Description
Q1
The data in the file enzyme.csv gives the initial rate of reaction of an enzyme (y) and the substrate concentration (x). Consider the following nonlinear regression model:
y =
θ1x
θ2 + x
+ ϵ
where ϵi
iid∼ N
0, σ2
, θ1 > 0, and θ2 > 0. Assume noninformative priors for θ1, θ2, and σ
2
. You can choose
appropriate prior distributions.
(a) Plot the marginal posterior densities of θ1, θ2, and σ
2
. Use θ1 = 200, θ2 = 0.1, and σ
2 = 100
(equivalently, τ = 0.01) for initializing the MCMC chain.
(b) Provide evidence that your model has converged, whether it is a trace plot, lack of divergences, the
Gelman-Rubin statistic (Rhat), or something else.
(c) Compute 95% credible intervals, the mean, and the standard deviation for each of the three parameters.
(From now on, we will rarely specify which type of credible interval—you may use the default for your
chosen software.)
(d) Plot the posterior predictive distribution of y when x = 0.75 and provide the 95% credible intervals.
Q2
Walpole et al. (2007)1 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:
1Walpole, W. A., Myers, R. H., Myers, S. L., and Ye (2007). Probability and Statistics for Engineers and Scientists (9th
ed.). Pearson
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 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.
(c) In a standard one-way ANOVA, we assume constant variance σ
2
for each group. If you relax that
assumption and put a prior on each group’s standard deviation (σi for i = 1, . . . , 4), do the results
from (a) and (b) change? Do the contrasts between the posterior distributions of each σi show that
they were significantly different?
Q3
The data set (available as wolves.csv) described below provides skull morphometric measurements on
wolves (Canis lupus L.) coming from two geographic locations: Rocky Mountain (0) and Arctic (1). The
original source of the data is from Jolicoeur (1959)2
, and many authors have subsequently used this data to
illustrate various multivariate statistical procedures.
The goal of Jolicoeur’s study was to determine how location and gender affect skull shape among wolf
populations. There were 9 predictor variables measured (see Table 1).
2Jolicoeur, P. (1959). Multivariate geographical variation in the wolf Canis lupus L. Evolution, 13(3), 283–299. Data here
are given in inches.
2
Table 1: Wolf skull morphometric data (in inches) from Jolicoeur (1959)3
.
Variable Description
location 0 = Rocky Mountain, 1 = Arctic
gender 0 = male, 1 = female
x1 Palatal length
x2 Postpalatal length
x3 Zygomatic width
x4 Palatal width (outside first upper molars)
x5 Palatal width (inside second upper molars)
x6 Width between postglenoid foramina
x7 Interorbital width
x8 Least width of the braincase
x9 Crown length of the first upper molar
(a) Try a frequentist logistic regression on the data (in Python, you can use the statsmodels package or
sklearn). What are the results?
(b) Set up a Bayesian logistic regression. Try at least three separate models, changing regression coefficient
variance to increasingly informative values for each. What do you observe in the results? How do they
differ from the frequentist model and from each other?
(c) Re-sample the model with only three predictors: gender, x3, and x7. Give an estimate and credible
interval of the probability that a female wolf with measures x3 = 5.28 and x7 = 1.78 comes from an
Arctic habitat.
