Decomposition of Generalized Asymmetry Model for Square Contingency Tables

For the analysis of square contingency tables with same row and column ordinal classifications, the present paper gives the decomposition of the generalized linear diagonals-parameter symmetry model using the diagonals-parameter symmetry model. Moreover, it gives the decomposition of the symmetry model using above the proposed decomposition.


Introduction
Consider an R R × square contingency table with the same row and column classifications.Let ij p denote the probability that an observation will fall in the ith row and jth column of the table ( 1, , ; 1, , i R j R = =   ).For square tables with ordered categories, Goodman [1] proposed the diagonals-parameter symmetry (DPS) model, defined by where ij ji ψ ψ = .Note that the DPS models with { } to the symmetry (S) (Bowker [2]), linear diagonals-parameter symmetry (LDPS) (Agresti [3]), and another LDPS (ALDPS) (Tomizawa [4]) models, respectively.
Yamamoto and Tomizawa [5] proposed the generalization of LDPS model.We will denote  as the set of integers.For a fixed K ∈  , the generalized LDPS (LDPS(K)) model is defined by where ij ji ψ ψ = . Note that the LDPS(K) model with 1 δ = is identical to the S model.Especially the LDPS(0) and LDPS(-R) models are equivalent to the LDPS and ALDPS models, respectively.
Tomizawa [6] gave the decomposition of the LDPS model using the DPS model, and showed that a test statistic for the LDPS model was equal to the sum of those for decomposed models.
For the analysis of square contingency tables with ordered categories, the purposes of this paper are (1) to give the decomposition of the LDPS(K) model using the DPS model, (2) to show that for the test statistic for the LDPS(K) model is equal to the sum of those for decomposed models, and (3) to give the decomposition of the S model using above the decomposition of the LDPS(K) model.

Decomposition of the Generalized Asymmetry Model
Tomizawa [6] proposed the linear diagonals-parameter marginal symmetry (LDPMS) model, defined by .
Let X and Y denote the row and column variables, respectively.The LDPMS model indicates that ( ) > .Also, Tomizawa [6] gave the decomposition of the LDPS model using the DPS and LDPMS models, and showed that a test statistic for the LDPS model is equal to the sum of those for the DPS and LDPMS models.
To consider the decomposition of the LDPS(K) model, we shall introduce a new model.For a fixed K ∈  , the generalized LDPMS (LDPMS(K)) model is defined by .
We obtain the following theorem.Theorem 1.For a fixed K ∈  , the LDPS(K) model holds if and only if both the DPS and LDPMS(K) models hold.
Proof.If the LDPS(K) model holds, then the DPS and LDPMS(K) models hold.Assuming that both the DPS and LDPMS(K) models hold, then we shall show that the LDPS(K) model holds.
From the LDPMS(K) model holds, we obtain ( ) . Namely, the LDPS(K) model holds.The proof is completed.

Orthogonality of Test Statistic and Model Selection
Assume that a multinomial distribution applies to the R R × table.Let ij n denote the observed frequency in the ith row and jth column of the R R = ∑∑ .The maximum likelihood estimates (MLEs) of expected frequencies under the model could be obtained by using, e.g., the Newton-Raphson method in the log-likelihood equation.Each model can be tested for goodness-of-fit by, e.g., the likelihood ratio chi-square statistic (denoted by 2 G ) with the corresponding degrees of freedom (df).The test statistic 2  G of model M is given by ( ) Thus, for the data, the model with the minimum AIC + (i.e., the minimum AIC) is the best-fitting model.For the analysis of contingency tables, Read [9] discussed the orthogonality, which is equivalent to the asymptotic separability in Aitchison [10] and the independence in Darroch and Silvey [11] of test statistic for goodness-of-fit of two models.
On the orthogonality of test statistic for models in Theorem 1, we obtain the following theorem.Theorem 2. For a fixed K ∈  , the following equation holds: The number of df for the LDPS(K) model equals the sum of number of df for the DPS and LDPMS(K) models.
Proof.First, we consider that the MLEs of expected frequencies { } ij m under the LDPS(K) model are given by ( ) ( ) for 1 for , where δ is the solution of the following equation for , Last, we consider that the MLEs of expected frequencies { } ij m under the LDPMS(K) model are given by where ∆ is the solution of the Equation (3.1) with δ replaced by ∆ .Thus, we see that ( ) îj ij n m under the LDPS(K) model is equal to the product of ( ) îj ij n m under the DPS model and that under the LDPMS(K) model.Therefore, the test statistic for goodness-of-fit for LDPS(K) model is equal to the sum of those for two models.The proof is completed.

