Survival/Failer Time Analysis in Clinical Research

Transcription

1 Vanderbilt Clinical Research Center Research Skills Workshop Survival/Failer Time Analysis in Clinical Research Zhiguo (Alex) Zhao Division of Cancer Biostatistics Department of Biostatistics Vanderbilt University School of Medicine October 29, 2010 Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

2 Outline 1 Introduction What is survival analysis? Why do we need survival analysis? 2 Terms You Want to Know Censoring Survival & hazard 3 Methods Widely Used Estimate and interpret survival characteristics Compare survival in different groups Assess the relationship between explanatory variables and survival time 4 A Case Study Study introduction KM method Log-rank test Cox proportional hazard regression 5 Advanced Topics 6 Questions? Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

3 Introduction What is survival analysis? What is survival analysis? Generally defined as a set of methods for analyzing data where the outcome variable is the time until the occurrence of an event of interest. Also called time-to-event analysis time to cardiovascular death after some treatment intervention time until a response (10% decrease in SBP) time until tumor recurrence time until AIDS for HIV patients time until infection time until pregnancy Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

4 Introduction Why do we need survival analysis? Why do we need survival analysis? Why not use linear regression to model the survival time as a function of a set of predictor variables? Time to event is restricted to be positive, and has a skewed distribution Change of interest (probability of surviving past a certain point in time) Cannot effectively handle the censoring of the observations Censoring (incomplete observations of the survival time, partial information) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

5 Introduction Why do we need survival analysis? Why do we need survival analysis? How about a logistic regression? Change of interest (status at certain time point) Lower power Censoring problem Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

6 Introduction Why do we need survival analysis? Why do we need survival analysis? Example: We want to predict 2-year cancer recurrence rate using patient characteristics, such as patient demo, tumor histology, gene profile. Logistic regression. If the only interest is the status at the end of 2-year follow-up, and such info is available for all subjects. Questions: If the result from another study is 1-year recurrence rate, and you want to compare the 2-year study to it. If some subject drop out. If subject A has recurrence at 2.1 years and subject B has recurrence at 5 years, should the two subjects be treated the same in your analysis? Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

7 Terms You Want to Know Censoring What is censoring? In statistics, engineering, and medical research, censoring occurs when the value of an observation is only partially known. Different from missing Type of censoring: Left, Right, Interval fixed type I random type I type II Assume that it is non-informative about the event Result from: loss of follow-up drop out ALL patient dies in automobile accident before relapsing Bone marrow transplant patient dies of opportunistic infection before engraftment termination of the study (follow-up ends before event occurs) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

8 What is censoring? Terms You Want to Know Censoring Figure: Understand censoring Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

9 Terms You Want to Know Survival & hazard Survival and hazard functions The survival function is: S(t) = Pr(T > t) Probability that a subject will survive past time t Non-increasing Smooth in theory. In practice, we see step functions. The hazard function, h(t), is the instantaneous rate at which events occur, given no previous events. Pr(t < T t + t T > t) h(t) = lim = f (t) t 0 t S(t) The cumulative hazard function, H(t), is the accumulated risk up to time t. Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

10 Terms You Want to Know Survival and hazard functions Survival & hazard If we know any one of these three functions, we can derive the other two. h(t) = log(s(t)) t H(t) = log(s(t)) S(t) = exp( H(t)) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

11 Goals and methods Methods Widely Used Estimate and interpret survival characteristics Kaplan-Meier plots Parametric survival functions Median survival time 5-year survival rate Confidence intervals (CI) Compare survival in different groups Log-rank test Assess the relationship between explanatory variables and survival time Proportional hazards models Accelerated failure time models Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

12 Kaplan-Meier estimator Methods Widely Used Estimate and interpret survival characteristics Also called product-limit estimator Non-parametric estimation of S Step-wise, not smooth, left closed Any jumping point is a failure time point Figure: Kaplan-Meier estimator calculation Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

13 Kaplan-Meier estimator Methods Widely Used Estimate and interpret survival characteristics Median survival time, 1-year survival rate, CIs can be estimated. s^(t) Time (days) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

