Survival Analysis 生存分析

,Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,| Basic Statistics in Clinical Trials | IIS China | Analysis of Survival Data | All Rights Reserved,#,Lecture 9Analysis of Survival Data,Outline,Introduction,Basic,Parametric Models,Estimation,Comparison of Groups,Summary and Conclusions,2,Learning Objectives,Understand survival data (,生存数据,),and its special,features, in particular non-normality and censoring (,删失,),Know how to perform basic survival time,(,生存,时间,),analyses, including Kaplan-Meier plots and log-rank tests,Be able to apply the,basic methods,to real-world,problems,3,Introduction,COPD Case Study,Basic Concepts,Basic,Parametric Models,Estimation,Comparison of Groups,Summary,and Conclusions,4,COPD Study Example,Background,Design,:,double-blind, parallel-group study to assess a new drug A in patients with Chronic Obstructive Pulmonary Disease (COPD,慢性阻塞性肺病,),48,patients,were,randomized to either drug,A or,placebo at 1 : 1 ratio,25 patients received drug A; 23 patients received placebo,Patients were followed up to 52 weeks,Objective,:,compare the time to first,exacerbation (,病情加重,),between two treatment groups (new drug,A versus,placebo,),5,COPD Study Example,Data Structure,Censor (,删失,): if patient has not had the event at the time of analysis,Data,are a mixture of,censored,and uncensored,observations,PID,Group,E,vent,Censoring,Survival time,(in weeks),1,Drug,YES,NO,8,3,Drug,NO,YES,12,6,Drug,NO,YES,52,.,.,.,.,.,2,Placebo,YES,NO,15,4,Placebo,NO,YES,44,5,Placebo,NO,YES,2,.,.,.,.,.,PID = Patient identification,6,COPD Study Example,Boxplots,The survival times are not normally distributed,How to analyze the data?,9,Survival Data,Special Features,Survival data are usually continuous, but often highly skewed (偏斜的,偏态的) and non-normal,Observations may be censored,Event may not be observed for some patients at the time of analysis,Some patients may drop out in the middle of the study and all we know about them is the last time when they were still “free of the event,Unequal follow-up time for each patient,Patients may enter the study at different time point and are followed for different length of time,Need special statistical methods to analyse survival data,10,COPD Study Example,Censoring Illustration,S = Study begin,E = Event,C = Censored,11,Survival Data,Remarks on Censoring,Censoring means that some of the survival times will be unknown,Main assumption in this lecture:,t,he,censored and uncensored patients,are alive,at,a specific time are,a,homogeneous (,同质的,),sample of the total population,That is, a censored patient is assumed to be representative of those comparable patients at risk,Censoring at random,Censoring at fixed time of analysis,Censored due to termination of treatment as a result of deteriorating,(,恶化,) condition,12,Basic Concepts,Survival Function,(,生存函数,) ,Definition,13,Basic Concepts,Survival Function Properties,14,Basic Concepts,Survival Function Percentiles,15,Basic Concepts,Hazard,Function,(,风险函数,) Definition,(David Clayton),16,Basic Concepts,Hazard,Function Possible Shapes,17,Example,: recovery after surgery (e.g. high risk to die at the beginning, but once critical time passed, the risk is small),Example,: advanced chronic disease (e.g. stable risk to die),Example,: cancer patients not responding to treatment (e.g. risk to die increases gradually),Basic Concepts,Survival, Hazard and Cumulative Hazard Functions,18,Summary,Survival analysis is used to analyze data in which the time to event is of interest (different from binary responses),Survival data are typically non-normal and observations may be censored,The two main functions of interest are,Survival function = probability that a patient will survive past time 𝑡,Hazard function = likelihood for an event (e.g. death) soon after time 𝑡, given that the patient has survived until,𝑡,The two functions can be estimated in two ways,Specify a parametric model, i.e., a particular distribution,Develop an empirical estimate, i.e., non-parametric estimation,19,Introduction,Basic,Parametric Models,Exponential Survival Model,Weibull Survival Model,Estimation,Comparison of Groups,Summary and Conclusions,20,Exponential Survival Model,Survival Function,21,Exponential Survival Model,Median Survival