The SPP Procedure

Testing Covariate Dependency with EDF Tests

In a test for covariate dependency, the goal is to test the null hypothesis $H_0$ that the point process is independent of the covariate. PROC SPP tests $H_0$ by interpolating the covariate values at the event locations. The EDF is weighted by the intensity at the corresponding locations (Baddeley and Turner, 2005). PROC SPP performs this weighted EDF test for covariates that are defined in a TREND statement.

Weighted EDF Tests

To test dependence on a trend covariate, PROC SPP initially computes the covariate EDF. The EDF is weighted by using intensity-based weights to account for the current intensity model. For example, under the CSR assumption the intensity $\lambda $ is constant across the study area; hence, the weight for each of the M observations of a covariate is $\lambda _ i/\sum _ i{M}\lambda _ i = \lambda /M\lambda = 1/M$. This weighted EDF is the predicted distribution that any other set of independent covariate observations should follow under the assumed intensity model.

Next, the covariate is interpolated at the n event locations $\bm {s}_ i, i=1,\ldots ,n$, using ordinary kriging; kriging analysis assumes a linear semivariance correlation function and considers the four closest covariate observations for each event location. The outcome is a set of covariate values $X_ i$. With the $X_ i$ in hand, PROC SPP assumes that the probability integral transform $Z=F(X)$ is the linear interpolation of the weighted EDF at the covariate values $X_ i$, and it produces the transformed EDF Z in $[0,1]$. If the intensity model assumption is correct, then Z follows a uniform distribution $U(0,1)$. Finally, PROC SPP uses EDF tests to examine the fit of the EDF $F_ n(x)=F_ n(z)$ to a standard uniform EDF.