The Importance of Null Hypotheses: Understanding Differences in Local Moran’s under Heteroskedasticity

A recent review noted important differences in the results of the local Moran's statistic depending on the inference method. These differences had significant practical implications. In closing, the authors speculated the differences may be due to local spatial heterogeneity. In this article, we propose that different null hypotheses, not heteroskedasticity, generate these differences. To test this, we examine the null hypotheses implicit in common statistical significance tests of local Moran’s . We design an experiment to assess the impact of local heterogeneity on tests conducted under the two most common null hypotheses. 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 statistic. We run this experiment with controlled synthetic heteroskedastic data and with uncontrolled real-world data with varying degrees and patterns of local heteroskedasticity. We show that, in both situations, estimates that use the same null are extremely similar, regardless of estimation method. In contrast, all estimates (regardless of the null) are moderately affected by spatial heteroskedasticity. Ultimately, this article demonstrates that there are important conceptual and computational differences about null hypothesis in local testing frameworks, and these differences can have significant practical implications. Therefore, researchers must be aware as to how their choices may shape the observed spatial patterns.