Controlling the False Discovery Rate: A New Application to Account for Multiple and Dependent Tests in Local Statistics of Spatial Association
de Castro, Marcia Caldas, Singer, Burton H., Geographical Analysis
Assessing the significance of multiple and dependent comparisons is an important, and often ignored, issue that becomes more critical as the size of data sets increases. If not accounted for, false-positive differences are very likely to be identified. The need to address this issue has led to the development of a myriad of procedures to account for multiple testing. The simplest and most widely used technique is the Bonferroni method, which controls the probability that a true null hypothesis is incorrectly rejected. However, it is a very conservative procedure. As a result, the larger the data set the greater the chances that truly significant differences will be missed. In 1995, a new criterion, the false discovery rate (FDR), was proposed to control the proportion of false declarations of significance among those individual deviations from null hypotheses considered to be significant. It is more powerful than all previously proposed methods. Multiple and dependent comparisons are also fundamental in spatial analysis. As the number of locations increases, assessing the significance of local statistics of spatial association becomes a complex matter. In this article we use empirical and simulated data to evaluate the use of the FDR approach in appraising the occurrence of clusters detected by local indicators of spatial association. Results show a significant gain in identification of meaningful clusters when controlling the FDR, in comparison to more conservative approaches. When no control is adopted, false clusters are likely to be identified. If a conservative approach is used, clusters are only partially identified and true clusters are largely missed. In contrast, when the FDR approach is adopted, clusters are fully identified. Incorporating a correction for spatial dependence to conservative methods improves the results, but not enough to match those obtained by the FDR approach.
Tobler's First Law of Geography says that "everything is related to everything else, but near things are more related than distant things" (Tobler 1979). This law applies to any phenomena that have a spatial nature, with considerable implications for studies in disciplines such as sociology, demography, economics, epidemiology, urban planning, ecology, biology, archeology, and, of course, geography. The statistical investigation of these phenomena has been called spatial data analysis (Bailey and Gatrell 1995). The objectives are identification of the spatial distribution of the data, spatial patterns, and the occurrence of outliers (Anselin 1996). A spatial arrangement can be clustered, dispersed, or random depending on the observed spatial dependence (also referred to as spatial autocorrelation or spatial association). Measures of spatial association can be global or local. Global measures consider all available locations simultaneously, utilizing a single statistic that summarizes the spatial pattern. However, the larger the number of locations, the less will be the interpretability of the statistic, as a spatial pattern can vary substantially by location. Local measures represent the association between each location and its neighbors based on defined distances. One statistic is provided for each location, facilitating the identification of clusters, testing of stationarity assumptions, and inference about distances over which spatial association occurs (Getis and Ord 1996). Anselin (1995) proposed criteria to classify a statistic within a class of local indicators of spatial association (LISA).
Local statistics rely on tests of spatial association for each location in the data, and the issue of multiple comparisons is a concern when assessing their significance (Kurtz et al. 1965; Miller 1981; Tukey 1991). In other words, if multiple inferences (tests) are drawn from a given data set, the selection of statistically significant effects/differences is carried out utilizing formal multiple comparison methods. …