Measure of Departure from Point Symmetry and Decomposition of Measure for Square Contingency Tables

For square contingency tables with ordered categories, Tomizawa, Biometrica J. 28 (1986), 387 – 393, considered the conditional point symmetry model. Kurakami et al ., J. Stat. Adv. Theory Appl. 17 (2017), 33 – 42, considered the another point symmetry and the reverse global symmetry model. The present paper proposes Kullback – Leibler information type measures to represent the degree of departure from each of the models. Also this paper shows a theorem that the measure for the another point symmetry modelisequaltothesumofthemeasuresforthereverseglobalsymmetrymodelandfortheconditionalpointsymmetrymodel. © 2021 The Authors . Published by Atlantis Press B.V. This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).


INTRODUCTION
For an r × r contingency table with the same row and column ordinal classifications, let X and Y denote the row and column variables, respectively.Also let Pr (X = i, Y = j) = p ij (1 ≤ i, j ≤ r).The symmetry (S) model (Bowker, [1]) is defined by p ij = p ji (i < j); see also Bishop et al. ([2], p.282).The S model indicates a structure of symmetry of the probabilities {p ij } with respect to the main diagonal of the table.The global symmetry (GS) model (Read,[3]) is defined by where The conditional symmetry (CS) model (Read,[3]; McCullagh, [4]) is defined by where  is unknown parameter; see also Agresti ([5], p.361) and Tomizawa [6].We note that the CS model is also expressed as *Corresponding author.Email: iki.kiyotaka@nihon-u.ac.jpPdf_Folio:526 So, the CS model indicates the CS.A special case of this model obtained by putting  = 1 is the S model.Read [3] gave the theorem that the S model holds if and only if both the GS and CS models hold.
Wall and Lienert [7] defined the point symmetry (PS) model defined by where i * = r + 1 − i and j * = r + 1 − j.This model indicates a structure of PS of the probabilities { p ij } with respect to the center cell when r is odd or the center point when r is even in square tables.Kurakami et al. [8] considered the another point symmetry (APS) model defined by The APS model has less restrictions than the PS model by excluding the restrictions imposed on reverse diagonal probabilities.Kurakami et al. [8] considered reverse global symmetry (RGS) model defined by where Tomizawa [9] considered the conditional point symmetry (CPS) model defined by where  is unknown parameter.The CPS model indicates that Kurakami et al. [8] gave the theorem that the APS model holds if and only if the RGS and CPS models hold.For more details on contingency tables analysis, see also Rao [10], Mosteller [11] and Wang [12].
By the way, when a model does not hold, we are interested in measuring the degree of departure from the model.Tomizawa [13], Tomizawa [14] and Tomizawa and Saitoh [15] considered the measures which indicate the degree of departure from S, GS and CS, respectively.Tomizawa and Saitoh [15] gave the theorem that the measure from S is equal to the sum of the measure from GS and the measure from CS. Now, we are interested in proposing measures which indicate the degree of departure from APS, RGS and CPS, and showing the theorem that the measure from APS is equal to the sum of the measure from RGS and the measure from CPS. Section 2 proposes the new measures which represent the degree of departure from APS, RGS and CPS (denoted by Φ APS , Φ RGS and Φ CPS ), and show that the value of Φ APS is equal to the sum of the value of Φ RGS and the value of Φ CPS .Section 3 gives an approximate standard error and large-sample confidence intervals for the proposed measures.Section 4 describes the relationship between the proposed measures and likelihood ratio statistic.Section 5 gives an example.Section 6 provides some concluding remarks.

MEASURES FROM MODELS AND DECOMPOSITION OF MEASURE
We assume that p ij + p i * j * > 0, for 1 ≤ i, j ≤ r; i We propose the measure for indicating how degree the departure from the APS model is as follows: where Pdf_Folio:527 Note that I APS is the Kullback-Leibler information between {p c ij } and {p APS ij }.The measure Φ APS has characteristics that, (i) 0 ≤ Φ APS ≤ 1, (ii) Φ APS = 0 if and only if p ij = p i * j * for i + j < r + 1, and (iii) Φ APS = 1 if and only if p ij = 0 (then p i * j * > 0) or p i * j * = 0 (then p ij > 0) for i + j < r + 1.

