Various extensions of the isotonic regression problems and its applications

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1.A unifying approach to non-inferiority, equivalence and superiority tests via multiple decision processesIn the comparative clinical study of a new drug against an active control the simultaneous testing of the non-inferiority hypothesis and the superiority hypothesis has been of great interest. In Japan the significance level of these two tests was set commonly at α=0.05 but the α for the non-inferiority hypothesis has been altered to 0.025 by the International Guideline introduced in 1999. It naturally gives a very big impact on the Japanese practice of clinical trial since according to the International Guideline approximately 1.3 times the sample size is required as compared with the old Japanese practice for proving the non-inferiority of a test drug which is truly equivalent to the control. This is indeed a very serious problem to be discussed carefully, see also Hirotsu and Hothorn (2003). In this research we succeeded in unifying the old Japanese Guideline and the International Guideline by applying the partitioning principle of the parameter space. This method is particularly attractive changing the strength of the evidence of relative efficacy of the test drug against a control according to the achievement of the clinical trial. It includes the decisions 1. weak non-inferiority, 2. strong non-inferiority, 3. at least equivalence, and 4. superiority. The weak non-inferiority corresponds to the Japanese practice and the strong non-inferiority to the International Guideline. Yet another approach has also been proposed taking the advantages of both of the Japanese and International practices.2.Dose - response analysisIn the dose-response analysis a new approach of estimating the dose - response patterns is proposed replacing the classical estimation of dose-response function. This idea is very useful in the phase 2 clinical trial whose purpose is to decide the recommended dose for the clinical treatment.