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.
