EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6439 ISSN 1307-5543 – ejpam.com Published by New York Business Global Anti-Linear-Diagonals-Parameter Symmetry Model and Orthogonal Decomposition of Anti-Symmetry for Square Contingency Tables with Ordinal Classifications Shuji Ando Department of Information Sciences, Faculty of Science and Technology, Tokyo University of Science, Noda City, Chiba, