STAT 432/532 - Survival Analysis Final
STAT 432/532 - Survival Analysis
Lingzhu Li
Intructions: The questions are equally weighted. If you have multiple PDFs or
photos, try to combine them into one document. When you answer the questions, please
present enough evidence. If you use R, please also include the R code in .r or .rmd,
and highlight/indicate your answers clearly in your handwritten copy or the
PDF generated by R markdown. Good Luck!
1. (a) What is the difference between Type I censoring and the Type II censoring.
(b) Suppose X is a continuous positive random variable. Its survival function,
hazard function are denoted by S(x) and b(x).
i. Show that E(X) =
∫∞
0
S(x)dx.
ii. Obtain b(x) in term of the mean residual life.
2. Consider a small study with 8 subjects. The event times were recorded as follows:
1, 4, 2+, 4+, 2, 6+, 7+, 4
Here “+” denotes the censoring times.
(a) Calculate the Kaplan-Meier estimate for the survival function S(t), and draw
a plot of the estimated Sˆ(t).
(b) Derive a 100(1 − α)% confidence interval for Sˆ(2.5) using the delta method
with a logit transformation of survival function. A logit function is defined as
logit(x) = log( x
1−x).
(c) Suppose you want to test whether these 8 subjects are from a population with
hazard function
h0(t) =
{
1, 0 < t ≤ 3;
2, t > 3.
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STAT 432/532 - Survival Analysis Final
How to do this test? What is your conclusion? [ Please state clearly the null
hypothesis, the alternatives, and the test statistics you use. ]
3. To explore the efficacy of triple-drug combinations of antiretroviral therapy for
treatment of HIV-infected patients, investigators performed a randomized study
comparing AZT + zalcitabine (ddC) versus AZT+ zalcitabine (ddC)+ saquinavir.
The data, time from administration of treatment (in days) until the CD4 count
reached a prespecified level, is given below for the two groups.
AZT + zalcitabine (ddC): 85, 32, 38+, 45, 4+, 84, 49, 180+, 87, 75, 102, 39, 12, 11, 80, 35, 6
AZT + zalcitabine (ddC) + saquinavir: 22, 2, 48, 85, 160, 238, 56+, 94+, 51+, 12, 171
80, 180, 4, 90, 180+, 3
(a) Use the log rank statistic to test if there is a difference in the distribution of
the times at which patient’s CD4 reaches the prespecified level for the two
treatments.
(b) Build a Cox proportional model, then test again. Compare the result with
(3a).
4. Suppose we are interested in relationship between the disease-free survival func-
tions of 137 patients given a bone marrow transplant. Three groups were con-
sidered: Group 1 consisting of 38 ALL patients; Group 2 consisting of 54 AML
low-risk patients and Group 3 consisting of 45 AML high-risk patients. We want
to test the hypothesis that the disease-free survival functions of these three pop-
ulations are the same over the range of observation, τ < 2204 days, versus the
alternative that at least one of the populations has a different survival rate.
(a) Denote Zj(τ) =
∑D
i=1W (ti)
{
dij − Yij diYi
}
, j = 1, 2, . . . , K. If one wants to
test if there is any differences in the survival rates for the long term, which
weight function will you suggest?
(b) What test statistic you can use for this test? Write down the formula and
calculate its value.
(c) If you choose 0.05 as the accepted type I error for this test, will you reject
the null hypothesis? Interpret the result.
5. Suppose Z1, . . . , Zp are some factors in a data set and you are now interested in
building a Cox model based on the data.
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STAT 432/532 - Survival Analysis Final
(a) If you do not have any preference for some factors at the beginning, state
your steps and criteria in detail. [Do not copy from the slides. Use your own
words.]
(b) Is it possible to build a model using other methods? Would the models
selected by different methods be the same?
6. A study was performed to determine the effiacy of boron neutron capture therapy
(BNCT) in treating the therapeutically refractory F98 glioma, using boronopheny-
lalanine (BPA) as the capture agent. F98 glioma cells were implanted into the
brains of rats. Three groups of rats were studied. One group went untreated,
another was treated only with radiation, and the third group received radiation
plus an appropriate concentration of BPA.
You can find the data by “library(KMsurv);data(bnct)” in R.
(a) Using the group factor, fit a Cox proportional model. What method is used
to estimate the parameters in the model? What is the reference group in your
answer?
(b) Find an estimate and a 95% confidence interval for the relative risk of death
for a radiation plus BPA animal as compared to a radiated only animal.
(c) Is delta-method needed in (6b)? Why?
(d) Using an appropriate set of time-dependent covariates, test that the hazard
rates of the three groups are proportional.
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