Decomposition of Point-Symmetry Using Ordinal Quasi Point-Symmetry for Ordinal MultiWay Tables

For multi-way tables with ordered categories, the present paper gives a decomposition of the point-symmetry model into the ordinal quasi point-symmetry and equality of point-symmetric marginal moments. The ordinal quasi point-symmetry model indicates asymmetry for cell probabilities with respect to the center point in the table.


Introduction
and k = 1, ,T  , and let i p denote the probability that an observation will fall in ith cell of the table.Let k X denote the kth variable of the table for 1, , k T =  .Denote the hth-order ( 1, , In the case of 1 1 , , T j j j =  of i (Bhapkar and Darroch, [1]; Agresti, [2], p. 439).We may also refer to this model as the permutation-symmetry model.
The hth-order marginal symmetry ( MS T h ) model is defined by, for a fixed h ( 1, , ( )

, , for any ,
where i j ψ ψ = for any permutation j of i (Bhapkar and Darroch, [1]).Bhapkar and Darroch [1] gave the theorem that: 1) For the T R table and a fixed h ( 1, , ), the S T model holds if and only if both the QS T h and MS T h models hold.Tahata, Yamamoto and Tomizawa [3] considered the hth-linear ordinal quasi symmetry ( LQS T h ) model, which was defined by, for a fixed h ( 1, , for any , where i j ψ ψ = for any permutation j of i.This model is a special case of the QS T h model.The LQS T h model is the ordinal quasi symmetry model when 1 h = (Agresti, [4], p. 244).Tahata et al. [3] also considered the hth-order marginal moment equality ( MME T h ) model, which was expressed as, for a fixed h ( 1, , , , Tahata et al. [3] obtained the theorem that: 2) For the T  R table and a fixed h ( 1, , ), the S T model holds if and only if both the LQS T h and MME T h models hold.Various decompositions of the symmetry model are given by several statisticians, e.g.Caussinus [5], Bishop, Fienberg and Holland ( [6], Ch.8), Read [7], Kateri and Papaioannou [8], and Tahata and Tomizawa [9].
For the 1 ( ) (Wall and Lienert, [10]; Tomizawa, [11]).This model indicates the point-symmetry of cell probabilities with respect to the center point of multi-way table.
For the T R table, Tahata and Tomizawa [12] considered the hth-order marginal point-symmetry ( MP T h ) model defined by, for a fixed h ( 1, , ) ( ) Tahata and Tomizawa [12] also considered the hth-order quasi point-symmetry where * i i γ γ = .Tahata and Tomizawa [12] gave the theorem that: 3) For the T R table and a fixed h ( 1, , ), the P T model holds if and only if both the QP T h and MP T h models hold.Theorem 3) is Theorem 1) with structures in terms of permutation-symmetry, i.e. the S T , QS T h and MS T h models, replaced by structures in terms of point-symmetry, i.e. the P T , QP T h and MP T h models.However, a theorem in terms of point-symmetry corresponding to Theorem 2) is not obtained yet.So we are now interested in the decomposition of the P T model.
In the present paper, Section 2 proposes three models.Section 3 gives a new decomposition of the P T model.Section 4 provides the concluding remarks.

Models
, where x     denotes the largest integer less than or equal to x.
Consider the model defined by, for a fixed odd number h ( h S ∈ ), ( ) ; 1, 3, , , , , Then we obtain, for any 1 k and 2 k ( Then we obtain, for any 1 k , 2 k , 3 k and 4 k ( We shall refer to this model as the ordinal quasi point-symmetry (OQP T ) model.In the case of 2 T = , this model is identical to the model proposed by Tahata and Tomizawa [13].The special case of the OQP T model obtained by putting 1 1 is the P T model.Also the OQP T model is the special case of the β can be interpreted as the effect of a unit increase in the kth variable on the log-odds.Consider the model being more general than the OQP T model as follows, for a fixed odd number h ( h S ∈ ), . Figure 1 shows the relationships among models.

Decomposition of Point-Symmetry
We obtain the following theorem:  = denote cell probabilities which satisfy both the Then the LQP T h model is also expressed as The MMP T h model is expressed as ( ) , , ; 1, 3, , , where , .( ) ; 1, 3, , , where , .
For the analysis of data, the test of goodness-of-fit of the LQP T h model is achieved based on, e.g., the likelihood ratio chi-square statistic which has a chi-square distribution with the number of degrees of freedom ( ) ( )

∑
We point out that, for a fixed h, the number of degrees of freedom for the P T model is equal to sum of those for the LQP T h and MMP T h models.

Concluding Remarks
For multi-way contingency tables, we have proposed the MMP T h , OQP T and LQP T h models.Under the OQP T model, the log-odds that an observation falls in a cell instead of in its point-symmetric cell is described as a linear combination of category indicators.For a fixed odd number h ( h S ∈ ), the LQP T h model implies the QP T h model.We have gave the theorem that the P T model holds if and only if both the LQP T h and MMP T h models.For the analysis of data, the decomposition given in the present paper may be useful for determining the when the P T model fits data poorly.
the symmetry (S T ) model is defined by for any , the point-symmetry (P T ) model is defined by for any ,

1 QP
this equation, we can see the log-odds that an ob-servation falls in ith cell instead of in the point-symmetric i * th cell, i.e. is described as a linear combination with intercept 0 β and slope k β for the category indicator k i under the OQP T model.Thus the parameter k shall refer to this model as the hth-linear ordinal quasi point-symmetry ( LQP T h ) model.Especially, when 1 h = , the LQP T h model is identical to the OQP T model.Also the LQP T h model is the special case of the QP T h model obtained by putting models hold, then we shall show the P T model holds.Let { } i q q


Also the number of degrees of freedom for the MMP T h , , , We shall refer to this model as the hth-order marginal moment point-symmetry ( MMP T h ) model.Note that if the MP T If the P T model holds, then both the