MCMChierEI             package:MCMCpack             R Documentation

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_I_n_f_e_r_e_n_c_e _M_o_d_e_l

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

     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, ...)
        

_A_r_g_u_m_e_n_t_s:

      r0: (ntables * 1) vector of row sums from row 0.

      r1: (ntables * 1) vector of row sums from row 1.

      c0: (ntables * 1) vector of column sums from column 0.

      c1: (ntables * 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 'verbose'th 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 mu0 parameter.

      M0: Prior variance of the mu0 parameter.

      m1: Prior mean of the mu1 parameter.

      M1: Prior variance of the mu1 parameter.

      a0: 'a0/2' is the shape parameter for the inverse-gamma prior on
          the sigma^2_0 parameter.

      b0: 'b0/2' is the scale parameter for the inverse-gamma prior on
          the sigma^2_0 parameter.

      a1: 'a1/2' is the shape parameter for the inverse-gamma prior on
          the sigma^2_1 parameter.

      b1: 'b1/2' is the scale parameter for the inverse-gamma prior on
          the sigma^2_1 parameter.

     ...: further arguments to be passed

_D_e_t_a_i_l_s:

     Consider the following partially observed 2 by 2 contingency table
     for unit t where t=1,...,ntables:


                  | Y=0      | Y=1      |
       - - - - -  - - - - -  - - - - -  - - - - -
       X=0        | Y0[t]    |          |r0[t]
       - - - - -  - - - - -  - - - - -  - - - - -
       X=1        | Y1[t]    |          | r1[t]
       - - - - -  - - - - -  - - - - -  - - - - -
                  | c0[t]    | c1[t]    | N[t]

     Where r0[t], r1[t], c0[t], c1[t], and N[t]  are non-negative
     integers that are observed. The interior cell entries are not
     observed. It is assumed that Y0[t]|r0[t] ~ Binomial(r0[t], p0[t])
     and  Y1[t]|r1[t] ~ Binomial(r1[t],p1[t]). Let theta0[t] =
     log(p0[t]/(1-p0[t])), and  theta1[t] = log(p1[t]/(1-p1[t])).

     The following prior distributions are assumed: theta0[t] ~
     Normal(mu0, sigma^2_0), theta1[t] ~ Normal(mu1, sigma^2_1).
     theta0[t] is assumed to be a priori independent of theta1[t] for
     all t. In addition, we assume the following hyperpriors: mu0 ~
     Normal(m0, M0), mu1 ~ Normal(m1, M1), sigma^2_0 ~ InvGamma(a0/2,
     b0/2), and sigma^2_1 ~ InvGamma(a1/2, b1/2).

     The default priors have been chosen to make the implied prior
     distribution for p0 and p1 _approximately_ uniform on (0,1). 

     Inference centers on p0, p1, mu0, mu1, 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.

_V_a_l_u_e:

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

_R_e_f_e_r_e_n_c_e_s:

     Jonathan C. Wakefield.  2003. ``Ecological inference for 2x2
     tables."  Read  before the Royal Statistical Society, on November
     12th, 2003.

     Radford Neal. 2003. ``Slice Sampling" (with discussion). _Annals
     of Statistics_, 31: 705-767. 

     Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein.  2004. 
     _Scythe Statistical  Library 1.0._ <URL: http://scythe.wustl.edu>.

     Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002.
     _Output Analysis and Diagnostics for MCMC (CODA)_. <URL:
     http://www-fis.iarc.fr/coda/>.

_S_e_e _A_l_s_o:

     'MCMCdynamicEI', 'plot.mcmc','summary.mcmc'

_E_x_a_m_p_l_e_s:

        ## 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)

