# Homework 5 ISyE 6420 solution

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Original Work ?

## 1. Blood Volume in Infants.

The total blood volume of normal newborn infants was
estimated by Sch¨ucking (1879) who took into account the addition of placental blood to the
circulation of the newborn infant when clamping of the umbilical cord is delayed. Demarsh
et al. (1942) further studied the importance of early and late clamping.

For 16 babies in
whom the cord was clamped early the total blood, as a percentage of weight, on the third
day is listed below:
13.8 8.0 8.4 8.8 9.6 9.8 8.2 8.0
10.3 8.5 11.5 8.2 8.9 9.4 10.3 12.6

For 16 babies in whom the cord was not clamped until the placenta began to descend,
the corresponding figures are listed below:
10.4 13.1 11.4 9.0 11.9 16.2 14.0 8.2
13.0 8.8 14.9 12.2 11.2 13.9 13.4 11.9

Using WinBUGS, find the posterior distribution for difference in mean blood percentages
for the two procedures. Assume gamma likelihoods with different shape/rate parameters for
the two procedures. For both the shape and rate parameters use noninformative gamma
priors, say Ga(0.001, 0.001).

Does the 95% Credible Set for the difference of means contain 0? Comment.
Hint: Starter file is babies0.odc. From the posterior simulations of parameters in gamma
likelihoods you will need to calculate the means for the two groups as well as their difference.

## 2. Can Skull Variations in Canis lupus L.

Predict Habitat? Data set described
below provides skull morphometric measurements on wolves (Canis lupus L.) coming from
two geographic locations: Rocky Mountain (0) and Arctic (1). Original source of data
is Jolicoeur (1959) 1
, but subsequently many authors used this data to illustrate various
multivariate statistical procedures.

The goal of Jolicoeur’s study was to determine how the location and gender affect the
skull shape among the wolf populations.

There were 9 predictor variables:

x1 = palatal length;
x2 = postpalatal length;
x3 = zygomatic width;
x4 = palatal width outside the first upper molars;
x5 = palatal width inside the second upper molars;
x6 = width between the postglenoid foramina;
x7 = interorbital width;
x8 = least width of the braincase;
x9 = crown length of the first upper molar.

These 9 measurements (Columns 3-11) and gender (Column 2: male=0, female=1) are
associated to one of the two locations (Column 1: Rocky Mountain = 0, Arctic = 1)
0 0 4.96 4.09 5.55 3.19 1.25 2.59 2.00 1.73 0.72
0 0 5.04 4.37 5.94 3.17 1.33 2.75 2.07 1.70 0.73
0 0 4.96 4.25 5.98 3.37 1.37 2.72 1.94 1.80 0.70
0 0 4.92 4.29 5.55 3.27 1.34 2.68 1.90 1.72 0.72
0 0 4.96 4.21 5.63 3.22 1.34 2.60 1.93 1.67 0.70
0 0 5.04 4.33 5.63 3.17 1.30 2.56 1.83 1.58 0.72
0 1 4.57 4.02 5.16 3.02 1.24 2.56 1.79 1.54 0.66
0 1 4.72 4.06 5.12 2.96 1.19 2.51 1.75 1.62 0.67
0 1 4.57 4.06 4.92 2.94 1.24 2.46 1.63 1.74 0.67
1 0 4.61 3.90 5.28 3.28 1.37 2.68 1.60 1.46 0.68
1 0 4.53 3.94 5.87 3.19 1.30 2.63 1.86 1.59 0.70
1 0 4.61 4.17 5.59 3.23 1.28 2.60 1.77 1.50 0.72
1 0 4.61 3.98 5.67 3.24 1.29 2.66 1.78 1.63 0.75
1 0 4.61 4.06 5.87 3.26 1.38 2.77 1.90 1.72 0.70
1 0 4.69 3.98 5.63 3.21 1.34 2.72 1.97 1.62 0.74
1 0 4.53 4.02 5.75 3.20 1.33 2.61 1.88 1.65 0.72
1 0 4.61 3.94 5.67 3.20 1.46 2.63 1.63 1.48 0.70
1 0 4.49 4.02 5.55 3.31 1.25 2.67 1.88 1.49 0.68
1 0 4.33 3.70 5.20 3.03 1.19 2.44 1.65 1.59 0.71
1 1 4.41 3.70 5.28 3.13 1.26 2.49 1.77 1.68 0.70
1 1 4.29 3.58 5.24 3.07 1.20 2.44 1.78 1.62 0.67
1 1 4.41 3.90 5.47 3.04 1.29 2.65 1.85 1.61 0.72
1 1 4.41 3.90 5.24 3.09 1.28 2.58 1.74 1.34 0.69

1Jolicoeur, P. (1959). Multivariate geographical variation in the wolf Canis lupus L. Evolution, 13, 3,
283–299.
Data here are given in inches.
2
1 1 4.45 3.82 5.75 3.32 1.39 2.70 2.01 1.72 0.68
1 1 4.21 3.82 5.39 3.07 1.21 2.43 1.77 1.47 0.65

What is the probability that a female wolf with measures x3 = 5.28 and x7 = 1.78, comes
from Arctic habitat? Set a Bayesian logistic regression with two predictors, x3 and x7, and
ignore other variables. A starter file is wolves0.odc.

Hint: If all predictor variables are used, the classical logistic regression will fail since 0’s
and 1’s are perfectly separated and iterative reweighed least squares algorithm produces an
error. Bayesian logistic regression with all variables will work (since it behaves as iterative
penalized least squares), but the model is not stable.

Take relatively high precision in priors
for regression coefficients, say dnorm(0, 0.01). Expect large standard deviations for the
regression coefficients (and wide 95% credible sets).

## 3. Micronuclei.

The Micronuclei (MN) Assay procedure involves breaking the DNA
of lymphocytes in a blood sample with a powerful dose of radiation, then measuring the
efficiency of its ability to repair itself.

Micronuclei are fragments of DNA that have not
healed back into either of two daughter nuclei after irradiation. The MN assay entails
scoring the number of micronuclei; the higher the number, the less efficient is the subjects
DNA repair system.

The dose response of the number of micronuclei in cytokinesis-blocked lymphocytes after
in-vitro irradiation of whole blood with x-rays in the dose range 0-4 Gy was studied by
Thierens et al, 1991.2

The data provided in table are from one patient (male, 54 y.o.) and represent the frequency of micronuclei numbers for six levels of radiation, 0, 0.5, 1, 2, 3, and 4 Gy.

Number of micronuclei
0 1 2 3 4 5 6
0 976 21 3 0 0 0 0
0.5 936 61 3 0 0 0 0
Dose 1 895 94 11 0 0 0 0
(in Gy) 2 760 207 32 1 0 0 0
3 583 302 97 12 6 0 0
4 485 319 147 35 11 2 1

(a) Fit a Poisson regression in which the number of micronuclei is the response (y) and
dose is a covariate (x).

2Thierens, H., Vral, A., and de Ridder, L. (1991). Biological dosimetry using the micronucleus assay for
lymphocytes: interindividual differences in dose response. Health Phys., 61, 5, 623–630.
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(b) What is the average number of micronuclei for dose of 3.5 Gy?

Hint: The composite data from the table are entered to WinBUGS in form of individual
pairs (x, y).

There are 6000 pairs in DATA in the starter file micronuclei0.odc. Since 6000
pairs slow down WinBUGS, do not simulate more than 3,000 instances from corresponding
posteriors. The burn in of 500 is fine.
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