Decomposition of Marginal Homogeneity into Logit and Mean Ridits Equality for Square Contingency Tables

Problem statement: For square contingency tables with ordered categor i s, if the Marginal Homogeneity (MH) model holds, then the probability that X selected at random from the row marginal distribution is less than Y selected independently a random from the column marginal distribution equals the probability that X is greater than Y. Ho wever, the converse does not hold. We are intereste d in the condition in order that the converse holds. Approach: This study gave a decomposition theorem that the MH model holds if and only if both the mar ginal cumulative logistic model and the model of equality of mean ridits for the row and column marg inal distributions hold. Examples are given. Results: For the data of cross-classification of ewes accor ding to number of lambs born in consecutive years and cross-classification of unaided distance vision of women in British, the decomposition of th e MH model is applied and the detailed analysis is gi ven. Conclusion: When the MH model fits the data poorly, this decomposition is useful for seeing whi ch of decomposed two models influences stronger.


INTRODUCTION
Consider an R×R square contingency table with the same row and column ordinal classifications. Let X and Y denote the row and column variables, respectively and let ij Pr(X i Y j) p = , = = (i 1 R j 1 R). = , , ; = , , … … The Marginal Homogeneity (MH) model is defined by (Stuart, 1955;Bhapkar, 1966;Bishop et al., 1975;Caussinus, 1965): This indicates that the row marginal distribution is identical to the column marginal distribution. The MH model also may be expressed as: L denote the marginal cumulative logit of X and Y, respectively. These are given as: The MH model may be further expressed as: As an extension of the MH model, the Marginal cumulative Logistic (ML) model is defined by McCullagh (1977) and Agresti (1984): This model states that the odds that X is i or below instead of i+1 or above, is exp(∆) times higher than the odds that Y is i or below instead of i+1 or above, for i 1 R 1 = , , − … . If ∆>0, X rather than Y tends to be i or below instead of i+1 or above, for . A special case of this model obtained by putting ∆ = 0 is the MH model. Miyamoto et al. (2005) and Tahata et al. (2007) gave the theorem that the MH model holds if and only if both the ML model and the marginal mean equality model (i.e., E(X) E(Y) = ) hold (Tahata et al., 2008). On the other hand, Agresti (1983;1984) considered the comparison between the marginal distributions using the measure defined by: Where: X = Selected at random from the row marginal distribution Y = Selected independently at random from the column marginal distribution This is a population value of the difference between discrete analogs of the Mann-Whitney statistics. This measure is positive when X is stochastically less than Y and negative when X is stochastically greater than Y. Note that τ can be expressed using the mean ridits; as described later.
We note that the MH model implies the structure of τ = 0, thus, the MH model is not equivalent to τ = 0. We are now interested in what structure is necessary to obtain the MH model in addition to the structure of τ = 0.
The purpose of this study is to give a theorem that the MH model holds if and only if both the ML model and the structure of τ = 0 hold.

MATERIALS AND METHODS
Let: The { X i r } and { Y i r } are the marginal ridits; Bross (1958). The mean ridit for the distribution of Y when the distribution of X is the identified distribution for calculating the ridits is:

∑ ∑
Then, τ is expressed as Agresti (1984): We shall refer to the structure of τ = 0 as the marginal Mean Ridits equality (MR) model. We see: . Therefore, the MH model may be expressed as: This indicates that the probability that the row variable X selected at random from the row marginal distribution is in category i or below and the column variable Y selected independently at random from the column marginal distribution is in category i+1 or above is equal to the probability that such X is in category i+1 or above and such Y is in category i or below. Using {H 1(i) } and {H 2(i) }, the ML model may also be expressed as: This model with θ = 1 is the MH model. We obtain the following theorem. Since the ML model holds, we have: If θ = 1, we see that the MH model holds. If θ>1, we have: we have: Thus, if θ<1, we have τ<0. Since the MR model holds, i.e., τ = 0, we obtain θ = 1. Namely, the MH model holds. The proof is completed.
Assume that a multinomial distribution is applied to the R×R table. The maximum likelihood estimates of expected frequencies under the MH, ML and MR models could be obtained using the Newton-Raphson method in the log-likelihood equation. The numbers of degrees of freedom for testing the goodness-of-fit of the MH, ML and MR models are R-1, R-2 and 1, respectively.

RESULTS
Example 1: The data in Table 1, are taken directly from Bishop et al. (1975). These data indicate that ewes are cross-classified by the number of lambs born to them in two consecutive years, 1952 and 1953. The MH model fits these data poorly, yielding the likelihood ratio chi-squared statistic G 2 = 18.65 with 2 degrees of freedom. Also, the ML model fits these data poorly, yielding G 2 = 18.55 with 1 degree of freedom. However, the MR model fits these data well, yielding G 2 = 0.78 with 1 degree of freedom.
From Theorem 1, we see that the poor fit of the MH model is caused by the influence of the lack of structure of the ML model rather than the MR model.
The MH model fits these data poorly, yielding the likelihood ratio chi-squared statistic G 2 = 11.99 with 3 degrees of freedom. Also, the MR model fits these data poorly, yielding G 2 = 11.94 with 1 degree of freedom. However, the ML model fits these data well, yielding G 2 = 0.39 with 2 degrees of freedom.
We see from Theorem 1 that the poor fit of the MH model is caused by the influence of the lack of structure of the MR model rather than the ML model.     1943-1946from Stuart (1955) Left eye grade Best (