`MCMChierEI' is used to fit Wakefield's hierarchical ecological inference model for partially observed 2 x 2 contingency tables.
MCMChierEI(
r0,
r1,
c0,
c1,
burnin = 5000,
mcmc = 50000,
thin = 1,
verbose = 0,
seed = NA,
m0 = 0,
M0 = 2.287656,
m1 = 0,
M1 = 2.287656,
a0 = 0.825,
b0 = 0.0105,
a1 = 0.825,
b1 = 0.0105,
...
)An mcmc object that contains the sample from the posterior distribution. This object can be summarized by functions provided by the coda package.
\((ntables \times 1)\) vector of row sums from row 0.
\((ntables \times 1)\) vector of row sums from row 1.
\((ntables \times 1)\) vector of column sums from column 0.
\((ntables \times 1)\) vector of column sums from column 1.
The number of burn-in scans for the sampler.
The number of mcmc scans to be saved.
The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value.
A switch which determines whether or not the progress of the
sampler is printed to the screen. If verbose is greater than 0 then
every verboseth iteration will be printed to the screen.
The seed for the random number generator. If NA, the Mersenne
Twister generator is used with default seed 12345; if an integer is passed
it is used to seed the Mersenne twister. The user can also pass a list of
length two to use the L'Ecuyer random number generator, which is suitable
for parallel computation. The first element of the list is the L'Ecuyer
seed, which is a vector of length six or NA (if NA a default seed of
rep(12345,6) is used). The second element of list is a positive
substream number. See the MCMCpack specification for more details.
Prior mean of the \(\mu_0\) parameter.
Prior variance of the \(\mu_0\) parameter.
Prior mean of the \(\mu_1\) parameter.
Prior variance of the \(\mu_1\) parameter.
a0/2 is the shape parameter for the inverse-gamma
prior on the \(\sigma^2_0\) parameter.
b0/2 is the scale parameter for the inverse-gamma
prior on the \(\sigma^2_0\) parameter.
a1/2 is the shape parameter for the inverse-gamma
prior on the \(\sigma^2_1\) parameter.
b1/2 is the scale parameter for the inverse-gamma
prior on the \(\sigma^2_1\) parameter.
further arguments to be passed
Consider the following partially observed 2 by 2 contingency table for unit \(t\) where \(t=1,\ldots,ntables\):
| | \(Y=0\) | | \(Y=1\) | | | |
| --------- | ------------ | ------------ | ------------ |
| \(X=0\) | | \(Y_{0t}\) | | | | \(r_{0t}\) |
| --------- | ------------ | ------------ | ------------ |
| \(X=1\) | | \(Y_{1t}\) | | | | \(r_{1t}\) |
| --------- | ------------ | ------------ | ------------ |
| | \(c_{0t}\) | | \(c_{1t}\) | | \(N_t\) |
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.
Jonathan C. Wakefield. 2004. ``Ecological Inference for 2 x 2 Tables.'' Journal of the Royal Statistical Society, Series A. 167(3): 385445.
Radford Neal. 2003. ``Slice Sampling" (with discussion). Annals of Statistics, 31: 705-767.
Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. ``MCMCpack: Markov Chain Monte Carlo in R.'', Journal of Statistical Software. 42(9): 1-21. tools:::Rd_expr_doi("10.18637/jss.v042.i09").
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. ``Output Analysis and Diagnostics for MCMC (CODA)'', R News. 6(1): 7-11. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf.
MCMCdynamicEI,
plot.mcmc,summary.mcmc
if (FALSE) {
## simulated data example
set.seed(3920)
n <- 100
r0 <- round(runif(n, 400, 1500))
r1 <- round(runif(n, 100, 4000))
p0.true <- pnorm(rnorm(n, m=0.5, s=0.25))
p1.true <- pnorm(rnorm(n, m=0.0, s=0.10))
y0 <- rbinom(n, r0, p0.true)
y1 <- rbinom(n, r1, p1.true)
c0 <- y0 + y1
c1 <- (r0+r1) - c0
## plot data
tomogplot(r0, r1, c0, c1)
## fit exchangeable hierarchical model
post <- MCMChierEI(r0,r1,c0,c1, mcmc=40000, thin=5, verbose=100,
seed=list(NA, 1))
p0meanHier <- colMeans(post)[1:n]
p1meanHier <- colMeans(post)[(n+1):(2*n)]
## plot truth and posterior means
pairs(cbind(p0.true, p0meanHier, p1.true, p1meanHier))
}
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