Correction: ‘Generating survival times to simulate Cox proportional hazards models with time‐varying covariates’

Abstract

In a recent paper, we derived closed-form expressions for simulating survival times from a Cox proportional hazards model with time-dependent covariates 1. We considered three different distributions for the distribution of event times: exponential, Weibull and Gompertz. We considered three different types of time-dependent covariates: (i) a dichotomous time-varying covariate that can change at most once from untreated to treated; (ii) a dichotomous time-varying covariate with a subject being able to move repeatedly between treatment states; (iii) a continuous time-varying covariate of the form kt, where t denotes time. For each of the nine different scenarios considered, we derived a closed-form expression that allows one to simulate survival times from the specified distribution of event times, given the specified type of time-varying covariate. Each derivation involved determining the cumulative hazard function and the inverse of the cumulative hazard function. Unfortunately, we made two errors in the derivations 2. We made a minor error in deriving the expression for the cumulative hazard function for a Weibull distribution of event times under the first type of time-varying covariate (Section 3.1.2). In the final expression for the cumulative hazard function, H(t,x,z(t)), for the case when t ⩾ t0, we inadvertently did not cancel the υ from the denominator when it was canceled from the numerator. However, we correctly cancelled this term from the denominator in the subsequent derivations in this section when the inverse of the cumulative hazard function was determined. Thus, the final closed-form expression for generating survival times in this scenario is correct. A second error occurred in deriving a closed-form expression for the cumulative hazard function for a Weibull distribution of event times with a continuous time-varying covariate (Section 3.2.2). When determining the cumulative hazard function, we incorrectly evaluated the following integral: . The correct integration is , where Γ(x) denotes the gamma function and Γ(x,b) the upper incomplete gamma function. Accordingly, the cumulative hazard function involves the lower incomplete gamma function. As such, a closed-form expression for the cumulative hazard function does not exist in this particular scenario. Thus, a closed-form expression for generating survival times from a Weibull distribution with a continuous time-varying covariate does not exist. As noted elsewhere, one can evaluate this integral numerically 2. Alternatively, researchers wanting to simulate survival times with continuous time-dependent covariates are encouraged to consider either an exponential distribution of event times (Section 3.2.1, formula (4)) or a Gompertz distribution of event times (Section 3.2.3, formula (6)).