Measure of Departure from Partial Symmetry for Square Contingency Tables

Corresponding Author: Yusuke Saigusa Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City, Chiba, 278-8510, Japan Email: saigusaysk@gmail.com Abstract: For square contingency tables, the present paper newly considers the partial symmetry model which indicates that there is a symmetric structure of probabilities for at least one of pairs of symmetric cells. It also proposes the measure to express the degree of departure from the partial symmetry model. Examples are given.


Introduction
Consider a square r×r contingency table with the same row and column classifications. Let p ij denote the cell probability that an observation falls in its (i, j) cell (i = 1,…,r; j = 1,…,r). Consider the Symmetry (S) model as follows: where, ψ ij = ψ ji for i≠j (Bowker, 1948;Bishop et al., 1975, p. 282). For the analysis of data, the S model may fit the data poorly because it has the strong restriction. When the S model fits the data poorly, many statisticians may be interested in applying some models which have weaker restriction than the S model. There are some symmetry or asymmetry models; for instance, the marginal homogeneity model (Stuart, 1955), the quasi-symmetry model (Caussinus, 1965), the conditional symmetry model (McCullagh, 1978), the diagonals-parameter symmetry model (Goodman, 1979) and the cumulative diagonals-parameter symmetry model (Tomizawa, 1993;Tahata and Tomizawa, 2014), etc.
On the other hands, some statisticians may be interested in measuring the degree of departure from the S model when the model fits the data poorly.
Assume that p ij + p ji >0 for i≠j. Let for i≠j with δ = ΣΣ i≠j p ij . Tomizawa et al. (1998) gave the measure to express the degree of departure from the S model as follows: and the value at λ = 0 is taken to be the limit as λ→0, where λ is a real-valued parameter which is chosen by the user. Note that ( ) ij H λ is the Patil and Taillie (1982) diversity index of degree λ which includes the Shannon entropy. We point out that ( ) S λ Φ is expressed as the weighted arithmetic mean of the diversity index ( ) ij H λ . Table 1a is taken from Hashimoto (1999, p. 151) and Tables 1b and 1c are taken from Bishop et al. (1975, p. 100). These data relate father's and son's occupational status categories in Japan, in Denmark and in British. The smaller category number means the higher status in each of Tables 1a-c. For example, on the father's and son's occupational status in Denmark, it seems that there is a symmetric structure of probabilities for several pairs although there is not a symmetric structure of probabilities for all pairs.  (Hashimoto, 1999, p.151), (b) in Denmark and (c) in British (Bishop et al., 1975, p   where, ψ st = ψ ts for at least one (s, t) with s≠t. We shall refer to this model as the Partial Symmetry (PS) model. Since the S model indicates that p ij equals p ji for all (i, j), the PS model is implied by the S model. For each of Tables 1a-c, the PS model means that the probability that a father's occupational status is i and son's occupational status is j, equals the probability that the father's occupational status is j and son's occupational status is i for at least one (i, j), i = 1,…,5; j = 1,…,5; i ≠ j.
We are now interested in measuring the degree of departure from the PS model than the S model.
By the way, Tomizawa et al. (2004) gave the measure in the form of geometric mean, which describes the strength of association between the row and column variables for two-way contingency table, although the detail is omitted. In order to express the degree of departure from the PS model, we shall consider the geometric mean type measure.
In the present paper, section 2 proposes a new measure which expresses the degree of departure from the PS model. Section 3 gives the approximate confidence interval of the proposed measure. Section 4 gives Examples. Section 5 compares two measures and shows that the proposed measure is appropriate for measuring the degree of departure from PS. Section 6 presents concluding remarks.

Measure
Assume that p ij + p ji >0 for i≠j. Consider the measure defined by: and the value at λ = 0 is taken to be the limit as λ→0 and λ is a real-valued parameter which is chosen by the user.
Note that ( ) It is easily seen that the value of ( )

Approximate Confidence Interval of Measure
Assume that a multinomial distribution applies to the r×r table. We shall obtain the approximate standard error and the large-sample confidence interval of ( ) P λ Φ . Let n ij denote the observed frequency of (i, j) cell in the table  ( 1  1  ), i r j r = , , ; = , , … … and let n denote the total number of observations, i.e., n = ΣΣn ij . The sample version of ( )

Examples
Consider the data in Table 1 again. Tables 2 and 3 Table 3, for any λ(>-1), the confidence interval of ( ) P λ Φ applied to the data in Table 1a does not include 0. So there would not be the structure of PS in Table 1a. On the other hand, for any λ(>-1), the confidence intervals of ( ) P λ Φ applied to the data in each of Table 1b and 1c include 0. So there may be the structure of PS in each of Tables 1b and 1c.
We shall further compare the degrees of departure from PS for Tables 1a-c using ( ) P λ Φ . Comparing the confidence intervals of ( ) P λ Φ for Tables 1a-c, for any λ(>-1), it is inferred that the degree of departure from PS for Table 1a is larger than that for each of Tables 1b and 1c. In a similar way, from Table 2, it is inferred that the degree of departure from S for Table 1a is larger than that for each of Tables 1b and 1c.  We point out that, for any λ(>-1) the estimated value of ( ) P λ Φ applied to each of Tables 1a-c is less than that of ( ) S λ Φ .

Concluding Remarks
For an r×r square contingency table, we have considered the PS model which has weaker restriction than the S model. The PS model indicates symmetry of probabilities for at least a pair of symmetric cells instead of all pairs of symmetric cells. We have proposed the measure to express the degree of departure from PS. The measure enables us to see how far cell probabilities are distant from those with a PS structure.
The readers may be interested in the relationship between the proposed measure and the goodness-of-fit test for the PS model. However it may be difficult to discuss the relationship.
We also have shown with Examples that ( ) P λ Φ is useful for expressing and comparing the degree of departure from the partial symmetry toward the complete asymmetry between different tables.