14 Methods Widely Used Estimate and interpret survival characteristics Parametric survival functions With more assumptions, we may model the data in more detail. Easily compute selected quantiles of the distribution Estimate the expected event time Estimate survival function more precisely than KM Popular distributions for estimating survival curves: Exponential Weibull Log-normal Gamma Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

15 Methods Widely Used Estimate and interpret survival characteristics Exponential survival curve s^(t) Time (days) Figure: Kaplan-Meier and exponential survival curves Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

16 Log-rank test Methods Widely Used Compare survival in different groups Idea: If survival is independent of group effect, then at each time point, roughly the same proportion in each group will have an event. Two-sample log-rank test: Group 1: Survival function S 1 (t) Group 2: Survival function S 2 (t) Statistical hypothesis: H 0 : S 1 (t) = S 2 (t) H A : S 1 (t) S 2 (t) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

17 Log-rank test Methods Widely Used Compare survival in different groups Proportion in Remission mercaptopurine (6 MP) Placebo Time since Enrollment (weeks) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

18 Methods Widely Used Compare survival in different groups Log-rank test Median survival time is 22.5 months for 6-MP group and 8 months for placebo group. The KM curve for 6-MP group (superior) lies above that for the placebo The gap seems to become bigger as time progresses. Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

19 Methods Widely Used Compare survival in different groups Log-rank test Log-rank test in R: Call: survdiff(formula = Surv(WeeksinRemission, status) ~ treatment, data = leuk) N Observed Expected (O-E)^2/E (O-E)^2/V treatment=6-mp treatment=placebo Chisq= 16.8 on 1 degrees of freedom, p= 4.17e-05 The p value of the test is p<0.001, which implies a statistically significant difference in the survival of the two groups. Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

20 Methods Widely Used Compare survival in different groups Log-rank test The method falls short in the following situations: Not work with continuous variables Cannot handle multiple factors Cannot quantify the differences Bad performance when two survival curves cross Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

21 Methods Widely Used Accelerated failure time models Assess the relationship between explanatory variables and survival time AFT model assumes that the effect of a covariate is to multiply the predicted event time by some constant. AFT models can therefore be framed as linear models for the logarithm of the survival time. S(t X ) = ψ((log(t) X β)/σ) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

22 Methods Widely Used Assess the relationship between explanatory variables and survival time Proportional hazards models Modeling: h(t X ) = h(t)exp(x β) The most widely used survival regression Predictors act on a subject s hazard h(t) is underling hazard function exp(xβ) is called a relative hazard function The effect of the predictors is the same for all values of t. Any parametric hazard function can be used for h(t) h(t) can be left completely unspecified Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

23 Methods Widely Used Proportional hazards models Assess the relationship between explanatory variables and survival time Figure: Proportional hazards Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

24 Methods Widely Used Proportional hazards models Assess the relationship between explanatory variables and survival time Interpretation of coefficients: The regression coefficient for X j is the increase in log hazard at any time point if X j is increased by one unit and all other predictors are held constant. Interpretation of exp(β): The effect of increasing X j by 1 unit is to increase the hazard of the event by a factor of exp(β) at all points in time. What if X j increase from X 1 j to X 2 j? Hazard ratio. The ratio of hazard for an subject with predictor values X 2 j compared to an subject with predictor values X 1 j is exp((x 2 j X 1 j )β). What if X j is a binary predictor? X j = 1 if subject is male. X j = 0 if subject is female. The hazard of the event for male is exp(β) times that for female. (Assuming female as reference group.) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

25 Methods Widely Used Assess the relationship between explanatory variables and survival time Cox proportional hazards model A semiparametric model Makes no assumptions about the underling survival function Assumes parametric form for the effect of the predictors on the hazard More interested in the parameter estimates than the shape of the hazard. Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

26 A Case Study Study introduction An Eastern Cooperative Oncology Group study A randomized trial comparing two treatments for ovarian cancer. Data dictionary: Variable Explanation Coding futime survival or censoring time in days fustat censoring status 1=death, 0=censoring age in years resid.ds residual disease present 1=No, 2=Yes rx treatment group ecog.ps ECOG performance status 1 is better Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

