An Empirical Investigation of the Merits of a Class of Analytically Tractable Matern Covariance Structures in Spatial Data Analysis

I investigate, using the R package spaMM, the effect of misspecification of the smoothing parameter, Q, of the Matern covariance structure on the mean part of hierarchical generalised linear models (HGLMs) with spatially correlated Gaussian Matern random effects. In particular, by restricting Q to the set {0.5, 1.5, 2.5} I examine via a simulation study the amount of bias introduced on the fixed effects estimates in which the data used to fit the model was generated with different values to the aforementioned set. The effect of misspecification was found to be minimal. By restricting the smoothing parameter, Q, to the set {0.5, 1.5, 2.5} I utilise the R package hglm, to develop a procedure (MaternHGLM) for fitting spatial Matern HGLMs. In particular, I constructed a hierarchical likelihood (h-likelihood) function with given correlation parameters which thus enabled me to Choleski decompose the Matern covariance matrix and utilise hglm to estimate fixed and random effects along with dispersion parameters. Using the above estimated parameters I then formed an adjusted profile h-likelihood for the estimation of the Matern scaling parameter, U, using the Newton-Raphson procedure. Simulation studies were carried out to assess the computational efficiency of MaternHGLM compared to spaMM. I found that, on average, MaternHGLM was 136% faster than spaMM. I also analysed two real world datasets using both spaMM and MaternHGLM. By fixing Q at the most appropriate value from the set {0.5, 1.5, 2.5} I examined to what extent, if any, did the conclusions drawn differ from those in the original study. I found that in general the conclusions were the same, however, on one of the datasets spaMM’s conclusion didn’t align at all with the original analysis even with Q estimated from the data