The impact of parametric assumptions on the bias-variance tradeoff for causal analysis of time-to-event outcomes

Background: Parametric assumptions have the potential to reduce the variance of effect estimates, but making such assumptions comes with the cost of the potential for bias. Though the bias/variance tradeoff is well described in many settings, it has not been well characterized for the causal analysis of time-to-event outcomes.

Objectives: To evaluate how parametric assumptions affect the bias and variance of causal estimation of time-to-event outcomes.

Methods: We conducted a simulation study to explore how non-parametric and semi-parametric estimators perform when estimating a causal cumulative risk difference. The simulated data included a binary treatment and Weibull-distributed censoring and potential outcome times. Two settings were assessed, one in which the hazard ratio was constant across time (proportional hazards) and one in which it was not (non-proportional hazards). The bias, variance, and mean squared error (MSE) of the risk difference estimates were then assessed for each estimator.

Results: In the proportional hazards scenario, the MSE of the semi-parametric estimator was at least 50% lower compared to non-parametric estimator across most scenarios. With a sample size of 1000, the semi-parametric estimator had a 95% lower MSE compared to non-parametric models. However, when the data did not have proportional hazards, the MSE of the semi-parametric estimator was at least twice that of the non-parametric model and went as high as 9.5 times the MSE for with a sample size of 1000.

Conclusions: When hazards are proportional, making parametric assumptions nearly halved the MSE compared to a non-parametric estimate, but almost doubled the MSE when the parametric assumptions were not valid. Thus, the causal cumulative risk difference is sensitive to model misspecification and assuming any semi-parametric model on the data should be conducted cautiously when a causal interpretation of cumulative risk is desired.