On Statistical Methods For Survival Data With A Cured Subgroup Based On Different Processing Strategies Of Intermediate Events

In studies with survival endpoints,intermediate events(IE)happen between the initiation of follow-up and the occurrence of endpoint events.Examples of IEs include the progressive disease(PD),adverse events,the treatment switching,and so on.The occurrence status of IEs is time-dependent.Like time-independent baseline variables,IEs influence the hazard of endpoint events.Different processing strategies of IEs are adopted in studies of different types.1)In observational studies,the IE is often treated as a study factor and its effect on endpoint events needs to be estimated.2)In randomized controlled trials(RCT),IE(also known as intercurrent event)is not the study factor.It is necessary to adjust the influence of the IE to accurately estimate the effect of the study factor on the endpoint events.An example is the treatment switching adjustment analysis.When estimating the effect of IEs on endpoint events in observational studies,timedependent Cox regression and landmark method could be used to avoid the survivor bias.In RCTs allowing the patients progressed in control group to switch onto the experimental group,there are many methods to adjust the impact of the treatment switching on the overall survival(OS)estimation and the treatment effect estimation,including rank preserving structural failure time model with grid estimation or iterative parameter estimation,two stage estimation method,inverse probability of censoring weights,semi-competing risks model,and other methods.Different methods are adopted according to the study design and purpose.However,existing methods assume that all subjects are susceptible to the IE without considering that some subjects in study population will never suffer the IE.Take the PD as an example,with the development of medicine,especially the targeted therapy and immunotherapy,many diseases could be clinically curable.On the one hand,the cured subgroup is insusceptible to some IEs,such as PD.On the other hand,the cured subgroup will experience the endpoint event with a hazard different from that of uncured subgroup.In this study,we define the cured subgroup as the group of subjects who are insusceptible to IE but will experience endpoint events in the study population.For studies with the survival endpoint affected by IEs,ignoring the existence of the cured subgroup may lead to bias of results and even wrong conclusions.When there is a cured subgroup in study population,based on observational studies and RCTs,respectively,we explore new IE processing strategies in this paper,with the main contents and results as follows.1.The effect estimation of IEs in observational studies with a cured subgroup.Inspired by the idea of enrichment,we propose the susceptible pre-identification-based method to estimate the effect of IEs in observational studies with a cured subgroup.Considering the cured subgroup is insusceptible to IEs,the proposed method excludes the cured subgroup from the analysis set to eliminate its impact on effect estimation results.There are three steps in the proposed method: 1)modelling the observed IE occurrence states and time with the mixture cure model(MCM);2)with fitted MCM,pre-identifying the susceptibility to IE of patients with censored IE time by the proposed residual intermediate-event time imputation(RITI)method;3)estimating the effect of IE on endpoint events with extended Cox regression or landmark method within the pre-identified susceptible subgroup,i.e.,the pre-identified uncured subgroup.With Monte Carlo simulations,the proposed IE effect estimation method based on the susceptible subgroup pre-identified by RITI was compared,in different scenarios,with the IE effect estimation method based on the susceptible subgroup pre-identified by logistic regression model(LRM)and the IE effect estimation method neglecting the susceptibility to IE.Then,the comparison of proposed method with existing methods was conducted in the background of mycosis fungoides.The simulation study and case study results show that the proposed IE effect estimation method based on the susceptible subgroup pre-identified by RITI method greatly reduces the effect estimation bias of existing method when there is a cured subgroup insusceptible to the IE.The effect estimation performance of proposed method is also better than that of the effect estimation method based on the susceptible subgroup pre-identified by LRM.Besides,in scenarios with varying IE effects(protective or harmful),covariates in MCM,hazard gaps of endpoint event between the cured and uncured subgroups,landmark times,and sample sizes,the proposed method maintains its superiority on the effect estimation accuracy.When there are more continuous covariates in MCM,the superiority of the proposed method is larger.2.The treatment effect estimation in RCTs with treatment switching and a cured subgroup.In RCTs with treatment switching permit for patients progressed in control group and with OS as the endpoint,we propose a multistate transition model to deal with the impact of the cured subgroup on the treatment effect estimation.In the proposed model,there are five states including cured,uncured,PD,treatment switching and death.The death hazard of the cured is assumed to be the same as that of the healthy people who have never suffered the disease,so it is small and has nothing to do with the treatment received.For those who are not cured,the semi-competing risks model with shared frailty is used to model