In Example 5.6 (popularity data), and retaining separate univariate normal priors (i.e. model 1) on the cluster effects at stage 2 , assess the impact on inferences, such as on posterior means of \(\left\{\sigma, \tau_{1}, \tau_{2}, \tau_{3}\right\}\), of adopting uniform priors on the shrinkage parameters \(\sigma^{2} /\left(\sigma^{2}+\tau_{p}^{2}\right)\).

Data from Example 5.6

Consider two level continuous data, namely pupil popularity scores y;; for N = 2000 pupils in J = 100 school classes (the clusters). At level 1 there are P = 2 predictors, gender xiij (1=girl, 0=boy), and a measure of extraversion X2ij. At level 2 there is a single predictor Z.;,