Abstract
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.
Highlights
Suppose that an R × R square contingency table has the same categories in the row classification as in the column classification
Agresti (2002, p.429) considered the ordinal quasi-symmetry (OQS) model defined by pij =
If it is reasonable to assume an underlying three-variate normal distribution which does not require the equality of marginal variances, the extended LDPS (ELDPS)-3 model rather than the linear diagonals-parameter symmetry (LDPS)-3 model may be appropriate for an ordinal three-way table
Summary
Suppose that an R × R square contingency table has the same categories in the row classification as in the column classification. For square tables with ordered categories, Agresti (1984, p.203) proposed the linear diagonals-parameter symmetry (LDPS) model defined by pij =. Agresti (2002, p.429) considered the ordinal quasi-symmetry (OQS) model defined by pij =. When σ12 = σ22, the f (u, v)/f (v, u) has the form ξv−u for some constant ξ, and the 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. Tomizawa (1991) described that the ELDPS model rather than the LDPS model would be appropriate if it is reasonable to assume an underlying bivariate normal distribution which does not require the equality of marginal variances. Note that the LDPS (OQS) and ELDPS models are special cases of the QS model. Theorems 2.1 and 2.2 may be useful for seeing the reason for the poor fit when the S model fits the data poorly
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