the PD and death time.The transition hazards between different states are modelled by exponential distributions and extended to the case of Weibull distributions.With the proposed multistate transition model and observed data,the likelihood function is established.Then the particle swarm optimization(PSO)algorithm is used to estimate the parameters that represent the treatment effect on transition hazards between states.Monte Carlo simulation study was performed to compare the treatment effect estimation performances of the proposed multistate transition model,semi-competing risks model and other simple treatment switching adjustment methods without considering the cured subgroup.At last,the performance comparison was conducted in the background of diffuse large B-cell lymphoma.The simulation study and case study results show that the proposed multistate transition model could provide more accurate treatment effect estimates by correcting the impact of the treatment switching and the cured subgroup.Compared with the semi-competing risks model and other simple treatment switching adjustment methods that fail to consider the cured subgroup,including intention-to-treat analysis,per-protocol analysis,grid estimation,iterative parameter estimation,and two-stage estimation methods,the proposed multistate transition model shows great superiority in scenarios of different cure rates,treatment switching proportions,treatment effects,switching effects and sample sizes.In scenarios where the transition hazards between states are modelled by exponential and Weibull distributions,respectively,the effect estimation performances of the proposed multistate transition model are robust.By contrast,existing methods,which ignore the cured subgroup,lead to large biases to the treatment effect estimation,and the biases change greatly with the cure rates and switching proportions.Besides,the proposed multistate transition model is sensitive to the sample size due to the large number of parameters to be estimated.In the scenario of small sample sizes,the accuracy of the effect estimate via proposed model decreases slightly,but it is still better than that of existing methods.3.The sample size estimation for RCTs with treatment switching and a cured subgroup.For RCTs with treatment switching permit,a cured subgroup,and the purpose of comparing the effects of different treatments on OS,we propose a sample size estimation method based on simulations.With the proposed multistate transition model in last section,the median OS of each group is calculated taking consideration of the cured subgroup and the treatment switching.Then,calculate the initial sample size under the assumption of proportional hazard.Next,obtain the empirical power with simulations and adjust the sample size until the expected power is achieved.In the simulated trial,factors such as the enrollment rate,cure rate,switching proportion,and competing risks of death and PD are considered to make the simulated trial closer to the reality and obtain a more accurate sample size estimate.In addition,the intention-to-treat analysis method is used in simulations considering the acceptance by regulators.That is,the log-rank test is performed on the OS between groups without adjusting the cured subgroup and the treatment switching so as to control the type I error rate.The proposed sample size estimation method has been compiled into a SAS macro %n＿Rctcc.Users could obtain the sample size estimate quickly with parameters of the designed trial.With the proposed sample size estimation method,the sample sizes under different cure rates,treatment switching proportions,switching effects,switching hazards,competing risk ratios of PD and death,and enrollment rates are calculated to explore the impact of different factors on the sample size.The trend of the sample size estimate change with respect to factors above are exactly consistent with the theoretical speculation,demonstrating that the proposed sample size estimation method is scientific and rational.This study takes IEs in survival studies as the object.When there is a cured subgroup insusceptible to IE in the study population,new IE processing strategies are proposed based on observational studies and RCTs with treatment switching permit,respectively.Both of the simulation study and case study results show that the proposed IE processing strategies could correct the impact of the cured subgroup and greatly reduce the conclusion bias.The main innovations and contributions of this study are as follows: 1)In observational studies,when there is a cured subgroup insusceptible to IE,a new effect estimation method for IE is proposed based on the susceptibility pre-identification via RITI.The proposed method could greatly reduce the effect estimation bias brought by existing methods and provide a more accurate effect estimate.2)In RCTs with treatment switching permit and a cured subgroup,a new multistate transition model to estimate the treatment effect on OS is proposed.A simply applied method,i.e.,PSO algorithm,is innovatively used to estimate the model parameters.On the one hand,the proposed method could provide a more accurate treatment effect estimate.On the other hand,the PSO algorithm provides a new way to estimate parameters in statistical models.3)When OS is the primary endpoint in RCTs with treatment switching permit and a cured subgroup,a sample size estimation method based on the multistate transition model and simulated trial is proposed.The method has been compiled as a SAS macro to provide convenience for clinical practice.