Homework 6 ISyE 6420 solution


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1. Rinderpest Virus in Rabbits with Missing Data.

Temperatures (temp) were
recorded in a rabbit at various times (time) after the rabbit was inoculated with rinderpest
virus (the data modified from Carter and Mitchell, 1958). Rinderpest (RP) is an infectious
viral disease of cattle, domestic buffalo, and some species of wildlife; it is commonly referred
to as cattle plague.

It is characterized by fever, oral erosions, diarrhea, lymphoid necrosis,
and high mortality.

Time after injection Temperature
(time in hrs) (temp in ◦ F)
24 102.8
32 104.5
48 106.5
56 107.0
NA 107.1
70 105.1
72 103.9
75 NA
80 103.2
96 102.1

(a) Using WinBUGS and properly accounting for the missing data demonstrate that
a linear regression with one predictor (time) gives relatively low Bayesian R2
. What are
estimators of the missing data? Does the 95% Credible Set for the slope contain 0? Comment.

(b) Include time2 (squared time) as the second predictor, making the regression quadratic
in variables, but still linear in coefficients. Show that this regression has a larger Bayesian
. What are the estimators of missing data? Do the 95% Credible Sets for parameters in
the quadratic model contain 0? Comment.

Hint: To have cleaner codes, do the modeling in (a) and (b) in two separate WinBUGS

2. Bladder Cancer Data.

An exercise in the book Pagano and Gauvreau (2000) 1
features data on 86 patients who after surgery were assigned to placebo or chemotherapy

Endpoint was the time to cancer recurrence (in months).
Variables are: time, group (0 – placebo, 1- chemotherapy), and observed (0 – recurrence
not observed, 1 – recurrence observed). This data is given in files bladerc.csv|dat|xlsx.

Data are given in WinBUGS format bladderBUGS.csv|dat|xlsx. The starter file bladderc0.odc
contains data and also initial values for parameters and censored observations.

Assume that observed times are exponentially distributed with the rate parameter λi
depending on the covariate group, as
λi = exp{β0 + β1 × groupi}

After β0 and β1 are estimated, since the variable group takes values 0 or 1, the means for
the placebo and treatment times become
µ0 =
= exp{−β0}
µ1 =
exp{β0 + β1}
= exp{−β0 − β1},

The censored data are modeled as exponentials left truncated by the censoring
time. Use noninformative priors on β0 anmd β1.
(a) Is the 90% Credible Set for µ1 − µ0 all positive?

(b) What is the posterior probability of hypothesis H : µ1 > µ0?

(c) Comment on the benefits of the treatment (a paragraph).

1Bladder cancer data from M Pagano and K Gauvreau, ”Principles of Biostatistics, 2nd Ed. Duxbury
2000. Chapter 21, Exercise 9, page 512.