Fig. 4

Simulation results with different sample sizes = 200 or 500 and different censoring rates = 25% or 50% in the scenario where proportional hazard assumption dose not hold based on 1000 replications when PS models misspecified. The red dotted line represents the true value, the true values of \(\Delta \left(t=10\right)\) or \(\Delta \left(t=20\right)\) are 0.2524 and 0.3816, respectively. IPW: inverse probability weighting; OR: outcome regression; MR, multiply robust; MR estimators are denoted as “MR-0000”, where each digit of the four numbers, from left to right, indicates if \({\pi }^{1}\left({\varvec{X}};{{\varvec{\beta}}}^{1}\right)\), \({\pi }^{2}\left({\varvec{X}};{{\varvec{\beta}}}^{2}\right)\), \({m}^{1}\left({\varvec{X}},Z;{{\varvec{\gamma}}}^{1}\right)\) or \({m}^{2}\left({\varvec{X}},Z;{{\varvec{\gamma}}}^{2}\right)\) is included in the estimator (“1” means yes and “0” means no)