Likelihood ratio tests involving a bivariate trend in two-factor designs: the level probabilities

Abstract

Two-factor experiments in which both factors are ordinal are considered. If it is believed apriorithat the mean response is nondecreasing in each factor with the other held fixed, then one may test for a treatment effect by testing homogeneitywith the appropriate ordered alternative. However, if this ordering were in question, one might test it as the null hypothesis. The likelihood ratio tests for both of these situations have been developed, but the level probabilities needed to implement these tests have only been determined in a few special cases. Monte Carlo estimates of the level probabilities are given here for balanced designs with no more than nine levels on each factor. These level probabilities are also useful when testing these hypotheses in one-parameter exponential families and when testing hypotheses involving a bivariate stochastic ordering. Monte Carlo estimates of the constants which are needed to estimate certain powers in these two-factor experiments are also given.

Keywords:

• blvariate stochastic orderings
• chi-bar-squared and E-bar-squared distributions
• level probabilities
• matrix ordering
• order restricted tests