Interactions in Fine-Gray Regression

Why Interactions Need Careful Interpretation

An interaction in Fine-Gray regression allows the subdistribution-hazard association for one covariate to vary with another. For example, a treatment’s subdistribution hazard ratio (SHR) may differ across values of a continuous biomarker. The model is easy to write, but its coefficients are not all effects that apply uniformly to every patient: the treatment contrast depends on the biomarker value.

The essential task is to define the covariate coding, write the linear predictor, and subtract the predictors for the two profiles being compared. Exponentiating that difference gives the SHR contrast. This calculation describes the subdistribution-hazard scale; it does not, by itself, give a ratio or difference in cumulative incidence.

Definition: For event type \(k\), let \(F_k(t\mid x)\) be the cumulative incidence of that event by time \(t\), conditional on covariates \(x\). The subdistribution hazard is \(\tilde{\lambda}_k(t\mid x)=f_k(t\mid x)/\{1-F_k(t-\mid x)\}\), where \(f_k\) is the cause-\(k\) event density. Fine-Gray regression models this hazard. Its risk-set construction retains people who have experienced a competing event, with censoring handled through the method’s weighting.

Specifying the Interaction Model

For an event of interest \(k\), a Fine-Gray proportional subdistribution-hazards model with binary treatment \(A\) and continuous covariate \(Z\) can be written as

\tilde{\lambda}_k(t\mid A,Z) = \tilde{\lambda}_{k0}(t) \exp\{\beta_A A+\beta_Z Z+\beta_{AZ}AZ\}.

Here \(A=0\) denotes the reference treatment group and \(A=1\) the other group. The coefficient \(\beta_{AZ}\) is the interaction coefficient. When it is zero, the treatment contrast does not vary with \(Z\) on the log subdistribution-hazard scale. When it is nonzero, the treatment contrast changes with \(Z\). Centering \(Z\), for example by defining it as the number of ten-unit increments from a specified reference value, makes the main treatment coefficient interpretable at that reference.

Key result: At a fixed value \(Z=z\), the log SHR comparing \(A=1\) with \(A=0\) is \(\beta_A+\beta_{AZ}z\). Thus the treatment SHR is \(\exp(\beta_A+\beta_{AZ}z)\). The main-effect coefficient \(\beta_A\) alone is the log SHR only when \(z=0\).

More generally, if the model’s linear predictor is \(\eta(x)\), comparing covariate profiles \(x_1\) and \(x_0\) gives

\log\{\mathrm{SHR}(x_1:x_0)\}=\eta(x_1)-\eta(x_0), \qquad \mathrm{SHR}(x_1:x_0)=\exp\{\eta(x_1)-\eta(x_0)\}.

The baseline subdistribution hazard cancels because both profiles are evaluated at the same time and share the same baseline in this model. With a vector of fitted coefficients \(\hat{\beta}\), write the contrast as \(c^\mathsf{T}\hat{\beta}\), where \(c\) contains the differences in the model terms between profiles. If \(\widehat{\mathrm{Var}}(\hat{\beta})=\widehat{\Sigma}\), its estimated standard error is \(\sqrt{c^\mathsf{T}\widehat{\Sigma}c}\). A Wald confidence interval for the log contrast is the estimate plus or minus the appropriate normal critical value times this standard error; exponentiating both endpoints gives an interval for the SHR.

Worked Contrasts

Worked Example: Treatment SHR at Different Biomarker Values

Suppose a hypothetical Fine-Gray model for a specified event has estimated coefficients \(\hat{\beta}_A=0.30\), \(\hat{\beta}_Z=0.10\), and \(\hat{\beta}_{AZ}=-0.10\). Let \(Z\) be measured in ten-unit increments above a clinically chosen reference, so \(Z=1\) means ten units above that reference. Assume the proportional subdistribution-hazards model is a reasonable working model and that the data include adequate representation of both treatment groups across the biomarker values being compared.

At \(Z=1\), the treatment log SHR is

\hat{\beta}_A+\hat{\beta}_{AZ}Z =0.30+(-0.10)(1)=0.20.

Therefore the treatment SHR is \(\exp(0.20)=1.221\). At this biomarker value, the fitted subdistribution hazard for the event of interest is 1.221 times as high for \(A=1\) as for \(A=0\), under the model.

