Geometric Mean Type Measure of Marginal Homogeneity for Square Contingency Tables with Ordered Categories

Corresponding Authors: Tomoyuki Nakagawa Department of Information Sciences, Tokyo University of Science, Chiba, Japan Email: t_nakagawa@rs.tus.ac.jp Abstract: For square contingency tables, some studies have developed the weighted arithmetic mean type measure to represent the degree of departure from the marginal homogeneity. The present paper proposes (1) the cumulative partial marginal homogeneity model which has the weaker restriction than the marginal homogeneity model and (2) the measure to represent the degree of departure from the proposed model. The measure is expressed as a weighted geometric mean of the diversity index. Finally, numerical studies are presented.


Introduction
Consider an r × r square contingency table with the same row and column classifications. Let pij denote the probability that an observation will fall in the ith row and jth column of the table ( 1,..., : 1,..., ) i r j r  . Stuart (1955)  The PMH model indicates that the row marginal distribution is identical to the column marginal distribution for at least one i. In addition, Saigusa et al. (2020) also proposed the measure to represent the degree of departure from the PMH model. Assuming pi + pi  0 for all i = 1,…, r, the measure is defined by: where, πi = (pi·+ p·i)/2, p1(i) = pi·/(pi· + p·i) and p2(i) = p·i/(pi· + p·i) and: The value at λ = 0 is taken to be the limit as λ → 0. It should be noted that Φ (λ) is expressed the weighted geometric mean of     i   . We also remark that the   i I  is the diversity index given by (Patil and Taillie, 1982).
Over the past few years, such partial structure and geometric mean type measure have been developed by many studies (Saigusa et al., 2016;2019).
Let X and Y denote the row and column variables, respectively. By considering the difference between cumulative probabilities G1(i) and G2(i), the MH model is also expressed as: where the cumulative probabilities are defined as follows: Tomizawa et al. (2003) proposed the measure to represent the degree of departure from the MH model.

Model and Measure
In this section, we propose a new model which has the structure of the cumulative partial marginal homogeneity for an r × r contingency table with ordered categories. In addition, we also propose the geometric mean type measure to represent the degree of departure from the new model.

New Model
A new model is proposed as:   We refer to this model as the Cumulative Partial Marginal Homogeneity (CPMH) model herein. It should be noted that the CPMH model has a different structure from the MH and PMH models. It is easy to see that the MH model has the structure of CPMH. Consider the artificial probability in Table 1 and the marginal cumulative probability for in Table 2. Tables 2a and 2b give the cumulative probability calculated from Tables 1a and 1b, respectively. Table 1a has the structure of MH and CPMH models, while Table 1b does not have that of the MH model. Therefore the CPMH model is not equivalent to the MH model.

Measure
Assume that G1(i) + G2(i)  0 for i = 1,···, r − 1. The measure to represent the degree of departure from the CPMH model is proposed as follows: Note that λ is a real value chosen by users. The measure holds the following properties. For any λ > −1: It should be noted that the measure   M   is expressed as the weighted geometric mean of the diversity index whereas   M   is the weighted arithmetic mean.

Approximate Confidence Interval
In this section, nij denotes the observed frequency in the ith row and jth column of the

Numerical Studies
This section presents the results of applying the model and the measure to some examples and real data.

Real Data
Consider the data in Tables 6a and 6b taken from (Tominaga, 1979). These data describe the crossclassifications of father's and his son's occupational status categories in Japan, which were examined in 1955 and in 1965, respectively. We regard as the smaller category number means the higher status herein. We are interested in whether there is the structure of CPMH in each table. Table 7 gives the estimated values of the measure   M   applied to Tables 6a and 6b.     Table 3a  0  0  0  Table 3b 0.159 0.245 0.309 Table 3c 0.182 0.280 0.353 Table 3d 0.199 0.306 0.384 Table 3e 0.215 0.328 0.409 Table 3f 1 1 1  (1)  We shall compare the values of measure which represents the degree of departure from the CPMH model for Tables 6a and 6b. From Table 7, the values in the confidence interval of   M   are greater for Table  6b than for Table 6a. Therefore, it is inferred that the degree of departure from the CPMH model for fatherson pairs is larger in 1965 than in 1955.

Concluding Remarks
For an r × r square contingency table with ordered categories, we have proposed the CPMH model which has weaker restriction than that of the MH model. We also have proposed the measure to represent the degree of departure from the CPMH model. The proposed measure  