Adjacency matrix for the infant mortality data.

Description

A contains the adjacency matrix for the infant mortality data analyzed in (Hughes and Haran, 2013).

Usage

data(A)

Source

The data were obtained from the 2008 Area Resource File (ARF), a county-level database maintained by the Bureau of Health Professions, Health Resources and Services Administration, US Department of Health and Human Services.

Examples

data(A)

Return an adjacency matrix for a square lattice.

Description

Return an adjacency matrix for a square lattice.

Usage

adjacency.matrix(m, n = NULL)

Arguments

Details

This function builds the adjacency matrix for the m by n square lattice.

Value

A matrix A of 0s and 1s, where A_{ij} is equal to 1 iff vertices i and j are adjacent.

Fit a centered autologistic model using maximum pseudolikelihood estimation or MCMC for Bayesian inference.

Description

Fit a centered autologistic model using maximum pseudolikelihood estimation or MCMC for Bayesian inference.

Usage

autologistic(formula, data, A, method = c("PL", "Bayes"), model = TRUE,
  x = FALSE, y = FALSE, verbose = FALSE, control = list())

Arguments

Details

This function fits the centered autologistic model of Caragea and Kaiser (2009) using maximum pseudolikelihood estimation or MCMC for Bayesian inference. The joint distribution for the centered autologistic model is

\pi(Z\mid\theta)=c(\theta)^{-1}\exp\left(Z^\prime X\beta - \eta Z^\prime A\mu + \frac{\eta}{2}Z^\prime AZ\right),

where \theta = (\beta^\prime, \eta)^\prime is the parameter vector, c(\theta) is an intractable normalizing function, Z is the response vector, X is the design matrix, \beta is a (p-1)-vector of regression coefficients, A is the adjacency matrix for the underlying graph, \mu is the vector of independence expectations, and \eta is the spatial dependence parameter.

Maximum pseudolikelihood estimation sidesteps the intractability of c(\theta) by maximizing the product of the conditional likelihoods. Confidence intervals are then obtained using a parametric bootstrap or a normal approximation, i.e., sandwich estimation. The bootstrap datasets are generated by perfect sampling (rautologistic). The bootstrap samples can be generated in parallel using the parallel package.

Bayesian inference is obtained using the auxiliary variable algorithm of Moller et al. (2006). The auxiliary variables are generated by perfect sampling.

The prior distributions are (1) zero-mean normal with independent coordinates for \beta, and (2) uniform for \eta. The common standard deviation for the normal prior can be supplied by the user as control parameter sigma. The default is 1,000. The uniform prior has support [0, 2] by default, but the right endpoint can be supplied (as control parameter eta.max) by the user.

The posterior covariance matrix of \theta is estimated using samples obtained during a training run. The default number of iterations for the training run is 100,000, but this can be controlled by the user (via parameter trainit). The estimated covariance matrix is then used as the proposal variance for a Metropolis-Hastings random walk. The proposal distribution is normal. The posterior samples obtained during the second run are used for inference. The length of the run can be controlled by the user via parameters minit, maxit, and tol. The first determines the minimum number of iterations. If minit has been reached, the sampler will terminate when maxit is reached or all Monte Carlo standard errors are smaller than tol, whichever happens first. The default values for minit, maxit, and tol are 100,000; 1,000,000; and 0.01, respectively.

Value

autologistic returns an object of class “autologistic”, which is a list containing the following components.

References

Caragea, P. and Kaiser, M. (2009) Autologistic models with interpretable parameters. Journal of Agricultural, Biological, and Environmental Statistics, 14(3), 281–300.

Hughes, J., Haran, M. and Caragea, P. C. (2011) Autologistic models for binary data on a lattice. Environmetrics, 22(7), 857–871.

Moller, J., Pettitt, A., Berthelsen, K., and Reeves, R. (2006) An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika, 93(2), 451–458.

See Also

rautologistic, residuals.autologistic, summary.autologistic, vcov.autologistic

Examples

# Use the 20 x 20 square lattice as the underlying graph.

n = 20
A = adjacency.matrix(n)

# Assign coordinates to each vertex such that the coordinates are restricted to the unit square
# centered at the origin.

x = rep(0:(n - 1) / (n - 1), times = n) - 0.5
y = rep(0:(n - 1) / (n - 1), each = n) - 0.5
X = cbind(x, y)                                 # Use the vertex locations as spatial covariates.
beta = c(2, 2)                                  # These are the regression coefficients.
col1 = "white"
col2 = "black"
colfunc = colorRampPalette(c(col1, col2))

# Simulate a dataset with the above mentioned regression component and eta equal to 0.6. This
# value of eta corresponds to dependence that is moderate in strength.

theta = c(beta, eta = 0.6)
set.seed(123456)
Z = rautologistic(X, A, theta)

# Find the MPLE, and do not compute confidence intervals.

fit = autologistic(Z ~ X - 1, A = A, control = list(confint = "none"))
summary(fit)

# The following examples are not executed by default since the computation is time consuming.

# Compute confidence intervals based on the normal approximation. Estimate the "filling" in the
# sandwich matrix using a parallel parametric bootstrap, where the computation is distributed
# across six cores on the host workstation.