Time,22,Exponential Survival Model,Hazard,Function,23,Example,: advanced chronic disease (e.g. stable risk to die),Weibull Survival Model,Hazard and Survival Functions,24,Weibull Survival Model,Advantages,25,Weibull Survival Model,Graphical Display,Example,: Hazard depends on age (the older, the higher the risk),Example,: Recovery after surgery (high risk to die at,the,beginning, but once critical time passed, the risk is small),26,Weibull Survival Model,Diagnostic,27,Summary,The exponential distribution is the simplest parametric survival model with a constant hazard function,The Weibull distribution is a parametric survival model with a flexible hazard function,Weibull model includes the exponential model as a special case,Parameters can be estimated using,maximum,likelihood,estimation,(not covered in this lecture,),However, assuming an inappropriate parametric model may result in incorrect inference,Non-parametric approaches require fewer assumptions and are more robust,28,Introduction,Basic,Parametric Models,Estimation,Comparison of Groups,Summary and,Conclusions,29,Estimation,Overview,30,Estimation,Kaplan-Meier (KM) Estimate,31,Estimation,Kaplan-Meier (KM) Estimate Properties,32,COPD Example,Revisited,Background,Recall:,study to compare,the time to first exacerbation,between drug and placebo,in,48 patients,with,COPD (followed up to 52 weeks),Individual outcomes for each treatment group,E,= Event; C = Censored observation,33,COPD Example,Revisited,Kaplan-Meier (KM) Estimates for,Placebo,Group Tabular Form,Weeks,Number of censored observations,0,23,0,1,1,1,23,1,0.9565,0.9565,2,1,22,4,21,1,0.9524,0.9110,5,20,1,0.95,0.8654,6,19,2,0.8947,0.7743,8,17,1,0.9412,0.7288,9,1,16,10,15,1,0.9333,0.6802,12,14,2,0.8571,0.5830,15,12,2,0.8333,0.4859,15,1,10,.,.,.,.,.,.,At week 15 the KM estimate drops below 50% for the first time.,34,2 events and 1 censored observation occur at week 15. Consider first the event and then the consoring.,COPD Example,Revisited,Kaplan-Meier,Plot for,Placebo,Group,Median survival time,is defined as the first time point when the survival curve drops below 50%,Median survival time is 15 weeks,Survial estimate for Placebo group,with censoring and number of patients at risk,35,15,Estimation in SAS,PROC LIFETEST,The,LIFETEST,procedure can be used to compute nonparametric,estimates of the,survival function,Basic,SAS code for,survival analysis,PROC,LIFETEST,DATA,=data ;,TIME,variable,;,RUN,;,TIME,statement,indicates,the,survival time variable,variable,and,the censoring variable,censor,36,COPD Example,Revisited,Survival Analysis SAS,Code,The,LIFETEST,procedure,applied to the COPD example,PROC,LIFETEST,DATA,=copd,PLOTS,=(SURVIVAL LLS);,TIME,wks*c(1,);,RUN,;,Each,statement allows some options, providing various,outputs,This displays the estimated survival and LCH functions,wks,is the survival time variable (time measured in weeks),c is the censoring variable (where “1 denotes the censoring value),37,COPD Example Revisited,Survival Analysis ,SAS Output,Median survival (exacerbation) time is 15 weeks,95%,confidence interval,is 8,to,27 weeks,Kaplan-Meier,estimates,for,placebo group,In total 23 patients are on,placebo,group,17 patients had COPD exacerbation,6 patients were censored,38,COPD Example,Revisited,Survival Analysis ,LCH Plot,Mostly linear shape supports a Weibull parametric model,Slight deviation from linearity in the upper right,plot of the log,of the negative log of the estimated,survival,functions against the log,of time,=,plot,of the log-cumulative hazard,function (LCH) against the log of time,39,Summary,The Kaplan-Meier curve is the standard estimate of a survival function,Useful,for the description and,display of survival data,The plot,of,the estimated log-cumulative,hazard,function (LCH) can be used to assess the suitability of a parametric model,(i.e.,Weibull survival model),The,LIFETEST,procedure,in SAS can,be used to compute,the,Kaplan-Meier estimate of,a,survival function,40,Introduction,Basic,Parametric Models,Estimation,Comparison of,Groups,Graphical Methods,Statistical Tests,Summary and,Conclusions,41,Comparison of,Groups,Overview,42,Comparison of,Groups,Comparison of Two Weibull LCH Functions,43,Comparison of,Groups,Hazard Ratio,44,Comparison of,Groups,Proportional Hazard