Second‐Order Analysis of Inhomogeneous Spatial Point Processes Using Case–Control Data

Abstract

Methods for the statistical analysis of stationary spatial point process data are now well established, methods for nonstationary processes less so. One of many sources of nonstationary point process data is a case-control study in environmental epidemiology. In that context, the data consist of a realization of each of two spatial point processes representing the locations, within a specified geographical region, of individual cases of a disease and of controls drawn at random from the population at risk. In this article, we extend work by Baddeley, Møller, and Waagepetersen (2000, Statistica Neerlandica54, 329-350) concerning estimation of the second-order properties of a nonstationary spatial point process. First, we show how case-control data can be used to overcome the problems encountered when using the same data to estimate both a spatially varying intensity and second-order properties. Second, we propose a semiparametric method for adjusting the estimate of intensity so as to take account of explanatory variables attached to the cases and controls. Our primary focus is estimation, but we also propose a new test for spatial clustering that we show to be competitive with existing tests. We describe an application to an ecological study in which juvenile and surviving adult trees assume the roles of controls and cases.

A typology of distance-based measures of spatial concentration

“…If the kernel's bandwidth is very small, intensity is highly variable and independence is found, while a wide kernel results in more stationarity and dependence. In other words, the results are highly dependent on the arbitrary choice of the estimation kernel bandwidth (Diggle et al, 2007). If it is not guided by additional knowledge supporting it, results may be arbitrary.…”

“…They may be excluded (Diggle et al, 2007;Marcon et al, 2012), for instance if the sampling design is such that all points of interest (the cases), but only a sample of the benchmark distribution (the controls) are recorded. Then, the whole distribution of points is better represented by the controls alone.…”

“…K inhom was employed by Diggle et al (2007) and Arbia et al (2012) in a case-control design: a number of points, the controls, were used to estimate the point process intensity (assuming they are approximately independently distributed), while the different pattern of cases was attributed to dependence. This approach is quite similar to that of M where all points weight 1: estimating λ (x j ) in table 1 for K inhom with a simple box kernel with bandwidth r is not different from z(r) in table 3 for M. The main difference (normalization apart) is that all reference points have the same weight in M, while each piece of space has in K inhom (table 2).…”

Spatial attraction in migrants' settlement patterns in the city of Catania

“…To distinguish between these two sources of clustering is challenging, as many spatial processes are "equifinal" and hence unidentifiable (Harvey 1966), i.e., one realization of a point process may be consistent with underlying processes involving clustering due to either spatial inhomogeneity or spatial attraction, as demonstrated in Bartlett (1963). The risk of confounding the two sources of clustering is a fundamental limitation on the scope of statistical inference from a spatial point pattern, assuming we have access to only a single realization of the underlying process (Diggle et al 2007;Baddeley 2010;Baddeley, Rubak, and Turner 2015) . Diggle et al 2007 outline three possible solutions.…”

Technical note: A bootstrapped LOESS regression approach for comparing soil depth profiles

“…Overall P values for the difference between depth profiles can be obtained by taking normalised test statistics across the full set of bootstrap samples and taking the percentile of these values corresponding to the same statistic obtained from the LU 2 data. This is a similar approach to that adopted in the spatial statistics literature when analysing K functions under resampling as demonstrated in Diggle et al (2007) and Henrys and Brown (2009) …”