Consider the following partially observed 2 by 2 contingency table
for unit \(t\) where \(t=1,\ldots,ntables\):
|
|
| \(Y=0\) |
| \(Y=1\) |
| |
|
--------- |
------------ |
------------ |
------------ |
|
\(X=0\) |
| \(Y_{0t}\) |
| |
| \(r_{0t}\) |
|
--------- |
------------ |
------------ |
------------ |
|
\(X=1\) |
| \(Y_{1t}\) |
| |
| \(r_{1t}\) |
|
--------- |
------------ |
------------ |
------------ |
Where \(r_{0t}\), \(r_{1t}\), \(c_{0t}\), \(c_{1t}\), and
\(N_t\) are non-negative integers that are observed. The interior
cell entries are not observed. It is assumed that
\(Y_{0t}|r_{0t} \sim \mathcal{B}inomial(r_{0t}, p_{0t})\) and
\(Y_{1t}|r_{1t} \sim \mathcal{B}inomial(r_{1t}, p_{1t})\). Let
\(\theta_{0t} = log(p_{0t}/(1-p_{0t}))\), and \(\theta_{1t} =
log(p_{1t}/(1-p_{1t}))\).
The following prior distributions are assumed: \(\theta_{0t}
\sim \mathcal{N}(\mu_0, \sigma^2_0)\), \(\theta_{1t} \sim
\mathcal{N}(\mu_1, \sigma^2_1)\). \(\theta_{0t}\) is assumed to
be a priori independent of \(\theta_{1t}\) for all t. In
addition, we assume the following hyperpriors: \(\mu_0 \sim
\mathcal{N}(m_0, M_0)\), \(\mu_1 \sim \mathcal{N}(m_1, M_1)\),
\(\sigma^2_0 \sim \mathcal{IG}(a_0/2, b_0/2)\), and
\(\sigma^2_1 \sim \mathcal{IG}(a_1/2, b_1/2)\).
The default priors have been chosen to make the implied prior
distribution for \(p_{0}\) and \(p_{1}\) approximately
uniform on (0,1).
Inference centers on \(p_0\), \(p_1\), \(\mu_0\),
\(\mu_1\), \(\sigma^2_0\), and \(\sigma^2_1\). Univariate
slice sampling (Neal, 2003) along with Gibbs sampling is used to
sample from the posterior distribution.
See Section 5.4 of Wakefield (2003) for discussion of the priors
used here. MCMChierEI departs from the Wakefield model in
that the mu0 and mu1 are here assumed to be drawn
from independent normal distributions whereas Wakefield assumes
they are drawn from logistic distributions.