Testing for and against a set of linear inequality constraints in a multinomial setting

There are numerous situations in categorical data analysis where one wishes to test hypotheses involving a set of linear inequality constraints placed upon the cell probabilities. For example, it may be of interest to test for symmetry in k × k contingency tables against one-sided alternatives. In this case, the null hypothesis imposes a set of linear equalities on the cell probabilities (namely p_ij = P_ji ×i > j), whereas the alternative specifies directional inequalities. Another important application (Robertson, Wright, and Dykstra 1988) is testing for or against stochastic ordering between the marginals of a k × k contingency table when the variables are ordinal and independence holds. Here we extend existing likelihood-ratio results to cover more general situations. To be specific, we consider testing H_t,0 against H₁ - H₀ and H₁ against H₂ - H ₁ when H₀:k × i=1 p_ix_ji = 0, j = 1,…, s, H₁:k × i=1 p_ix_ji × 0, j = 1,…, s, and does not impose any restrictions on p. The x_ji's are known constants, and s × k - 1. We show that the asymptotic distributions of the likelihood-ratio tests are of chi-bar-square type, and provide expressions for the weighting values.