Description
1. Suppose
Z ∼ N(0, 1)
ε1 ∼ N(0, σ2
1
)
ε2 ∼ N(0, σ2
2
)
independent
and let
Y1 = Z + ε1 Y2 = Z + ε2
(a) [2 pts] Under what conditions (if any) are Y1 and Y2 exchangeable? Justify your answer.
(b) [2 pts] Determine the covariance of Y1 and Y2. Under what conditions (if any) are they
(marginally) independent?
2. Twelve separate case-control studies were run to investigate the potential link between
presence of a certain genetic trait (the PlA2 polymorphism of the glycoprotein IIIa subunit
of the fibrinogen receptor) and risk of heart attack.1 For the i
th study, an estimated
log-odds ratio, ψˆ
i
, and its (estimated) standard error, σi
, were computed:
i ψˆ
i σi i ψˆ
i σi i ψˆ
i σi
1 1.055 0.373 5 1.068 0.471 9 0.507 0.186
2 -0.097 0.116 6 -0.025 0.120 10 0.000 0.328
3 0.626 0.229 7 -0.117 0.220 11 0.385 0.206
4 0.017 0.117 8 -0.381 0.239 12 0.405 0.254
Consider this Bayesian hierarchical model:
ψˆ
i
| ψi ∼ indep. N(ψi
, σ2
i
) i = 1, …, 12
ψi
| ψ0, σ2
0 ∼ indep. N(ψ0, σ2
0
) i = 1, …, 12
ψ0 ∼ N(0, 1000)
τ
2
0 = 1/σ2
0 ∼ gamma(0.001, 0.001)
with ψ0 and τ
2
0
independent, and the values σ
2
i
, i = 1, . . . , 12, regarded as fixed and known.
(a) [2 pts] Draw a directed acyclic graph (DAG) appropriate for this model. (Use the
notation introduced in lecture, including “plates”.)
(b) [3 pts] Using the template prob2template.bug provided on the Compass course web
site, form a JAGS model statement (corresponding to your graph). [ Remember: JAGS
“dnorm” uses precisions, not variances! ]
1From Burr, et al. (2003), Statistics in Medicine, 22: 1741–1760.
(c) [3 pts] Perform a preliminary run of your model using rjags, to check for convergence
of your sampler. Use three chains, separately initialized. Include the lists of the initial
values you used. Show the plots you used to monitor convergence. Explicitly determine
how many iterations to burn (i.e. exclude from inference), and justify the number you
chose.
(d) [3 pts] Perform at least 100000 iterations (after burn-in), and produce a summary of
your inference results, for both ψ0 and σ
2
0
.
Be sure to include estimates of the posterior
expected value and standard deviation, the Monte Carlo error, and a 95% posterior
credible interval. Also include graphical estimates of the densities.