At \(Z=3\), the log SHR is \(0.30+(-0.10)(3)=0.00\), giving an SHR of \(\exp(0)=1.000\). At \(Z=0\), it is \(0.30\), giving \(\exp(0.30)=1.350\). The interaction also has a direct ratio interpretation: for each one-unit increase in \(Z\), the treatment SHR is multiplied by \(\exp(-0.10)=0.9048\). In other words, the fitted treatment SHR decreases by about 9.52% per ten-unit increase in the biomarker. This is a change in the SHR, not a claim that the cumulative incidence decreases by that percentage.

Worked Example: Binary Treatment-by-Group Interaction

Consider hypothetical estimates from a model with binary treatment \(A\) and binary group indicator \(G\), where \(G=0\) is the reference group and \(G=1\) is the other group:

\hat{\eta}=\hat{\beta}_A A+\hat{\beta}_G G+\hat{\beta}_{AG}AG, \qquad \hat{\beta}_A=0.25,\quad \hat{\beta}_G=-0.15,\quad \hat{\beta}_{AG}=0.40.

Assume the fitted model and covariate support are adequate for both groups. In group \(G=0\), the treatment log SHR is \(\hat{\beta}_A=0.25\), so the SHR is \(\exp(0.25)=1.284\). In group \(G=1\), it is \(\hat{\beta}_A+\hat{\beta}_{AG}=0.25+0.40=0.65\), so the SHR is \(\exp(0.65)=1.916\). The ratio of these two treatment SHRs is \(\exp(0.65-0.25)=\exp(0.40)=1.492\), matching the interaction coefficient on the exponentiated scale.

For an uncertainty calculation in group \(G=1\), suppose the estimated variances of \(\hat{\beta}_A\) and \(\hat{\beta}_{AG}\) are 0.0400 and 0.0225, and their covariance is 0.0100. The variance of their sum is

0.0400+0.0225+2(0.0100)=0.0825, \qquad SE=\sqrt{0.0825}=0.2872.

A 95% Wald interval for the group-specific log SHR is \(0.65\pm1.96(0.2872)=[0.0871,1.213]\). Exponentiating gives an SHR interval of approximately \([1.091,3.364]\). Under the model and usual large-sample Wald approximation, this interval is for the group-\(1\) treatment SHR, not for the interaction coefficient or a cumulative-incidence contrast.

To test whether the treatment SHR differs between groups, test \(H_0:\beta_{AG}=0\). If the interaction coefficient’s standard error is 0.20, the Wald statistic is \(0.40/0.20=2.00\), with a two-sided normal-approximation p-value of about 0.0455. This tests the interaction, not whether treatment has an effect in either group considered separately.

Worked Example: Comparing Two Complete Covariate Profiles

Suppose a hypothetical fitted model uses the same binary-treatment and continuous-covariate specification, with \(\hat{\beta}_A=-0.12\), \(\hat{\beta}_Z=0.08\), and \(\hat{\beta}_{AZ}=0.05\). Compare a treated individual with \(Z=2\) to a control individual with \(Z=-1\). Assume both profiles are within the observed covariate support and the model assumptions are tenable over the comparison.

The treated profile has linear predictor \(-0.12+0.08(2)+0.05(1)(2)\). The control profile has linear predictor \(0.08(-1)\). Subtracting the latter from the former gives

\begin{aligned} \hat{\eta}(1,2)-\hat{\eta}(0,-1) &=\hat{\beta}_A+\hat{\beta}_Z\{2-(-1)\}+\hat{\beta}_{AZ}(2)\\ &=-0.12+0.08(3)+0.05(2)\\ &=-0.12+0.24+0.10=0.22. \end{aligned}

The estimated SHR comparing these profiles is \(\exp(0.22)=1.246\). This is a joint profile contrast: it changes both treatment and \(Z\). It is not the treatment SHR at one fixed value of \(Z\), so the covariate difference must be included in the contrast.

Inference, Assumptions, and Diagnostics

The model must be specified before interpreting its contrasts. At minimum, the event of interest and competing event types must be defined, the covariates and their reference coding stated, and the interaction form selected. Fine-Gray regression assumes proportional subdistribution hazards for the included covariate pattern: covariate effects in the linear predictor do not vary with time. It also relies on appropriate handling of censoring for the weighted partial-likelihood procedure. These assumptions do not make the model causal; causal interpretation requires additional design and identification assumptions.

