BIOS 924 Biostatistical Theory and Models for Survival Analysis
Homework 2
Due by 11:59 PM on Wednesday, 03/15/2020
Student Name: ____________________________________________
Total points: ______(out of 105)
N.B. (i) The values (in points) are shown for each problem, and for some individual parts. It is
difficult for this HW to provide the value of each portion of the individual parts, but the rule for
grading will be that the more essential work you show, the more credit you will get.
(ii) Please answer each question by adding text to this MS Word file, immediately after the
respective question, using BLUE INK. Please don’t use red ink or black ink, with one exception -
the R/SAS code and output could be in BLACK INK, but all explanations should be in BLUE INK.
(iii) For each problem (when relevant), include ALL of your R (and, if applicable, SAS) code, but
only the relevant parts of the R (and/or SAS) output.
(iv) When saving this file (before submitting it on Canvas), include your LAST_NAME, HW#, and
DATE in the file name (for us not to confuse your work with someone else’s).
Example: DOE-HW#2-031520
(v) You could use LaTeX (rather than MS Word) as your text editor, but you should provide your
original LaTeX file, in addition to a PDF.
1. [15 pts] Suppose that a continuous, positive random variable T has a hazard function defined by
h(t) = hj when aj-1 ≤ t < aj, j = 1, 2, …, k,
where a0 = 0 and ak = ∞, a0 < a1 < … < ak and hj > 0, j = 1, 2, …, k.
(a) Express S(t), the survivor function of T, in terms of the hj’s.
(b) Express f(t), the probability density function of T, in terms of the hj’s.
(c) If you observe a random sample of size n of the form (t1, δ1), …, (tn, δn), where δi = 1 if ti is a
failure time, and 0 if ti is a right-censoring time, derive the maximum likelihood estimates of the
hj’s and, hence, the maximum likelihood estimate of S(t).
(d) What happens to the maximum likelihood estimate of S(t) as k becomes “large” and the distances
aj – aj-1 get “small”?
SOLUTION:
2. [15 pts] (a) Formulate the Cochran-Mantel-Haenszel (CMH) test.
(b) The log-rank test (sometimes called Cox-Mantel test) for comparison of two survivor curves is
obtained by constructing a 2x2 table at each distinct death time, and comparing the death rates
