Recursive computation for evaluating the exact $p$-values of temporal and spatial scan statistics

Abstract

Let $V$ be a finite set of indices, and let $B_i$, $i=1,\dots ,m$, be subsets of $V$ such that $V={\bigcup }_{i=1}^m{B}_i$. Let $X_i$, $i\in V$, be independent random variables, and let $X_{B_i}=(X_j{)}_{j\in B_i}$. In this paper, we propose a recursive computation method to calculate the conditional expectation $E[{\prod }_{i=1}^m{\chi }_i(X_{B_i})|N]$ with $N={\sum }_{i\in V}{X}_i$ given, where ${\chi }_i$ is an arbitrary function. Our method is based on the recursive summation/integration technique using the Markov property in statistics. To extract the Markov property, we define an undirected graph whose cliques are $B_j$, and obtain its chordal extension, from which we present the expressions of the recursive formula. This methodology works for a class of distributions including the Poisson distribution (that is, the conditional distribution is the multinomial). This problem is motivated from the evaluation of the multiplicity-adjusted $p$-value of scan statistics in spatial epidemiology. As an illustration of the approach, we present the real data analyses to detect temporal and spatial clustering.

23 pages, 7 figures, 3 tables

References in corpus (2)

• A Localization Approach to Improve Iterative Proportional Scaling in Gaussian Graphical Models
• Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph