Multiple Dependent Hypothesis Tests in Geographically Weighted Regression

Geographically weighted regression (Fotheringham et al., 2002) is a method of modelling spatial variability in regression coefficients. The procedure yields a separate model for each spatial location in the study area with all models generated from the same data set using a differential weighting scheme. The weighting scheme, which allows for spatial variation in the model parameters, involves a bandwidth parameter which is usually deter- mined from the data using a cross-validation procedure. Part of the main output is a set of location-specific parameter estimates and associated t statistics which can be used to test hypotheses about individual model parameters. If there are n spatial locations and p parameters in each model, there will be up to np hypotheses to be tested which in most applications defines a very high order multiple inference problem. Solutions to problems of this type usually involve an adjustment to the decision rule for individual tests designed to contain the overall risk of mistaking chance variation for a genuine effect. An undesir- able by-product of achieving this control is a reduction in statistical power for individual tests, which may result in genuine effects going undetected. These two competing aspects of multiple inference have become known as the multiplicity problem. In this paper we develop a simple Bonferroni style adjustment for testing multiple hypotheses about GWR model coefficients. The adjustment takes advantage of the intrinsic dependency between local GWR models to contain the overall risk mentioned above, without the large sacrifice in power associated with the traditional Bonferroni correction. We illustrate this adjustment and a range of other corrective procedures on two data sets. The first models the determinants of educational attainment in the counties of Georgia USA. Using area based census data we examine the links between levels of educational attainment and four potential predictors: the proportion of elderly, the proportion who are foreign born, the proportion living below the poverty line and the proportion of ethnic blacks. The second model is a geographically weighted hedonic house price model based on individual mortgage records in Greater London in 1990. In both models we show how the various corrections can be used to guide the interpretation of the spatial variations in the parameter estimates. Finally we compare the statistical power of the proposed method with Bonferroni/Sidak corrections and those based on false discovery rate control