The Latent Scale Covariogram: A Tool for Exploring the Spatial Dependence Structure of Nonnormal Responses

Abstract

ABSTRACT When modeling spatially distributed normal responses Yi in terms of vectors xi of explanatory variables, one may fit a linear model assuming independence, and then use the empirical variogram of the residuals to determine an appropriate parametric form for the autocorrelation function. Suppose, however, that the responses are not normally distributed—for example, Poisson or Bernoulli. One may model spatial dependence using a hierarchical generalized linear model in which, conditional on a latent Gaussian field Z = {Zi}, the Yi have independent distributions from the exponential family, with an appropriate link function connecting their conditional means with the linear predictors xtiβ + Zi. The question then is how to determine an appropriate model for the autocorrelation function of Z. The empirical variogram of the Yi is no longer appropriate, since (unless the link function is the identity) it is on the wrong scale. We propose here an alternative, the latent scale covariogram, whose graph reflects the autocorrelation structure of the underlying normal field. We illustrate its use on several real datasets, together with a simulated dataset, and obtain results quite different from those obtained using the variogram. Supplementary materials for this article are available online.