contrast is conditional on covariates or standardized over them.

• Wrong: “The case fraction in the sample is the disease risk.” Correction: Case-control sampling fixes or alters the case fraction; it does not generally preserve source-population prevalence.
• Wrong: “The logistic odds ratio is the risk ratio, so the odds-ratio RERI is the risk-scale interaction.” Correction: Odds and risks differ. Convert odds ratios using a compatible baseline risk, or estimate population risks with calibration or valid sampling weights.
• Wrong: “An external overall prevalence is enough to determine every cell risk.” Correction: Calibration also needs a risk model and target-population predictor distribution (or other information that identifies the cell risks).
• Wrong: “A positive additive contrast proves biological or causal interaction.” Correction: It establishes departure from additivity for the estimated risks. Causal claims require additional design and identification assumptions.