Analysis of odds ratios in 2 - n ordinal contingency tables

The set of all bivariate probability distributions with support contained in {(i,j); I=1, 2 and J = 1,2,...,n} which are totally positive of order two is shown to be a convex set under some conditions on one of the marginal distributions. The extreme points of this compact convex set are explicitly enumerated. Using the structure of this convex set, we show that the power function of any test for testing the hypothesis of independence against the hypothesis of strict total positivity of order two in 2 - n ordinal contingency tables has a simple form in terms of the extreme points. A numerical illustration is provided.