# set.seed(123456)
# fit = autologistic(Z ~ X - 1, A = A, verbose = TRUE,
#                   control = list(confint = "sandwich", nodes = 6))
# summary(fit)

# Compute confidence intervals based on a parallel parametric bootstrap. Use a bootstrap sample
# of size 500, and distribute the computation across six cores on the host workstation.

# set.seed(123456)
# fit = autologistic(Z ~ X - 1, A = A, verbose = TRUE,
#                   control = list(confint = "bootstrap", bootit = 500, nodes = 6))
# summary(fit)

# Do MCMC for Bayesian inference. The length of the training run will be 10,000, and
# the length of the subsequent inferential run will be at least 10,000.

# set.seed(123456)
# fit = autologistic(Z ~ X - 1, A = A, verbose = TRUE, method = "Bayes",
#                   control = list(trainit = 10000, minit = 10000, sigma = 1000))
# summary(fit)

Infant mortality data.

Description

infant contains the infant mortality data analyzed in (Hughes and Haran, 2013).

Usage

data(infant)

Source

The data were obtained from the 2008 Area Resource File (ARF), a county-level database maintained by the Bureau of Health Professions, Health Resources and Services Administration, US Department of Health and Human Services.

Examples

data(infant)

Family function for negative binomial GLMs.

Description

Provides the information required to apply a sparse SGLMM to conditionally negative binomial outcomes.

Usage

negbinomial(theta = stop("'theta' must be specified."), link = "log")

Arguments

Value

An object of class “family”, a list of functions and expressions needed to fit a negative binomial GLM.

Return a perfect sample from a centered autologistic model.

Description

Return a perfect sample from a centered autologistic model.

Usage

rautologistic(X, A, theta)

Arguments

Details

This function implements a perfect sampler for the centered autologistic model. The sampler employs coupling from the past.

Value

A vector that is distributed exactly according to the centered autologistic model with the given design matrix and parameter values.

References

Moller, J. (1999) Perfect simulation of conditionally specified models. Journal of the Royal Statistical Society, Series B, Methodological, 61, 251–264.

Propp, J. G. and Wilson, D. B. (1996) Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms, 9(1-2), 223–252.

Examples

# Use the 20 x 20 square lattice as the underlying graph.

n = 20
A = adjacency.matrix(n)

# Assign coordinates to each vertex such that the coordinates are restricted to the unit square
# centered at the origin.

x = rep(0:(n - 1) / (n - 1), times = n) - 0.5
y = rep(0:(n - 1) / (n - 1), each = n) - 0.5
X = cbind(x, y)                                 # Use the vertex locations as spatial covariates.
beta = c(2, 2)                                  # These are the regression coefficients.

# Simulate a dataset with the above mentioned regression component and eta equal to 0.6. This
# value of eta corresponds to dependence that is moderate in strength.

theta = c(beta, eta = 0.6)
set.seed(123456)
Z = rautologistic(X, A, theta)

Extract model residuals.

Description

Extract model residuals.

Usage

## S3 method for class 'autologistic'
residuals(object, type = c("deviance", "pearson",
  "response"), ...)

Arguments

Value

A vector of residuals.

See Also

autologistic, residuals.glm

Extract model residuals.

Description

Extract model residuals.

Usage

## S3 method for class 'sparse.sglmm'
residuals(object, type = c("deviance", "pearson",
  "response"), ...)

Arguments

Value

A vector of residuals.

See Also

sparse.sglmm, residuals.glm

Fit a sparse SGLMM.

Description

Fit a sparse SGLMM.

Usage

sparse.sglmm(formula, family = gaussian, data, offset, A, method = c("BSF",
  "RSR"), attractive = 50, repulsive = 0, tol = 0.01, minit = 10000,
  maxit = 1e+06, tune = list(), hyper = list(), model = TRUE,
  x = FALSE, y = FALSE, verbose = FALSE, parallel = FALSE)

Arguments

Details

This function fits the sparse restricted spatial regression model of Hughes and Haran (2013), or the Bayesian spatial filtering model of Hughes (2017). The first stage of the model is

g(\mu_i)=x_i^\prime\beta+m_i^\prime\gamma\hspace{1 cm}(i=1,\dots,n)

or, in vectorized form,

g(\mu)=X\beta+M\gamma,

where X is the design matrix, \beta is a p-vector of regression coefficients, the columns of M are q eigenvectors of the Moran operator, and \gamma are spatial random effects. Arguments attractive and repulsive can be used to control the number of eigenvectors used. The default values are 50 and 0, respectively, which corresponds to pure spatial smoothing. Inclusion of some repulsive eigenvectors can be advantageous in certain applications.

