STT520-420: BIOSTATISTICS ANALYSIS

Dr. Cuixian Chen

Chapter 7: Parametric Survival Models under Censoring

STT520-420 1

Recall Chapter 5 and 6: Estimating the survival function with right censoring

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To estimate the survival function in the presence of right censoring:Method 1: Cohort Lifetable method (Recall advantage/ disadvantage).Method 2: Product-limit estimator (PLE), (also call Kaplan-Meier estimator (1958)). Both are univariate Non-parametric/distribution-free estimate of survivor function.Lifetable method uses fixed endpoints from intervals, while PLE used observed data points.

Consider survival r.v. Y follows certain survival model with parameters.

It is called parametric survival models under censoring.

The aim is to estimate these parameters by maximizing the (log) likelihoods.

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Parametric Survival Models (ch. 7)

Parametric Survival Models

notations… Y = survival r.v.; S=survival function,

f=density (pdf); F=cdf=1-S. All these functions involve the unknown parameter (or vector of parameters) theta (.

The 2 examples we’ll consider are: exponential with

Weibull with

S(y;) exp( y /), so E(Y) MTTF

S(y;) exp( (y /) ), so (,)T

The maximum likelihood estimator of

is a function of the observed Y’s which maximizes the likelihood:

Def 7.1 : L(a constant multiple of the joint distribution of the observed data. So theta-hat maximizes L over all possible values of theta…

The mathematical method is discussed at the top of page 120 - note that usually the log of the likelihood is maximized…

Example: Find the log-likelihood function for Y ~ exp( .

is ˆ

Parametric Survival Models

Important results given in this section: define the score function as the derivative

of the log-likelihood define Fisher’s information as then

at the bottom of p. 120 and top of p.121, the method of actually finding maximum likelihood estimators is discussed and it is shown that

U() l'()

I() E[ l' '()]

" "

)),(,0( )(

asddistributeallyasymptoticmeanswhere

INU

ˆ N(, 1/I() )

Parametric Survival Models

Example: For Exponential model With all exact observations, assume Y ~

exp( . (1) Find the log-likelihood function. (2) Find the maximum likelihood estimate. (3) Find the Fisher information. (4) Find the 95% Confidence Interval for (5) Find the asymptotic distribution of MLE

of

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U() l'()

I() E[ l' '()]

ˆ N(, 1/I() )

now go to page 131… Maximum likelihood under Type I censoring .

Suppose we have a survival variable Y ~exp(), subject to Type I censoring.

Beta is the MTTF and we may use maximum likelihood methods to estimate it…(p.131 and 132)

The Exponential Model

Maximizing the likelihood equation gives the MLE of as

so we may use this to construct a 95% CI for the MTTF

ˆ total of observed data (lifetimes)

total # of uncensored data values

valuesuncensoredtotal

valuesdatauncensoredtotalsqrtofesestimatedThe

whereIN

#

ˆ

ˆ #

/1 ˆ ..

)),(/1,(ˆ

2

The Exponential Model

Example: For Exponential model With RC observations, assume Y ~ exp( . (1) Find the log-likelihood function. (2) Find the maximum likelihood estimate. (3) Find the Fisher information. (4) Find the 95% Confidence Interval for (5) Find the asymptotic distribution of MLE

of

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ˆ total of observed data (lifetimes)

total # of uncensored data values

valuesuncensoredtotal

valuesdatauncensoredtotalsqrtofesestimatedThe

whereIN

#

ˆ

ˆ #

/1 ˆ ..

)),(/1,(ˆ

2

Assume the survival r.v. Y follows Exp(). The observed data is given as following:

Flight control package failure time (mins): (n=14). Observed: 1, 8, 10 Alive: 59, 72, 76, 113, 117, 124, 145, 149, 153, 182,

320.

Q: (a) Determine the maximum likelihood estimate of (b) Determine the standard error of the estimate of (c) Determine a 95% confidence interval of

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Example 7.3, page 132

Then hypothesis tests about can be based on any of the 3 statistics below.

To test use one of :

Wald test:

LR test:

Score test:

H0 : 02

2 200 1

ˆˆ ˆ( ) ( )

ˆ( )aI

SE

201

( )2 log

ˆ( )e a

L

L

2201

0

( )

( ) a

U

I

Parametric Survival Models

~

~

~

Recall: Weibull Prob Plots (chap4) Recall Power hazard model:

Substitute , then take logarithm:

Take logarithm of base 10 again to create log-life variable:

We now have:

Note: base-10 log are traditionally used in lifetime, but natural log is equivalent with a difference of scale.

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)()(log )(

)(j

je

yyS

))(exp()(

y

yS

]log[log)](log[log 10)(10)(10 jje yyS

)( jyy

)](log[log1

loglog )(1010)(10 jej ySy

Recall: Weibull Prob Plots

If data fit Weibull model:

Weibull Prob plot is:

It follows a straight line with slope and intercept .

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)]5.0

1(log[log1

loglog 1010)(10 n

jy ej

njyn

jje ,...,2,1),log),

5.01(log[(log )(1010

/1 10log

ˆ b 1/ ˆ ; ˆ u loge ( ˆ )

Weibull model: assume Y ~ Weibull( subject to Type I censoring. More generally, take log of survival data, X=loge(Y).

X=loge(Y) follows Extremevalue(u, b):

The survival function is given by

The original Weibull parameters can be estimated if u and b are estimated by u-hat and b-hat:

f (x) 1

bexp(

x u

b)exp( exp(

x u

b)),

x , u , b 0.

S(x) exp( exp(x u

b))

ˆ b 1/ ˆ ; ˆ u loge ( ˆ )

The Weibull Model

Use SAS to read in the switch failure time data (see website). Then get estimates of the Weibull parameters for both the log-transformed and non-transformed data - use PROC LIFEREG:

Notice that we may use the NOLOG option in the MODEL statement to not take logs of the data…

proc lifereg data=switch;

model u*censor(0)= /nolog dist=weibull;

title 'Modeling u=log(Y) w/ NOLOG option';

proc lifereg data=switch;

model y*censor(0)= /dist=weibull;

title 'Modeling non-transformed Y'; run; quit;

Return to Section 4.4 (p. 61-62) and use SAS to get a probability plot (formula 4.4) of this data…

ˆ b 1/ ˆ ; ˆ u loge ( ˆ )

Example 7.4, page 135

Modeling u=log(Y) w/ NOLOG option