Inference for spatial regression models with functional response using a permutational approach

The aim of this work is to introduce an approach to null hypothesis significance testing in a functional linear model for spatial data. The proposed method is capable of dealing with the spatial structure of data by building a permutation testing procedure on spatially filtered residuals of a spatial regression model. Indeed, due to the spatial dependence existing among the data, the residuals of the regression model are not exchangeable, breaking the basic assumptions of the Freedman and Lane permutation scheme. Instead, it is proposed here to estimate the variance–covariance structure of the residuals by variography, remove this correlation by spatial filtering residuals and base the permutation test on these approximately exchangeable residuals. A simulation study is conducted to evaluate the performance of the proposed method in terms of empirical size and power, examining its behavior under different covariance settings. We show that neglecting the residuals spatial structure in the permutation scheme (thus permuting the correlated residuals directly) yields a very liberal testing procedures, whereas the proposed procedure is close to the nominal size of the test. The methodology is demonstrated on a real world data set on the amount of waste production in the Venice province of Italy.