ISYE6420 Homework 6 Solution

ISyE 6420
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 an MCMC modeling library such as BUGS or PyMC and properly accountingfor 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 on how you chose your priors and the results of your model.
(b) Include time2 (squared time) as the second predictor, making the regression quadratic in variables, but still linear in coefficients. What are the estimators of missing data? Do the 95% Credible Sets for parameters in the quadratic model contain 0? Comment on how you chose your priors and the results of your model. Is the Bayesian R better or worse than the model from 1(a)?
Hint: If using BUGS, it is recommended to do the modeling in (a) and (b) in two separate BUGS programs. The linear regression is not good for part (a). Given the missing data, the estimator for σ2 is large, and SSE = (n − p) · σ2 is bigger than SST, making the R2 negative. For the single predictor model, please consider BR2=max(0, 1-SSE/SST) in your BUGS code. The quadratic regression is fine without any modification of BR2.
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. Students should understand that these formats are equivalent, and be able to convert one into the other as needed.
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
,
respectively. 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).
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