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 x 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), For All 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 x 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-0 against H-1 - H-0 and H-1 against H-2 - H-1 when H-0 : Sigma(i=1)(k)P(i)X(ji) = 0, j = 1,...,s, H-1 : Sigma(i=1)(k)P(i)X(ji) less than or equal to 0, j = 1,...,s, and H-2 does not impose any restrictions on P. The x(ji)'s are known constants, and s less than or equal to k - 1. We show that the asymptotic distributions of the likelihead-ratio tests are of chi-bar-square type, and provide expressions for the weighting values.