Testing against umbrella or tree orderings for binomial proportions with an adaptation of an insect resistance case.

Alternative hypotheses for order restrictions, such as umbrella or inverse umbrella (a.k.a tree) orderings, have been studied extensively in the literature, although less so when the studied response for each individual is the presence or absence of the event of interest. Two families of test statistics for solving the problem of testing against an umbrella or a tree ordering when the responses are binomial proportions are studied in this work and their asymptotic distributions are derived. A simulation study is conducted to compare the empirical power of some members of the derived families of test statistics with competing approaches. The methodology developed here was driven by an applied problem arising in stored products research where despite universal mortality in the case of doses of 1000 ppm of the insecticide phosphine, unexpected survival was noted at higher doses.

Extensions of empirical likelihood and chi-squared-based tests for ordered alternatives.

Several methods for comparing k populations have been proposed in the literature. These methods assess the same null hypothesis of equal distributions but differ in the alternative hypothesis they consider. We focus on two important alternative hypotheses: monotone and umbrella ordering. Two new families of test statistics are proposed, including two known tests, as well as two new powerful tests under monotone ordering. Furthermore, these families are adapted for testing umbrella ordering. We compare some members of the families with respect to power and Type I errors under different simulation scenarios. Finally, the methods are illustrated in several applications to real data.