Interactions for Patient-Specific Landmark Risk

Why Include an Interaction?

A biomarker may carry different prognostic information for different patients. For example, the association between a biomarker measured at a landmark and the probability of an event over the next year might differ by treatment group or by another patient characteristic. A model without an interaction imposes the same biomarker association across those groups. An interaction allows that association to vary.

Here the focus is a fixed-horizon landmark risk model. The analysis population consists of patients who are eligible and event-free at landmark time \(t_0\). Let \(Y_i(\tau)\) indicate whether patient \(i\) experiences the event of interest during the interval from \(t_0\) to \(t_0+\tau\), where \(\tau\) is the prediction horizon. Let \(X_i\) be the biomarker value, expressed in a chosen unit, and let \(G_i\) be a binary patient characteristic coded 0 or 1. The model describes risk conditional on being in the landmark population and on the included predictors; it does not by itself establish a causal effect.

A fixed-horizon logistic model can be written as

Here \(\operatorname{logit}(p)=\log\{p/(1-p)\}\), and \(\mathbf{Z}_i\) contains any other covariates in the model. The probability is the patient’s modeled risk over the specified horizon. The coefficient \(\beta_3\) is the interaction coefficient. When \(G_i=0\), the biomarker slope on the log-odds scale is \(\beta_1\); when \(G_i=1\), it is \(\beta_1+\beta_3\). Thus \(\beta_3\) is a difference between slopes, not the biomarker slope for either group by itself.

Assumptions and Meaning of the Model

The model requires a clearly defined landmark cohort, a specified horizon, and a consistently defined event outcome. For ordinary logistic regression on \(Y_i(\tau)\), each patient’s outcome must be ascertainable through the horizon, or the analysis must use a suitable method that accounts for censoring before the horizon. Simply treating a patient censored before \(t_0+\tau\) as event-free can misclassify the outcome. The model also assumes independent patient-level observations, a correctly specified conditional mean structure, and adequate information to estimate the coefficients. If patients are clustered, or if the landmark data contain repeated records per patient, the analysis must account for that dependence.

The linear predictor is assumed linear in the included terms. If the biomarker association is nonlinear, a linear interaction may not be enough; the nonlinear biomarker representation established in the earlier material on nonlinear biomarker effects should be retained and, where appropriate, interacted with the covariate. Coding and units matter: \(\beta_1\) is the biomarker association when \(G=0\), and its numerical interpretation depends on the unit used for \(X\). Centering a biomarker changes the interpretation of the main effect for \(G\), but not the fitted predictions when the model is reparameterized consistently.

Deriving the Subgroup Associations

The interaction’s meaning follows directly by substituting each group value into the model. Hold the other covariates fixed. For \(G=0\), the linear predictor involving \(X\) is \(\beta_0+\beta_1X\). For \(G=1\), it is \((\beta_0+\beta_2)+(\beta_1+\beta_3)X\). Therefore, a one-unit increase in \(X\) changes the log odds by \(\beta_1\) in group 0 and by \(\beta_1+\beta_3\) in group 1. Exponentiating gives the corresponding odds ratios.

The third expression is the ratio of the two biomarker odds ratios. It is not a ratio of risks. Logistic models are nonlinear when converted to probabilities, so a constant log-odds slope does not imply a constant absolute risk difference across patients.

Use Predicted Risks to Show the Difference

Subgroup-specific predicted risks make the model’s implications easier to communicate in context. For the binary covariate model, first calculate the linear predictor for the patient profile, then apply the inverse-logit transformation \(p=1/\{1+\exp(-\eta)\}\). Predictions should use the same values for other covariates when the goal is to compare only \(X\) and \(G\).

A useful additional quantity is a risk difference between two biomarker values within a subgroup. In subgroup 0, the predicted change from \(X=-1\) to \(X=1\) is \(0.2315-0.0573=0.1742\), or 17.42 percentage points. In subgroup 1, it is \(0.2142-0.1545=0.0597\), or 5.97 percentage points. These risk differences summarize the fitted predictions at those particular values; they are not constant effects implied by the logistic model.

Testing and Quantifying Uncertainty

A formal test of whether the biomarker association differs by \(G\) is a test of \(H_0:\beta_3=0\). A Wald test divides the estimated interaction coefficient by its standard error and compares the result with a standard normal reference distribution in a sufficiently large sample. A confidence interval for \(\beta_3\), or for \(\exp(\beta_3)\), conveys the estimated magnitude and precision. Because subgroup slopes involve combinations of coefficients, uncertainty for \(\beta_1+\beta_3\) must use their covariance as well as their individual standard errors.

A small p-value does not measure clinical importance, and a large p-value does not establish that the subgroup associations are identical. Report the estimated subgroup odds ratios and predicted risks, with uncertainty where available, rather than relying only on the interaction test.

Diagnostics, Alternatives, and Limitations

Interaction terms increase model flexibility and can be difficult to estimate when few events occur in one subgroup or at the extremes of the biomarker distribution. Inspect the distribution of biomarker values and event counts within subgroups, look for sparse cells or separation, and assess whether the fitted probabilities are plausible. Compare predicted-risk curves with uncertainty intervals across the observed range; avoid emphasizing predictions far outside that range. Check calibration and other model diagnostics in a way that reflects the landmark population and horizon. Missing biomarker values and censoring before the horizon also require appropriate handling rather than silent exclusion or incorrect outcome coding.

The main alternative is a model without an interaction, which is more parsimonious but assumes a common biomarker slope on the log-odds scale. Separate subgroup models allow different associations but do not directly pool information or provide the same single interaction parameter; estimates can be unstable in small groups. A landmark Cox model can include a biomarker-by-covariate product too, but that interaction concerns log hazard, not fixed-horizon log odds. Translating it to absolute risk requires additional survival-model quantities, so the interpretations should not be interchanged.

A frequent mistake is to say, “The interaction coefficient is negative, so subgroup 1 has lower risk at every biomarker value.” The coefficient only describes a difference in slopes on the log-odds scale. The subgroup’s level also depends on the intercept and main effect, and absolute risks depend nonlinearly on the entire linear predictor. Calculate and present predicted risks for meaningful patient profiles before making a statement about which subgroup has greater risk.