R: Multinomial Dirichlet model for Ecological Inference in RxC...

ei.MD.bayes {eiPack}

R Documentation

Multinomial Dirichlet model for Ecological Inference in RxC tables

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 `verbose`th 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 \times C tables suggested by Rosen et al. (2001).

Let r = 1, \ldots, R index rows, C = 1, \ldots, C index columns, and i = 1, \ldots, n index units. Let N_{\cdot 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:

(N_{\cdot 1i}, \ldots, N_{\cdot Ci}) {\sim} {\rm Multinomial}(N_i,\sum_{r=1}^R \beta_{r1i}X_{ri}, \dots, \sum_{r=1}^R \beta_{rCi}X_{ri})

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 :

(\beta_{r1i}, \dots, \beta_{rCi}) {\sim} {\rm Dirichlet}(\alpha_{r1}, \dots, \alpha_{rC})

In this case, the prior on each \alpha_{rc} is assumed to be:

\alpha_{rc} \sim {\rm Gamma}(\lambda_1, \lambda_2)

If the model is fit with a covariate, the distribution of the vector \beta_{ri} is :

(\beta_{r1i}, \dots, \beta_{rCi}) {\sim} {\rm Dirichlet}(d_r\exp(\gamma_{r1} + \delta_{r1}Z_i), d_r\exp(\gamma_{r(C-1)} + \delta_{r(C-1)}Z_i), d_r)

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:

d_r \sim {\rm Gamma}(\lambda_1, \lambda_2)

while \gamma_{rC} and \delta_{rC} are given improper uniform priors if covariate.prior.list = NULL or have independent normal priors of the form:

\delta_{rC} \sim {\rm N}(\mu_{\delta_{rC}}, \sigma_{\delta_{rC}}^2)

\gamma_{rC} \sim {\rm N}(\mu_{\gamma_{rC}}, \sigma_{\gamma_{rC}}^2)

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.delta an R \times (C-1) matrix of prior means for Delta
sigma.delta an R \times (C-1) matrix of prior standard deviations for Delta
mu.gamma an R \times (C-1) matrix of prior means for Gamma
sigma.gamma an R \times (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.dr a vector of length R of starting values for Dr. Starting values for Dr must be greater than zero.
start.betas an R \times C by precincts array of starting values for Beta. Each row of every precinct must sum to 1.
start.gamma an R \times C matrix of starting values for Gamma. Values in the right-most column must be zero.
start.delta an R \times 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.alphas an R \times C matrix of starting values for Alpha. Starting values for Alpha must be greater than zero.
start.betas an R \times C \times 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.dr a vector of length R of tuning parameters for Dr
tune.beta an R \times (C-1) by precincts array of tuning parameters for Beta
tune.gamma an R \times (C-1) matrix of tuning parameters for Gamma
tune.delta an R \times (C-1) matrix of tuning parameters for Delta

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

tune.alpha an R \times C matrix of tuning parameters for Alpha
tune.beta an R \times (C-1) by precincts array of tuning parameters for Beta

Value

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: `Dr` Posterior draws for Dr parameters as an R `\times`sample matrix. If `ret.mcmc = TRUE`, `Dr` is an `mcmc` object. `Beta` Posterior draws for beta parameters. Only returned if `ret.beta = TRUE`. If `ret.mcmc = TRUE`, a (R * C * units) `\times` sample matrix saved as an `mcmc` object. Otherwise, a R `\times` C `\times` units `\times` sample array `Gamma` Posterior draws for gamma parameters. If `ret.mcmc = TRUE`, a (R * (C - 1)) `\times` sample matrix saved as an `mcmc` object. Otherwise, a R `\times` (C - 1) `\times` sample array `Delta` Posterior draws for delta parameters. If `ret.mcmc = TRUE`, a (R * (C - 1)) `\times` sample matrix saved as an `mcmc` object. Otherwise, a R `\times`(C - 1) `\times` sample array `Cell.count` Posterior draws for the cell counts, summed across units. If `ret.mcmc = TRUE`, a (R * C) `\times` sample matrix saved as an `mcmc` object. Otherwise, a R `\times` C `\times` sample array If the model is fit without a covariate, the list includes: `Alpha` Posterior draws for alpha parameters. If `ret.mcmc = TRUE`, a (R * C) `\times` sample matrix saved as an `mcmc` object. Otherwise, a R `\times` C `\times` sample array `Beta` Posterior draws for beta parameters. If `ret.mcmc = TRUE`, a (R * C * units) `\times` sample matrix saved as an `mcmc` object. Otherwise, a R `\times` C `\times` units `\times` sample array `Cell.count` Posterior draws for the cell counts, summed across units. If `ret.mcmc = TRUE`, a (R * C) `\times` sample matrix saved as an`mcmc` object. Otherwise, a R `\times` C `\times` sample array
`acc.ratios`	A list containing acceptance ratios for the parameters. If the model includes a covariate, the list includes: `dr.acc` A vector of acceptance ratios for `Dr` draws `beta.acc` A vector of acceptance ratios for `Beta` draws `gamma.acc` A vector of acceptance ratios for `Gamma` and `Delta` draws If the model is fit without a covariate , the list includes: `alpha.acc` A vector of acceptance ratios for `Alpha` draws `beta.acc` A vector of acceptance ratios for `Beta` draws
`usrfun`	Output from the optional `usrfn`
`call`	Call to `ei.MD.bayes`

Author(s)

Michael Kellermann <mrkellermann@gmail.com> and Olivia Lau <olivia.lau@post.harvard.edu>

References

Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). https://CRAN.R-project.org/package=coda.

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