Next, assume that Δ
We propose the measure for indicating how degree the departure from the RGS model is as follows: where Moreover, assuming that Δ U > 0, Δ L > 0, and p ij + p i * j * > 0 for 1 ≤ i, j ≤ r; i + j ≠ r + 1, we propose the measure for indicating how degree the departure from the CPS model is as follows: where Note that I CPS is the Kullback-Leibler information between {p c ij } and {p CPS ij }.We obtain the theorem as follows: Theorem 1.The value of Φ APS is equal to the sum of the value of Φ RGS and the value of Φ CPS .
Proof.We see that For i + j > r + 1, we see Pdf_Folio:528 From equations ( 1) and ( 2), we see The proof is completed.
Then, we obtain 0 ≤ Φ APS ≤ 1 and 0 ≤ Φ RGS < 1 (note that Φ RGS ≠ 1 because of both Δ U > 0 and Δ L > 0).Since Φ CPS ≥ 0, we obtain 0 ≤ Φ CPS ≤ 1. Besides, (i) Φ CPS = 0 if and only if there is a structure of CPS in the square table, and (ii) Φ CPS = 1 if and only if Φ APS = 1 and Φ RGS = 0; i.e., p ij = 0 (then p i * j * > 0) or p i * j * = 0 (then p ij > 0) for i + j < r + 1 and Consider the artificial probabilities in Table 1.We see in Table 1a that there is the structures of p i * j * = 0 (then p ij > 0) for all i + j < r + 1 and Δ L = 0 (then Δ U > 0).Since the degrees of departure from APS (RGS) are largest, the values of Φ APS and Φ RGS are both 1.We also see in Table 1a that there is not the structure of Δ L > 0. Thus, the value of Φ CPS is not definded.We see in Table 1b that there is the structure of p ij = 0 or p i * j * = 0 for all i + j < r + 1.Since the degrees of departure from APS are largest, the value of Φ APS is 1.Also, we see in Table 1b that there is the structure of Δ U = Δ L (= 0.3).Thus, the value of Φ RGS is 0. From Theorem 1, we obtain the value of Φ CPS is 1.We see in Table 1c that there is the structure of p ij = 3p i * j * for all i + j < r + 1, thus, the CPS model hold.The value of Φ CPS is 0 and the values of the Φ APS and Φ RGS are both 0.189.

APPROXIMATE CONFIDENCE INTERVALS FOR MEASURES
Let n ij denote the observed frequency in the ith row and jth column of the table (1 ≤ i, j ≤ r).Assuming that a multinomial distribution applies to the r × r table, we shall consider approximate standard errors and large-sample confidence intervals for Φ APS , Φ RGS and Φ CPS using the delta method of which descriptions are given by Bishop et al. (2, Sec.14.6) and Agresti (5, Sec.12.1).The sample version of Φ APS (Φ RGS , Φ CPS ), i.e., ΦAPS ( ΦRGS , ΦCPS ), is given by Φ APS (Φ RGS , Φ CPS ) with { p ij } replaced by { p ij }, where p ij = n ij /n and n = Σ r i=1 Σ r j=1 n ij .Using the delta method, each of √n( ΦAPS − Φ APS ), √n( ΦRGS − Φ RGS ) and √n( ΦCPS − Φ CPS ) has asymptotically (as n → ∞) a normal distribution with mean zero and the corresponding variance, as , where and , where .  ]/√n is an approximate 100(1 − p) percent confidence interval for Φ APS , where z p/2 is the percentage point from the standard normal distribution corresponding to a two-tail probability equal to p.In a similar way, approximate confidence intervals for Φ RGS and Φ CPS are given.

RELATIONSHIPS BETWEEN MEASURE AND LIKELIHOOD RATIO STATISTIC
Let G 2 APS denote the likelihood ratio chi-squared statistic for testing the goodness-of-fit of the APS model, i.e., Note that { p APS ij } are the maximum likelihood estimates of {p ij } under the APS model.Then it is that the estimated measure ΦAPS is equal to Next, let G 2 RGS denote the likelihood ratio chi-squared statistic for testing the goodness-of-fit of the RGS model, i.e., Note that { p RGS ij } are the maximum likelihood estimates of {p ij } under the RGS model.Then it is that the estimated measure ΦRGS is equal to CPS denote the likelihood ratio chi-squared statistic for testing the goodness-of-fit of the CPS model, i.e., and ΔU , ΔL and Δ denote Δ U , Δ L and Δ with {p ij } replaced by { p ij }, respectively.Note that { p CPS ij } are the maximum likelihood estimates of {p ij } under the CPS model.Then it is that the estimated measure ΦCPS is equal to G 2 CPS /n † .

