Improved precision in the analysis of randomized trials with survival outcomes, without assuming proportional hazards

Abstract

We present a new estimator of the restricted mean survival time in randomized trials where there is right censoring that may depend on treatment and baseline variables. The proposed estimator leverages prognostic baseline variables to obtain equal or better asymptotic precision compared to traditional estimators. Under regularity conditions and random censoring within strata of treatment and baseline variables, the proposed estimator has the following features: (i) it is interpretable under violations of the proportional hazards assumption; (ii) it is consistent and at least as precise as the Kaplan-Meier and inverse probability weighted estimators, under identifiability conditions; (iii) it remains consistent under violations of independent censoring (unlike the Kaplan-Meier estimator) when either the censoring or survival distributions, conditional on covariates, are estimated consistently; and (iv) it achieves the nonparametric efficiency bound when both of these distributions are consistently estimated. We illustrate the performance of our method using simulations based on resampling data from a completed, phase 3 randomized clinical trial of a new surgical treatment for stroke; the proposed estimator achieves a 12% gain in relative efficiency compared to the Kaplan-Meier estimator. The proposed estimator has potential advantages over existing approaches for randomized trials with time-to-event outcomes, since existing methods either rely on model assumptions that are untenable in many applications, or lack some of the efficiency and consistency properties (i)-(iv). We focus on estimation of the restricted mean survival time, but our methods may be adapted to estimate any treatment effect measure defined as a smooth contrast between the survival curves for each study arm. We provide R code to implement the estimator.

Statistical inference for data‐adaptive doubly robust estimators with survival outcomes

“…The estimator is constructed by tilting an initial estimate toward a solution of the relevant estimating equation, by means of an empirical risk minimizer in a parametric submodel. The interested reader is referred to the works of Díaz et al and Moore and van der Laan for more details on the construction of a TMLE for survival analysis. The preliminary estimator , or the component necessary to evaluate , may be obtained based on data‐adaptive regression methods.…”

“…, we refer to n 1/4 -rate convergence as the property that n 1∕4 ||̂− 0 || = o P (1). For a collection of functions ( f 1 , … , f k ), the notation ||( f 1 , … , f k )|| is used to denote the vector of element-wise norms.…”

“…where G 1 and S 1 denote the censoring and survival probabilities under the limits g R, 1 and h 1 of the estimators (C1). Define the following weighted error functions:…”

Targeted learning ensembles for optimal individualized treatment rules with time-to-event outcomes

“…The subscript c denotes a causal parameter, that is, a parameter of the distribution of the potential outcomes T 1 and T 0 . It can be shown (see Díaz et al, 2015) that…”

“…Our ensembles are linear combinations of estimators in a user-supplied library, where the coefficients in the linear combination are chosen to minimize the crossvalidated risk. We propose to use a doubly robust loss function with roots in efficient estimation theory for marginal causal effects (Moore and van der Laan, 2009;Díaz et al, 2015). In our context, double robustness means that the estimated rules will have certain optimality properties under consistent estimation of at least one of two nuisance parameters: (a) the hazard of the outcome at each time point conditional on covariates and treatment, and (b) the hazard of censoring and the treatment mechanism.…”