Recursive computation for evaluating the exact $p$-values of temporal and spatial scan statistics
Recursive computation for evaluating the exact $p$-values of temporal and spatial scan statistics
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发表时间：
2015-10
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通讯作者：
S. Kuriki;Kunihiko Takahashi;Hisayuki Hara
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作者：
S. Kuriki;Kunihiko Takahashi;Hisayuki Hara
设$V$是有限指数集，$B_i$，$i=1，\ldots，m$是$V$的子集，使得$V=\Bigcup_{i=1}^{m}B_i$。设$X_i$，$i\in V$是独立的随机变量，且$X_{B_i}=(X_J)_{j\in B_i}$。本文提出了一种递归计算方法来计算给定条件期望$E\Bigl[\prod_{i=1}^m\chi_i(X_{B_i})\，|\，N\BiGR]$其中$N=\sum_i\in V}X_i$。我们的方法是基于递归求和/积分技术，利用统计学中的马尔可夫性质。为了提取马氏性，我们定义了一个团为$B_j$的无向图，并得到了它的弦扩张，由此给出了递推公式的表达式。这种方法适用于一类分布，包括泊松分布(即条件分布是多项式分布)。这个问题的起因是对空间流行病学中扫描统计的多重性调整的$p$值进行了评估。作为对该方法的说明，我们给出了检测时间和空间聚集的真实数据分析。
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.