Exercises February 23
STA 4508S (Spring, 2016)
1. The data set heart in the R library MASS has the Stanford hearttransplant data, a famous data set from the early 70s, that also appears in the survival data text by Kalbfleisch & Prentice. The treatment variable of interest is “transplant” (yes or no), and it is a time-dependent covariate, because patients have to live long enough for a suitable donor to be found. The R library survival fits a proportional hazards model with time-dependent or time-independent covariates. In From Modern Applied Statistics with S, Ch. 13.4, this library is used to develop models for the heart transplant data. The exercise is to reproduce the analysis on pp 377 and 378, including the figures. The text is available on-line through the library, and the relevant two pages are posted on the course web site. 2. Suppose in a proportional hazards regression that we have a single covariate xi that takes the value 1 or 0, depending as unit i is in the treatment group, or the control group, respectively. (a) Show that `0part (β) simplifies to n X
di {xi −
i=1
m1i eβ }, m1i eβ + m0i
where m1i is the number of treatment group units available to fail at time yi , and m0i is the number of control group units available to fail at time yi . (b) Find an expression for j(β) = −`00part (β), and show that at β = 0 this simplifies to j(0) =
n X i=1
di
m0i m1i . (m0i + m1i )2
Reference: Statistical Models, §10.8
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