Using a GLM to Decompose the Symmetry Model in Square Contingency Tables with Ordered Categories

Area of Science:

• Statistics
• Statistical Modeling
• Categorical Data Analysis

Background:

• Symmetry models are crucial for analyzing contingency tables.
• Existing methods may not fully capture complex symmetrical structures.
• Decomposition of symmetry models offers a more nuanced approach.

Purpose of the Study:

• To employ generalized linear models (GLM) for decomposing complete symmetry models.
• To develop and implement factor and regression variables for SAS and SPSS.
• To analyze real-world contingency table data, including Japanese vision and Brazilian social mobility.

Main Methods:

• Utilized a generalized linear model (GLM) formulation: 𝓁ij = Xλ.
• Assumed an underlying Poisson distribution for observed counts (fij).
• Developed factor and regression variables for implementation in SAS PROC GENMOD and SPSS PROC GENLOG.

Main Results:

• The methodology was successfully applied to Japanese Unaided distance vision data.
• Quasi linear diagonal-parameters symmetry (QLDPS) and quasi 2-ratios parameter symmetry (Q2RPS) models showed excellent fit for Brazilian social mobility data.
• Identified QLDPS and quasi-conditional symmetry (QCS) as parsimonious models.

Conclusions:

• The GLM-based decomposition of symmetry models is a viable and effective analytical approach.
• The developed methods and software implementations (SAS/SPSS) facilitate practical application.
• The study provides valuable insights into the structure of contingency tables, particularly for social mobility data.