An Evaluation of Spatial Autocorrelation and Heterogeneity in the Residuals of Six Regression Models

Abstract

Abstract Spatial effects include spatial autocorrelation and heterogeneity. Ignoring spatial effects in a modeling process causes misleading significance tests and suboptimal model prediction. In this study, we used three forest plots with different spatial patterns of tree locations (i.e., clustered, random, and regular patterns) to investigate the spatial distributions and heterogeneity in the model residuals from six regression models with the ordinary least squares (OLS) as the benchmark. Our results revealed that when significant spatial autocorrelations and variations existed in the relationship between tree height and diameter, as in the softwood plot (clustered) and hardwood plot (random), OLS was not appropriate for modeling the relationship between tree variables. Spatial regression models (i.e., spatial lag and spatial error models) were effective for accounting for spatial autocorrelation in the model residuals, but they were insufficient to deal with the problem of spatial heterogeneity. It was evident that the model residuals in both spatial lag and spatial error models had a similar pattern and magnitudes of spatial heterogeneity at spatial scales different from those of the OLS model. In contrast, the linear mixed model and geographically weighted regression incorporated the spatial dependence and variation into modeling processes, and consequently, fitted the data better and predicted the response variable more accurately. The model residuals from both the linear mixed model and geographically weighted regression had desirable spatial distributions, meaning fewer clusters of similar or dissimilar model residuals over space.