Pour l'analyse des tableaux carres, Caussinus (1965) a propose le modele de quasi-symetrie et montre qu'un tableau est symetrique si et seulement s'il satisfait a la fois quasi-symetrie et egalite des distributions marginales. Bishop, Fienberg et Holland (1975, p. 307) ont note qu'un theoreme semblable valait pour les tableaux a trois dimensions, tandis que Bhapkar et Darroch l'ont donne pour des tableaux de dimension quelconque. Le but de cet article est (1) de passer en revue les questions de symetrie, les modeles eux-memes, leur decomposition et les mesures d'ecart au modele pour divers concepts de symetrie et asymetrie, (2) de montrer que, pour les tableaux multiples, la statistique du rapport de vraisemblance pour tester la symetrie est asymptotiquement equivalente a la somme des statistiques analogues testant respectivement la quasi-symetrie d'un certain ordre et l'egalite des marges pour l'ordre correspondant.

The analysis of symmetry and asymmetry : orthogonality of decomposition of symmetry into quasi-symmetry and marginal symmetry for multi-way tables

For the analysis of square contingency tables with nominal categories, Tomizawa and coworkers have considered measures that represent the degree of departure from symmetry. This paper proposes a measure that represents the degree of asymmetry for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed using the Cressie–Read power-divergence or Patil–Taillie diversity index, defined for the cumulative probabilities that an observation falls in row (column) category i or below and column (row) category j(> i) or above. The measure depends on the order of listing the categories. It should be useful for comparing the degree of asymmetry in several tables with ordered categories. The relationship between the measure and the normal distribution is shown.

Theory & Methods: Measure of Asymmetry for Square Contingency Tables Having Ordered Categories

For the analysis of square contingency tables with nominal categories, Tomizawa (1995a) and Tomizawa and Makii (2001a) considered the measures to represent the de gree of departure from the marginal homogeneity (MH). This paper proposes a measure to represent the degree of departure from MH for square contingency tables with ordered categories (instead of those with nominal categories). The measure proposed is expressed by using the Cressie and Read’s (1984) power-divergence or Patil and Tail-lie’s (1982) diversity index, which is defined for the cumulative probablities that an observation will fall in row (column) category i or below and column (row) category i + 1 or above. The measure depends on the order of listing the categories. It would be useful for comparing the degree of departure from MH in several tables with ordered categories. The relationships between the measure and the normal distribution (also some models) are also shown. Examples are given.

Measure of departure from marginal homogeneity for square contingency tables having ordered categories

For square contingency tables with ordered categories, Tomizawa (Calcutta Stat Assoc Bull 43:123–125, 1993a; Sankhyā Ser B 60:293–300, 1998) gave theorems that the marginal homogeneity (MH) model is equivalent to certain two or three models holding simultaneously. This paper proposes a generalized MH model, which describes a structure of the odds that an observation will fall in row category i or below and column category i + 1 or above, instead of in column category i or below and row category i + 1 or above. In addition, this paper gives the theorems that the MH model is equivalent to the generalized MH model and some models holding simultaneously whose each indicates: (1) the equality of m-order moment of row and column variables, (2) the equality of skewness of them and (3) the equality of kurtosis of them. When the MH model fits the data poorly, these may be useful for seeing the reason for the poor fit; for instance, the poor fit of the MH model is caused by the poor fit of the equality of row and column means rather than the generalized MH model. Examples are given.

Generalized marginal homogeneity model and its relation to marginal equimoments for square contingency tables with ordered categories

Asymptotic Approximations for the Distributions of the Multinomial Goodness-of-Fit Statistics under Local Alternatives

For square contingency tables that have nominal categories, Tomizawa considered two kinds of measure to represent the degree of departure from symmetry. This paper proposes a generalization of those measures. The proposed measure is expressed by using the average of the power divergence of Cressie and Read, or the average of the diversity index of Patil and Taillie. Special cases of the proposed measure include Tomizawa's measures. The proposed measure would be useful for comparing the degree of departure from symmetry in several tables.

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

For the analysis of square contingency tables with ordered categories, Agresti (1983) introduced the linear diagonals-parameter symmetry (LDPS) model Tomizawa (1991) considered an extended LDPS (ELDPS) model, which has one more parameter than the LDPS model These models are special cases of Caussinus (1965) quasi-symmetry (QS) model Caussinus showed that the symmetry (S) model is equivalent to the QS model and the marginal homogeneity (MH) model holding simultaneously For square tables with ordered categories, Agresti (2002, p430) gave a decomposition for the S model into the ordinal quasi-symmetry and MH models This paper proposes some decompositions which are different from Caussinus’ and Agresti’s decompositions It gives (i) two kinds of decomposition theorems of the S model for two-way tables, (ii) extended models corresponding to the LDPS and ELDPS, and the generalized model further for multi-way tables, and (iii) three kinds of decomposition theorems of the S model into their models and marginal equimoment models for multi-way tables The proposed decompositions may be useful if it is reasonable to assume the underlying multivariate normal distribution

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Decomposition of Symmetry into Ordinal Quasi-Symmetry and Marginal Equimoment for Multi-way Tables

Yamamoto, Iwashita, and Tomizawa (2007) gave the decompositions of the symmetry (S) model into the extended S and two or three marginal equimoment models for multi-way tables with ordered categories. However the goodness-of-fit test statistic for the S model is not asymptotically equivalent to the sum of those for the decomposed three or four models. This paper gives, for multi-way tables, the modification of their decompositions such that the test statistic for the S model is asymptotically equivalent to the sum of those for the extended S model and one modified marginal equimoment model, which is the constraints combined the marginal means equality and variances equality or combined the marginal means equality, variances equality and correlations equality. Examples are given.

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Orthogonality of Decompositions of Symmetry into Extended Symmetry and Marginal Equimoment for Multi-way Tables with Ordered Categories

The challenges and potential benefits of incorporating biomarkers into clinical trial designs have been increasingly discussed, in particular to develop new agents for immune‐oncology or targeted cancer therapies. To more accurately identify a sensitive subpopulation of patients, in many cases, a larger sample size—and consequently higher development costs and a longer study period—might be required. This article discusses a biomarker‐based Bayesian (BM‐Bay) randomized clinical trial design that incorporates a predictive biomarker measured on a continuous scale with pre‐determined cutoff points or a graded scale to define multiple patient subpopulations. We consider designing interim analyses with suitable decision criteria to achieve correct and efficient identification of a target patient population for developing a new treatment. The proposed decision criteria allow not only the take‐in of sensitive subpopulations but also the ruling‐out of insensitive ones on the basis of the efficacy evaluation of a time‐to‐event outcome. Extensive simulation studies are conducted to evaluate the operating characteristics of the proposed method, including the probability of correct identification of the desired subpopulation and the expected number of patients, under a wide range of clinical scenarios. For illustration purposes, we apply the proposed method to design a randomized phase II immune‐oncology clinical trial.

Hideharu Yamamoto

Papers

Power-divergence-type measure of departure from symmetry for square contingency tables that have nominal categories

Decomposition of Symmetry into Ordinal Quasi-Symmetry and Marginal Equimoment for Multi-way Tables

Orthogonality of Decompositions of Symmetry into Extended Symmetry and Marginal Equimoment for Multi-way Tables with Ordered Categories

Biomarker‐based Bayesian randomized clinical trial design for identifying a target population