Bootstrapping complex time-to-event data without individual patient data, with a view toward time-dependent exposures

Abstract

We consider nonparametric and semiparametric resampling of multistate event histories by simulating multistate trajectories from an empirical multivariate hazard measure. One advantage of our approach is that it does not necessarily require individual patient data, but may be based on published information. This is also attractive for both study planning and simulating realistic real-world event history data in general. The concept extends to left-truncation and right-censoring mechanisms, nondegenerate initial distributions, and nonproportional as well as non-Markov settings. A special focus is on its connection to simulating survival data with time-dependent covariates. For the case of qualitative time-dependent exposures, we demonstrate that our proposal gives a more natural interpretation of how such data evolve over the course of time than many of the competing approaches. The multistate perspective avoids any latent failure time structure and sampling spaces impossible in real life, whereas its parsimony follows the principle of Occam's razor. We also suggest empirical simulation as a novel bootstrap procedure to assess estimation uncertainty in the absence of individual patient data. This is not possible for established procedures such as Efron's bootstrap. A simulation study investigating the effect of liver functionality on survival in patients with liver cirrhosis serves as a proof of concept. Example code is provided.

Keywords: bootstrap; internal time-dependent covariates; joint model; simulation; survival analysis.

Figure 1

A joint model to assess the effect of a time‐dependent covariate Y(t) ∈ {0,1} on survival: illness‐death multistate model without recovery treating the two covariate levels as separate transient states. Transition hazards α ₀₁(t),α ₀₂(t), and α ₁₂(t) are included

Figure 2

Relation between the multistate process X _t and covariate process Y(t). Both T ₀ and T are random variables. X is right‐continuous, but Y is left‐continuous

Figure 3

Comparison of the survival hazards $\tilde{\alpha }(t)$ for t ₀ = 2 specified a priori (dashed line) and the population survival hazard α(t) resulting from the illness‐death model without recovery of Figure 1 (solid line) in the presence of a time‐dependent covariate with, at most, one change over time. Details are given in Appendix B

Figure 4

Simulation algorithm—proof of concept. Solid black lines are the “true” (study‐based) state occupation probabilities, and solid gray lines are 300 randomly selected state occupation probabilities from the simulation. The averages of the simulated state occupation probabilities are drawn as black dots. PI, prothrombin index