# Lecture 15 Introduction to Survival Analysis

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1 Lecture 15 Introduction to Survival Analysis BIOST 515 February 26, 2004 BIOST 515, Lecture 15

2 Background In logistic regression, we were interested in studying how risk factors were associated with presence or absence of disease. Sometimes, though, we are interested in how a risk factor or treatment affects time to disease or some other event. Or we may have study dropout, and therefore subjects who we are not sure if they had disease or not. In these cases, logistic regression is not appropriate. Survival analysis is used to analyze data in which the time until the event is of interest. The response is often referred to as a failure time, survival time, or event time. BIOST 515, Lecture 15 1

3 Examples Time until tumor recurrence Time until cardiovascular death after some treatment intervention Time until AIDS for HIV patients Time until a machine part fails BIOST 515, Lecture 15 2

4 The survival time response Usually continuous May be incompletely determined for some subjects i.e.- For some subjects we may know that their survival time was at least equal to some time t. Whereas, for other subjects, we will know their exact time of event. Incompletely observed responses are censored Is always 0. BIOST 515, Lecture 15 3

5 Analysis issues If there is no censoring, standard regression procedures could be used. However, these may be inadequate because Time to event is restricted to be positive and has a skewed distribution. The probability of surviving past a certain point in time may be of more interest than the expected time of event. The hazard function, used for regression in survival analysis, can lend more insight into the failure mechanism than linear regression. BIOST 515, Lecture 15 4

6 Censoring Censoring is present when we have some information about a subject s event time, but we don t know the exact event time. For the analysis methods we will discuss to be valid, censoring mechanism must be independent of the survival mechanism. There are generally three reasons why censoring might occur: A subject does not experience the event before the study ends A person is lost to follow-up during the study period A person withdraws from the study These are all examples of right-censoring. BIOST 515, Lecture 15 5

7 Types of right-censoring Fixed type I censoring occurs when a study is designed to end after C years of follow-up. In this case, everyone who does not have an event observed during the course of the study is censored at C years. In random type I censoring, the study is designed to end after C years, but censored subjects do not all have the same censoring time. This is the main type of right-censoring we will be concerned with. In type II censoring, a study ends when there is a prespecified number of events. BIOST 515, Lecture 15 6

8 Regardless of the type of censoring, we must assume that it is non-informative about the event; that is, the censoring is caused by something other than the impending failure. BIOST 515, Lecture 15 7

9 Terminology and notation T denotes the response variable, T 0. The survival function is S(t) = P r(t > t) = 1 F (t). S(t) t BIOST 515, Lecture 15 8

10 The survival function gives the probability that a subject will survive past time t. As t ranges from 0 to, the survival function has the following properties It is non-increasing At time t = 0, S(t) = 1. In other words, the probability of surviving past time 0 is 1. At time t =, S(t) = S( ) = 0. As time goes to infinity, the survival curve goes to 0. In theory, the survival function is smooth. In practice, we observe events on a discrete time scale (days, weeks, etc.). BIOST 515, Lecture 15 9

11 The hazard function, h(t), is the instantaneous rate at which events occur, given no previous events. h(t) = lim t 0 P r(t < T t + t T > t) t = f(t) S(t). The cumulative hazard describes the accumulated risk up to time t, H(t) = t 0 h(u)du. H(t) H(t) t t BIOST 515, Lecture 15 10

12 If we know any one of the functions S(t), H(t), or h(t), we can derive the other two functions. h(t) = log(s(t)) t H(t) = log(s(t)) S(t) = exp( H(t)) BIOST 515, Lecture 15 11

13 Survival data How do we record and represent survival data with censoring? T i denotes the response for the ith subject. Let C i denote the censoring time for the ith subject Let δ i denote the event indicator δ i = { 1 if the event was observed (Ti C i ) 0 if the response was censored (T i > C i ). The observed response is Y i = min(t i, C i ). BIOST 515, Lecture 15 12

14 Example T i C i Y i δ i Termination of study BIOST 515, Lecture 15 13

15 Estimating S(t) and H(t) If we are assuming that every subject follows the same survival function (no covariates or other individual differences), we can easily estimate S(t). We can use nonparametric estimators like the Kaplan-Meier estimator We can estimate the survival distribution by making parametric assumptions exponential Weibull Gamma log-normal BIOST 515, Lecture 15 14

