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 pij = Pji ×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 Ht,0 against H1 - H0 and H1 against H2 - H 1 when H0:k × i=1 pixji = 0, j = 1,…, s, H1:k × i=1 pixji × 0, j = 1,…, s, and does not impose any restrictions on p. The xji'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.
Il y a plusieurs situations dans l'analyse de categories de donnees ou l'on veut tester des hypotheses impliquant un ensemble de contraintes ayant la forme d'inegalites lineaires placees sur les cellules de probabilites. Par exemple il peut ětre interessant de tester la symetrie de tables de contingence de dimension k × k contre des alternatives avec assymetries. Dans ce cas, l'hypothese nulle impose un ensemble d'egalites sur les cellules de probabilites (ou plus precisement pij = pji, ×i > j) alors que les hypotheses alternatives specifies certaines inegalites directionnelles. Une autre application importante (Robertson, Wright and Dykstra, 1988) est de tester pour ou contre l'existence d'un arrangement stochastique des marginales d'une table de contingence k × k lorsque les variables sont de type ordinal et independantes. Ici nous etendons des resultats existants sur les ratio de vraisemblance, pour couvrir des situations plus generales. Pour ětre plus specifique, nous testons H0 contre H1 - H0 et H1 contre H2 - H1 ou H0:k × i=1 pixji = 0, j = 1,…, s, H1:k × i=1 pixji × 0, j = 1,…, s, et H2 n'impose pas de restrictions sur P. Les sont connus constants et s × k - 1. Nous montrons que les distributions asymptotiques des tests de ratio de vraisemblance sont de type Chi-deux, et donnent des expressions pour les valeurs ponderantes.
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