Exact null and alternative distributions of the two-way maximally selected x' for interaction between the ordered rows and columns are derived for each of the normal and Poisson models, respectively. The method is one of the multiple comparison procedures for ordered parameters and is useful for defining a block interaction or a two-way change-point model as a simple alternative to the two-way additive model. The construction of a confidence region for the two-way change-point is then described. An important application is found in a dose-response clinical trial with ordered categorical responses, where detecting the dose level which gives significantly higher responses than the lower doses can be formulated as a problem of detecting a change in the interaction effects.
Concavity and sigmoidicity hypotheses are developed as a natural extension of the simple ordered hypothesis in normal means. Those hypotheses give reasonable shape constraints for obtaining a smooth response curve in the non-parametric input±output analysis. The slope change and in¯ection point models are introduced correspondingly as the corners of the polyhedral cones de®ned by those isotonic hypotheses. Then a maximal contrast type test is derived systematically as the likelihood ratio test for each of those changepoint hypotheses. The test is also justi®ed for the original isotonic hypothesis by a complete class lemma. The component variables of the resulting test statistic have second or third order Markov property which, together with an appropriate nonlinear transformation, leads to an exact and very ef®cient algorithm for the probability calculation. Some considerations on the power of the test are given showing this to be a very promising way of approaching to the isotonic inference.
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