16 Non-parametric estimation of S When no event times are censored, a non-parametric estimator of S(T ) is 1 F n (t), where F n (t) is the empirical cumulative distribution function. When some observations are censored, we can estimate S(t) using the Kaplan-Meier product-limit estimator. BIOST 515, Lecture 15 15

17 t No. subjects Deaths Censored Cumulative at risk survival /26 = / = / = / = / = / = / = / = / = / = BIOST 515, Lecture 15 16

18 How can we get this in R? > library(survival) > data(ovarian) > S1=Surv(ovarian\$futime,ovarian\$fustat) > S1 [1] [13] [25] BIOST 515, Lecture 15 17

19 > fit1=survfit(s1) > summary(fit1) Call: survfit(formula = S1) time n.risk n.event survival std.err lower 95% CI upper 95% CI BIOST 515, Lecture 15 18

20 >plot(fit1,xlab="t",ylab=expression(hat(s)*"(t)")) S^(t) t BIOST 515, Lecture 15 19

21 Parametric survival functions The Kaplan-Meier estimator is a very useful tool for estimating survival functions. Sometimes, we may want to make more assumptions that allow us to model the data in more detail. By specifying a parametric form for S(t), we can easily compute selected quantiles of the distribution estimate the expected failure time derive a concise equation and smooth function for estimating S(t), H(t) and h(t) estimate S(t) more precisely than KM assuming the parametric form is correct! BIOST 515, Lecture 15 20

22 Appropriate distributions Some popular distributions for estimating survival curves are Weibull exponential log-normal (log(t ) has a normal distribution) log-logistic BIOST 515, Lecture 15 21

23 Estimation for parametric S(t) We will use maximum likelihood estimation to estimate the unknown parameters of the parametric distributions. If Y i is uncensored, the ith subject contributes f(y i ) to the likelihood If Y i is censored, the ith subject contributes P r(y > Y i ) to the likelihood. The joint likelihood for all n subjects is L = n f(y i ) n S(Y i ). i:δ i =1 i:δ i =0 BIOST 515, Lecture 15 22

24 The log-likelihood can be written as log L = n log(h(y i )) n H(Y i ). i:δ i =1 i=1 BIOST 515, Lecture 15 23

25 Example Let s look at the ovarian data set in the survival library in R. Suppose we assume the time-to-event follows an distribution, where h(t) = λ and S(t) = exp( λt). > s2=survreg(surv(futime, fustat)~1, ovarian, dist= exponential ) > summary(s2) Call: survreg(formula = Surv(futime, fustat) ~ 1, data = ovarian, dist = "exponential" Value Std. Error z p (Intercept) e-136 Scale fixed at 1 BIOST 515, Lecture 15 24

26 Exponential distribution Loglik(model)= -98 Loglik(intercept only)= -98 Number of Newton-Raphson Iterations: 4 n= 26 In the R output, λ = exp( (Intercept)) = exp( 7.17) Therefore, S(t) = exp( exp( 7.17)t). BIOST 515, Lecture 15 25

27 plot(t,1-pexp(t,exp(-7.169)),xlab="t",ylab=expression(hat(s)*"(t)")) S^(t) t BIOST 515, Lecture 15 26

28 Example Let s look at the ovarian data set in the survival library in R. Suppose we assume the time-to-event follows a Weibull distribution, where h(t) = αγt γ 1 and S(t) = exp( αt γ ). > s1=survreg(surv(futime, fustat)~1, ovarian, dist= weibull,scale=0) > summary(s1) Call: survreg(formula = Surv(futime, fustat) ~ 1, data = ovarian, dist = "weibull", scale = 0) Value Std. Error z p (Intercept) e-130 Log(scale) e-01 Scale= BIOST 515, Lecture 15 27

29 Weibull distribution Loglik(model)= -98 Loglik(intercept only)= -98 Number of Newton-Raphson Iterations: 5 n= 26 To match the notation above, γ = 1/Scale and α = exp( (Intercept)γ). This gives us the following survival function, S(t) = exp( exp[ 7.111/.902])t 1/.902 ). BIOST 515, Lecture 15 28

30 plot(t,1-pweibull(t,1/0.902,exp(7.111))) S^(t) t BIOST 515, Lecture 15 29

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