Simulation of the K-function in the analysis of spatial clustering for non-randomly distributed locations--exemplified by bovine virus diarrhoea virus (BVDV) infection in Denmark.

The K-function is often used to detect spatial clustering in spatial point processes, e.g. clustering of infected herds. Clustering is identified by testing the observed K-function for complete spatial randomness modelled, e.g. by a homogeneous Poisson process. The approach provides information about spatial clustering as well as the scale of distances of clustering. However, there are several problems related to applying the K-function, e.g. estimation of the size of the study area and the assumption about modelling spatial random distribution of the events by, e.g. a homogeneous Poisson process. The objective of the present study was to develop a null hypothesis version of the K-function that overcomes the assumption about a specific underlying spatial distribution characterising complete spatial randomness. Furthermore, the objective was to develop an approach that does not include the estimation of the size of the study area. The paper presents a simulation procedure to derive the null hypothesis version of the K-function. The null hypothesis version of the K-function is simulated by random sampling of N(+) locations from the distribution of N observed locations (infected (N(+)) and non-infected (N-N(+))). The differences between the empirical and the estimated null-hypothesis version of the K-function are plotted together with the 95% simulation envelopes versus the distance, h. In this way we test if the spatial distribution of the infected herds differs from the spatial distribution of the herd locations in general. The approach also overcomes edge effects and problems with complex shapes of the study region. An application to bovine virus diarrhoea virus (BVDV) infection in Denmark is described.