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ei.MD.bayes {eiPack}

Multinomial Dirichlet model for Ecological Inference in RxC tables
Package: 
eiPack
Version: 
0.1-7

Description

Implements a version of the hierarchical model suggested in Rosen et al. (2001)

Usage

ei.MD.bayes(formula, covariate = NULL, total = NULL, data, 
            lambda1 = 4, lambda2 = 2, covariate.prior.list = NULL,
            tune.list = NULL, start.list = NULL, sample = 1000, thin = 1, 
            burnin = 1000, verbose = 0, ret.beta = 'r', 
            ret.mcmc = TRUE, usrfun = NULL)

Arguments

formula
A formula of the form cbind(col1, col2, ...) ~       cbind(row1, row2, ...). Column and row marginals must have the same totals.
covariate
An optional formula of the form ~ covariate. The default is covariate = NULL, which fits the model without a covariate.
total
if row and/or column marginals are given as proportions, total identifies the name of the variable in data containing the total number of individuals in each unit
data
A data frame containing the variables specified in formula and total
lambda1
The shape parameter for the gamma prior (defaults to 4)
lambda2
The rate parameter for the gamma prior (defaults to 2)
covariate.prior.list
a list containing the parameters for normal prior distributions on delta and gamma for model with covariate. See `details' for more information.
tune.list
A list containing tuning parameters for each block of parameters. See `details' for more information. Typically, this will be a list generated by tuneMD. The default is NULL, in which case fixed tuning parameters are used.
start.list
A list containing starting values for each block of parameters. See `details' for more information. The default is start.list = NULL, which generates appropriate random starting values.
sample
Number of draws to be saved from chain and returned as output from the function (defaults to 1000). The total length of the chain is sample*thin + burnin.
thin
an integer specifying the thinning interval for posterior draws (defaults to 1, but most problems will require a much larger thinning interval).
burnin
integer specifying the number of initial iterations to be discarded (defaults to 1000, but most problems will require a longer burnin).
verbose
an integer specifying whether the progress of the sampler is printed to the screen (defaults to 0). If verbose is greater than 0, the iteration number is printed to the screen every verboseth iteration.
ret.beta
A character indicating how the posterior draws of beta should be handled: `r'eturn as an R object, `s'ave as .txt.gz files, `d'iscard (defaults to r).
ret.mcmc
A logical value indicating how the samples from the posterior should be returned. If TRUE (default), samples are returned as coda mcmc objects. If FALSE, samples are returned as arrays.
usrfun
the name of an optional a user-defined function to obtain quantities of interest while drawing from the MCMC chain (defaults to NULL).

Details

ei.MD.bayes implements a version of the hierarchical Multinomial-Dirichlet model for ecological inference in R x C tables suggested by Rosen et al. (2001).

Let r = 1, ..., R index rows, C = 1, ..., C index columns, and i = 1, ..., n index units. Let N_.ci be the marginal count for column c in unit i and X_ri be the marginal proportion for row r in unit i. Finally, let beta_rci be the proportion of row r in column c for unit i.

The first stage of the model assumes that the vector of column marginal counts in unit i follows a Multinomial distribution of the form:

The second stage of the model assumes that the vector of beta for row r in unit i follows a Dirichlet distribution with C parameters. The model may be fit with or without a covariate.

If the model is fit without a covariate, the distribution of the vector beta_ri is :

In this case, the prior on each alpha_rc is assumed to be:

If the model is fit with a covariate, the distribution of the vector beta_ri is :

The parameters gamma_rC and delta_rC are constrained to be zero for identification. (In this function, the last column entered in the formula is so constrained.)

Finally, the prior for d_r is:

while gamma_rC and delta_rC are given improper uniform priors if covariate.prior.list = NULL or have independent normal priors of the form:

If the user wishes to estimate the model with proper normal priors on gamma_rC and delta_rC, a list with four elements must be provided for covariate.prior.list:

  • mu.deltaan R x (C-1) matrix of prior means for Delta
  • sigma.deltaan R x (C-1) matrix of prior standard deviations for Delta
  • mu.gammaan R x (C-1) matrix of prior means for Gamma
  • sigma.gammaan R x (C-1) matrix of prior standard deviations for Gamma

Applying the model without a covariate is most reasonable in situations where one can think of individuals being randomly assigned to units, so that there are no aggregation or contextual effects. When this assumption is not reasonable, including an appropriate covariate may improve inferences; note, however, that there is typically little information in the data about the relationship of any given covariate to the unit parameters, which can lead to extremely slow mixing of the MCMC chains and difficulty in assessing convergence. Because the conditional distributions are non-standard, draws from the posterior are obtained by using a Metropolis-within-Gibbs algorithm. The proposal density for each parameter is a univariate normal distribution centered at the current parameter value with standard deviation equal to the tuning constant; the only exception is for draws of gamma_rc and delta_rc, which use a bivariate normal proposal with covariance zero. The function will accept user-specified starting values as an argument. If the model includes a covariate, the starting values must be a list with the following elements, in this order:

  • start.dra vector of length R of starting values for Dr. Starting values for Dr must be greater than zero.
  • start.betasan R x C by precincts array of starting values for Beta. Each row of every precinct must sum to 1.
  • start.gammaan R x C matrix of starting values for Gamma. Values in the right-most column must be zero.
  • start.deltaan R x C matrix of starting values for Delta. Values in the right-most column must be zero.

If there is no covariate, the starting values must be a list with the following elements:

  • start.alphasan R x C matrix of starting values for Alpha. Starting values for Alpha must be greater than zero.
  • start.betasan R x C x units array of starting values for Beta. Each row in every unit must sum to 1.

The function will accept user-specified tuning parameters as an argument. The tuning parameters define the standard deviation of the normal distribution used to generate candidate values for each parameter. For the model with a covariate, a bivariate normal distribution is used to generate proposals; the covariance of these normal distributions is fixed at zero. If the model includes a covariate, the tuning parameters must be a list with the following elements, in this order:

  • tune.dra vector of length R of tuning parameters for Dr
  • tune.betaan R x (C-1) by precincts array of tuning parameters for Beta
  • tune.gammaan R x (C-1) matrix of tuning parameters for Gamma
  • tune.deltaan R x (C-1) matrix of tuning parameters for Delta

If there is no covariate, the tuning parameters are a list with the following elements:

  • tune.alphaan R x C matrix of tuning parameters for Alpha
  • tune.betaan R x (C-1) by precincts array of tuning parameters for Beta

Values

A list containing

draws
A list containing samples from the posterior distribution of the parameters. If a covariate is included in the model, the list contains:
  • DrPosterior draws for Dr parameters as an R xsample matrix. If ret.mcmc = TRUE, Dr is an mcmc object.
  • BetaPosterior draws for beta parameters. Only returned if ret.beta = TRUE. If ret.mcmc =  TRUE, a (R * C * units) x sample matrix saved as an mcmc object. Otherwise, a R x C x units x sample array
  • GammaPosterior draws for gamma parameters. If ret.mcmc = TRUE, a (R * (C - 1)) x sample matrix saved as an mcmc object. Otherwise, a R x (C - 1) x sample array
  • DeltaPosterior draws for delta parameters. If ret.mcmc =                  TRUE, a (R * (C - 1)) x sample matrix saved as an mcmc object. Otherwise, a R x(C - 1) x sample array
  • Cell.countPosterior draws for the cell counts, summed across units. If ret.mcmc =     TRUE, a (R * C) x sample matrix saved as an mcmc object. Otherwise, a R x C x sample array

If the model is fit without a covariate, the list includes:

  • AlphaPosterior draws for alpha parameters. If ret.mcmc =     TRUE, a (R * C) x sample matrix saved as an mcmc object. Otherwise, a R x C x sample array
  • BetaPosterior draws for beta parameters. If ret.mcmc =     TRUE, a (R * C * units) x sample matrix saved as an mcmc object. Otherwise, a R x C x units x sample array
  • Cell.countPosterior draws for the cell counts, summed across units. If ret.mcmc =     TRUE, a (R * C) x sample matrix saved as anmcmc object. Otherwise, a R x C x sample array

acc.ratios
A list containing acceptance ratios for the parameters. If the model includes a covariate, the list includes:
  • dr.accA vector of acceptance ratios for Dr draws
  • beta.accA vector of acceptance ratios for Beta draws
  • gamma.accA vector of acceptance ratios for Gamma and Delta draws

If the model is fit without a covariate , the list includes:

  • alpha.accA vector of acceptance ratios for Alpha draws
  • beta.accA vector of acceptance ratios for Beta draws

usrfun
Output from the optional usrfn
call
Call to ei.MD.bayes

References

Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). .

     Ori Rosen, Wenxin Jiang, Gary King, and Martin A. Tanner.      2001. “Bayesian and Frequentist Inference for Ecological      Inference: The R x (C-1) Case.”      Statistica Neerlandica 55: 134-156.

See Also

lambda.MD, cover.plot, density.plot, tuneMD, mergeMD

Author(s)

Michael Kellermann <kellerm@post.harvard.edu> and Olivia Lau <olivia.lau@post.harvard.edu>

Documentation reproduced from package eiPack, version 0.1-7. License: GPL (>= 2)