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A network algorithm for performing Fisher’s exact test in r$$\times c$$ contingency tables. (English) Zbl 0545.62039
Let $$X=(x_{ij})$$ be an $$r\times c$$ contingency table with $$R_ i=\sum^{c}_{j=1}x_{ij}$$, $$C_ j=\sum^{r}_{i=1}x_{ij}$$ and $$x_{ij}\geq 0$$. Denote by $${\mathcal T}$$ the reference set of all possible $$r\times c$$ contingency tables with the same marginal totals as X, i.e., $${\mathcal T}=\{Y:$$ Y is $$r\times c$$, $$\sum^{c}_{j=1}y_{ij}=R_ i$$, $$\sum^{r}_{i=1}y_{ij}=C_ j\}$$. The authors assume that the probability of observing any $$Y\in {\mathcal T}$$ can be expressed as a product of multinomial coefficients $P(Y)=(\prod^{c}_{j=1}C_ j!/y_{1j}!...y_{ij}!)/(T!/R_ 1!...R_ r!),\quad T=\sum^{r}_{i=1}R_ i.$ For testing the hypothesis that the row and column effects are independent, the exact significance level or p value associated with the observed table X is defined as the sum of the probabilities of all the tables in $${\mathcal T}$$ that are no more likely than X. A network algorithm to evaluate p and some numerical results are given. Some new optimization theorems are developed also.
Reviewer: K.T.Fang

##### MSC:
 62H17 Contingency tables 65C99 Probabilistic methods, stochastic differential equations
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