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% condence interval for the relevant relative risk and checking whether 1 falls in the
interval.

 

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