Statistical Methods For Survival Data By Incorporating Short-term Outcome And Its Application In Seamless Phase ?/? Design

BackgroundIn oncology research,the most effective measure of treatment effective is overall survival?OS?or disease-free survival?DFS?.However,it usually takes a long time to observe the two survival outcomes.Indicators of decreased tumor burden after treatment,such as objective response?OR?,complete remission?CR?and partial remission?PR?,are often observed faster than DFS.Besides,tumor load can be effectively predict survival outcomes.Therefore,tumor burden reduction is often used as an efficacy evaluation indicator in the phase ? oncology clinical trials.In phase ? oncology clinical trials,survival outcomes are the gold standard for evaluating outcomes;however,short-term outcome response rates are not usually considered when analyzing survival outcomes.Thall?2008?believes that short-term response rates should be considered in the analysis of long-term survival outcomes,because those who respond usually have longer survival times.Neglecting the short-term outcome response rate not only loses important information,but also ignores the mixed distribution of the survival time of response and non-response survivors.Inoue?2002?and Lai?2012?proposed methods for inferring long-term outcomes by incorporating short-term outcomes in a group sequential design setting.Assuming that the survival time under the exponential distribution,Inoue?2002?proposed a Bayesian parametric model.However,the survival time may not be an exponential distribution in clinical setting,while its parameters are difficult to estimate.Thus,the usage of Bayesian model is limited in practice.Lai?2012?proposed a semi-parametric model based on the Cox model.However,Lai?2012?included interaction items between treatment groups and short-term outcomes,leading to the difficulty in explaining the result.In addition,the method proposed Lai?2012?is in the setting of group sequential design and the measure of association was not fully evaluated.ObjectiveBased on the Lai?2012?model,we first proposed a measure that incorporating short-term outcome when inferring long-term outcome in a non-group sequential design.Subsequently,based on the likelihood theory,we proposed the corresponding hypothesis methods?e.g.,likelihood ratio test,Wald test,and Score test?,and then the statistical performance of the proposed measure,including the accuracy of parameter estimation,distribution of test statistics,type I errors rate and statistical power,were evaluated comprehensively through simulation.Finally,the proposed method was extended to the seamless phase ?/? design where subgroup selection was based on the short-term outcome.The type I error rate and statistical power were evaluated under this setting.MethodsBased on the Lai et al.?2012?model,this study proposed the following semi-parametric model for inferring long-term outcomes by incorporating short-term outcomes ??t|Y,Z?=?0?t?exp??Y?,?Z.When a?1,the hazard ratio of the experimental group to the control group can be approximated as 1-R??,??,where?=??0,?1?,? =?a,b?,a = e?,b = e?,and R??,??={?0a+{1-?0?}-{?1?b+?1-?1?b}.R??,??>0 indicating that experimental group have better outcome than the control group.The partial likelihood estimation method was used to estimate a and R.The Wald test,likelihood ratio test,and score test were proposed to test the hypothesis H0:R = 0?corresponding toH0:1-R = 1?based on likelihood theory.The corresponding test statistics areXW2=?R-R?[IRR??,R?]-1?R-R?for Wald test,XLR2=2{LL??,R?-LL???R?,R?} for Likelihood Ratio Test,and XSC2 =U???R?,R?[IRR???R?,R?]-1U???R?,R?for Score test,respectively.The estimation accuracy of R as well as the distribution of those test statistics,type I error rate,and statistical power were evaluated through the simulation study.Finally,the proposed measure was applied to the seamless phase ?/? design where subgroup selection was based on the short-term outcome.Also,the type I error and statistical power were evaluated through simulation.Results?1?Statistical methods for inferring survival outcomes by incorporating short-term outcomesBoth bias and Mean Square Error?MSE?were used to evaluate the accuracy of parameter estimation.1)Bias:The new measure?1-R?and the HR estimation considering short-term outcomes are all close to 0,and the larger sample size and the stronger the effect,the closer the bias is to 0.However,it is slightly higher than 0,showing the tendence to reduce the effect of intervention.Thus,the results are somewhat conservative.2)MSE:The root mean square error of each measures decreases as the sample size increases,and decreases as the intervention effect increases.The root mean square error of the new measure?1-R?and the HR estimation considering short-term outcomes are all smaller than the HR estimates without considering short-term outcomes.Statistical distribution of the statistics:The test statistics of Wald test,LR test and Score test presented as the form of chi-square distribution in general.The likelihood ratio test?LRT?fitted the right tail the best.Type I error?significant level at 0.05?:The Type I error of the three tests and both Cox models are well controlled in general.On the whole,the Score test has the highest type I error rate than other methods,while the Wald test ranks the second.The LRT is basically the same as the two Cox models,and was controlled at the nominal significant level.Statistical power:When ppvT is 0.7,the Score test has higher power than the LRT and Cox₁?excluding short-term outcome?.The latter two are almost the same.As ppvT increases to more than 0.8,the power of LRT and Cox₁ increases gradually.At this point,the LRT has higher power than the Score test,while the latter has higher power than Cox₁.In all cases,Cox₂?including short-term outcomes?had the lowest statistical power.?2?Application in the seamless ?/? designType I error?one-sided significant level at 0.025?:Both the Wald test and the Score test have expansion in Type I error.The type I error rate of LRT was consistent with the Cox₁,Cox₂,and log-rank tests,and was slightly lower than the nominal significant level.Statistical power:Overall,the statistical power of LRT is similar to that of Cox₁.As ppvT increased to more than 0.8,the power of LRT is larger than that of Cox₁.The power of LRT and the logrank test are almost the same except under some setting where the LRT has higher power than the logrank test.Cox₂ has the lowest power.ConclusionsThis study proposes a measure incorporating short-term outcomes when inferring survival outcomes in a non-group sequential design setting.Then,the proposed measure was extended to the seamless phase ?/? design.The LRT has well controlled overall type I error rate,and the power of LRT is the same as or slightly higher than the existing methods?such as the Cox model and the log rank test?.In practice,it is recommended LRT should be used to analyze the survival outcomes by incorporating short-term outcomes.