An interaction coefficient tests whether a specified contrast varies with the modifier on the log subdistribution-hazard scale. It is not a test that the treatment effect is zero for all values of the modifier. For a treatment effect at \(Z=z\), use the full contrast \(\beta_A+z\beta_{AZ}\) and its variance, including the covariance between the coefficient estimates. A confidence interval for that contrast can include the null value even when the interaction test rejects, or vice versa.

Useful checks include plotting or tabulating the covariate distributions by treatment group, especially near the values where contrasts will be reported; inspecting whether a continuous modifier’s functional form is plausible; and assessing proportionality using suitable diagnostics. Sparse observations in a treatment-by-modifier region can make the fitted contrast unstable even if the model converges. Large standard errors, extreme estimates, or sensitivity to a few observations are warnings, not evidence of a genuine strong interaction.

A model without an interaction is more parsimonious and imposes a common treatment SHR across modifier values. Separate models fitted within subgroups allow more flexibility but can be less precise and do not automatically provide a direct test of whether subgroup effects differ. Cause-specific Cox regression addresses a different hazard and therefore answers a different question; its interaction coefficients should not be substituted for Fine-Gray interaction contrasts.

Why the Scale Matters

Under the Fine-Gray model, the cumulative incidence and subdistribution hazard are related by

F_k(t\mid x)=1-\exp\{-\tilde{\Lambda}_k(t\mid x)\}.

Consequently, even when the SHR between two profiles is constant over time under proportional subdistribution hazards, the difference or ratio of their cumulative incidences generally varies with time and with the baseline cumulative subdistribution hazard. A reported SHR should therefore be described as a relative subdistribution-hazard contrast. If the clinical question concerns absolute risk by a specified time, calculate and report model-based cumulative incidences at that time rather than relabeling an SHR as a risk ratio.

Common pitfall: Reading \(\beta_A\) as the treatment effect for everyone in an interaction model is incorrect. It is the log treatment SHR only at the modifier’s reference value (such as \(Z=0\)). The correction is to substitute the covariate value into the entire treatment contrast, \(\beta_A+\beta_{AZ}z\), and exponentiate that result.

Practical Calculation Sequence

Write the fitted linear predictor.
Include the main effects and the interaction terms, using the stated reference coding and units.

Define the comparison.
List the values of every covariate in profile \(x_1\) and profile \(x_0\); do not leave the modifier value implicit.

Subtract predictors and exponentiate.
Compute \(\eta(x_1)-\eta(x_0)\), then exponentiate to obtain the estimated SHR.

Quantify uncertainty and check interpretation.
Use the full coefficient covariance matrix for the contrast, and state that the result is on the subdistribution-hazard scale.

Check Your Understanding

Work each contrast from the model terms, not from a single main-effect coefficient. Solutions are included below.

In a model with \(\hat{\beta}_A=0.20\) and \(\hat{\beta}_{AZ}=0.05\), what is the treatment SHR at \(Z=2\)?
For binary treatment and group indicator, suppose \(\hat{\beta}_A=-0.10\) and \(\hat{\beta}_{AG}=0.30\). What are the treatment SHRs in groups \(G=0\) and \(G=1\), and what is their ratio?
For \(\hat{\beta}_A=0.10\), \(\hat{\beta}_Z=0.04\), and \(\hat{\beta}_{AZ}=-0.02\), compare \((A=1,Z=1)\) with \((A=0,Z=0)\). Give the log contrast and SHR.
What does the null hypothesis \(H_0:\beta_{AZ}=0\) test in a treatment-by-continuous-covariate model?
Why is it incorrect to report an SHR of 1.2 as a 20% increase in cumulative incidence?

Solutions. (1) The log treatment SHR is \(0.20+0.05(2)=0.30\), so the SHR is \(\exp(0.30)=1.350\). (2) For \(G=0\), the SHR is \(\exp(-0.10)=0.9048\). For \(G=1\), it is \(\exp(-0.10+0.30)=\exp(0.20)=1.221\). Their ratio is \(\exp(0.30)=1.350\). (3) The log contrast is \(0.10+0.04(1)+(-0.02)(1)(1)=0.12\), so the SHR is \(\exp(0.12)=1.127\). (4) It tests whether the treatment log SHR changes with the continuous covariate on the model’s subdistribution-hazard scale; it does not test whether the treatment SHR equals one at every covariate value. (5) The SHR is a relative subdistribution-hazard contrast. Cumulative incidence is related nonlinearly to the cumulative subdistribution hazard, so its relative or absolute difference depends on time and baseline risk.