Estimating Cumulative Treatment Effects in the Presence of Nonproportional Hazards

Abstract

Often in medical studies of time to an event, the treatment effect is not constant over time. In the context of Cox regression modeling, the most frequent solution is to apply a model that assumes the treatment effect is either piecewise constant or varies smoothly over time, i.e., the Cox nonproportional hazards model. This approach has at least two major limitations. First, it is generally difficult to assess whether the parametric form chosen for the treatment effect is correct. Second, in the presence of nonproportional hazards, investigators are usually more interested in the cumulative than the instantaneous treatment effect (e.g., determining if and when the survival functions cross). Therefore, we propose an estimator for the aggregate treatment effect in the presence of nonproportional hazards. Our estimator is based on the treatment-specific baseline cumulative hazards estimated under a stratified Cox model. No functional form for the nonproportionality need be assumed. Asymptotic properties of the proposed estimators are derived, and the finite-sample properties are assessed in simulation studies. Pointwise and simultaneous confidence bands of the estimator can be computed. The proposed method is applied to data from a national organ failure registry.

Utilization, Outcomes, and Retransplantation of Liver Allografts From Donation After Cardiac Death

“…Cox regression modeling of the entire cohort was performed to identify predictors associated with allograft loss and compare time to allograft failure between groups. 25,26 Hazards ratios (HR) and 95% confidence intervals (CI) are presented. Time to allograft failure was defined as the shortest interval between LTX and ReTx or death.…”

Cumulative Hazard Ratio Estimation for Treatment Regimes in Sequentially Randomized Clinical Trials

“…This approach is similar to that advocated by Wei and Schaubel [8] for a single-stage randomization. Comparisons among treatment regimes were performed by testing the log ratio of the estimated cumulative hazards.…”

“…4, [11]). Hence, a straightforward application of the argument therein leads to $$\underset{t\in [0,L]}{\sup }true\mid {\hat{\Lambda }}_{\mathrm{italicjk}}(t,\widehat{\beta })-{\Lambda }_{\mathit{jk}0}\left(t\right)true\mid \stackrel{\mathrm{a}.\mathrm{s}.}{true\to }0.$$ With the above results (1–3), it is now straightforward to apply the arguments in the Web Appendix A of Wei and Schaubel [8] to establish that ${\hat{\theta }}_{{\mathrm{italicjkj}}^{\prime }{k}^{\prime }}\left(t\right)$ is uniformly consistent and asymptotically normal. …”

“…Let us denote the corresponding estimator by ${\hat{\theta }}_{{\mathrm{italicjkj}}^{\prime }{k}^{\prime }}\left(t\right)$. For unweighted group comparisons, Wei and Schaubel [8] showed that such a ratio converges asymptotically to a Gaussian process. A similar argument outlined in Appendix A can be used to show that ${\hat{\theta }}_{{\mathrm{italicjkj}}^{\prime }{k}^{\prime }}\left(t\right)$ follows a Gaussian process with mean θjkj’k’ ( t ) and variance function ${\sigma }_{{\mathrm{italicjkj}}^{\prime }{k}^{\prime }}^2\left(t\right)$, where $$\begin{array}{cc}right{\sigma }_{{\mathit{jkj}}^{\prime }{k}^{\prime }}^2\left(t\right)& left=E\left\{{\xi }_{{\mathit{jkj}}^{\prime }{k}^{\prime }i}^2(t,\beta )\right\},\\ right{\xi }_{{\mathit{jkj}}^{\prime }{k}^{\prime }i}(t,\beta )& left=\frac{{\Phi }_{\mathit{jki}}(t,\beta )}{{\Lambda }_{j^{\prime }{k}^{\prime }0}\left(t\right)}-\frac{{\Lambda }_{\mathrm{italicjk}0}\left(t\right){\Phi }_{j^{\prime }{k}^{\prime }i}(t,\beta )}{{\Lambda }_{j^{\prime }{k}^{\prime }0}^2\left(t\right)},\\ right{\Phi }_{\mathit{jki}}(t,\beta )& left=h_{\mathit{jk}}^T(t,\beta ){\Omega }^{-1}\left(\beta \right){\Psi }_i\left(\beta \right)+{\int }_0^t{y}_{\mathit{jk}}^{\left(0\right)}{(s,\beta )}^{-1}{\mathit{dM}}_{\mathit{jki}}(s,\beta ),\\ right{h}_{\mathit{jk}}(t,\beta )& left=-{\int }_0^t{\overline{v}}_{\mathit{jk}}(s\end{array}$$…”

“…Therefore, comparisons among treatment regimes in the presence of non-proportional hazards is of importance to health research communities. Wei and Schaubel [8] proposed cumulative treatment effect estimation based on treatment-specific cumulative baseline hazards using a stratified proportional hazards model. Their model assumed proportional hazards with respect to baseline covariates within each treatment group and non-proportional hazards across treatment groups.…”