4. Eight different training methods for a particular exam have been implemented in classes in different schools. In some

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4 Eight Different Training Methods For A Particular Exam Have Been Implemented In Classes In Different Schools In Some 1 (147.02 KiB) Viewed 14 times

4. Eight different training methods for a particular exam have been implemented in classes in different schools. In some preliminary data analysis, the overall effect yj on the exam result in the jth school (where training method j was used; j = 1,...,8) has been estimated along with an estimate of its accuracy (reflected in the estimated standard error, 0;), as reported in this table: 1 2 3 4 5 6 8 School i Effect of training Yj Standard error oj 28 15 8 10 -3 16 7 11 -1 9 1 11 7 18 10 12 18 The higher values of the effect correspond to better training performance. To explore the effects of the different training methods in general, a Bayesian normal hierarchical model is formulated and fitted in WinBUGS with the following model code: model { for (j in 1:8) { y[j] dnorm(theta[j], inv.sig2[j]) theta[j] - dnorm(mu, inv.tau2) inv.sig2[j] <- 1/ (sigma[j] * sigma [j]) } dnorm (0, 1.0E-6) inv.tau2 <-1/(tau*tau) tau dunif(0,100) } mu (Here y[j] corresponds to y; and sigma [j] to 0;). (a) Convert the WinBUGS model specification back into standard statistical notation, taking care to include distributions that fully characterise the likelihood and the prior distribution, and to state what distributions are independent of each other. Remember that WinBUGS function dnorm uses normal distribution precision as the second parameter. [4 MARKS]

(b) The table below presents posterior summaries produced by WinBUGS: node mean sd MC error theta [1] 11.25 8.221 0.09032 theta [2] 7.835 6.218 0.06468 theta [3] 6.106 7.692 0.06912 theta [4] 7.588 6.490 0.06546 theta [5] 5.112 6.344 0.06719 theta [6] 6.106 6.672 0.06461 theta [7] 10.59 6.759 0.07920 theta [8] 8.377 7.778 0.07115 7.862 5.149 0.06598 tau 6.478 5.576 0.08033 2.5% median 97.5% start sample -2.198 10.07 31.12 501 99500 -4.588 7.747 20.44 501 99500 -11.31 6.633 20.48 501 99500 -5.536 7.593 20.64 501 99500 -8.935 5.663 16.54 501 99500 -8.437 6.497 18.60 501 99500 -1.385 9.906 25.86 501 99500 -6.737 8.068 25.39 501 99500 -2.196 7.784 18.10 501 99500 0.2771 5.140 20.50 501 99500 mu In the remainder of this question, the symbols Oig ul and T will be used to represent the variables called theta, mu and tau, respectively, in WinBUGS. For each of the posterior summaries listed below, state whether it can be determined using the WinBUGS output above and why. If your answer is “yes”, compute the estimate; otherwise explain how you would compute it in WinBUGS. i. The central 95% posterior interval for 03. [2 MARKS] ។ ii. The posterior mean of 01 - 05. [2 MARKS]