Parsimonious Cumulative Odds-Symmetry Model and its Decomposition for Square Tables with Ordered Categories

For analyzing contingency tables to estimate a probability distribution from presented data, we often use a parsimonious model fitting the presented data. The cumulative odds-symmetry (COS) model indicates that the log odds based on cumulative probability depend only on the column category. Considering that the COS model has many parameters, this study considers a more parsimonious model. This study provides another expression for the COS model whereby the log odds have polynomial patterns depending only on the column category. This study also proposes the cumulative polynomial columns-parameter symmetry (CPCPS) model as a more parsimonious than the COS model. The CPCPS model bridges the gap between the symmetry and COS models. The COS model always holds true when the CPCPS model is true. However, the opposite does not necessarily hold true. This study identifies an additional model that must be satisfied in order to achieve equivalence with the CPCPS model. This detection is useful for explaining why CPCPS does not fit the presented data. The CPCPS model provides a better fit for application to a single occupational dataset from Japan.