A bivariate index vector for measuring departure from double symmetry in square contingency tables

For square contingency tables, a double symmetry model having a matrix structure that combines both symmetry and point symmetry was proposed. Also, an index which represents the degree of departure from double symmetry was proposed. However, this index cannot simultaneously characterize the degree of departure from symmetry and the degree of departure from point symmetry. For measuring the degree of departure from double symmetry, the present paper proposes a bivariate index vector that can simultaneously characterize the degree of departure from symmetry and the degree of departure from point symmetry.


Introduction
Consider an r × r square contingency table P = ( p i j ) with the same row and column nominal classifications. Here p i j denotes the probability that an observation will fall in the ith row and jth column of the table (i = 1, . . . , r ; j = 1, . . . , r ). For the analysis of square contingency tables, we are interested here in whether or not the row classification is symmetric or point-symmetric to the column classification (see Tahata and Tomizawa 2014). Bowker (1948) proposed the symmetry model defined by This model indicates that the probabilities are symmetric with respect to the main diagonal of the contingency table. Namely, this model indicates that the probability that an observation will fall in row category i and column category j is equal to the probability that the observation falls in row category j and column category i (i = j). Wall and Lienert (1976) proposed the point symmetry model defined by p i j = p r +1−i,r +1− j for i, j = 1, . . . , r.
This model indicates that the probabilities are point symmetric with respect to the center cell (when r is odd) or center point (when r is even) of the contingency table.
Consider the data in Table 1 taken from Andersen (1997, p. 226). These data show the forecasts for production and prices for the coming three year periods given by experts in July 1956 and the actual production figures for production and prices in May 1959 given from a sample of about 4000 Danish factories. For Table 1, we shall denote, for example, the probability that the forecast and the actual values are "Higher" and "Lower", respectively, by P(H, L), and the probability that those are "Higher" and "No change", respectively, by P(H, N). For these data, we are interested in whether (1) P(H, N) = P(N, H), P(H, L) = P(L, H), and P(L, N) = P(N, L), and (2) P(H, H) = P(L, L), P(H, N) = P(L, N), P(H, L) = P(L, H), and P(N, H) = P(N, L). Note that (1) means the symmetry of cell probabilities with respect to the main diagonal in the table, and (2) means the symmetry of cell probabilities with respect to the center point (the center cell) in the table. Tomizawa (1985) considered a matrix structure that combines both symmetry and point symmetry. The double symmetry model is defined by This model indicates that the probabilities are symmetric with respect to both the main diagonal and the secondary diagonal of the contingency table.
In case of unknown probabilities p i j these are estimated from an observed r × r contingency table N = (x i j ) obtained from random sampling. Typically, when a given model does not hold, we are interested in checking for an extended model or in analyzing the deviation from the model (from the residuals). On the other hand, we are also interested in measuring the degree of departure from the corresponding Table 1 The two tables below show the forecasts for production and prices for the coming three year periods given by experts in July 1956 and the actual production figures for production and prices in May 1959 given from a sample of about 4000 Danish factories; from Andersen (1997, p

Index Φ S for departure from symmetry
For square contingency tables with nominal categories, Tomizawa (1994) considered the index Φ S which represents the degree of departure from symmetry. The index Φ S is independent of the dimension r . Assume that p i j + p ji > 0 for all i = j. The index Φ S is expressed as follows: where Note that I S is the Kullback-Leibler information between a i j and b i j for all i = j. Note that (1) the index Φ S lies between 0 and 1, (2) Φ S = 0 if and only if the symmetry model holds, and (3) Φ S = 1 if and only if the degree of departure from symmetry is maximal, in the sense that for all i = j either p i j = 0 or p ji = 0. We point out that the index Φ S can be expressed as where minimization is constrained by 1.2 Index Φ P S for departure from point symmetry Tomizawa et al. (2007) proposed the index Φ P S which represents the degree of departure from point symmetry. The index Φ P S is independent of the dimension r . Let be the set of cells (i, j) outside of the center of the table. Assume that p i j + p r +1−i,r +1− j > 0 for all (i, j) ∈ D. The index Φ P S is expressed as follows: Note that I P S is the Kullback-Leibler information between A i j and B i j for all (i, j) ∈ D. Note that (1) the index Φ P S lies between 0 and 1, (2) Φ P S = 0 if and only if the point symmetry model holds, and (3) Φ P S = 1 if and only if the degree of departure from point symmetry is maximal, in the sense that for all (i, j) ∈ D either p i j = 0 or p r +1−i,r +1− j = 0. We point out that the index Φ P S can be expressed as where minimization is constrained by 1.3 Index Φ DS for departure from double symmetry Yamamoto et al. (2010) proposed the index Φ DS which represents the degree of departure from double symmetry. Let and, let E 1 is the set of cells (i, j) on the diagonal or on the secondary diagonal (without center point if any), while E 2 includes the remaining cells. Assume that The index Φ DS is expressed as follows: For the analysis of the data, when the double symmetry model does not hold, we may measure the degree of departure from the double symmetry model by using Φ DS . However, the Φ DS cannot simultaneously characterize the degree of departure from symmetry and the degree of departure from point symmetry (although the double symmetry model has a structure that combines both symmetry and point symmetry), see Sect. 4.

Index vector and a confidence region
The purpose of the present paper is to propose a bivariate index vector which represents the degree of departure from double symmetry. For measuring the degree of departure from double symmetry, the proposed index vector can simultaneously characterize the degree of departure from symmetry and the degree of departure from point symmetry. Also, the proposed index vector would be useful for visually comparing the degrees of departure from double symmetry using confidence regions.

