Testing global and local dependence of point patterns on covariates in parametric models

Abstract

Testing for a covariate effect in a parametric point process model is usually done through the Wald test, which relies on an asymptotic null distribution of the test statistic. We propose a Monte Carlo version of the test that also allows local investigation of the covariate effect in the globally fitted model. Two different test statistics are suggested for this purpose: the first, a spatial statistic computed at every location of the observation window, resembles the classical F-statistic that is usually used in general linear models (GLMs) to express the distance between a model and its sub model. This statistic allows one to detect locations where the smoothed point process residuals are reduced by adding the interesting covariates into the model. The second spatial statistic tries to capture local improvements in the shape of the predicted intensity caused by an interesting, continuous covariate. A simulation scheme resembling the permutation inference for GLMs is used to obtain the null distribution of the statistics. Thereafter, a Monte Carlo test with graphical interpretation (a global envelope test) is applied to the empirical and simulated statistic fields to determine the global significance of the covariate and the spatially significant areas. We study the empirical significance level and power of the test in different scenarios and, by applying the test to simulated and real point pattern data, show that the proposed statistics can be valuable for model construction.