Markov Chain Monte Carlo for Quinn's Dynamic Ecological Inference Model
MCMCdynamicEI is used to fit Quinn's dynamic ecological inference model for partially observed 2 x 2 contingency tables.
MCMCdynamicEI( r0, r1, c0, c1, burnin = 5000, mcmc = 50000, thin = 1, verbose = 0, seed = NA, W = 0, a0 = 0.825, b0 = 0.0105, a1 = 0.825, b1 = 0.0105, ... )
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
|
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
|
W |
Weight (not precision) matrix structuring the temporal dependence among elements of θ_{0} and θ_{1}. The default value of 0 will construct a weight matrix that corresponds to random walk priors for θ_{0} and θ_{1}. The default assumes that the tables are equally spaced throughout time and that the elements of r0, r1, c0, and c1 are temporally ordered. |
a0 |
|
b0 |
|
a1 |
|
b1 |
|
... |
further arguments to be passed |
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:
p(θ_0|σ^2_0) \propto σ_0^{-ntables} \exp ≤ft(-\frac{1}{2σ^2_0} θ'_{0} P θ_{0}\right)
and
p(θ_1|σ^2_1) \propto σ_1^{-ntables} \exp ≤ft(-\frac{1}{2σ^2_1} θ'_{1} P θ_{1}\right)
where P_{ts} = -W_{ts} for t not equal to s and P_{tt} = ∑_{s \ne t}W_{ts}. The θ_{0t} is assumed to be a priori independent of θ_{1t} for all t. In addition, the following hyperpriors are assumed: 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).
Inference centers on p_0, p_1, σ^2_0, and σ^2_1. Univariate slice sampling (Neal, 2003) together with Gibbs sampling is used to sample from the posterior distribution.
An mcmc object that contains the sample from the posterior distribution. This object can be summarized by functions provided by the coda package.
Kevin Quinn. 2004. “Ecological Inference in the Presence of Temporal Dependence." In Ecological Inference: New Methodological Strategies. Gary King, Ori Rosen, and Martin A. Tanner (eds.). New York: Cambridge University Press.
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/.
Radford Neal. 2003. “Slice Sampling" (with discussion). Annals of Statistics, 31: 705-767.
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.
Jonathan C. Wakefield. 2004. “Ecological Inference for 2 x 2 Tables.” Journal of the Royal Statistical Society, Series A. 167(3): 385445.
## Not run:
## simulated data example 1
set.seed(3920)
n <- 100
r0 <- rpois(n, 2000)
r1 <- round(runif(n, 100, 4000))
p0.true <- pnorm(-1.5 + 1:n/(n/2))
p1.true <- pnorm(1.0 - 1:n/(n/4))
y0 <- rbinom(n, r0, p0.true)
y1 <- rbinom(n, r1, p1.true)
c0 <- y0 + y1
c1 <- (r0+r1) - c0
## plot data
dtomogplot(r0, r1, c0, c1, delay=0.1)
## fit dynamic model
post1 <- MCMCdynamicEI(r0,r1,c0,c1, mcmc=40000, thin=5, verbose=100,
seed=list(NA, 1))
## fit exchangeable hierarchical model
post2 <- MCMChierEI(r0,r1,c0,c1, mcmc=40000, thin=5, verbose=100,
seed=list(NA, 2))
p0meanDyn <- colMeans(post1)[1:n]
p1meanDyn <- colMeans(post1)[(n+1):(2*n)]
p0meanHier <- colMeans(post2)[1:n]
p1meanHier <- colMeans(post2)[(n+1):(2*n)]
## plot truth and posterior means
pairs(cbind(p0.true, p0meanDyn, p0meanHier, p1.true, p1meanDyn, p1meanHier))
## simulated data example 2
set.seed(8722)
n <- 100
r0 <- rpois(n, 2000)
r1 <- round(runif(n, 100, 4000))
p0.true <- pnorm(-1.0 + sin(1:n/(n/4)))
p1.true <- pnorm(0.0 - 2*cos(1:n/(n/9)))
y0 <- rbinom(n, r0, p0.true)
y1 <- rbinom(n, r1, p1.true)
c0 <- y0 + y1
c1 <- (r0+r1) - c0
## plot data
dtomogplot(r0, r1, c0, c1, delay=0.1)
## fit dynamic model
post1 <- MCMCdynamicEI(r0,r1,c0,c1, mcmc=40000, thin=5, verbose=100,
seed=list(NA, 1))
## fit exchangeable hierarchical model
post2 <- MCMChierEI(r0,r1,c0,c1, mcmc=40000, thin=5, verbose=100,
seed=list(NA, 2))
p0meanDyn <- colMeans(post1)[1:n]
p1meanDyn <- colMeans(post1)[(n+1):(2*n)]
p0meanHier <- colMeans(post2)[1:n]
p1meanHier <- colMeans(post2)[(n+1):(2*n)]
## plot truth and posterior means
pairs(cbind(p0.true, p0meanDyn, p0meanHier, p1.true, p1meanDyn, p1meanHier))
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