scholarly journals Orthogonal decomposition of the sum-symmetry model for square contingency tables with ordinal categories: Use of the exponential sum-symmetry model

2021 ◽  
Vol 58 (2) ◽  
pp. 95-104
Author(s):  
Shuji Ando
Keyword(s):  
Contingency Tables ◽  
Decomposition Theorem ◽  
Real Data ◽  
Exponential Sum ◽  
Orthogonal Decomposition ◽  
Global Symmetry ◽  
Asymptotic Equivalence ◽  
Artificial Data ◽  
Test Statistic ◽  
Symmetry Model

Summary In the existing decomposition theorem, the sum-symmetry model holds if and only if both the exponential sum-symmetry and global symmetry models hold. However, this decomposition theorem does not satisfy the asymptotic equivalence for the test statistic. To address the aforementioned gap, this study establishes a decomposition theorem in which the sum-symmetry model holds if and only if both the exponential sum-symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic. We demonstrate the advantages of the proposed decomposition theorem by applying it to datasets comprising real data and artificial data.

scholarly journals Orthogonal decomposition of the sum-symmetry model using the two-parameters sum-symmetry model for ordinal square contingency tables

2021 ◽  
Vol 58 (2) ◽  
pp. 105-117
Author(s):  
Shuji Ando
Keyword(s):  
Likelihood Ratio ◽  
Decomposition Theorem ◽  
Asymmetric Structure ◽  
Global Symmetry ◽  
Asymptotic Equivalence ◽  
Test Statistic ◽  
Symmetric Structure ◽  
Symmetry Model ◽  
Chi Squared ◽  
Two Parameters

Summary Studies have been carried out on decomposing a model with symmetric structure using a model with asymmetric structure. In the existing decomposition theorem, the sum-symmetry model holds if and only if all of the two-parameters sum-symmetry, global symmetry and concordancediscordance models hold. However, this existing decomposition theorem does not satisfy the asymptotic equivalence for the test statistic, namely that the value of the likelihood ratio chi-squared statistic of the sum-symmetry model is asymptotically equivalent to the sum of those of the decomposed models. To address this issue, this study introduces a new decomposition theorem in which the sum-symmetry model holds if and only if all of the two-parameters sum-symmetry, global symmetry and weighted global-sum-symmetry models hold. The proposed decomposition theorem satisfies the asymptotic equivalence for the test statistic—the value of the likelihood ratio chi-squared statistic of the sum-symmetry model is asymptotically equivalent to the sum of those of the two-parameters sum-symmetry, global symmetry and weighted global-sum-symmetry models.

scholarly journals An anti-sum-symmetry model and its orthogonal decomposition for ordinal square contingency tables with an application to grip strength test data

2021 ◽  
Vol 58 (1) ◽  
pp. 59-68
Author(s):  
Shuji Ando
Keyword(s):  
Grip Strength ◽  
Contingency Tables ◽  
Decomposition Theorem ◽  
Strength Test ◽  
Global Symmetry ◽  
High Confidence ◽  
Simple Interpretation ◽  
Asymmetric Structures ◽  
Symmetry Model ◽  
Chi Squared

Summary For the analysis of R × R square contingency tables, we need to estimate an unknown probability distribution with high confidence from obtained observations. For that purpose, we need to perform the analysis using a statistical model that fits the data well and has a simple interpretation. This study proposes two original models that have symmetric and asymmetric structures between the probability with which the sum of row and column variables is t, for t = 2, . . ., R, and the probability with which the sum of row and column variables is 2(R + 1) − t. The study also reveals that it is necessary to satisfy the anti-global symmetry model, in addition to the proposed asymmetry model, in order to satisfy the proposed symmetry model. This decomposition theorem is useful to explain why the proposed symmetry model does not hold. Moreover, we show that the value of the likelihood ratio chi-squared statistic of the proposed symmetry model is equal to the sum of those of the decomposed models. We evaluate the utility of the proposed models by applying them to real-world grip strength data.

scholarly journals Asymmetry models based on ordered score and separations of symmetry model for square contingency tables

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Shuji Ando
Keyword(s):  
Asymmetry Parameter ◽  
Contingency Tables ◽  
Real Data ◽  
Symmetry Model ◽  
Th Cell ◽  
Proposed Model ◽  
Log Odds ◽  
The Difference ◽  
Mean Equality ◽  
Column Variable

Summary This study proposes two original asymmetry models based on ordered scores for square contingency tables with the same row and column ordinal classifications. The proposed models can be applied to cases in which the scores of all categories are known or unknown. In the proposed models, the log odds for an observation falling in the (i, j)th cell instead of the (j, i)th cell are inversely proportional to the difference of the ordered scores corresponding to categories i and j. The asymmetry parameter of the proposed model can be useful for inferring whether the row variable is stochastically greater than the column variable or vice versa. The proposed models constantly hold when the symmetry model holds, but the converse is not necessarily true. This study also examines what is necessary for a model, in addition to the proposed models, to satisfy the symmetry model, and gives separations of the symmetry model using the proposed and marginal mean equality models. We apply real data to show the utility of the proposed models. The proposed models provide a better fit than that of the existing models.

scholarly journals Exact tests to compare contingency tables under quasi-independence and quasi-symmetry

2019 ◽  
Vol 10 (1) ◽  
pp. 13-29 ◽  
Author(s):  
Cristiano Bocci ◽  
Fabio Rapallo
Keyword(s):  
Linear Models ◽  
Contingency Tables ◽  
Real Data ◽  
Exact Tests ◽  
Exact Test ◽  
Markov Bases ◽  
Symmetry Model ◽  
Log Linear

In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the use of these models in different fields of applications.

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