zbMATH — the first resource for mathematics

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

62H17 Contingency tables
65C99 Probabilistic methods, stochastic differential equations
Full Text: DOI