Randomized block design: Researchers interested in identifying the optimal planting density for a type of perennial grass performed the following randomized experiment: Ten different plots of land were each divided into eight subplots, and planting densities of 2, 4, 6 and 8 plants per square meter were randomly assigned to the subplots, so that there are two subplots at each density in each plot. At the end of the growing season the amount of plant matter yield was recorded in metric tons per hectare. These data appear in the file pdensity.dat. The researchers want to fit a model like y = β₁ + β₂x + β₃x² + ε, where y is yield and x is planting density, but worry that since soil conditions vary across plots they should allow for some across-plot heterogeneity in this relationship. To accommodate this possibility we will analyze these data using the hierarchical linear model described in Section 11.1.

a) Before we do a Bayesian analysis we will get some ad hoc estimates of these parameters via least squares regression. Fit the model y = β1+β2x+β3x 2+ε using OLS for each group, and make a plot showing the heterogeneity of the least squares regression lines. From the least squares coefficients find ad hoc estimates of θ and Σ. Also obtain an estimate of σ² by combining the information from the residuals across the groups.

b) Now we will perform an analysis of the data using the following distributions as prior distributions: 

Σ −1 ∼ Wishart(4, Σˆ−1 ) θ ∼ multivariate normal(θˆ, Σˆ) σ 2 ∼ inverse − gamma(1, σˆ 2 ) where θˆ, Σ, ˆ σˆ 2 are the estimates you obtained in a). Note that this analysis is not combining prior information with information from the data, as the“prior” distribution is based on the observed data.

However, such an analysis can be roughly interpreted as the Bayesian analysis of an individual who has weak but unbiased prior information.

c) Use a Gibbs sampler to approximate posterior expectations of β for each group j, and plot the resulting regression lines. Compare to the regression lines in a) above and describe why you see any differences between the two sets of regression lines.

d) From your posterior samples, plot marginal posterior and prior densities of θ and the elements of Σ. Discuss the evidence that the slopes or intercepts vary across groups.

e) Suppose we want to identify the planting density that maximizes average yield over a random sample of plots. Find the value x_max of x that maximizes expected yield, and provide a 95% posterior predictive interval for the yield of a randomly sampled plot having planting density x_max.