STT : BIOSTATISTICS ANALYSIS Dr. Cuixian Chen Chapter 7: Parametric Survival Models under Censoring STT

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

 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 note that usually the log of the likelihood is maximized…  Example: Find the log-likelihood function for Y ~ exp( . 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 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  STT

 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  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  STT

 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  STT 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: 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. STT

Recall: Weibull Prob Plots  If data fit Weibull model:  Weibull Prob plot is:   It follows a straight line with slope and intercept. STT

 Weibull model: assume Y ~ Weibull(  subject to Type I censoring. More generally, take log of survival data, X=log e (Y).  X=log e (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: 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 ) and use SAS to get a probability plot (formula 4.4) of this data… Example 7.4, page 135 Modeling u=log(Y) w/ NOLOG option