ei.MD.bayes | R Documentation |
Implements a version of the hierarchical model suggested in Rosen et al. (2001)
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)
formula |
A formula of the form |
covariate |
An optional formula of the form |
total |
if row and/or column marginals are given as proportions,
|
data |
A data frame containing the variables specified in
|
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 |
start.list |
A list containing starting values for each block of
parameters. See ‘details’ for more information. The default is
|
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 |
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 |
ret.beta |
A character indicating how the posterior draws of beta should be
handled: ' |
ret.mcmc |
A logical value indicating how the samples from the posterior
should be returned. If |
usrfun |
the name of an optional a user-defined function to obtain quantities of
interest while drawing from the MCMC chain (defaults to |
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:
(N_.1i,..., N_.Ci) ~ Multinomial(N_i,sum_{r=1}^R(beta_r1i*X_ri), ..., 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, ..., beta_rCi) ~ Dirichlet(alpha_r1, ..., alpha_rC)
In this case, the prior on each alpha_rc is assumed to be:
alpha_rc ~ Gamma(lambda_1, lambda_2
If the model is fit with a covariate, the distribution of the vector beta_ri is :
(beta_r1i, ..., beta_rCi) ~ 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 ~ 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} ~ N(mu_{delta_{rC}}, sigma_{delta_{rC}}^2)
gamma_{rC} ~ 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 x (C-1) matrix of
prior means for Delta
sigma.delta
an R x (C-1) matrix of
prior standard deviations for Delta
mu.gamma
an R x (C-1) matrix of
prior means for Gamma
sigma.gamma
an 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.dr
a vector of length R of starting values for Dr.
Starting values for Dr must be greater than zero.
start.betas
an R x C by precincts array
of starting values for Beta. Each row of every precinct must sum to 1.
start.gamma
an R x C matrix of starting
values for Gamma. Values in the right-most column must be zero.
start.delta
an 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.alphas
an R x C matrix of starting values for Alpha. Starting values for Alpha must be greater than zero.
start.betas
an 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.dr
a vector of length R of tuning parameters for Dr
tune.beta
an R x (C-1) by precincts array
of tuning parameters for Beta
tune.gamma
an R x (C-1) matrix of tuning
parameters for Gamma
tune.delta
an 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.alpha
an R x C matrix of tuning parameters for Alpha
tune.beta
an R x (C-1) by precincts array
of tuning parameters for Beta
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:
If the model is fit without a covariate, the list includes:
|
acc.ratios |
A list containing acceptance ratios for the parameters. If the model includes a covariate, the list includes:
If the model is fit without a covariate , the list includes:
|
usrfun |
Output from the optional |
call |
Call to |
Michael Kellermann <mrkellermann@gmail.com> and Olivia Lau <olivia.lau@post.harvard.edu>
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 x (C-1) Case.” Statistica Neerlandica 55: 134-156.
lambda.MD
, cover.plot
,
density.plot
, tuneMD
,
mergeMD
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