Reference-based sensitivity analysis for time-to-event data

Abstract

The analysis of time-to-event data typically makes the censoring at random assumption, ie, that-conditional on covariates in the model-the distribution of event times is the same, whether they are observed or unobserved (ie, right censored). When patients who remain in follow-up stay on their assigned treatment, then analysis under this assumption broadly addresses the de jure, or "while on treatment strategy" estimand. In such cases, we may well wish to explore the robustness of our inference to more pragmatic, de facto or "treatment policy strategy," assumptions about the behaviour of patients post-censoring. This is particularly the case when censoring occurs because patients change, or revert, to the usual (ie, reference) standard of care. Recent work has shown how such questions can be addressed for trials with continuous outcome data and longitudinal follow-up, using reference-based multiple imputation. For example, patients in the active arm may have their missing data imputed assuming they reverted to the control (ie, reference) intervention on withdrawal. Reference-based imputation has two advantages: (a) it avoids the user specifying numerous parameters describing the distribution of patients' postwithdrawal data and (b) it is, to a good approximation, information anchored, so that the proportion of information lost due to missing data under the primary analysis is held constant across the sensitivity analyses. In this article, we build on recent work in the survival context, proposing a class of reference-based assumptions appropriate for time-to-event data. We report a simulation study exploring the extent to which the multiple imputation estimator (using Rubin's variance formula) is information anchored in this setting and then illustrate the approach by reanalysing data from a randomized trial, which compared medical therapy with angioplasty for patients presenting with angina.

Keywords: MNAR; missing data; multiple imputation; sensitivity analysis; time to event.

Figure 1

RITA‐2 trial–Nelson‐Aalen cumulative hazard survival plots for all cause mortality (up to 8 y only 18 patients lost to follow‐up)

Figure 2

Time‐to‐event data–Jump to Reference

Figure 3

Proportionate increase in variance as censoring increases under (a) censoring at random and (b) Jump to Reference

Figure 4

Simulation results: exploration of information anchoring for two sample sizes and two hazard ratios. For each scenario, as the proportion of active arm censoring increases, each panel shows the evolution of the variance of the estimated hazard ratio calculated in four ways: (a) −+− information anchored variance; (b) −∘− Rubin's MI variance under Jump to Reference; (c) −×− $\widehat{\mathrm{E}}[{\widehat{\mathrm{V}}}_{inf}(\widehat{\beta })]$ when censored data recreated under Jump to Reference; and (d) −⋄− ${\widehat{\mathrm{V}}}_{emp}({\widehat{\beta }}_{MI})$ under Jump to Reference

Figure 5

Plot of the log cumulative hazard against time with Nelson‐Aalen estimates for the PTCA arm (upper dashed, red) and medical arm (lower dashed, black). The solid (red) line shows the estimated Weibull model log cumulative hazard for the medical arm when patients are censored at their first nonrandomized intervention and “Jump to PTCA arm”