Description
1. Vasoconstriction.
The data give the presence or absence (yi = 1 or 0) of vasoconstriction in the skin of the fingers following inhalation of a certain volume of air (vi) at a certain
average rate (ri). Total number of records is 39. The candidate models for analyzing the
relationship are the usual logit, probit, cloglog, loglog, and cauchyit models.
Data are given as follows.
y:1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,
0,1,0,0,0,0,1,0,1,0,1,0,1,0,0,1,1,1,0,0,1
v:3.7, 3.5, 1.25, 0.75, 0.8, 0.7, 0.6, 1.1, 0.9, 0.9,
0.8, 0.55, 0.6, 1.4, 0.75, 2.3, 3.2, 0.85, 1.7, 1.8,
0.4, 0.95, 1.35, 1.5, 1.6, 0.6, 1.8, 0.95, 1.9, 1.6,
2.7, 2.35, 1.1, 1.1, 1.2, 0.8, 0.95, 0.75, 1.3
r: 0.825, 1.09, 2.5, 1.5, 3.2, 3.5, 0.75, 1.7, 0.75,
0.45, 0.57, 2.75, 3, 2.33, 3.75, 1.64, 1.6, 1.415,
1.06, 1.8, 2, 1.36, 1.35, 1.36, 1.78, 1.5, 1.5, 1.9,
0.95, 0.4, 0.75, 0.3, 1.83, 2.2, 2, 3.33, 1.9, 1.9, 1.625
(a) Transform covariates v and r as
x1 = log(10 × v), x2 = log(10 × r).
(b) Estimate posterior means for coefficients in the logit model. Use noninformative
priors on all coefficients.
(c) For a subject with v = r = 1.5, find the probability of vasoconstriction.
(d) Compare with the result of probit model. Which has smaller deviance?
2. Magnesium Ammonium Phosphate and Chrysanthemums. Walpole et al.
(2007) provide data from a study on the effect of magnesium ammonium phosphate on the
height of chrysanthemums, which was conducted at George Mason University in order to
determine a possible optimum level of fertilization, based on the enhanced vertical growth
response of the chrysanthemums.
Forty chrysanthemum seedlings were assigned to 4 groups,
each containing 10 plants. Each was planted in a similar pot containing a uniform growth
medium. An increasing concentration of MgNH4PO4, measured in grams per bushel, was
added to each plant. The 4 groups of plants were grown under uniform conditions in a
greenhouse for a period of 4 weeks.
The treatments and the respective changes in heights,
measured in centimeters, are given in the following table:
Treatment
50 g/bu 100 g/bu 200 g/bu 400 g/bu
13.2 16.0 7.8 21.0
12.4 12.6 14.4 14.8
12.8 14.8 20.0 19.1
17.2 13.0 15.8 15.8
13.0 14.0 17.0 18.0
14.0 23.6 27.0 26.0
14.2 14.0 19.6 21.1
21.6 17.0 18.0 22.0
15.0 22.2 20.2 25.0
20.0 24.4 23.2 18.2
Solve the problem as a Bayesian one-way ANOVA. Use STZ constraints on treatment
effects.
(a) Do different concentrations of MgNH4PO4 affect the average attained height of
chrysanthemums? Look at the 95% credible sets for the differences between treatment effects.
(b) Find the 95% credible set for the contrast µ1 − µ2 − µ3 + µ4.
3. Hocking–Pendleton Data. This popular data set was constructed by Hocking and
Pendelton (1982) to illustrate influential and outlier observations in regression. The data are
organized as a matrix of size 26 ×4; the predictors x1, x2, and x3 are the first three columns,
and the response y is the fourth column. The data are given in hockpend.dat.
(a) Fit the linear regression model with the three covariates, report the parameter estimates and Bayesian R2
.
(b) Is any of the 26 observations influential or outlier (in the sense of CPO and comulative)?
(c) Find the mean response and prediction response for a new observation with covariates
x
∗
1 = 10, x∗
2 = 5, and x
∗
3 = 5. Report the corresponding 95% credible sets.
