  Several interesting models for contingency tables are defined by a system of
equality and inequality constraints on a suitable set of marginal log-linear
parameters. After reviewing the most common difficulties which are intrinsic to
order restricted testing problems, we propose two new families of testing
procedures, based on similar attempts appeared in the econometric literature,
in order to increase the probability of detecting several relevant violations
of the supposed order relations. One set of procedures is based on the
decomposition of the log-likelihood ratio when testing the given set of
inequalities and the nested model derived by forcing inequalities into strict
equalities. The other set uses the asymptotic joint normal distribution of the
estimates of the marginal log-linear parameters to be constrained.
