R统计代写-STAT 432/532
时间:2021-03-16
STAT 432/532 - Survival Analysis Midterm
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) Explain right and left censored observations in survival data.
(b) State the difference between cencoring and truncation, illustrate it by prac-
tical example.
(c) Describe the interval censoring and illustrate it by a practical example.
2. (a) The mean residual life is
mrl(t) = E(T − t|T > t).
Obtain S(t) in term of mrl(t).
(b) Derive the likelihood function of Type II censoring data.
3. To estimate the distribution of the ages at which postmenopausal woman develop
breast cancer, a sample of eight 50-year-old women were given yearly mammograms
for a period of 10 years. At each exam, the presence or absence of a tumor was
recorded. In the study, no tumors were detected by the women by self-examination
between the scheduled yearly exams, so all that is known about the onset time of
breast cancer is that it occurs between examinations. For four of the eight women,
breast cancer was not detected during the 10 year study period. The age at onset
of breast cancer for the eight subjects was in the following intervals:
(55, 56], (58, 59], (52, 53], (59, 60],≥ 06,≥ 06,≥ 06,≥ 06.
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STAT 432/532 - Survival Analysis Midterm
(a) What type of censoring or truncation is represented in this sample?
(b) Assuming that the age at which breast cancer develops follows a Weibull
distribution with parameters λ and θ, construct the likelihood function.
4. A group of individuals with a sexually transmitted disease (STD), all initially HIV-
free, was monitored for 12 years until HIV infection diagnosis. The numbers of
those either diagnosed with HIV or censored are presented in the table below, where
the term ‘censored’ refers to individuals who withdrew from the study prematurely
for unrelated reasons, or who did not become HIV-infected during the course of
the study.
(a) How many individuals were involved in the study?
(b) Treating times to be coming HIV-infected as ‘survival’ times, estimate the
survival function S(t) and sketch its plot.
(c) Estimate a 10-year survival rate.
(d) Construct a 95% confidence interval for your answer in 4c.
5. (a) Write down Nelson–Aalen estimator H˜(t) of the cumulative hazard rate func-
tion H(t).
(b) Derive an approximate expression for the variance of S˜(t) estimated by H˜(t).
[Hint: Delta method.]
6. In a study (Nahman et al., 1992) designed to assess the time to first exitsite infec-
tion (in months) in patients with renal insufficiency, 43 patients utilized a surgically
placed catheter (Group 1), and 76 patients utilized a percutaneous placement of
their catheter (Group 2). Cutaneous exitsite infection was defined as a painful
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STAT 432/532 - Survival Analysis Midterm
cutaneous exit site and positive cultures, or peritonitis, defined as a presence of
clinical symptoms, elevated peritoneal dialytic fluid, elevated white blood cell count
(100 white blood cells 1.55 µl with > 50% neutrophils), and positive peritoneal di-
alytic fluid cultures. The data is in the following table. Hint: You can import the
data in R by library(“KMsurv”);data(kidney).
(a) For each group plot the estimated survival function. Which technique seems
better in delaying the time to infection?
(b) Estimate the cumulative hazard rate for each group of patients. Provide a
crude estimate of the hazard rate at 5 months after placement of the catheter
in each group.
(c) Find a 95% confidence interval for the mean time to first exit site infection
restricted to 36 months for both groups.
7. In the bone marrow transplant study, we have three groups: ALL patients, AML
high-risk and AML low-risk patients. The data can be imported by library(“KMsurv”);
data(bmt). Because acute graft-versus-host (aGVHD) disease is considered to have
an anti-leukemic effect, one would expect lower relapse rates for patients who have
developed aGVHD than for those that do not develop aGVHD. Examine the va-
lidity of this finding by
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STAT 432/532 - Survival Analysis Midterm
(a) testing if the hazard rate for the occurrence of aGVHD is the same for the
three groups,
(b) testing if the hazard rate for relapse is the same in all three groups, and
(c) testing if the hazard rate for relapse in the three disease groups is the same
for patients who have developed aGVHD. Hint: For this test, the data is
left-truncated at the time of aGVHD.
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