Clock-reset “log-rank-type” test to assess the impact of the intermediate event in a non-markovian illness-death model

The simplest multistate model is named "illness-death" because 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 is a therapeutic intervention (treatment), administered after waiting some time since the initial event. Within such context, it could be of interest to compare the hazards of mortality without vs after treatment administration. In the Mantel-Byar test, an extension of the log-rank test accounting for the time-dependent nature of treatment indicator, all patients initially belong to the "untreated" group but some are shifted to the "treated" group at the time of treatment administration. Using this approach, the hazard of death is always defined as function of time since origin, also after treatment ("clock-forward" time scale). Using theoretical arguments and simulations, we show that this method is valid only under the Markov assumption but it is not suitable when the process is not markovian. This happens when the hazard of death after treatment depends: (i) on time since treatment (semi-Markov process) or (ii) on time since treatment and also on the waiting time to treatment (extended semi-Markov process). We propose two modifications of the Mantel-Byar test suitable for semi-Markov and extended semi- Markov scenarios, respectively. These procedures are both 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.