27 Life table A Case Study KM method > fit1=survfit(surv(futime,fustat)~1, data=ovarian) > summary(fit1) Call: survfit(formula = Surv(futime, fustat) ~ 1, data = ovarian) time n.risk n.event survival std.err lower 95% CI upper 95% CI Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

28 Overall KM curve A Case Study KM method Proportion of survival Time (days) Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

29 A Case Study KM method Zhiguo (Alex) Zhao (VU) Figure: Kaplan-Meier Survival curves Analysisby treatment group Vanderbilt CRC, Oct. 29, / 36 KM curve by treatment group > fit2=survfit(surv(futime,fustat)~rx, data=ovarian) > plot(fit2,lty = 1:2,lwd=2,ylim=c(0.3,1.0),xlab="Time (days)", ylab="proportion of survival",col=1:2) > legend("topright", legend=c("treatment 1","Treatment 2"), lty = 1:2,col=1:2) Proportion of survival Treatment 1 Treatment Time (days)

30 A Case Study KM curve by treatment group KM method > fit2 Call: survfit(formula = Surv(futime, fustat) ~ rx, data = ovarian) records n.max n.start events median 0.95LCL 0.95UCL rx= NA rx= NA 475 NA Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

31 A Case Study Log-rank test Use log-rank test to compare two survival curves > survdiff(surv(futime,fustat)~rx, data=ovarian) Call: survdiff(formula = Surv(futime, fustat) ~ rx, data = ovarian) N Observed Expected (O-E)^2/E (O-E)^2/V rx= rx= Chisq= 1.1 on 1 degrees of freedom, p= Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

32 A Case Study Assess effect of age on survival Cox proportional hazard regression > summary(fit3 <- coxph(surv(futime,fustat)~age, data=ovarian)) Call: coxph(formula = Surv(futime, fustat) ~ age, data = ovarian) n= 26 coef exp(coef) se(coef) z Pr(> z ) age ** --- Signif. codes: 0 Ś***Š Ś**Š 0.01 Ś*Š 0.05 Ś.Š 0.1 Ś Š 1 exp(coef) exp(-coef) lower.95 upper.95 age Rsquare= (max possible= ) Likelihood ratio test= on 1 df, p= Wald test = on 1 df, p= Score (logrank) test = on 1 df, p= Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

33 Rsquare= (max possible= ) Likelihood ratio test= on 2 df, p= Wald test = on 2 df, p= Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36 A Case Study Cox proportional hazard regression Assess effect of treatment while age was adjusted > summary(fit4 <- coxph(surv(futime,fustat)~rx+age, data=ovarian)) Call: coxph(formula = Surv(futime, fustat) ~ rx + age, data = ovarian) n= 26 coef exp(coef) se(coef) z Pr(> z ) rx age ** --- Signif. codes: 0 Ś***Š Ś**Š 0.01 Ś*Š 0.05 Ś.Š 0.1 Ś Š 1 exp(coef) exp(-coef) lower.95 upper.95 rx age

34 Checking PH assumption A Case Study Cox proportional hazard regression > cox.zph(fit4) rho chisq p rx age GLOBAL NA Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

35 Advanced Topics Advanced topics need further discussing Left truncation: Selective sampling, i.e. patient is included in the sample if a specific condition (e.g. T > t 0 ) is satisfied. Interval censoring: Information about the survival time is in the form t 1 < T < t 2. Competing risk: Involve multiple causes of failure. Time-dependent covariates: Covariates change with time. Dependent survival times: Bivariate survival models (e.g. correlated frailty models) can be used to analyze survival data on twins and relatives. Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

36 Questions? Questions? The slides will be available at Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

37 Questions? Questions? The slides will be available at If you have any questions, feel free to send me an at Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36

38 Questions? Questions? The slides will be available at If you have any questions, feel free to send me an at Thank you! Zhiguo (Alex) Zhao (VU) Survival Analysis Vanderbilt CRC, Oct. 29, / 36