Definition of the index vector
Assume that p i j + p ji > 0 for all i = j, and p i j + p r +1−i,r +1− j > 0 for all (i, j) ∈ D. In order to represent the degree of departure from double symmetry, we propose in this paper the bivariate index vector It has the properties: (1) Ψ = (0, 0) if and only if both symmetry and point symmetry models hold (namely, the double symmetry model holds), (2) Ψ = (1, 1) if and only if the degrees of departure from both symmetry and point symmetry are maximal (i.e., the degree of departure from double symmetry is maximal), in the sense that p i j = p r +1− j,r +1−i = 0 (then p ji > 0 and p r +1−i,r +1− j > 0) or p ji = p r +1−i,r +1− j = 0 (then p i j > 0 and p r +1− j,r +1−i > 0) for all i = j, and either p ii = 0 or p r +1−i,r +1−i = 0 for i = 1, . . . , r/2 (when r is even) or i = 1, . . . , (r − 1)/2 (when r is odd). Note that the definition of the maximum degree of departure from double symmetry for the proposed index vector Ψ is different from that for the index Φ DS . We assume that x has a multinomial distribution Multi(n; p) with sample size n and probability vector p. Then √ n(p − p) has asymptotically a normal distribution with zero mean and covariance matrix Diag( p) − pp , wherep = x/n and Diag( p) is diagonal matrix with the elements of p on the main diagonal (see, e.g., Agresti 2013, p. 590). In order to estimate the indexes,Φ S andΦ P S are given by Φ S and Φ P S with { p i j } replaced by {p i j }, respectively. Therefore, the sample version of Ψ , i.e., Ψ , is given by Ψ with Φ S and Φ P S replaced byΦ S andΦ P S , respectively. Let (∂Ψ /∂ p ) denote the 2 × r 2 matrix for which the entry in row k and column l is ∂Ψ k ( p)/∂ p l , where Ψ 1 and Ψ 2 denote Φ S and Φ P S , respectively, and p l denotes the lth element of p. For n approaching infinity, the estimated index vector can be approximated by

A confidence region for the index vector
where o( p − p ) tends to (0, 0) . Using the delta method (see Agresti 2013, Sect. 16.1), √ n( Ψ − Ψ ) has asymptotically a bivariate normal distribution with zero mean and covariance matrix with σ 12 = σ 21 . The elements σ 11 , σ 12 and σ 22 are expressed as follows: where Ω i j = 1 log 2 log 2 p i j p i j + p ji , W i j = 1 log 2 log 2 p i j p i j + p r +1−i,r +1− j .
Note that the previous formulas for the asymptotic variances σ 11 and σ 22 of Φ S and Φ P S , respectively, have been derived by Tomizawa (1994) and Tomizawa et al. (2007). Therefore, an approximate bivariate 100(1 − α)% confidence region for the index vector Ψ is given by where χ 2 (1−α;2) is the 1 − α quantile of the chi-square distribution with two degrees of freedom and is given by with { p i j } replaced by {p i j }.

Examples
Consider the data in Table 1 taken from Andersen (1997, p. 226). For these data, we are interested in whether (1)  For Table 1a, b, the estimated index vectors arê Ψ = 0.0770 0.0887 andΨ = 0.0148 0.0604 , respectively (see Table 2). We shall compare the degrees of departure from double symmetry in Table 1a, b using the confidence region for Ψ . From Table 1a respectively. Figure 1 shows that the confidence ellipsoids for Ψ do not overlap for the data from Table 1a (prices) and b (production). Therefore, we see that Table 1a, b have a different structure with respect to the degree of departure from double symmetry, in the sense that Table 1a, b have a different structure with respect to the degree of departure from symmetry or point symmetry. From Fig. 1, we can conclude that the degree of departure from symmetry in Table 1a is greater than that in Table 1b. But, we cannot conclude that the degree of departure from point symmetry in Table 1a is greater than that in Table 1b. Thus, it may be difficult to judge whether the degree of departure from double symmetry in Table 1a is greater than that in Table 1b. Also, we shall compare the degrees of departure from double symmetry in Table 1a, b using the confidence interval of Φ DS . From Table 2, we see that the confidence intervals overlap for Table 1a, b. Thus, we cannot judge whether the degree of departure from double symmetry in Table 1a is greater than that in Table 1b. Table 2 Estimates of Φ S , Φ P S and Φ DS , approximate standard errors forΦ S ,Φ P S andΦ DS , and approximate 95% confidence intervals for Φ S , Φ P S and Φ DS , for the data of  Table 3a is not equal to those for Table 3b, c. Also, from Fig. 2, we see the degree of departure from double symmetry, while distinguishing the degree of departure from symmetry and the degree of departure from point symmetry. Thus, we see that (1) for Table 3a, the degree of departure from symmetry is large but the degree of departure from point symmetry is small, (2) for Table 3b, the degree of departure from symmetry is small but the degree of departure

Concluding remarks
This paper proposed a bivariate index vector that simultaneously characterizes the degree of departure from symmetry and the degree of departure from point symmetry.
Since Φ S and Φ P S are not independent as shown in Sect. 2.2, we believe that it is important to simultaneously characterizes the degree of departure from symmetry and the degree of departure from point symmetry. In addition, the proposed index  Table 3a-c vector would be useful for visually comparing the degrees of departure from double symmetry using confidence regions. For comparing the degrees of departure from double symmetry in several square contingency tables, we consider that with the proposed index vector it becomes easier to understand the characteristics of the data.