Received 12 April 2016; accepted 5 June 2016; published 8 June 2016
Consider an table with ordered categories. Let for and , and let denote the probability that an observation will fall in ith cell of the table. Let denote the kth variable of the table for. Denote the hth-order () marginal probability
In the case of , the symmetry (ST) model is defined by
where for any permutation of i (Bhapkar and Darroch,  ; Agresti,  , p. 439). We may also refer to this model as the permutation-symmetry model.
The hth-order marginal symmetry () model is defined by, for a fixed h (),
where is any permutation of, and for any and (Bhapkar and Darroch,  ). The hth-order quasi symmetry () model is defined by, for a fixed h (),
where for any permutation j of i (Bhapkar and Darroch,  ). Bhapkar and Darroch  gave the theorem that:
1) For the table and a fixed h (), the ST model holds if and only if both the and models hold.
Tahata, Yamamoto and Tomizawa  considered the hth-linear ordinal quasi symmetry () model, which was defined by, for a fixed h (),
where for any permutation j of i. This model is a special case of the model. The model is the ordinal quasi symmetry model when (Agresti,  , p. 244). Tahata et al.  also considered the hth-order marginal moment equality () model, which was expressed as, for a fixed h (),
where for. Tahata et al.  obtained the theorem that:
2) For the table and a fixed h (), the ST model holds if and only if both the and models hold.
Various decompositions of the symmetry model are given by several statisticians, e.g. Caussinus  , Bishop, Fienberg and Holland (  , Ch.8), Read  , Kateri and Papaioannou  , and Tahata and Tomizawa  .
For the table, the point-symmetry (PT) model is defined by
where and with for (Wall and Lienert,  ; Tomizawa,  ). This model indicates the point-symmetry of cell probabilities with respect to the center point of multi-way table.
For the table, Tahata and Tomizawa  considered the hth-order marginal point-symmetry () model defined by, for a fixed h (),
Tahata and Tomizawa  also considered the hth-order quasi point-symmetry () model defined by, for a fixed h (),
where. Tahata and Tomizawa  gave the theorem that:
3) For the table and a fixed h (), the PT model holds if and only if both the and models hold.
Theorem 3) is Theorem 1) with structures in terms of permutation-symmetry, i.e. the ST, and models, replaced by structures in terms of point-symmetry, i.e. the PT, and 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 PT model.
In the present paper, Section 2 proposes three models. Section 3 gives a new decomposition of the PT model. Section 4 provides the concluding remarks.
Let, where denotes the largest integer less than or equal to x.
Consider the model defined by, for a fixed odd number h (),
and for. We shall refer to this model as the hth-order marginal moment point-symmetry () model. Note that if the model holds then the model holds. Under the model, we see, for any k (),
Then we obtain, for any and (),
Under the model, we see, for any, and (),
Then we obtain, for any, , and (),
Thus we are not interested in the model with h being even. Therefore we shall consider the model with h being odd.
Consider the model defined by
where. We shall refer to this model as the ordinal quasi point-symmetry (OQPT) model. In the case of, this model is identical to the model proposed by Tahata and Tomizawa  . The special case of the OQPT model obtained by putting is the PT model. Also the OQPT model is the special case of the model obtained by putting. The OQPT model may be expressed as
with and. From 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 and slope for the category indicator under the OQPT model. Thus the parameter 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 OQPT model as follows, for a fixed odd number h (),
where. We shall refer to this model as the hth-linear ordinal quasi point-symmetry () model. Especially, when, the model is identical to the OQPT model. Also the model is the special case of the model obtained by putting, and
Figure 1 shows the relationships among models.
3. Decomposition of Point-Symmetry
We obtain the following theorem:
Theorem 1. For the table and a fixed odd number h (), the PT model holds if and only if both the and models hold.
Proof. If the PT model holds, then both the and models hold. Assuming that both the and models hold, then we shall show the PT model holds. Let denote cell pro- babilities which satisfy both the and models. The model is expressed as
Note that satisfy, and. Then the model is also ex-pressed as
Figure 1. Relationships among various models. Note: “” indicates that model implies model.
The model is expressed as
Then we denote () by.
Consider arbitrary cell probabilities which satisfy the model and
From (1), (2) and (3),
Let denote the Kullback-Leibler information, e.g., it between q and is
Thus, for fixed,
and then q uniquely minimize (see Darroch and Ratcliff,  ).
Let. Then, in a similar way as described above, we obtain
and then uniquely minimize, hence. Namely q satisfy the PT model. The proof is completed.
For the analysis of data, the test of goodness-of-fit of the 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
Also the number of degrees of freedom for the model is
We point out that, for a fixed h, the number of degrees of freedom for the PT model is equal to sum of those for the and models.
4. Concluding Remarks
For multi-way contingency tables, we have proposed the, OQPT and models. Under the OQPT 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 (), the model implies the model.
We have gave the theorem that the PT model holds if and only if both the and models. For the analysis of data, the decomposition given in the present paper may be useful for determining the reason when the PT model fits data poorly.
The authors thank the editor and the referees for their helpful comments.
 Bhapkar, V.P. and Darroch, J.N. (1990) Marginal Symmetry and Quasi Symmetry of General Order. Journal of Multivariate Analysis, 34, 173-184.
 Tahata, K., Yamamoto, H. and Tomizawa, S. (2011) Linear Ordinal Quasi-Symmetry Model and Decomposition of Symmetry for Multi-Way Tables. Mathematical Methods of Statistics, 20, 158-164.
 Agresti, A. (2010) Analysis of Ordinal Categorical Data. 2nd Edition, Wiley, Hoboken.
 Caussinus, H. (1965) Contribution à l’analyse statistique des tableaux de corrélation. Annales de la Faculté des Sciences de l’Université de Toulouse, 29, 77-182.
 Read, C.B. (1977) Partitioning Chi-Square in Contingency Tables: A Teaching Approach. Communications in Statistics, Theory and Methods, 6, 553-562.
 Kateri, M. and Papaioannou, T. (1997) Asymmetry Models for Contingency Tables. Journal of the American Statistical Association, 92, 1124-1131.
 Tomizawa, S. (1985) The Decompositions for Point Symmetry Models in Two-Way Contingency Tables. Biometrical Journal, 27, 895-905.
 Tahata, K. and Tomizawa, S. (2008) Orthogonal Decomposition of Point-Symmetry for Multiway Tables. Advances in Statistical Analysis, 92, 255-269.
 Tahata, K. and Tomizawa, S. (2015) Ordinal Quasi Point-Symmetry and Decomposition of Point-Symmetry for Cross-Classifications. Journal of Statistics: Advances in Theory and Applications, 14, 181-194.
 Darroch, J.N. and Ratcliff, D. (1972) Generalized Iterative Scaling for Log-Linear Models. Annals of Mathematical Statistics, 43, 1470-1480.