func.cle.ord {clespr}

Composite Likelihood Estimation for Spatial Ordinal Data

Description

func.cle.ord performs composite likelihood estimation of parameters and their standard errors in a spatial ordered probit model by maximizing its composite log-likelihood.

Usage

func.cle.ord(vec.yobs, mat.X, mat.lattice, radius, n.cat, n.sim = 100,
  parallel = TRUE, n.core = max(detectCores()/2, 1), output = TRUE)

Arguments

vec.yobs

a vector of observed responses for all N sites.

mat.X

regression (design) matrix, including intercepts.

mat.lattice

a data matrix containing geographical information of sites. The ith row constitutes a set of geographical coordinates.

radius

weight radius.

n.cat

number of categories.

n.sim

number of simulations used for calculate the Godambe matrix (default: 100).

parallel

logical flag indicates using parallel processing (default: TRUE).

n.core

number of physical cores used for parallel processing (when parallel is TRUE, default value is max(detectCores()/2,1)).

output

logical flag indicates whether printing out result (default: TRUE).

Details

Given the design matrix, the vector of observed responses, spatial lattice data, weight radius, number of categories, and the prespecified number of simulated vectors of responses used in estimating the Godambe information, this function assumes initial values of cutoff points and \beta as the estimates from the standard ordered probit regression with independent responses. After initial reparameterization, it first estimates parameters of interest by maximizing the composite log-likelihood using optim, then computes the reparameterized sample covariance matrix and the set of standard errors, and finally reverse the reparameterization to obtain estimates corresponding to the original parameterization.

Value

func.cle.ord returns a list containing:

vec.par: a vector of estimator for \theta=(cutoff,\beta,\sigma^2,\rho);

vec.se: a vector of standard error for the estimator;

mat.asyvar: estimated asymptotic covariance matrix H^{-1}(\theta)J(\theta)H^{-1}(\theta) for the estimator; and

vec.comp: a vector of computational time for parameter and standard error estimation.

CLIC: Composite likelihood information criterion proposed by Varin and Vidoni (2005), i.e. -2*logCL(\theta) + 2*trace(H^{-1}(\theta)J(\theta))

References

Feng, Xiaoping, Zhu, Jun, Lin, Pei-Sheng, and Steen-Adams, Michelle M. (2014) Composite likelihood Estimation for Models of Spatial Ordinal Data and Spatial Proportional Data with Zero/One values. Environmetrics 25(8): 571–583.

Examples

# Example of n.cat = 3 (Spatial ordinal regression)
# True parameter
vec.cutoff <- 2; vec.beta <- c(1, 2, 1, 0, -1); sigmasq <- 0.8; rho <- 0.6; radius <- 5
vec.par <- c(vec.cutoff, vec.beta, sigmasq, rho)

# Coordinate matrix
n.cat <- 3; n.lati <- 30; n.long <- 30
n.site <- n.lati * n.long
mat.lattice <- cbind(rep(1:n.lati, n.long), rep(1:n.long, each=n.lati))
mat.dist <- as.matrix(dist(mat.lattice, upper=TRUE, diag=TRUE))
mat.cov <- sigmasq * rho^mat.dist

set.seed(1228)
# Generate regression (design) matrix with intercept
mat.X <- cbind(rep(1, n.site),scale(matrix(rnorm(n.site*(length(vec.beta)-1)),nrow=n.site)))
vec.Z <- t(chol(mat.cov)) %*% rnorm(n.site) + mat.X %*% vec.beta
vec.epsilon <- diag(sqrt(1-sigmasq), n.site) %*% rnorm(n.site)
vec.ylat <- as.numeric(vec.Z + vec.epsilon)

# Convert to the vector of observation
vec.yobs <- func.obs.ord(vec.ylat, vec.alpha=c(-Inf,0,vec.cutoff,Inf))

# With parallel computing

## Not run: 
ord.example <- func.cle.ord(vec.yobs, mat.X, mat.lattice, radius, n.cat,
n.sim=100, parallel = TRUE, n.core = 2)

round(ord.example$vec.par,4)
# alpha1   beta0   beta1   beta2   beta3   beta4 sigma^2     rho
# 1.8395  0.9550  1.9690  0.9565  0.0349 -1.0398  0.8200  0.5578

round(ord.example$vec.se,4)
# alpha1   beta0   beta1   beta2   beta3   beta4 sigma^2     rho
# 0.1602  0.1222  0.1463  0.0916  0.0485  0.0889  0.1935  0.1267

## End(Not run)

# Without parallel computing

## Not run: 
ord.example2 <- func.cle.ord(vec.yobs, mat.X, mat.lattice, radius,
n.cat, n.sim=100, parallel = FALSE)

## End(Not run)

# Example for n.cat = 2 (i.e. Spatial probit regression)
# True parameter
vec.beta <- c(1, 2, 1, 0, -1); sigmasq <- 0.5; rho <- 0.6; radius <- 5
vec.par <- c(vec.beta, sigmasq, rho)

# Coordinate matrix
n.cat <- 2 ; n.lati <- n.long <- 40
n.site <- n.lati * n.long
mat.lattice <- cbind(rep(1:n.lati, n.long), rep(1:n.long, each=n.lati))
mat.dist <- as.matrix(dist(mat.lattice, upper=TRUE, diag=TRUE))
mat.cov <- sigmasq * rho^mat.dist

set.seed(123)
# Generate regression (design) matrix with intercept
mat.X <- cbind(rep(1, n.site),scale(matrix(rnorm(n.site*(length(vec.beta)-1)),nrow=n.site)))
vec.Z <- t(chol(mat.cov)) %*% rnorm(n.site) + mat.X %*% vec.beta
vec.epsilon <- diag(sqrt(1-sigmasq), n.site) %*% rnorm(n.site)
vec.ylat <- as.numeric(vec.Z + vec.epsilon)
# Convert to the vector of observation
vec.yobs <- func.obs.ord(vec.ylat, vec.alpha=c(-Inf,0,Inf))

## Not run: 
probit.example <- func.cle.ord(vec.yobs, mat.X, mat.lattice, radius, n.cat,
n.sim=100, parallel = TRUE, n.core = 4)

round(probit.example$vec.par,4)
# beta0   beta1   beta2   beta3   beta4 sigma^2     rho
# 1.0427  2.2250  1.0422  0.0156 -1.1489  0.4402  0.6636

round(probit.example$vec.se,4)
# beta0   beta1   beta2   beta3   beta4 sigma^2     rho
# 0.1198  0.1413  0.0863  0.0523  0.0935  0.1600  0.1263

## End(Not run)