Using a GLM to Decompose the Symmetry Model in Square Contingency Tables with Ordered Categories
In this paper, we are employing the generalized linear model (GLM) in the form ij =Xλ to decompose the symmetry model into the class of models discussed in Tomizawa ( 1992 ). In this formulation, the random component would be the observed counts f ij with an underlying Poisson distribution. This app...
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| Published in | Journal of applied statistics Vol. 31; no. 3; pp. 279 - 303 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
England
Taylor & Francis Ltd
01.04.2004
Taylor and Francis Journals |
| Series | Journal of Applied Statistics |
| Subjects | |
| Online Access | Get full text |
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| Abstract | In this paper, we are employing the generalized linear model (GLM) in the form
ij
=Xλ to decompose the symmetry model into the class of models discussed in Tomizawa (
1992
). In this formulation, the random component would be the observed counts f
ij
with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C. |
|---|---|
| AbstractList | In this paper, we are employing the generalized linear model (GLM) in the form
ij
=Xλ to decompose the symmetry model into the class of models discussed in Tomizawa (
1992
). In this formulation, the random component would be the observed counts f
ij
with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C. In this paper, we are employing the generalized linear model (GLM) in the form 𝓁 ij = Xλ to decompose the symmetry model into the class of models discussed in Tomizawa ( 1992 ). In this formulation, the random component would be the observed counts f ij with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C. In this paper, we are employing the generalized linear model (GLM) in the form lij=Xλ to decompose the symmetry model into the class of models discussed in Tomizawa (1992). In this formulation, the random component would be the observed counts fij with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C. In this paper, we are employing the generalized linear model (GLM) in the form 𝓁 = to decompose the symmetry model into the class of models discussed in Tomizawa (1992). In this formulation, the random component would be the observed counts with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C. In this paper, we are employing the generalized linear model (GLM) in the form ij=Xλ to decompose the symmetry model into the class of models discussed in Tomizawa (1992). In this formulation, the random component would be the observed counts f ij with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C.In this paper, we are employing the generalized linear model (GLM) in the form ij=Xλ to decompose the symmetry model into the class of models discussed in Tomizawa (1992). In this formulation, the random component would be the observed counts f ij with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4×4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6×6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C. In this paper, we are employing the generalized linear model (GLM) in the form lij=Xlambda to decompose the symmetry model into the class of models discussed in Tomizawa (1992). In this formulation, the random component would be the observed counts fij with an underlying Poisson distribution. This approach utilizes the non-standard log-linear model and our focus in this paper therefore relates to models that are decompositions of the complete symmetry model. That is, models that are implied by the symmetry models. We develop factor and regression variables required for the implementation of these models in SAS PROC GENMOD and SPSS PROC GENLOG. We apply this methodology to analyse the three 4 x 4 contingency table, one of which is the Japanese Unaided distance vision data. Results obtained in this study are consistent with those from the numerous literature on the subject. We further extend our applications to the 6 x 6 Brazilian social mobility data. We found that both the quasi linear diagonal-parameters symmetry (QLDPS) and the quasi 2-ratios parameter symmetry (Q2RPS) models fit the Brazilian data very well. Parsimonious models being the QLDPS and the quasi-conditional symmetry (QCS) models. The SAS and SPSS programs for implementing the models discussed in this paper are presented in Appendices A, B and C. [PUBLICATION ABSTRACT] |
| Author | Lawal, H. Bayo |
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| Cites_doi | 10.1080/00949650108812134 10.1177/0008068319900106 10.1093/biomet/66.3.413 10.18637/jss.v007.i08 10.1289/ehp.8563235 10.1086/228281 10.1080/00031305.1996.10473556 10.1093/biomet/65.2.413 10.1002/bimj.4710350211 10.1002/bimj.4710290109 10.1080/00324728.1958.10405012 10.1007/BF00209547 10.1086/226862 |
