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

TL;DR

This paper introduces a recursive method to compute exact p-values for scan statistics in spatial and temporal epidemiology, leveraging Markov properties and graph theory for efficient calculations.

Contribution

It presents a novel recursive computation technique for conditional expectations related to scan statistics, applicable to various distributions including Poisson, improving p-value evaluation accuracy.

Findings

Effective recursive formulas for p-value computation

Application to real spatial-temporal clustering data

Method extends to multinomial and Poisson distributions

Abstract

Let $V$ be a finite set of indices, and let $B_i$, $i=1,\ldots,m$, 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 $E\bigl[\prod_{i=1}^m\chi_i(X_{B_i}) \,|\, N\bigr]$ 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.