QR-Adjustment for Clustering Tests Based on Nearest Neighbor Contingency   Tables
  The spatial interaction between two or more classes of points may cause spatial clustering patterns such as segregation or association, which can be tested using a nearest neighbor contingency table (NNCT). A NNCT is constructed using the frequencies of class types of points in nearest neighbor (NN) pairs. For the NNCT-tests, the null pattern is either complete spatial randomness (CSR) of the points from two or more classes (called CSR independence) or random labeling (RL). The distributions of the NNCT-test statistics depend on the number of reflexive NNs (denoted by $R$) and the number of shared NNs (denoted by $Q$), both of which depend on the allocation of the points. Hence $Q$ and $R$ are fixed quantities under RL, but random variables under CSR independence. Using their observed values in NNCT analysis makes the distributions of the NNCT-test statistics conditional on $Q$ and $R$ under CSR independence. In this article, I use the empirically estimated expected values of $Q$ and $R$ under CSR independence pattern to remove the conditioning of NNCT-tests (such a correction is called the \emph{QR-adjustment}, henceforth). I present a Monte Carlo simulation study to compare the conditional NNCT-tests and QR-adjusted tests under CSR independence and segregation and association alternatives. I demonstrate that QR-adjustment does not significantly improve the empirical size estimates under CSR independence and power estimates under segregation or association alternatives. For illustrative purposes, I apply the conditional and empirically corrected tests on two example data sets. 
Elvan Ceyhan