EXAMPLE
Consider the data in Tables 2 and 3, taken from Tomizawa [16].Table 2 is constructed from the data of the unaided distance vision of 4746 students aged 18 to about 25, including about 10% of the women of the Faculty of Science and Technology, Science University of Tokyo in Japan examined in April, 1982.Table 3 is constructed from the data of the unaided distance vision of 3168 pupils aged 6-12, including about half the girls at elementary schools in Tokyo, Japan examined in June, 1984.In Tables 2 and 3 the row variable is the right eye grade and the column variable is the left eye grade with the categories ordered from the lowest grade (1) to the highest grade (4).For Tables 2 and 3, we are interested in whether models of various PS hold.For example, when the RGS model does not hold, the probability that the sum of the right eye grade and left eye grade is 4 or less, is not equal to the probability is 6 or above.When the model does not hold, we are interested in measuring and comparing the degrees of departure from the models for Tables 2 and 3. Table 4 gives the estimates of the measures Φ APS , Φ RGS and Φ CPS , the estimated approximate standard errors for ΦAPS , ΦRGS and ΦCPS and the approximate 95% confidence intervals for Φ APS , Φ RGS and Φ CPS .   2 and 3. From Table 4, when the degrees of departure from APS for Tables 2 and 3 are compared using the confidence interval for Φ APS , it would be greater in Table 3 than in Table 2.The same can be said about the degrees of departure from RGS.However, the comparison between degrees of departure from CPS in Tables 2 and 3 may be impossible.Because the values in the confidence interval for Table 3 are not always greater than the values in the confidence interval for Table 2.

CONCLUDING REMARKS
The measures Φ APS , Φ RGS and Φ CPS always range between 0 and 1 independent of the dimension r and sample size n.So, these measures may be useful for comparing the degrees of departure from APS, RGS and CPS in several tables, respectively.
As is well known, in general, the absolute value of the correlation coefficient between two variables is theoretically 0 or more and 1 or less.However in many actual data, the estimated absolute value of the correlation coefficient is 0 or more and less than 1.Similarly, each of the proposed measures theoretically ranges between 0 and 1.However, when the value of the proposed measure is 1, it has some structures of probability zero.We note that in many actual data, the estimated value of the measures is 0 or more and less than 1.
The measure Φ APS is used to measure what degree the departure from the APS model is toward the maximum departure of APS defined in Section 2. Similarly, the measure Φ RGS (Φ CPS ) is used to measure what degree the departure from the RGS model (the CPS model) is toward the maximum departure of RGS (CPS) defined in Section 2. We note that the definitions of the three models and the corresponding maximum departures are different.That is, we point out that the purpose of using each measure is different.Also, from Theorem 1, note that the values of the three measures are related to each other.
The CPS model imposes no restriction on the reverse diagonal cell probabilities { p ii * }.Thus, the structure of CPS based on the probabilities { p ij }, i.e., p ij /p i * j * =  (i + j < r + 1), is also expressed as p c ij /p c i * j * = , using the conditional cell probabilities { p c ij }, i + j ≠ r + 1.So, it seems natural that the measure of the degree of departure from CPS and their ranges do not depend on the reverse diagonal cell probabilities.In the sample versions, it may seem to many readers that both measures G 2 CPS /n and ΦCPS are reasonable measures for representing the degree of departure from CPS.However, ΦCPS rather than G 2 CPS /n would be useful for comparing the degree of departure from CPS in several tables.Because the range of G 2 CPS /n depends on the reverse diagonal proportions, i.e., 0 ≤ (G 2 CPS /n) ≤ (n † /n)[= (2 log 2)(1 − Σ r i=1 n ii * /n)], but ΦCPS always ranges between 0 and 1 without depending on the reverse diagonal proportions.By a similar reason, ΦCPS may also be preferable to G 2  CPS for comparing them.The same can be said about ΦAPS and ΦRGS .Note that the proposed three measures cannot be used to test the goodness-of-fit of each model.Also note that the three measures have different purposes and it is meaningless to compare the values of the three measures.Kurakami et al. [8] gave the orthogonality of likelihood ratio chi-square statistics for testing the goodness-of-fit, i.e., G 2 APS = G 2 CPS +G 2 RGS .We note that Theorem 1 is corresponding to the population version of this orthogonality. Pdf_Folio:532

Table 2
[16]ded distance vision of 4746 students aged 18 to about 25 including about 10% women in Faculty of Science and Technology, Science University of Tokyo in Japan examined in April 1982; from Tomizawa[16].

Table 3
[16]ded distance vision of 3168 pupils aged 6-12 including about half girls at elementary schools in Tokyo examined in June 1984; from Tomizawa[16].

Table 4
Estimate of Φ APS , Φ RGS and Φ CPS , estimated approximate standard error for ΦAPS , ΦRGS and ΦCPS , and approximate 95% confidence interval for Φ APS , Φ RGS and Φ CPS , applied to Tables