cle.rord {clordr}

Composite Likelihood Estimation for Replciations of Spatial Ordinal Data

Description

cle.rord Estimate parameters (including regression coefficient and cutoff) for replications of spatial ordinal data using pairwise likelihood approach.

Usage

cle.rord(
  response,
  covar,
  location,
  radius = 4,
  n.sim = 100,
  output = TRUE,
  SE = TRUE,
  parallel = FALSE,
  n.core = max(detectCores()/2, 1),
  ini.sp = c(0.5, 0.5),
  est.method = TRUE,
  maxiter = 100,
  rtol = 1e-06,
  factr = 1e+07
)

Arguments

response

a matrix of observation (row: spatial site and column: subject).

covar

regression (design) matrix, including intercepts.

location

a matrix contains spatial location of sites within each subject.

radius

radius for selecting pairs for the composite likelihood estimation.

n.sim

number of simulation used for parametric bootstrapping (and hence used for asymptotic variance and standard error).

output

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

SE

logical flag for detailed output.

parallel

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

n.core

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

ini.sp

initial estimate for spatial parameter, \phi,\sigma^2 (default: c(0.5,0.5)).

est.method

logical flag (default) TRUE for rootsolve and FALSE for L-BFGS-B.

maxiter

maximum number of iterations in the root solving of gradient function (dafault: 100).

rtol

relative error tolerrance in the root solving of gradient function (default: 1e-6).

factr

reduction in the objective (-logCL) within this factor of the machine tolerance for L-BFGS-B (default: 1e7).

Details

Given vector of ordinal responses, the design matrix, spatial location for sites, weight radius (for pair selection), and the prespecified number of simulation used for estimating the Godambe information matrix. Initial estimate is obtained by fitting model without spatial dependence (using MASS::polr()) and optional guess of spatial parameters. The function first estimates parameters of interest by either solving the gradient of composite log-likelihood using rootSolve::multiroot() or maximize the composite log-likelihood by optim(..., method="L-BFGS-B"). The asymptotic covariance matrix and standard error of parameters are then estimated by parametric boostrapping. Although the default root solving option is typically more efficient, it may encounter runtime error if negative value of \phi is evaluated (and L-BFGS-B approach should be used).

Value

cle.rord returns a list contains:

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

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;

mat.Hessian: Hessian matrix at the parameter estimate;

mat.J: Sensitivity matrix estimated by parametric boostrapping; and

CLIC: Composite likelihood information criterion (see help manual of clic() for detail).

Examples

set.seed(1228)
n.subject <- 20
n.lat <- n.lon <- 10
n.site <- n.lat*n.lon

beta <- c(1,2,-1) # First 1 here is the intercept
midalpha <- c(1.15, 2.18) ; phi <- 0.6 ; sigma2 <- 0.7

true <- c(midalpha,beta,phi,sigma2)

Xi <- rnorm(n.subject,0,1) ; Xj <- rbinom(n.site,1,0.6)

 VV <- matrix(NA, nrow = n.subject*n.site, ncol = 3)

 for(i in 1:n.subject){ for(j in 1:n.site){
     VV[(i-1)*n.site+j,] <- c(1,Xi[i],Xj[j])
       }
 }

location <- cbind(rep(seq(1,n.lat,length=n.lat),n.lat),rep(1:n.lon, each=n.lon))
sim.data <- sim.rord(n.subject, n.site, n.rep = 2, midalpha, beta, phi, sigma2, covar=VV, location)


options(digits=3)
result <- cle.rord(response=sim.data[[1]], covar=VV,
          location = location ,radius = 4, n.sim = 100, output = TRUE, parallel=TRUE, n.core =2)
result$vec.par
# alpha2  alpha3   beta0   beta1   beta2     phi sigma^2
# 1.249   2.319   1.169   1.990  -1.000   0.668   0.678

result$vec.se
# alpha2  alpha3   beta0   beta1   beta2     phi sigma^2
# 0.0704  0.1201  0.1370  0.2272  0.0767  0.0346  0.1050