Comparing Survival Without vs After Heart Transplant: Non-Parametric Methods for Estimation and Testing of a Non-Markovian Process

Abstract

Introduction. The “illness-death model” describes a simple three-states process where subjects can move from the initial state to the final state (death) possibly transiting to an intermediate state (illness). This framework applies also when the intermediate state consists in a therapeutic intervention administered after waiting some time since the initial event, e.g. cardiopathic patients waiting for heart transplant.

Within such context, it could be of interest to compare the hazards of mortality without vs after treatment
administration. The problem could be tackled using multistate hazard based models, often relying on the Markov assumption, considering adjustment for waiting time to treatment start. However, from a clinical perspective, it is often more interesting to answer counterfactual questions such as: what is the survival of a patient alive at a certain time after entry on list and that will never be transplanted compared to a patient alive and transplanted at that time? These estimands are different from state-occupation probabilities based on the joint distribution between survival time and treatment administration provided by multistate models.

Aims. The aim of the work is to adapt and apply non-parametric methods to estimate and compare the survival of patients on the waiting list vs after being transplanted in a setting where the underlying process in non-markovian. We propose an estimator of survival curves and a log-rank-type test.

Methods. We show the importance of checking the validity of the Markov assumption in order to select the most appropriate time scale (clock-forward or clock-reset) guiding the mortality after transplant and to assess the role of the waiting time. We propose a non-parametric method to estimate survival in the presence of a time-dependent intervention, accounting for the impact of waiting time and a modification of the Mantel-Byar test suitable for semi-Markov and extended semi- Markov scenarios. All these procedures are based on the adoption of the “clock-reset” scale that allows the hazard of death for treated patients to depend on time since treatment administration.

Using theoretical arguments and simulations we show the validity of the proposed methods.

Results. We considered 9 simulation scenarios combining three type of effects for treatment and waiting time (no effect, constant effect, time-varying effect). In all scenarios we obtained negligible bias of survival estimate and good coverage of the confidence interval. Regarding the test, in all scenarios under the null hypothesis (no effect of treatment) the fraction of false rejections was close to the nominal level, while for scenarios under the alternative hypothesis (protective effect of treatment) the power of our test was higher than that of the Mantel-Byar test.

Finally, we analyzed data from a real cohort of nearly 1000 patients affected by severe cardiomyopathy included in a waiting list for heart transplant but with low priority where mortality rate after transplant tends to increase along with a longer waiting time to transplant.

Conclusions. We show an original approach for survival estimation and testing in the presence of a time-dependent treatment accounting for the waiting time until treatment switch in a non-markovian process.