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A unifying approach to the shape and change-point hypothesestled

Speaker: Professor Chihiro Hirotsu, Japan

Professor Chihiro Hirotsu will mainly talk about the first topic and briefly the second topic. The abstracts of the two topics can be read below.

Abstract

1. A unifying approach to the shape and change-point hypotheses

The shape hypothesis like monotone is essential in the dose-response analysis where a rigid parametric model is usually difficult to assume. Then the isotonic regression is the most well known approach. It has been introduced, however, rather intuitively and too complicated computationally to extend to the non-normal distributions, to other shape constraints and to two-way data. Instead the author’s approach starts from a complete class lemma for the tests against the general restricted alternative. It suggests the use of the singly-, doubly- and triplyaccumulated statistics for the monotone, convexity and sigmoidicity hypotheses, respectively. It should be stressed here that there is a close relationship between the monotone hypothesis and step change-point model. Actually each component of the step change-point model is a particular monotone contrast and every monotone contrast can be expressed by a unique and positive linear combination of the step change-point contrasts. The idea is extended to convexity and slope change-point models, and the sigmoidicity and inflection point models, thus giving a unifying approach to the shape and change-point hypotheses. The basic statistics of the newly proposed approach are very simple and have a very nice Markov property for an elegant and exact probability calculation not only for the normal distribution but also for the Poisson and multinomial distributions. The power of the proposed method has been compared repeatedly and shown excellent. The approach is nicely extended to the analysis of two-way interaction.

2. Analysis of two-way interaction with or without replication in the cells

The analysis of two-way interaction is one of the central topics of data science but receives much less attention than it deserves. There are several immanent problems in the analysis of two-way data which are not described everywhere.

(1) The characteristics of the rows and columns – such as controllable, indicative, variational, and response – should be taken into consideration.

(2) The degrees of freedom are often so large that an overall analysis can tell almost nothing about the details of the data. On the other hand the multiple comparisons based on one degree of freedom statistics is too less powerful and its interpretation is unclear even when the test result is significant.

(3) There is often natural ordering in the rows and/or columns, which should be taken into account in the analysis. The isotonic regression is, however, too complicated for the analysis of two-way interaction effects. On the other hand our approach to the shape and change-point analysis is of so simple a structure that many of the procedures for one-way data can be extended in a systematic way to the analysis of two-way interaction. There are variations that both or only one of row or column categories have natural ordering.

For accommodating to those problems row- and/or column-wise multiple comparison procedure is proposed which is applicable when there is no repetition in the cells. They are also extended to two-way contingency tables including the profile analysis of rows based on the ordered column responses. Indeed, the analysis of two-way data is a rich source of interesting theories and applications. In the latter half of this talk we give those interesting real examples.