MCMCpack (version 0.4-5)

MCMChierEI: Markov chain Monte Carlo for Wakefield's Hierarchial Ecological Inference Model

Description

`MCMChierEI' is used to fit Wakefield's hierarchical ecological inference model for partially observed 2 x 2 contingency tables.

Usage

MCMChierEI(r0, r1, c0, c1, burnin=1000, mcmc=50000, thin=1,
           m0=0, M0=10, m1=0, M1=10, nu0=1, delta0=0.5, nu1=1,
           delta1=0.5, verbose=FALSE, tune=2.65316, seed=0, ...)

Arguments

r0

$(ntables \times 1)$ vector of row sums from row 0.

r1

$(ntables \times 1)$ vector of row sums from row 1.

c0

$(ntables \times 1)$ vector of column sums from column 0.

c1

$(ntables \times 1)$ vector of column sums from column 1.

burnin

The number of burn-in scans for the sampler.

mcmc

The number of mcmc scans to be saved.

thin

The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value.

tune

Tuning parameter for the Metropolis-Hasting sampling.

verbose

A switch which determines whether or not the progress of the sampler is printed to the screen. Information is printed if TRUE.

seed

The seed for the random number generator. The code uses the Mersenne Twister, which requires an integer as an input. If nothing is provided, the Scythe default seed is used.

m0

Prior mean of the $\mu_0$ parameter.

M0

Prior variance of the $\mu_0$ parameter.

m1

Prior mean of the $\mu_1$ parameter.

M1

Prior variance of the $\mu_1$ parameter.

nu0

Shape parameter for the inverse-gamma prior on the $\sigma^2_0$ parameter.

delta0

Scale parameter for the inverse-gamma prior on the $\sigma^2_0$ parameter.

nu1

Shape parameter for the inverse-gamma prior on the $\sigma^2_1$ parameter.

delta1

Scale parameter for the inverse-gamma prior on the $\sigma^2_1$ parameter.

...

further arguments to be passed

Value

• An mcmc object that contains the posterior density sample. This object can be summarized by functions provided by the coda package.

Details

Consider the following partially observed 2 by 2 contingency table for unit $t$ where $t=1,\ldots,ntables$: llll{ | $Y=0$ | $Y=1$ | - - - - - - - - - - - - - - - - - - - - $X=0$ | $Y_{0t}$ | |$r_{0t}$ - - - - - - - - - - - - - - - - - - - - $X=1$ | $Y_{1t}$ | | $r_{1t}$ - - - - - - - - - - - - - - - - - - - - | $c_{0t}$ | $c_{1t}$ | $N_t$ } 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: $\mu_0 \sim \mathcal{N}(m_0, M_0)$, $\mu_1 \sim \mathcal{N}(m_1, M_1)$, $\sigma^2_0 \sim \mathcal{IG}(\nu_0/2, \delta_0/2)$, and $\sigma^2_1 \sim \mathcal{IG}(\nu_1/2, \delta_1/2)$.

Inference centers on $p_0$, $p_1$, $\mu_0$, $\mu_1$, $\sigma^2_0$, and $\sigma^2_1$. The Metropolis-Hastings algorithm is used to sample from the posterior density.

References

Jonathan Wakefield. 2001. ``Ecological Inference for 2 x 2 Tables." Center for Statistics and the Social Sciences Working Paper no. 12. University of Washington.

Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2003. Scythe Statistical Library 0.4. http://scythe.wustl.edu. Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.

See Also

MCMCbaselineEI, MCMCdynamicEI, plot.mcmc,summary.mcmc

Examples

c0 <- rpois(5, 500)
   c1 <- c(200, 140, 250, 190, 75)
   r0 <- rpois(5, 400)
   r1 <- (c0 + c1) - r0
   posterior <- MCMChierEI(r0,r1,c0,c1, mcmc=200000, thin=50)
   plot(posterior)
   summary(posterior)