Model,45,COPD Example,Revisited,Comparison of Two,Groups: SAS code,The,LIFETEST,procedure can,also be,used to,compare,two,groups,PROC,LIFETEST,DATA,=copd,PLOTS,=(SURVIVAL,LLS);,TIME,wks*c(1,);,STRATA,group;,RUN,;,STRATA,statement,indicates which variable determines strata levels for computation and,comparison,46,COPD Example,Revisited,Comparison of Two,Groups: Kaplan-Meier Plot,Median survival time for:,Placebo is 15 weeks,Drug is 35 weeks,Drug seems to be better than placebo, because of,larger median survival time,better survival function,Is this difference statistically significant?,47,35,15,COPD Example,Revisited,Survival Analysis ,LCH Plot,48,Comparison of Groups,Log-rank Test Notation and Basic Idea,49,Comparison of Groups,Log-rank Test Construction,Drug,A,B,Total,50,Total number of deaths in both groups,Proportion,of,patients at risk in,Group A among all patients at risk,Comparison of Groups,Log-rank Test,Statistic,51,Comparison of Groups,Log-rank Test, Simple,Example,Drug,A,B,Total,52,Comparison of Groups,Log-rank Test, Simple Example (contd),8,3,0,6,1,9,1,0.333,0.22,10,3,0,5,1,8,1,0.375,0.23,14,3,1,4,0,7,1,0.429,0.24,18,2,1,3,1,5,2,0.800,0.36,21,1,1,2,0,3,1,0.333,0.22,Total,3,3,6,2.270,1.27,53,Comparison of Groups,Log-rank Test Remarks,54,COPD Example Revisited,Comparison of Two,Groups: SAS Code,The same code as before with the,STRATA,statement can be used to perform the log-rank test in SAS,PROC,LIFETEST,DATA,=copd,;,TIME,wks*c(1,);,STRATA,group;,RUN,;,55,COPD Example Revisited,Comparison of Two Groups: SAS,Output,14 patients,on Drug had,COPD,exacerbation and 11,were,censored,17,patients,on Placebo had,COPD,exacerbation,and 6 were,censored,56,Summary,The log-rank,test is a hypothesis test to compare the survival distributions of two,samples,It,is a nonparametric test and appropriate to use when the data are,skewed and censored,It,is widely used in clinical trials to establish the efficacy of a new treatment in comparison with a control treatment,for survival data,57,Introduction,Basic,Parametric Models,Estimation,Comparison of Groups,Summary,and Conclusions,58,Summary and Conclusions,Introduced survival data and its special features, in particular non-normality and censoring,Parametric models can be used to model survival data, however, inappropriate model may result in incorrect inference,Non-parametric approaches, e.g., the KM estimate, are more robust,Log-rank test can be applied to compare the survival distributions of different groups,59,Extension,In reality, the COPD study example was more complex and included the measurement of several covariates,Covariates,that may influence survival time include,Gender,(binary variable),Disease severity,(,疾病的严重程,度,categorical variable(,分类变量,),Age (in years),Objective,: model dependence of survival time on these 3 covariates,Which covariate is most important?,Adjusting for these, is there a treatment effect difference?,Cox regression model,allows GLM-type analyses assuming a generalization of the proportional hazard model (1,),PROC,PHREG,in SAS can fit Cox,models,60,Q & A,Any question?,61,Appendix,Reference,Collet, D. (2003).,Modelling Survival Data in Medical Research,. Chapman and Hall, Boca Raton.,Cox,D.R.,(1972).,Regression Models and Life-Tables,.,Journal of the Royal Statistical Society, Series B 34 (2): 187220.,62,Appendix,Abbreviations,KM: Kaplan-Meier,Chronic Obstructive Pulmonary,Disease: COPD,L,og-cumulative,hazard: LCH,63,Appendix,Glossary,B,inary Response:,二,元响应,变量,Categorical,V,ariable:,分类,变量,Censoring/Censored:,删失,Covariate:,协,变量,Cumulative,Hazard Function:,累计风险函数,Empirical,C,umulative,D,istribution Function:,经验累计,分布函数,Hazard,F,unction,:,风险函,数,Hazard Ratio:,风险比,Homogeneous:,齐次的,Hypergeometric Distribution:,超几何分布,64,Appendix,Glossary,Log-rank Test:,对数,秩检验,Median Survival,T,ime,:,中,位生存时,间,Proportional,H,azard,M,odel,:,比例风险模,型,Skew:,偏斜,的,偏,态,的,Survival,A,nalysis,:,生存分析,Survival F,unction,:,生存函数,65,Appendix,COPD Study Example Study Time vs. Patient Time,Two options to measure “time in a survival time study,Study time: measured from start of study using calendar time,Patient time: measured from time of entry to study (every patient starts on week 0),In this lecture,time,always,mean,patient,time,S= Study begin,E= Event,C= Censored,66,