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Survival Analysis. A Brief Introduction. 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)

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survival analysis

Survival Analysis

A Brief Introduction

1 survival function hazard function
1. SurvivalFunction, 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:
hazard function rate
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:


2 the key word is censoring
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 account for censoring


key assumption independent 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
3a kaplan meier product limit estimator of the survival curve
3A. 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

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.)

3b comparing survival functions
3B. Comparing Survival Functions




Survival Distribution Function














log rank test
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.
wilcoxon test
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.
log rank versus wilcoxon test
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.
4 two parametric distributions
4. Two Parametric Distributions
  • Here we present two most notable models for the distribution of T.
  • Exponential distribution:
  • Weibull distribution:
    • Its survival function:
    • Thus:
5 regression models
5. Regression Models
  • The Exponential and the Weibull distribution inspired two parametric regression approaches:
  • Parametric proportional hazard model – this model can be generalized to a semi-parametric model: the Cox proportional hazard model
  • Accelerated failure time model
proportional hazard model
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.
proportional hazard model19
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:
accelerated failure time model
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
apply both models simultaneously
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
an example
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
parametric versus nonparametric models
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.
parametric versus nonparametric models26
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.
cox proportional hazards model


Linear function of a set of predictor variables - does not involve time

Baseline Hazard function - involves time but not predictor variables

Cox Proportional Hazards Model
  • β = 0 → hazard ratio = 1
    • Two groups have the same survival experience
popularity of the cox model
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.
partial likelihood
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.
partial likelihood30
Partial Likelihood


Survival Time

















partial likelihood33
Partial Likelihood
  • The overall likelihood is the product of the individual likelihood. That is:
7 sas programs for survival analysis
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.
proc lifetest
  • The Kaplan-Meier(K-M) survival curves and related tests (Log-Rank, Wilcoxon) can be generated using SAS PROC LIFETEST

PROC LIFETEST DATA=SAS-data-set <options>;

TIME variable <*censor(list)>;

STRATA variable<(list)> <...variable <(list)>>;

TEST variables;


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;


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;

selected references
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.