Survival Analysis Bandit Thinkhamrop, PhD. (Statistics) Department of Biostatistics and Demography Faculty of Public Health, Khon Kaen University

Begin at the conclusion 7

Type of the study outcome: Key for selecting appropriate statistical methods Study outcomeStudy outcome Dependent variable or response variableDependent variable or response variable Focus on primary study outcome if there are moreFocus on primary study outcome if there are more Type of the study outcomeType of the study outcome ContinuousContinuous Categorical (dichotomous, polytomous, ordinal)Categorical (dichotomous, polytomous, ordinal) Numerical (Poisson) countNumerical (Poisson) count Event-free durationEvent-free duration

The outcome determine statistics Continuous CategoricalCountSurvival Mean Median Proportion (Prevalence Or Risk) Rate per “space” Median survival Risk of events at T (t) Linear Reg.Logistic Reg.Poisson Reg.Cox Reg.

Statistics quantify errors for judgments Parameter estimation [95%CI] Hypothesis testing [P-value] Parameter estimation [95%CI] Hypothesis testing [P-value]

Back to the conclusion Continuous CategoricalCountSurvival Magnitude of effect 95% CI P-value Magnitude of effect 95% CI P-value Mean Median Proportion (Prevalence or Risk) Rate per “space” Median survival Risk of events at T (t) Answer the research question based on lower or upper limit of the CI Appropriate statistical methods

Study outcome Survival outcome = event-free durationSurvival outcome = event-free duration Event (1=Yes; 0=Censor)Event (1=Yes; 0=Censor) Duration or length of time between:Duration or length of time between: Start date ()Start date () End date ()End date () At the start, no one had event (event = 0) at time t (0)At the start, no one had event (event = 0) at time t (0) At any point since the start, event could occur, hence, failure (event = 1) at time t (t)At any point since the start, event could occur, hence, failure (event = 1) at time t (t) At the end of the study period, if event did not occur, hence, censored (event = 0)At the end of the study period, if event did not occur, hence, censored (event = 0) Thus, the duration could be either ‘time-to-event’ or ‘time-to-censoring’Thus, the duration could be either ‘time-to-event’ or ‘time-to-censoring’

Censoring Censored data = incomplete ‘time to event’ dataCensored data = incomplete ‘time to event’ data In the present of censoring, the ‘time to event’ is not knownIn the present of censoring, the ‘time to event’ is not known The duration indicates there has been no event occurred since the start date up to last date assessed or observed, a.k.a., the end date.The duration indicates there has been no event occurred since the start date up to last date assessed or observed, a.k.a., the end date. The end date could beThe end date could be End of the studyEnd of the study Last observed prior to the end of the study due toLast observed prior to the end of the study due to Lost to follow-upLost to follow-up Withdrawn consentWithdrawn consent Competing events occurred, prohibiting progression to the event under observationCompeting events occurred, prohibiting progression to the event under observation Explanatory variables changed, irrelevance to occurrence of event under observationExplanatory variables changed, irrelevance to occurrence of event under observation

Magnitude of effects Median survival Survival probability Hazard ratio

SURVIVAL ANALYSIS Study aims: Median survival Median survival of liver cancer Survival probability Five-year survival of liver cancer Five-year survival rate of liver cancer Hazard ratio Factors affecting liver cancer survival Effect of chemotherapy on liver cancer survival

SURVIVAL ANALYSIS Event Dead, infection, relapsed, etc Cured, improved, conception, discharged, etc Smoking cessation, ect Negative Positive Neutral

Natural History of Cancer

Accrual, Follow-up, and Event ID Begin the studyEnd of the study Dead Start of accrualEnd of accrualEnd of follow-up Recruitment period Follow-up period

Time since the beginning of the study ID Dead months 22 months 14 months 40 months 26 months 13 months The data : >48 > >26 >13

DATA 1 48Still aliveat the end of the study Censored 2 22Dead due to accident Censored 3 14Dead caused by the disease under investigationDead 4 40 Dead caused by the disease under investigation Dead 5 26 Still aliveat the end of the study Censored 6 13Lost to follow-up Censored ID SURVIVAL TIME OUTCOME AT THE END EVENT (Months)OF THE STUDY

DATA 1 48 Censored 2 22 Censored 3 14 Dead 4 40 Dead 5 26 Censored 6 13 Censored ID TIME EVENT ID TIME EVENT

ANALYSIS ID TIME EVENT Prevalence = 2/6 Incidence density = 2/163 person-months Proportion of surviving at month ‘t’ Median survival time

RESULTS ID TIME EVENT Incidence density = 1.2 per100 person-months (95%CI: 0.1 to 4.4) Proportion of surviving at 24 month = 80% (95%CI: 20 to 97) Median survival time = 40 Months (95%CI: 14 to 48)

Type of Censoring 1)Left censoring: When the patient experiences the event in question before the beginning of the study observation period. 2)Interval censoring: When the patient is followed for awhile and then goes on a trip for awhile and then returns to continue being studied. 3)Right censoring: 1)single censoring: does not experience event during the study observation period 2)A patient is lost to follow-up within the study period. 3)Experiences the event after the observation period 4)multiple censoring: May experience event multiple times after study observation ends, when the event in question is not death.

Summary description of survival data set stdes This command describes summary information about the data set. It provides summary statistics about the number of subjects, records, time at risk, failure events, etc.This command describes summary information about the data set. It provides summary statistics about the number of subjects, records, time at risk, failure events, etc.

Computation of S(t) 1)Suppose the study time is divided into periods, the number of which is designated by the letter, t. 2)The survivorship probability is computed by multiplying a proportion of people surviving for each period of the study. 3)If we subtract the conditional probability of the failure event for each period from one, we obtain that quantity. 4)The product of these quantities constitutes the survivorship function.

