On Tests Against One-Sided Hypotheses in Some Generalized Linear Models

One-sided hypotheses arise naturally in many situations. When testing against such hypotheses, it is desirable to take the available one-sided information into account, rather than simply applying a two-sided test. What we expect to gain by applying a one-sided test instead of a two-sided test is an increase in the power of the test. We consider various tests of one-sided hypotheses in a class of models that includes generalized linear and Cox regression models. The tests are likelihood ratio, Wald, score, generalized distance, and a Pearson chi-square. It is shown that these test statistics are asymptomatically equivalent in terms of local power; this is a generalization of the well-known corresponding result for two-sided alternatives. Two examples are also discussed. They are on (1) testing for interaction in binomial response models, and (2) comparison of treatments with ordinal categorical responses.