On Null Hypotheses and Heteroskedasticity
Bivand and Wong (2018), a recent review on spatial statistical software, noted important differences in the results of the local Moran’s Ii statistic depending on the method of inference. That review speculated the differences may be due to the presence of local spatial heterogeneity. In this paper we design an experiment to assess the impact of local heterogeneity on hypothesis testing for local statistics. In this experiment, we analyze the relationship between measures of local variance, such as the local spatial heteroskedasticity (LOSH) statistic, and components of the local Moran’s Ii statistic. We consider this experiment with controlled synthetic heteroskedastic data and with uncontrolled real world data. We show that in both situations the variance components of the local Moran’s Ii statistic demonstrate a varying correlation with alternative measures of local variance like LOSH. In addition, we resituate the available inferential methods and suggest an alternative explanation for the differences observed in Bivand and Wong 2018. Ultimately, this paper demonstrates that there are important conceptual and computational differences as to what constituents a null hypothesis in local testing frameworks. Therefore, researchers must be aware as to how their choices may shape the observed spatial patterns.