Kaplan-Meier Methods

Kaplan-Meier survival curve

Median survival time

Survival Function The number in the risk set is used as the denominator.The number in the risk set is used as the denominator. For the numerator, the number dying in period t is subtracted from the number in the risk set. The product of these ratios over the study time=For the numerator, the number dying in period t is subtracted from the number in the risk set. The product of these ratios over the study time=

Survival experience

Survival curve more than one group

Comparing survival between groups IDTIMEDEADDRUG

Kaplan-Meier surve Kaplan-Meier survival estimates, by drug analysis time drug 0 drug 1

Log-rank test t=Time n=Number at risk for both groups at time t n1=Number at risk for group 1 at time t n2=Number at risk for group 2 at time t d=Dead for both groups at time t c=Censored for both groups at time t O1=Number of dead for group 1 at time t O2=Number of dead for group 2 at time t E1=Number of expected dead for group 1 at time t E2=Number of expected dead for group 2 at time t

Log-rank test example DRUG1 = 48+, 22+, 26+, 13+,14,40 DRUG0 = 13+, 6, 12, 14, 22, 13

Hazard Function

Survival Function vs Hazard Function H(t) = -ln(S (t) ) (S (t) ) = EXP(-H (t) )

Hazard rate The conditional probability of the event under study, provided the patient has survived up to an including that time periodThe conditional probability of the event under study, provided the patient has survived up to an including that time period Sometimes called the intensity function, the failure rate, the instantaneous failure rateSometimes called the intensity function, the failure rate, the instantaneous failure rate

Formulation of the hazard rate The HR can vary from 0 to infinity. It can increase or decrease or remain constant over time. It can become the focal point of much survival analysis.

Cox Regression The Cox model presumes that the ratio of the hazard rate to a baseline hazard rate is an exponential function of the parameter vector.The Cox model presumes that the ratio of the hazard rate to a baseline hazard rate is an exponential function of the parameter vector. h(t) = h 0 (t)  EXP(b 1 X 1 + b 2 X 2 + b 3 X b p X p )

Hazard ratio

Testing the Adequacy of the model 1.We save the Schoenfeld residuals of the model and the scaled Schoenfeld residuals. 2.For persons censored, the value of the residual is set to missing. borrowed from Professor Robert A. Yaffee

A graphical test of the proportion hazards assumption A graph of the log hazard would reveal 2 lines over time, one for the baseline hazard (when x=0) and the other for when x = 1A graph of the log hazard would reveal 2 lines over time, one for the baseline hazard (when x=0) and the other for when x = 1 The difference between these two curves over time should be constant = BThe difference between these two curves over time should be constant = B If we plot the Schoenfeld residuals over the line y=0, the best fitting line should be parallel to y=0. borrowed from Professor Robert A. Yaffee

Graphical tests Criteria of adequacy:Criteria of adequacy: The residuals, particularly the rescaled residuals, plotted against time should show no trend(slope) and should be more or less constant over time. borrowed from Professor Robert A. Yaffee

Other issues Time-Varying CovariatesTime-Varying Covariates Interactions may be plottedInteractions may be plotted Conditional Proportional Hazards models:Conditional Proportional Hazards models: Stratification of the model may be performed. Then the stphtest should be performed for each stratum.Stratification of the model may be performed. Then the stphtest should be performed for each stratum. borrowed from Professor Robert A. Yaffee

Suggested Readings for beginners

Suggested Readings for advanced learners

Survival analysis in practice What is the type of research question that survival analysis should be used?What is the type of research question that survival analysis should be used?

Stata for one-group survival analysis stset time, failure(event)stset time, failure(event) stdescribestdescribe tab eventtab event stsumstsum stratestrate stcistci sts list, at(12 24)sts list, at(12 24)

Stata for one-group survival analysis (cont.) sts gsts g sts g, atrisksts g, atrisk sts g, loststs g, lost sts g, entersts g, enter sts g, risktablests g, risktable sts g, cumhazsts g, cumhaz sts g, cumhaz cists g, cumhaz ci sts g, hazardsts g, hazard

stset time, failure(event)stset time, failure(event) stdescribestdescribe stsum, by(group)stsum, by(group) sts test groupsts test group sts test group, wilcoxonsts test group, wilcoxon strate groupstrate group stci, by(group)stci, by(group) sts g, by(group) atrisksts g, by(group) atrisk sts g, by(group) risktablests g, by(group) risktable sts g, by(group) cumhaz loststs g, by(group) cumhaz lost sts g, by(group) hazard cists g, by(group) hazard ci Stata for multiple-group survival analysis

sts list,, by(group) at(12 24)sts list,, by(group) at(12 24) sts list,, by(group) at(12 24) comparests list,, by(group) at(12 24) compare ltable group, interval(#)ltable group, interval(#) ltable group, graphltable group, graph ltable group, hazardltable group, hazard stmh groupstmh group stmh group, by(strata)stmh group, by(strata) stmc groupstmc group stcox groupstcox group stir groupstir group Stata for multiple-group survival analysis

Stata for Model Fitting Continuous covariateContinuous covariate xtile newvar = varlist, nq(4)xtile newvar = varlist, nq(4) tabstat varlist, stat(n min max) by(newvar)tabstat varlist, stat(n min max) by(newvar) xi:stcox i.newvarxi:stcox i.newvar stsum, by(newvar)stsum, by(newvar) Categorical covariateCategorical covariate tab exposure outcome, coltab exposure outcome, col xi:stcox i.exposurexi:stcox i.exposure

Sample size for Cox Model stpower cox, failprob(.2) hratio( ) sd(.3) r2(.1) power( ) hrstpower cox, failprob(.2) hratio( ) sd(.3) r2(.1) power( ) hr