Generating Survival Times Using Cox Proportional Hazards Models with Cyclic and Piecewise Time-Varying Covariates

Abstract

Time-to-event outcomes with cyclic time-varying covariates are frequently encountered in biomedical studies that involve multiple or repeated administrations of an intervention. In this paper, we propose approaches to generating event times for Cox proportional hazards models with both time-invariant covariates and a continuous cyclic and piecewise time-varying covariate. Values of the latter covariate change over time through cycles of interventions and its relationship with hazard differs before and after a threshold within each cycle. The simulations of data are based on inverting the cumulative hazard function and a log link function for relating the hazard function to the covariates. We consider closed-form derivations with the baseline hazard following the exponential, Weibull, or Gompertz distribution. We propose two simulation approaches: one based on simulating survival data under a single-dose regimen first before data are aggregated over multiple-dosing cycles and another based on simulating survival data directly under a multiple-dose regimen. We consider both fixed intervals and varying intervals of the drug administration schedule. The method's validity is assessed in simulation experiments. The results indicate that the proposed procedures perform well in generating data that conform to their cyclic nature and assumptions of the Cox proportional hazards model.

Keywords: Correlates of risk; Joint modeling of longitudinal and survival data; Survival data simulations; Time-dependent covariate; Zero-protection threshold.

Fig. 1

Illustration—simulated VRC01 serum concentration over time following ten 8-weekly IV infusions at 10 mg/Kg and 30 mg/Kg dose levels with perfect study adherence, according to the pharmacokinetics model described in Huang et al. [2]. Solid lines are the medians; shaded areas are the 2.5% and 97.5% percentiles of the concentrations over 1000 simulated datasets. A body weight of 74.5 Kg is used in the simulations

Fig. 2

Distributions of simulated event times since prior infusion (a) and cumulative hazard of HIV infection since the first infusion (b) under imperfect study adherences in AMP-like trials. The single-dose approach is used in these simulations of 1000 trials, each with a total of $n=4500$ participants randomized to receive ten 8-weekly infusions of 10 mg/Kg VRC01, 30 mg/Kg VRC01, or placebo in a 1:1:1 ratio. The high and medium adherence scenarios assume 2% and 10% of infusion visits missed, respectively. Additional assumptions are as follows: annual HIV incidence rate $=4\%$ in the placebo group, $\beta =0.03$ or HR$=2.32$ per 28 days for both VRC01 dose groups, and zero-protection concentration threshold $s=$ 5 mcg/mL

Fig. 3

Cumulative hazard of HIV infection within each infusion interval following ten 8-weekly IV infusions of VRC01 under perfect study adherence in a simulated trial of 3000 VRC01 recipients. Red lines are for $\beta =0.01$ or $HR=1.32$ per 28 days; blue lines are for $\beta =0.03$ or HR$=2.32$ per 28 days (Color figure online)