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Stat 4100/5100 – Survival Analysis – Fall 2015

W11: Additional Notes

Text References and Additional Reading: x8.5-8.8,9.1-9.2

Assignment 8 (Due Wed, Nov 25):

1. Consider a censored sample from the exponential distribution with rate parameter and xed cen-

soring time c, common to all individuals that were in the study. That is, individual i is censored if

failure time, Xi > c. Otherwise Xi is observed.

(a) Suppose that only the number of failures, D, is observed. Show that D has a binomial distribu-

tion with parameters n and p = 1 exp[c]. Using the relationship between the variance of

maximum likelihood estimators and expected information obtain an expression for the variance

of the maximum likelihood estimate of . This should be a true variance not a standard error;

i.e. an expression that depends upon the parameters of the model, not data.

(b) Suppose now the usual case: (

P

i ti;D) are observed, where the ti are the failure and censoring

times. You showed on Assignment 4 that l00() = D=2 and ^

= D=

P

i ti. Using the relation-

ship between the variance of maximum likelihood estimators and expected information obtain

an expression for the variance of the maximum likelihood estimate of . Again, this should

depend on the parameters of the model, not data.

(c) The eciency of an estimator ^

1 to another estimator ^

2 is the ratio of the large sample variances

of the two estimators: Var(^

2)=Var(^

1). Obtain an expression for the eciency of the estimator

in (a) to the estimator in (b). Note that the eciency depends upon and c. In R, produce a

plot of the eciency as a function of for c = 1. For what values of does the method in (a)

give almost as good results as method (b), if any?

2. 8.6. The data is available on the web site. Be sure to use factor to ensure variables that are

supposed to have interpretations as categorical variables do have such interpretations. In (a), by

ANOVA table, the question means the output of summary in R. In (c) you can answer the question

by nding a 95% condence interval for the relevant relative risk and checking whether 1 falls in the

interval.

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