# MCMChierEI: Markov Chain Monte Carlo for Wakefield's Hierarchial... In MCMCpack: Markov Chain Monte Carlo (MCMC) Package

## Description

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

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 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, ... ) 

## 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. verbose 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. seed 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. m0 Prior mean of the μ_0 parameter. M0 Prior variance of the μ_0 parameter. m1 Prior mean of the μ_1 parameter. M1 Prior variance of the μ_1 parameter. a0 a0/2 is the shape parameter for the inverse-gamma prior on the σ^2_0 parameter. b0 b0/2 is the scale parameter for the inverse-gamma prior on the σ^2_0 parameter. a1 a1/2 is the shape parameter for the inverse-gamma prior on the σ^2_1 parameter. b1 b1/2 is the scale parameter for the inverse-gamma prior on the σ^2_1 parameter. ... further arguments to be passed

## Details

Consider the following partially observed 2 by 2 contingency table for unit t where t=1,…,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 θ_{0t} = log(p_{0t}/(1-p_{0t})), and θ_{1t} = log(p_{1t}/(1-p_{1t})).

The following prior distributions are assumed: θ_{0t} \sim \mathcal{N}(μ_0, σ^2_0), θ_{1t} \sim \mathcal{N}(μ_1, σ^2_1). θ_{0t} is assumed to be a priori independent of θ_{1t} for all t. In addition, we assume the following hyperpriors: μ_0 \sim \mathcal{N}(m_0, M_0), μ_1 \sim \mathcal{N}(m_1, M_1), σ^2_0 \sim \mathcal{IG}(a_0/2, b_0/2), and σ^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, μ_0, μ_1, σ^2_0, and σ^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.

## Value

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

## References

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. https://www.jstatsoft.org/v42/i09/.

Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.lsa.umich.edu.

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
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27  ## Not run: ## 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)) ## End(Not run)