STAT 432/532 Assignment 3 Due date: April 7, 2021 by 23:59
STAT 432/532 Assignment 3
Due Date: April 7, 2021 by 23:59
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 evi-
dence. 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.
1. In section 1.10, times to death or relapse (in days) are given for 23 nonHodgkins lymphoma
(NHL) patients, 11 receiving an allogenic (Allo) transplant from an HLA- matched sibling
donor and 12 patients receiving an autologous (Auto) transplant. Also, data on 20 Hodgkins
lymphoma (HOD) patients, 5 receiving an allogenic (Allo) transplant from an HLAmatched
sibling donor and 15 patients receiving an autologous (Auto) transplant are given.
(a) Treating NHL Allo as the baseline hazard function, state the appropriate coding which
would allow the investigator to test for any difference in survival functions for the four
groups, treating them as four independent groups.
(b) Treating NHL Allo as the baseline hazard function, state the appropriate coding which
would allow the investigator to test for an interaction between type of transplant and
disease type using main effects and interaction terms.
(c) Suppose that we have the following model for the hazard rates in the four groups:
h(t|NHL Allo) = h0(t)
h(t|HOD Allo) = h0(t) exp(2)
h(t|NHL Auto) = h0(t) exp(1.5)
h(t|HOD Auto) = h0(t) exp(.5)
What are the risk coeffcients, βi, i = 1, 2, 3; for the interaction model in part (b)?
2. Consider Cox proportional hazards model.
(a) What is the partial likelihoods without ties?
(b) (Breslow Method) Derive the partial likelihoods with ties in the data set according to
part (a).
3. In section 1.11, a study was conducted on the effects of ploidy on the prognosis of pa-
tients with cancer of the tongue. Tissue samples were examined to determine if the tu-
mor had a aneuploid or diploid DNA profile. Times to death for these two groups of
patients are recorded in the following table. To analyze this data create a single indi-
cator variable, Z, which reflects the type of tumor. You can import the data by “li-
brary(KMsurv);data(tongue)” in R.
Winter 2021 p.1/2
STAT 432/532 Assignment 3 Due date: April 7, 2021 by 23:59
(a) Find the p−value of a test of the hypothesis of no effect of ploidy on survival using
the score test and the Breslow method of handling ties.
(b) Estimate β – coefficient in Cox model – and its standard error using the Breslow
method of handling ties. Find a 95% confidence interval for the relative risk of death
of an individual with an aneuploid tumor as compared to an individual with a diploid
tumor.
(c) Repeat (a) using the likelihood test. Compare your answer to that of part (a).
(d) Repeat (a) using the Wald test. Compare your answer to those in parts (a) and (c).
4. Build a model for the rate of staphylococcus infection, assuming that you know the fol-
lowing three factors: treatment (routine bathing v.s. body cleansing initiated with 4%
chlorhexidine gluconate), the type of burn (chemical, scald, electric, flame) and gender.
The data can be found by “library(KMsurv);data(burn).”
(a) Use Wald test.
(b) Use likelihood ratio test.