Estimation and prediction for spatial generalized linear mixed models with parametric links via reparameterized importance sampling

TL;DR

This paper introduces a novel approach for estimating and predicting spatial generalized linear mixed models with flexible parametric link functions, improving fit and inference through reparameterized importance sampling and model selection techniques.

Contribution

It proposes a new generalized importance sampling estimator with reparameterization for empirical Bayes analysis of SGLMMs, enabling flexible link function estimation and model selection.

Findings

Enhanced model fit with parametric links over prescribed links.

Efficient estimation of link function parameters using reparameterized importance sampling.

Improved predictive performance demonstrated on simulated and real data.

Abstract

Spatial generalized linear mixed models (SGLMMs) are popular for analyzing non-Gaussian spatial data. These models assume a prescribed link function that relates the underlying spatial field with the mean response. There are circumstances, such as when the data contain outlying observations, where the use of a prescribed link function can result in poor fit, which can be improved by using a parametric link function. Some popular link functions, such as the Box-Cox, are unsuitable because they are inconsistent with the Gaussian assumption of the spatial field. We present sensible choices of parametric link functions which possess desirable properties. It is important to estimate the parameters of the link function, rather than assume a known value. To that end, we present a generalized importance sampling (GIS) estimator based on multiple Markov chains for empirical Bayes analysis of SGLMMs. The GIS estimator, although more efficient than the simple importance sampling, can be highly variable when used to estimate the parameters of certain link functions. Using suitable reparameterizations of the Monte Carlo samples, we propose modified GIS estimators that do not suffer from high variability. We use Laplace approximation for choosing the multiple importance densities in the GIS estimator. Finally, we develop a methodology for selecting models with appropriate link function family, which extends to choosing a spatial correlation function as well. We present an ensemble prediction of the mean response by appropriately weighting the estimates from different models. The proposed methodology is illustrated using simulated and real data examples.