Description
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Name
Problem 1 2 3 Total
Score /45 /30 /25 /100
1. Orthodontic Distance. A longitudinal study was conducted to understand the effect of age and sex on the orthodontic distance (y). Measurements on 27 children are given in the file ortho.csv. There are a total of 16 boys and 11 girls, which are identified in the dataset using the column Subject. Consider the following random effects model:
,
for i = 1,···,27 and j = 1,···,4. Here ui represents the random effect of the ith subject. The sex variable should be coded as −1 for female and 1 for male. Assume the following prior distributions:
where
1. Fit the random effects model and plot the posterior densities of the five parameters
, and . (use 100,000 samples with 10,000 burn-in.) 2. The intraclass correlation coefficient is defined as
.
Plot the posterior density of ρ. Does it appear to be significantly different from 0?
3. Fit the model ignoring the random effects (that is, set all the ui’s to be 0) and plot the posterior densities of the four parameters β0,β1,β2, and . What differences do you see from the previous analysis using random effects (compare the posterior means and credible intervals of the four parameters)?
2. Nanowire density. Consider the problem of predicting the density of nanowires (y) with respect to the thickness of polymer films (x) in a solution-based growth process (see Figure 1). Eight experiments were conducted with two replicates (except for one run). The data are in the file nanowire.csv. The density of nanowires is assumed to follow a Poisson distribution with mean:
µ(x) = θ1 exp(−θ2×2) + θ3{1 − exp(−θ2×2)}Φ(−x/θ4),
logθ1,logθ3,logθ4 ∼iid N(0,σ2 = 10) θ2 ∼ U(0,1).
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Figure 1: Scanning Electron Microscopy images showing nanowire density at different levels of polymer thickness.
If using pymc: when transforming your priors using another function, you must wrap the result in pm.Deterministic to properly use the transformed variable.
1. Obtain the posterior samples of the four parameters (θ1,…,θ4) using MCMC. Provide their mean and 95% credible intervals (use 100,000 samples with 10,000 burn-in).
2. Find the predictive distribution of the density of nanowires when the thickness is 2.0 nm.
3. Color Attraction for Oulema melanopus. Some colors are more attractive to insects than others. Wilson and Shade (1967) conducted an experiment aimed at determining the best color for attracting cereal leaf beetles (Oulema melanopus). Six boards in each of four selected colors (lemon yellow, white, green, and blue) were placed in a field of oats during summer time. The following table (modified from Wilson and Shade, 1967) gives data on the number of cereal leaf beetles trapped:
Board color Insects trapped
Lemon yellow 45 59 48 46 38 47
White 21 12 14 17 13 17
Green 16 11 20 21 14 7
Blue 37 32 15 25 39 41
(a) Use MCMC software to conduct ANOVA analysis of the color “treatments.” UseSTZ constraints.
(b) Based on MCMC software output, state your conclusions about the attractiveness ofthese colors to the beetles.
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