The second stage, i.e., the prior for \gamma, is

p(\gamma\mid\tau_s)\propto\tau_s^{q/2}\exp\left(-\frac{\tau_s}{2}\gamma^\prime M^\prime QM\gamma\right),

where \tau_s is a smoothing parameter and Q is the graph Laplacian.

The prior for \beta is spherical p-variate normal with mean zero and common standard deviation sigma.b, which defaults to 1,000. The prior for \tau_s is gamma with parameters 0.5 and 2,000. The same prior is used for \theta (when family is negbinomial).

When the response is normally distributed, the identity link is assumed, in which case the first stage is

\mu=X\beta+M\gamma+M\delta,

where \delta are heterogeneity random effects. When the response is Poisson distributed, heterogeneity random effects are optional. In any case, the prior on \delta is spherical q-variate normal with mean zero and common variance 1/\tau_h. The prior for \tau_h is gamma with parameters a_h and b_h, the values of which are controlled by the user through argument hyper.

If the response is Bernoulli, negative binomial, or Poisson, \beta and \gamma are updated using Metropolis-Hastings random walks with normal proposals. The proposal covariance matrix for \beta is the estimated asymptotic covariance matrix from a glm fit to the data (see vcov). The proposal for \gamma is spherical normal with common standard deviation sigma.s.

The updates for \tau_s and \tau_h are Gibbs updates irrespective of the response distribution.

If the response is Poisson distributed and heterogeneity random effects are included, those random effects are updated using a Metropolis-Hastings random walk with a spherical normal proposal. The common standard deviation is sigma.h.

If the response is normally distributed, all updates are Gibbs updates.

If the response is negative binomial, the dispersion parameter \theta is updated using a Metropolis-Hastings random walk with a normal proposal. Said proposal has standard deviation sigma.theta, which can be provided by the user as an element of argument tune.

Value

sparse.sglmm returns an object of class “sparse.sglmm”, which is a list containing the following components.

References

Hughes, J. and Haran, M. (2013) Dimension reduction and alleviation of confounding for spatial generalized linear mixed models. Journal of the Royal Statistical Society, Series B, 75(1), 139–159.

See Also

residuals.sparse.sglmm, summary.sparse.sglmm, vcov.sparse.sglmm

Examples

## Not run: 

The following code duplicates the analysis described in (Hughes and Haran, 2013). The data are
infant mortality data for 3,071 US counties. We do a spatial Poisson regression with offset.

data(infant)
infant$low_weight = infant$low_weight / infant$births
attach(infant)
Z = deaths
X = cbind(1, low_weight, black, hispanic, gini, affluence, stability)
data(A)
set.seed(123456)
fit = sparse.sglmm(Z ~ X - 1 + offset(log(births)), family = poisson, A = A, method = "RSR",
                   tune = list(sigma.s = 0.02), verbose = TRUE)
summary(fit)

## End(Not run) 

Print a summary of a centered autologistic model fit.

Description

Print a summary of a centered autologistic model fit.

Usage

## S3 method for class 'autologistic'
summary(object, alpha = 0.05, digits = 4, ...)

Arguments

Details

This function displays (1) the call to autologistic, (2) the values of the control parameters, (3) a table of estimates, and (4) the size of the bootstrap/posterior sample. Each row of the table of estimates shows a parameter estimate, the confidence or HPD interval for the parameter, and, where applicable, the Monte Carlo standard error.

See Also

autologistic

Print a summary of a sparse SGLMM fit.

Description

Print a summary of a sparse SGLMM fit.

Usage

## S3 method for class 'sparse.sglmm'
summary(object, alpha = 0.05, digits = 4, ...)

Arguments

Details

This function displays (1) the call to sparse.sglmm, (2) the values of the hyperparameters and tuning parameters, (3) a table of estimates, (4) the DIC value for the fit, and (5) the number of posterior samples. Each row of the table of estimates shows an estimated regression coefficient, the HPD interval for the coefficient, and the Monte Carlo standard error.

See Also

sparse.sglmm

Return the estimated covariance matrix for an autologistic model object.

Description

Return the estimated covariance matrix for an autologistic model object.

Usage

## S3 method for class 'autologistic'
vcov(object, ...)

Arguments

Value

An estimate of the covariance matrix of the parameters (in a Bayesian setting), or an estimate of the covariance matrix of the maximum pseudolikelihood estimator of the parameters.

Return the covariance matrix of the regression parameters of a sparse.sglmm model object.

Description

Return the covariance matrix of the regression parameters of a sparse.sglmm model object.

Usage

## S3 method for class 'sparse.sglmm'
vcov(object, ...)

Arguments

Value

An estimate of the posterior covariance matrix of the regression coefficients.