The generalized odds ratio is a measure of the handedness of a cyclic competition graph.
- matrix with one degree of freedom, coerced to a board
The generalized odds ratio is the product of the odds ratios of the edges in a cyclic competition graph.
In the two-by-two case, the maximum likelihood estimatore of the odds ratio is just $a[1,1]*a[2,2]/a[1,2]/a[2,1]$. The generalization to larger boards with exactly one degree of freedom is immediate: it is just the product of the entries on the diagonal, divided by the product of the off-diagonal elements.
Ranking permissable tables in order of increasing odds ratio thus orders the sample space and allows one to perform one-sided tests.
data(gear) odds.ratio(gear) aylmer.test(gear) aylmer.test(gear,alternative="less") aylmer.test(gear,alternative="greater")