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| Keywords | quasi-diagonal symmetry model Poisson factor regression |
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| References | e_1_3_2_27_1 e_1_3_2_28_1 Stuart A (e_1_3_2_20_1) 1955 Yamagushi K (e_1_3_2_29_1) 1990 Tomizawa S (e_1_3_2_23_1) 1985; 15 e_1_3_2_24_1 e_1_3_2_26_1 Tomizawa S (e_1_3_2_25_1) 1986; 4 Bishop YM (e_1_3_2_6_1) 1975 Tomizawa S (e_1_3_2_22_1) 1984; 14 Agresti A (e_1_3_2_4_1) 1996 Arminger G (e_1_3_2_5_1) 1995 e_1_3_2_16_1 e_1_3_2_9_1 e_1_3_2_17_1 e_1_3_2_8_1 e_1_3_2_19_1 Clogg CC (e_1_3_2_7_1) 1990; 2 Agresti A (e_1_3_2_2_1) 1983 e_1_3_2_11_1 e_1_3_2_12_1 e_1_3_2_13_1 e_1_3_2_14_1 e_1_3_2_15_1 Goodman LA (e_1_3_2_10_1) 1983; 13 Tomizawa S (e_1_3_2_21_1) 1983; 13 McCullagh P (e_1_3_2_18_1) 1989 Agresti A (e_1_3_2_3_1) 1989 |
| References_xml | – volume: 4 start-page: 487 year: 1986 ident: e_1_3_2_25_1 article-title: Four kinds of symmetry models and their decompositions in a square contingency table with ordered categories publication-title: Biom. J. – ident: e_1_3_2_14_1 doi: 10.1080/00949650108812134 – volume: 2 start-page: 409 year: 1990 ident: e_1_3_2_7_1 article-title: Models for the analysis of change in discrete variables, in: A. Von Eye (Ed.) Statistical Methods in Longitudinal Research publication-title: San Diego, CA: Academic Press – ident: e_1_3_2_27_1 doi: 10.1177/0008068319900106 – volume: 13 start-page: 127 year: 1983 ident: e_1_3_2_21_1 article-title: The models for symmetric cells in a 2 and a 3 contingency tables with ordered categories publication-title: J. Japan Statist. Soc. – start-page: 181 volume-title: Oxford: Basil Blackwell year: 1990 ident: e_1_3_2_29_1 – ident: e_1_3_2_9_1 doi: 10.1093/biomet/66.3.413 – ident: e_1_3_2_15_1 doi: 10.18637/jss.v007.i08 – ident: e_1_3_2_24_1 doi: 10.1289/ehp.8563235 – ident: e_1_3_2_19_1 doi: 10.1086/228281 – ident: e_1_3_2_28_1 doi: 10.1080/00031305.1996.10473556 – start-page: 313 volume-title: Statist. Prob. Letters year: 1983 ident: e_1_3_2_2_1 – volume: 15 start-page: 151 year: 1985 ident: e_1_3_2_23_1 article-title: Decompositions for odds-symmetry models in a square contingency table with ordered categories publication-title: J. Japan Statist. Soc. – volume-title: New York: Plenum year: 1995 ident: e_1_3_2_5_1 – ident: e_1_3_2_17_1 doi: 10.1093/biomet/65.2.413 – ident: e_1_3_2_13_1 doi: 10.1002/bimj.4710350211 – volume-title: New York: Wiley year: 1996 ident: e_1_3_2_4_1 – start-page: 412 volume-title: Biometrika year: 1955 ident: e_1_3_2_20_1 – ident: e_1_3_2_26_1 doi: 10.1002/bimj.4710290109 – year: 1989 ident: e_1_3_2_18_1 article-title: Generalized Linear Models publication-title: London: Chapman & Hall – volume-title: Cambridge, MA: MIT Press year: 1975 ident: e_1_3_2_6_1 – ident: e_1_3_2_12_1 doi: 10.1080/00324728.1958.10405012 – ident: e_1_3_2_16_1 doi: 10.1007/BF00209547 – volume: 14 start-page: 413 year: 1984 ident: e_1_3_2_22_1 article-title: Three kinds of decomposition for the conditional symmetry model in a square contingency table publication-title: J. Japan Statist. Soc. – ident: e_1_3_2_8_1 doi: 10.1086/226862 – ident: e_1_3_2_11_1 – volume-title: New York: Wiley year: 1989 ident: e_1_3_2_3_1 – volume: 13 start-page: 10 year: 1983 ident: e_1_3_2_10_1 article-title: The analysis of cross-classified data having ordered and/or unordered categories: association models, correlation models and asymmetry models for contingency tables with or without missing entries publication-title: The Annals of Statist. |
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| Snippet | In this paper, we are employing the generalized linear model (GLM) in the form
ij
=Xλ to decompose the symmetry model into the class of models discussed in... In this paper, we are employing the generalized linear model (GLM) in the form 𝓁 = to decompose the symmetry model into the class of models discussed in... In this paper, we are employing the generalized linear model (GLM) in the form lij=Xλ to decompose the symmetry model into the class of models discussed in... In this paper, we are employing the generalized linear model (GLM) in the form lij=Xlambda to decompose the symmetry model into the class of models discussed... In this paper, we are employing the generalized linear model (GLM) in the form ij=Xλ to decompose the symmetry model into the class of models discussed in... In this paper, we are employing the generalized linear model (GLM) in the form 𝓁 ij = Xλ to decompose the symmetry model into the class of models discussed in... |
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| SubjectTerms | Data analysis factor Generalized linear models Original Poisson Poisson distribution quasi-diagonal symmetry model regression Symmetry Variables |
| Title | Using a GLM to Decompose the Symmetry Model in Square Contingency Tables with Ordered Categories |
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