# Composite Likelihood Estimation for Spatial Proportional Data `func.cle.prop` performs composite likelihood estimation of parameters and their standard errors in a spatial Tobit model by maximizing its composite log-likelihood. ```r func.cle.prop(vec.yobs, mat.X, mat.lattice, radius, 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 i-th row constitutes a set of geographical coordinates. - `radius`: weight radius. - `n.sim`: number of simulations used for calculate the Godambe matrix (default: 100). - `parallel`: logical flag indicating 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`). ## Returns `func.cle.prop` returns a list containing: `vec.par`: a vector of estimator for $\theta=(\alpha,\beta,\sigma^2,\rho)$; `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. $-2*logCL(\theta) + 2*trace(H^{-1}(\theta)J(\theta))$ ## Details Given the design matrix, the vector of observed responses, spatial lattice data, weight radius, and the prespecified number of simulated vectors of responses used in estimating the Godambe information matrix, this function assumes initial values of $\beta$ as the estimates from the standard Type I Tobit model with independent responses. The initial value of $\alpha$ and the right limit of the Tobit model are equally set to 1. Since there is only one cutoff point to be estimated, reparameterization is unnecessary. The function first estimates parameters of interest by maximizing the composite log-likelihood using `optim(...,method = "L-BFGS-B")`, then computes the simulated based standard error and asymptotic covariance matrix. ## Examples ```r # True parameter alpha <- 4; vec.beta <- c(1, 2, 1, 0, -1); sigmasq <- 0.8; rho <- 0.6; radius <- 5 vec.par <- c(alpha, vec.beta, sigmasq, rho) # Coordinate matrix 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.prop(vec.ylat, alpha=alpha) # With parallel computing ## Not run: prop.example <- func.cle.prop(vec.yobs, mat.X, mat.lattice, radius, n.sim=100, parallel = TRUE, n.core = 2) round(prop.example$vec.par,4) # alpha beta0 beta1 beta2 beta3 beta4 sigma^2 rho # 3.8259 0.9921 1.9679 0.9455 0.0148 -0.9871 0.8386 0.5761 round(prop.example$vec.se ,4) # alpha beta0 beta1 beta2 beta3 beta4 sigma^2 rho # 0.1902 0.1406 0.1103 0.0744 0.0385 0.0652 0.1527 0.1151 ## End(Not run) # Without parallel computing ## Not run: prop.example2 <- func.cle.prop(vec.yobs, mat.X, mat.lattice, radius, n.sim=100, parallel = FALSE) ## End(Not run) ``` ## 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.

func.cle.prop function