Transcript Comparing Survival Functions
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:
t
exp
t
0 3
Hazard Function/Rate
Hazard Function (t): instantaneous failure rate at time t given that the subject has survived upto time t. That is lim
t
|
T
t
)
t t
) lim ) 1 Here f(t) is the probability density function of the survival time T. That is,
d
where F(t) is the cumulative distribution function of T:
t
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
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
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
S
ˆ (
t
)
r
1
d
1
r
1
r
2
d
2
r
2 ...
r i
d i r i
r k
is the number of subjects alive just before time
t k d k
denotes the number who died at time
t k
7
Kaplan-Meier Curve, Example
0 Time t i 5 6 10 13 # at risk 20 20 18 15 14 2 0 1 2 0 # events
S
ˆ 1.00 [1-(2/20)]*1.00=0.90
[1-(0/18)]*0.90=0.90
[1-(1/15)]*0.90=0.84
(1-(2/14)]*0.84=0.72
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Kaplan Meier Curve
0 5 10 Survival Time 15 20
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.)
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3B. Comparing Survival Functions
1.00
0.75
Survival Distribution Function 0.50
0.25
0.00
0 High Low 10 20 30 Time 40 Medium 50 60 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.
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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.
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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.
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4. Two Parametric Distributions
Here we present two most notable models for the distribution of T.
Exponential distribution:
(
t
)
Weibull distribution:
(
t
)
p
(
t
)
p
1
p
p
◦ ◦ Its survival function: Thus:
S
(
t
) ln exp ln(
S
(
t
0
t
p
p u p
1
du
) ln(
t
t p
1 ) exp ln( ) (
t
)
p
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Weibull Hazard Function, Plot
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5. Regression Models
1.
2.
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 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: 0
c
( 0 1
x i
1 2
x i
2 ...
k x ik
) 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:
c
(.) exp(.) The resulting proportional hazard model is: 0 exp( 0 1
x i
1 2
x i
2 ...
k x ik
) For the Weibull distribution we have:
p
0
p
t p
1 exp( 0 1
x i
1 2
x i
2 ...
k x ik
) For the Exponential distribution we have: 0 exp( 0 1
x i
1 2
x i
2 ...
k x ik
) 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: ln 0 1
x i
1 2
x i
2
k x ik
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
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An Example
In a recent paper (Reisberg et al., 2010), we applied both regression models to a dementia study conducted at NYU: 0
T
1
Group
2 *
Age
3 *
Gender
4 *
Education
5 *
FollowUp
) log
T
0 1 *
Group
2 *
Age
3 *
Gender
4 *
Education
5 *
FollowUp
The results are shown next 22
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6. Cox Proportional Hazards Model
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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.
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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
i
h t e
0 ( ) { 1
X i
1
k X ik
}
...
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.
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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.
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Partial Likelihood
Subject C B A D E Survival Time 2.0
3.0
4.0
5.0
6.0
Status 1 1 0 1 0 30
Partial Likelihood
L c
h c
( 2 )
h b
( 2 )
h h c
( 2 )
a
( 2 )
h d
( 2 )
h e
( 2 )
L b
h b
( 3 )
h a
( 3
h b
) ( 3 )
h d
( 3 )
h e
( 3 )
L d
h d
( 5
h d
) ( 5 )
h e
( 5 ) 31
Partial Likelihood
L d
h d
( 5
h d
) ( 5 )
h e
( 5 )
L d
h o
( 5 )
e
1
X d
1 2
h o X d
2 ( 5 )
e
....
1
X
k d
1
X d k
2
X d h o
2 ....
k X d k
( 5 )
e
1
X e
1 2
X e
2 ....
k X ek L d
e
1
X d
1 2
X d
2
e
1
X d
1 ....
k
2
X d
2
X d k
....
k X d k e
1
X e
1 2
X e
2 ....
k X ek
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Partial Likelihood
The overall likelihood is the product of the individual likelihood. That is:
L
L c
*
L b
*
L d
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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
PROC LIFETEST DATA=SAS-data-set
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;
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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.
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Questions?
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