Survival Analysis A Brief Introduction 2 1 Survival

Survival Analysis A Brief Introduction

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1. Survival Function, Hazard Function �In many medical studies, the primary endpoint is time until an event occurs (e. g. death, remission) �Data are typically subject to censoring (e. g. when a study ends before the event occurs) �Survival Function - A function describing the proportion of individuals surviving to or beyond a given time. Notation: ◦ T: survival time of a randomly selected individual ◦ t: a specific point in time. ◦ Survival Function: 3

Hazard Function/Rate � Hazard Function l(t): instantaneous failure rate at time t given that the subject has survived upto time t. That is Here f(t) is the probability density function of the survival time T. That is, � where F(t) is the cumulative distribution function of T: � 4

2. The Key Word is ‘Censoring’ �Because of censoring, many common data analysis procedures can not be adopted directly. �For example, one could use the logistic regression model to model the relationship between survival probability and some relevant covariates ◦ However one should use the customized logistic regression procedures designed to 5 account for censoring

Key Assumption: Independent Censoring �Those still at risk at time t in the study are a random sample of the population at risk at time t, for all t �This assumption means that the hazard function, λ(t), can be estimated in a fair/unbiased/valid way 6

3 A. Kaplan-Meier (Product-Limit) Estimator of the Survival Curve �The Kaplan–Meier estimator is the nonparametric maximum likelihood estimate of S(t). It is a product of the form is the number of subjects alive just before time � denotes the number who died at time � 7

Kaplan-Meier Curve, Example Time ti # at risk # events 0 20 0 1. 00 5 20 2 [1 -(2/20)]*1. 00=0. 90 6 18 0 [1 -(0/18)]*0. 90=0. 90 10 15 1 [1 -(1/15)]*0. 90=0. 84 13 14 2 (1 -(2/14)]*0. 84=0. 72 8

Kaplan Meier Curve 9

Figure 1. Plot of survival distribution functions for the NCI and the SCI Groups. The Y-axis is the probability of not declining to GDS 3 or above. The X-axis is the time (in years) to decline. (Barry Reisberg et al. , 2010; Alzheimer & Dementia; in press. ) 10

3 B. Comparing Survival Functions 1. 00 0. 75 Survival Distribution Function High 0. 50 Low 0. 25 Medium 0. 00 0 10 20 30 40 50 60 Time 11

Log-Rank Test The log-rank test • tests whether the survival functions are statistically equivalent • is a large-sample chi-square test that uses the observed and expected cell counts across the event times • has maximum power when the ratio of hazards is constant over time. 12

Wilcoxon Test The Wilcoxon test • weights the observed number of events minus the expected number of events by the number at risk across the event times • can be biased if the pattern of censoring is different between the groups. 13

Log-rank versus Wilcoxon Test Log-rank test • is more sensitive than the Wilcoxon test to differences between groups in later points in time. Wilcoxon test • is more sensitive than the log-rank test to differences between groups that occur in early points in time. 14

4. Two Parametric Distributions �Here we present two most notable models for the distribution of T. �Exponential distribution: �Weibull distribution: ◦ Its survival function: ◦ Thus: 15

Weibull Hazard Function, Plot 16

5. Regression Models The Exponential and the Weibull distribution inspired two parametric regression approaches: 1. Parametric proportional hazard model – this model can be generalized to a semi-parametric model: the Cox proportional hazard model 2. Accelerated failure time model � 17

Proportional Hazard Model �In a regression model for survival analysis one can try to model the dependence on the explanatory variables by taking the (new) hazard rate to be: �Hazard rates being positive it is natural to choose the function c such that c(β, x) is positive irrespective the values of x. 18

Proportional Hazard Model �Thus a good choice is: �The resulting proportional hazard model is: �For the Weibull distribution we have: �For the Exponential distribution we have: 19

Accelerated Failure Time Model �For the Weibull distribution (including the Exponential distribution), the proportional hazard model is equivalent to a log linear model in survival time T: �Here the error term can be shown to follow the 2 -parameter Extreme Vvalue distribution 20

Apply Both Models Simultaneously �If the underlying distribution for T is Weibull or Exponential, one can apply both regression models simultaneously to reflect different aspects of the survival process. That is �Prediction of degree of decline using the Weibull proportional hazard model �Prediction of time of decline using the accelerated failure time model 21

An Example � In a recent paper (Reisberg et al. , 2010), we applied both regression models to a dementia study conducted at NYU: � The results are shown next 22

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6. Cox Proportional Hazards Model 24

Parametric versus Nonparametric Models Parametric models require that • the distribution of survival time is known • the hazard function is completely specified except for the values of the unknown parameters. Examples include the Weibull model, the exponential model, and the log-normal model. 25

Parametric versus Nonparametric Models Properties of nonparametric models are • the distribution of survival time is unknown • the hazard function is unspecified. An example is the Cox proportional hazards model. 26

. . . Cox Proportional Hazards Model Baseline Hazard function - involves time but not predictor variables Linear function of a set of predictor variables - does not involve time β = 0 → hazard ratio = 1 Two groups have the same survival experience 27

Popularity of the Cox Model The Cox proportional hazards model • provides the primary information desired from a survival analysis, hazard ratios and adjusted survival curves, with a minimum number of assumptions • is a robust model where the regression coefficients closely approximate the results from the correct parametric model. 28

Partial Likelihood Partial likelihood differs from maximum likelihood because • it does not use the likelihoods for all subjects • it only considers likelihoods for subjects that experience the event • it considers subjects as part of the risk set until they are censored. 29

Partial Likelihood Subject Survival Time Status C 2. 0 1 B 3. 0 1 A 4. 0 0 D E 5. 0 6. 0 1 0 30

Partial Likelihood 31

Partial Likelihood 32

Partial Likelihood �The overall likelihood is the product of the individual likelihood. That is: 33

7. SAS Programs for Survival Analysis � There are three SAS procedures for analyzing survival data: LIFETEST, PHREG, and LIFEREG. � PROC LIFETEST is a nonparametric procedure for estimating the survivor function, comparing the underlying survival curves of two or more samples, and testing the association of survival time with other variables. � PROC PHREG is a semiparametric procedure that fits the Cox proportional hazards model and its extensions. � PROC LIFEREG is a parametric regression procedure for modeling the distribution of survival time with a set of concomitant variables. 34

Proc LIFETEST �The Kaplan-Meier(K-M) survival curves and related tests (Log-Rank, Wilcoxon) can be generated using SAS PROC LIFETEST DATA=SAS-data-set <options>; TIME variable <*censor(list)>; STRATA variable <(list)> <. . . variable <(list)>>; TEST variables; 35

Proc PHREG �The Cox (proportional hazards) regression is performed using SAS PROC PHREG proc phreg data=rsmodel. colon; model surv_mm*status(0, 2, 4) = sex yydx / risklimits; run; 36

Proc LIFEREG �The accelerated failure time regression is performed using SAS PROC LIFEREG proc lifereg data=subset outest=OUTEST(keep=_scale_); model (lower, hours) = yrs_ed yrs_exp / d=normal; output out=OUT xbeta=Xbeta; run; 37

Selected References PD Allison (1995). Survival Analysis Using SAS: A Practical Guide. SAS Publishing. � JD Kalbfleisch and RL Prentice (2002). The Statistical Analysis of Failure Time Data. Wiley. Interscience. � 38

�Questions? 39