Linear Diagonals-parameter Symmetry Research Articles

An Extended Bivariate T-Distribution Type Symmetry Model for Square Contingency Tables

The purpose of this paper is to propose a new model of asymmetry for square contingency tables with ordered categories. The new model may be appropriate for a square contingency table if it is reasonable to assume an underlying bivariate t-distribution with different marginal variances having any degrees of freedom. As the degrees of freedom becomes larger, the proposed model approaches the extended linear diagonals-parameter symmetry model, which may be appropriate for a square table if it is reasonable to assume an underlying bivariate normal distribution. The simulation study based on bivariate t-distribution is given. An example is given.

Scaled double symmetry model and decomposition of double symmetry for square tables

For square contingency tables with ordered categories, Tahata and Tomizawa (2010) considered the double linear diagonals-parameter symmetry (DLDPS) model. The present paper proposes the generalized DLDPS model, which implies the structure of both asymmetry with represent to the main diagonal and with respect to the reverse diagonal in the table. Also, the double symmetry (DS) model is separated into the generalized DLDPS model and the moment equality model. Also it is shown that the test statistic for the DS model is asymptotically equivalent to the sum of those for separated models. An example is given.

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

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, p.430) 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.

Decomposition of Generalized Asymmetry Model for Square Contingency Tables

For the analysis of square contingency tables with same row and column ordinal classifications, the present paper gives the decomposition of the generalized linear diagonals-parameter symmetry model using the diagonals-parameter symmetry model. Moreover, it gives the decomposition of the symmetry model using above the proposed decomposition.

Decompositions of Symmetry Using Generalized Linear Diagonals-Parameter Symmetry Model and Orthogonality of Test Statistic for Square Contingency Tables

For square contingency tables with ordered categories, the present paper gives several theorems that the symmetry model holds if and only if the generalized linear diagonals-parameter symmetry model for cell probabilities and for cumulative probabilities and the mean nonequality model of row and column variables hold. It also shows the orthogonality of statistic for testing goodness-of-fit of the symmetry model. An example is given.

Generalized Linear Asymmetry Model and Decomposition of Symmetry for Multiway Contingency Tables

A k-order generalization of the linear diagonals-parameter symmetry model is proposed, and related orthogonal decompositions of the generalization are inspected. Applications to randomized clinical trials are given.

Measure of Departure from Average Cumulative Symmetry for Square Contingency Tables with Ordered Categories

Problem statement: For square contingency tables with ordered categories, we are interested in considering a structure of weak symmetry when Bowker's symmetry model does not hold and in measuring the degree of departure from weak symmetry. Approach: The present study considered the average cumulative symmetry model that has a weaker restriction than the structure of symmetry. It also gave a measure to represent the degree of departure from average cumulative symmetry. When the conditional symmetry and the cumulative linear diagonals-parameter symmetry holds, the proposed measure can measure what degree of departure from the symmetry is. Results: The proposed model and the measure were applied and analyzed (1) for the data of 4×4 contingency table of unaided distance vision of 7477 women aged 30-39 employed in Royal Ordnance factories in Britain from 1943-1946 and (2) the data of 4×4 contingency table of the 59 matched pairs using from dose levels of conjugated oestrogen. Conclusion: The proposed model is useful when the symmetry model does not hold and the proposed measure is useful for comparing the degree of departure from the weak symmetry model in several tables. Especially the proposed measure is useful to measure the degree of departure from symmetry when the conditional symmetry (the cumulative diagonals-parameter symmetry) model holds.

Double linear diagonals-parameter symmetry and decomposition of double symmetry for square tables

For square contingency tables with ordered category, the present paper proposes the double linear diagonals-parameter symmetry (D-LDPS) model which implies the structure of both asymmetry with respect to the main diagonal and with respect to the reverse diagonal in the table. The D-LDPS model may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. The present paper also gives the orthogonal decomposition of the double symmetry model into the D-LDPS model and the double marginal mean equality model. An example is given.

Contaminated Normal Type Symmetry Model and Decomposition of Symmetry for Square Contingency Tables

For the analysis of square contingency tables with ordered categories, Agresti (1983) considered the linear diagonals-parameter symmetry (LDPS) model being an extension of the symmetry (S) model, which may be appropriate for a square ordinal table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. This paper proposes another extension of the S model which may be appropriate for a square ordinal table if it is reasonable to assume an underlying e-contaminated bivariate normal distribution with equal marginal variances included in the class of elliptical distributions. The proposed model has a structure of weighted mixture of two LDPS models. Moreover, this paper gives a theorem that the S model holds if and only if both the proposed model and the marginal means equality model hold. The simulation studies based on the e-contaminated bivariate normal distribution are given.

A measure of linear diagonals-parameter symmetry for square contingency tables with ordered categories

For square contingency tables with ordered categories, Agresti (1) considered the linear diagonals-parameter symmetry (LDPS) model being an extension of the symmetry model. This paper proposes a measure which represents the degree of departure from the LDPS model. The measure proposed is expressed using the Cressie-Read power-divergence or Patil-Taillie diversity index. The measure is useful for comparing the degree of departure from LDPS in several tables, and it may be appropriate for comparing the degrees of departure from the equality of the marginal variances for several square ordinal tables if it is reasonable to assume underlying bivariate normal distributions. Examples are also given.

Linear Diagonals‐Parameter Symmetry and Quasi‐Symmetry Models for Cumulative Probabilities in Square Contingency Tables with Ordered Categories

AbstractFor the analysis of square contingency tables with ordered categories, Caussinus (1965) considered the quasi‐symmetry (QS) model, Goodman (1979) considered the diagonals‐parameter symmetry (DPS) model, and Agresti (1983) considered the linear diagonals‐parameter symmetry (LDPS) model. These models show the structures of symmetry for cell probabilities. Tomizawa (1993) proposed another DPS model which has a similar multiplicative form for cumulative probabilities that an observation will fall in row (column) category i or below and column (row) category j (>i) or above. This paper proposes another LDPS and QS models that have the corresponding similar multiplicative forms for cumulative probabilities instead of cell probabilities. Special cases of the proposed models include symmetry. Two kinds of unaided distance vision data and endometrial cancer data are analyzed using these models. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Decomposition of Linear Diagonals‐Parameter Symmetry Model for Square Contingency Tables with Ordered Categories

AbstractFor square contingency tables with ordered categories, this paper gives a decomposition for AGRESTI'S (1983) linear diagonals‐parameter symmetry (LDPS) model into GOODMAN'S (1979) diagonals‐parameter symmetry model and the linear diagonals‐parameter marginal symmetry model introduced in this paper. It is also pointed out that the likelihood ratio statistic for the LDPS model is equal to the sum of those for the decomposed two models.

Diagonal, weighted marginal homogeneity model and decompositions for linear diagonals-parameter symmetry model

For square contingency tables rith ordered categories, this paper proposes two kinds of extensions of marginal homogeneity model and gives decompositions for the Liseer diagonals-parameter symmetry model considered by Agresti (1983a) using the proposed models- The proposed models are also applied to an unaided vision data.