Modeling and inference for spatial processes with ordinal data.

Abstract

This research deals with some methods for modeling and analyzing spatially dependent ordered categorical data. Such data arise in many areas of application. We view the ordinal data {<italic>Yi</italic>, <italic>i</italic> = 1, ..., <italic> n</italic>} as arising from a latent continuous-valued spatial process {<italic> X_i, i</italic> = 1, ..., <italic>n</italic>}. Specifically, the spatial data {<italic>Y_i</italic>} are assumed to be generated from {<italic>X_i</italic>} as follows: <italic>Y_i</italic> = <italic>k</italic> if theta<italic>_k</italic>_-1 < <italic>X_i</italic> &le; theta<italic>_k</italic>, where {theta<italic>_k</italic>} are unknown thresholds or cut points. A first-order Gaussian Markov random field is used to model the latent data; this induces spatial dependence in the ordinal data. Both the homogeneous case as well as multisample and spatial regression problems are considered. The goal is to estimate the underlying cell probabilities, the latent measure of dependence, and the regression parameters. Maximum likelihood estimation is computationally intractable, so we consider alternative methods based on a pseudo-likelihood (PLE) and two other approximations to the likelihood (MnE and MdE). The pseudo-likelihood method leads to unbiased estimating equations. Large-sample properties of the PLE are derived. Simulations results show that the performances of MnE and MdE are comparable to the PLE in terms of bias and relative efficiency. We also develop Bayesian methods of inference using Gibbs sampling and compare them to MLE's obtained through stochastic E-M. The results are applied to some data from semiconductor manufacturing.