Decomposition of the Symmetry Model
For square contingency tables with ordered categories, Kurakami, Yamamoto and Tomizawa [12] considered two models.One is the generalized exponential symmetry (GES) model defined by  [12] also gave the decomposition of the S model using the GES and GWGS models, and showed that a test statistic for the S model is approximately equivalent to the sum of those for the GES and GWGS models.
We will denote 4  as the set of non-negative integers.Yamamoto, Ohama and Tomizawa [13] gave the following theorems.
Theorem 3.For a fixed 4 K ∈  , the S model holds if and only if both the LDPS(K) and WGS(K) models hold.
Theorem 4. For a fixed 4 K ∈  , the following asymptotic equivalence holds: The number of df for the S model equals the sum of the number of df for the LDPS(K) and WGS(K) models.
From the theorems given by Kurakami et al. [12], we obtain the following theorems as extensions of Theorems 3 and 4 (because the 1 3 ∪   includes 4  ).Theorem 5.For a fixed

the S model holds if and only if both the LDPS(K) and WGS(K) models hold.
Theorem 6.For a fixed K ∈ ∪   , the following asymptotic equivalence holds: ( The number of df for the S model equals the sum of the number of df for the LDPS(K) and WGS(K) models.From Theorems 1 to 6, we obtain the following corollaries.Corollary 1.For a fixed , the S model holds if and only all the DPS, LDPMS(K) and WGS(K) models hold.
Corollary 2. For a fixed , the following asymptotic equivalence holds: The number of df for the S model equals the sum of the number of df for the DPS, LDPMS(K) and WGS(K) models.

An Example
Consider the data in Table 1, taken directly from Bishop, Fienberg and Holland ( [14], p. 100).From Table 2, all LDPS(K) models, the S model and DPS model give poor fits to these data.However, all LDPMS(K) models fit these data well.
The LDPMS(2) model is the best-fitting model among the other LDPMS(K) models because it has a minimum AIC + value.Under the LDPMS(2) model, the MLE of ∆ is ˆ= 1.04 ∆ .Thus, we see that the status category for a father tends to be less than that for his son.
Theorem 1 would be useful for seeing the reason for its poor fit when the LDPS(K) model fits the data poorly.Thus, for the data in Table 1, the poor fit of the LDPS(K) model is caused by the poor fit of the DPS model rather than the LDPMS(K) model.Also, Theorem 5 would be useful for seeing the reason for its poor fit when the S model fits the data poorly.From Table 2, WGS(K) models (except the WGS(−1) model) give poor fits to these data.Thus, when K is not equal to −1, we cannot see that the poor fit of the S model is caused by the poor fit of either LDPS(K) and WGS(K) models (although, we can see that the poor fit of the S model is caused by the poor fit of both LDPS(K) and WGS(K) models).However, using Corollary 1, we can see that the poor fit of  the S model is caused by the poor fit of DPS and WGS(K) models rather than the LDPMS(K) model.

Concluding Remarks
We have given the decomposition of the LDPS(K) model using the DPS model (namely, Theorem 1).Also, we have shown that the test statistic for the LDPS(K) is equal to the sum of those for the decomposed models (namely, Theorem 2).Moreover, we have given the decomposition of the S model using Theorem 1 (namely, Corollary 1), and shown that the test statistic for the S model is approximately equivalent to the sum of those for the decomposed models (namely, Corollary 2).Although details will be omitted, Yamamoto, Ohama and Tomizawa [15] gave the another decomposition of the the LDPS(K) model for a fixed 4 K ∈  .However, it does not hold the orthogonality of test statistic for models.Thus, Theorem 1 may be useful for analyzing the data than the decomposition by Yamamoto et al. [15].Because Theorem 1 shows the decomposition of LDPS(K) for a fixed K ∈  (because  includes 4  ), and also holds the orthogonality of test statistic for models.
We can solve (3.1) for δ by using the Newton-Raphson method.Second, we consider that the MLEs of expected frequencies { } ij m under the DPS model are given by

3 K 1 K 2 K
specified non-negative values.The other is the generalized weighted global symmetry (GWGS) model defined by .∈  , the GES model with non-negative values { } ij w K j i = + − is identical to the LDPS(K) model.For a fixed are non-negative values, the LDPS(K) model is included in the GES model.Note that for a fixed ∈  , the LDPS(K) model is not included in the the GES model, because { } ij w K j i = + − have both positive and negative values.the WGS(K) model.Kurakami et al.

Table 2 .
Likelihood ratio chi-square values G 2 and AIC + for models applied to the data in Table1.
*Means significant at the 0.05 level.