Survival analysis Brian Healy, PhD

Previous classes Regression Regression –Linear regression –Multiple regression –Logistic regression

What are we doing today? Survival analysis Survival analysis –Kaplan-Meier curve –Dichotomous predictor –How to interpret results Cox proportional hazards Cox proportional hazards –Continuous predictor –How to interpret results

Big picture In medical research, we often confront continuous, ordinal or dichotomous outcomes In medical research, we often confront continuous, ordinal or dichotomous outcomes One other common outcome is time to event (survival time) One other common outcome is time to event (survival time) –Clinical trials often measure time to death or time to relapse We would like to estimate the survival distribution We would like to estimate the survival distribution

Types of analysis-independent samples OutcomeExplanatoryAnalysis ContinuousDichotomoust-test, Wilcoxon test ContinuousCategoricalANOVA, linear regression Continuous Correlation, linear regression Dichotomous Chi-square test, logistic regression DichotomousContinuousLogistic regression Time to eventDichotomousLog-rank test

Definitions Survival time: time to event Survival time: time to event Survival function: probability survival time is greater than a specific value S(t)=P(T>t) Survival function: probability survival time is greater than a specific value S(t)=P(T>t) Hazard function: risk of having the event (t)=# who had event/# at risk Hazard function: risk of having the event (t)=# who had event/# at risk These two factors are mathematically related These two factors are mathematically related

Example An important marker of disease activity in MS is the occurrence of a relapse An important marker of disease activity in MS is the occurrence of a relapse –This is the presence of new symptoms that lasts for at least 24 hours Many clinical trials in MS have demonstrated that treatments increase the time until the next relapse Many clinical trials in MS have demonstrated that treatments increase the time until the next relapse –How does the time to next relapse look in the clinic? What is the distribution of survival times? What is the distribution of survival times?

Kaplan-Meier curve Each drop in the curve represents an event

Survival data To create this curve, patients placed on treatment were followed and the time of the first relapse on treatment was recorded To create this curve, patients placed on treatment were followed and the time of the first relapse on treatment was recorded –Survival time If everyone had an event, some of the methods we have already learned could be applied If everyone had an event, some of the methods we have already learned could be applied Often, not everyone has event Often, not everyone has event –Loss to follow-up –End of study

Censoring The patients who did not have the event are considered censored The patients who did not have the event are considered censored –We know that they survived a specific amount of time, but do not know the exact time of the event –We believe that the event would have happened if we observed them long enough These patients provide some information, but not complete information These patients provide some information, but not complete information

Censoring How could we account for censoring? How could we account for censoring? –Ignore it and say event occurred at time of censoring  Incorrect because this is almost certainly not true –Remove patient from analysis  Potential bias and loss of power –Survival analysis Our objective is to estimate the survival distribution of patients in the presence of censoring Our objective is to estimate the survival distribution of patients in the presence of censoring

Example For simplicity, let’s focus on 10 patients whose time to relapse is provided here For simplicity, let’s focus on 10 patients whose time to relapse is provided here We assume that no one is censored initially We assume that no one is censored initially We would like to estimate S(t) and (t) We would like to estimate S(t) and (t) PatientTime

What do we see from our curve? 1)Drops in the curve only occur at time of event 2)Between events, the estimated survival remains constant What is the size of the drops?

Calculating size of drop To calculate the hazard at each time point=# events/# at risk To calculate the hazard at each time point=# events/# at risk –If no event, hazard=0 To calculate estimated survival use: To calculate estimated survival use: 1/ /9 1/8 1/7 1/5 1/4 1/3 1/2 1/1 1/ PatientTime

Example-censoring For simplicity, let’s focus on 10 patients whose time to relapse is provided here For simplicity, let’s focus on 10 patients whose time to relapse is provided here We assume that no one is censored initially We assume that no one is censored initially We would like to estimate S(t) and (t) We would like to estimate S(t) and (t) PatientTime

What do we see from our curve? 1)Drops in the curve only occur at time of event 2)Between events, the estimated survival remains constant 3)Survival curve does not drop at censored times

PatientTime Calculating size of drop To calculate the hazard at each time point=# events/# at risk To calculate the hazard at each time point=# events/# at risk –If no event, hazard=0 To calculate estimated survival use: To calculate estimated survival use: 1/ /8 1/7 1/5 1/4 1/3 0 1/

Confidence interval for survival curve A confidence interval can be placed around the estimated survival curve A confidence interval can be placed around the estimated survival curve –Greenwood’s formula

Summary Kaplan-Meier curve represents the distribution of survival times Kaplan-Meier curve represents the distribution of survival times Drops only occur at event times Drops only occur at event times Censoring easily accommodated Censoring easily accommodated If last time is not event, curve does not go to zero If last time is not event, curve does not go to zero

Comparison of survival curve One important aspect of survival analysis is the comparison of survival curves One important aspect of survival analysis is the comparison of survival curves Null hypothesis: S 1 (t)=S 2 (t) Null hypothesis: S 1 (t)=S 2 (t) Method: log-rank test Method: log-rank test

Example Untreated PatientTime TreatedPatientTime

Log-rank test-technical To compare survival curves, a log-rank test creates 2x2 tables at each event time and combines across the tables To compare survival curves, a log-rank test creates 2x2 tables at each event time and combines across the tables –Similar to MH-test Provides a  2 statistic with 1 degree of freedom (for a two sample comparison) and a p-value Provides a  2 statistic with 1 degree of freedom (for a two sample comparison) and a p-value Same procedure for hypothesis testing Same procedure for hypothesis testing

Hypothesis test 1) H 0 : S 1 (t)=S 2 (t) 2) Time to event outcome, dichotomous predictor 3) Log rank test 4) Test statistic:  2 =4.4 5) p-value= ) Since the p-value is less than 0.05, we reject the null hypothesis 7) We conclude that there is a significant difference in the survival time in the treated compared to untreated

p-value

Notes Inspection of Kaplan-Meier curve will allow you to determine which of the groups had the significantly longer survival time Inspection of Kaplan-Meier curve will allow you to determine which of the groups had the significantly longer survival time Other tests are possible Other tests are possible –Gehan’s generalized Wilcoxon test –Tarone-Ware test –Peto-Peto-Prentice test Generally give similar results, but emphasize different parts of survival curve Generally give similar results, but emphasize different parts of survival curve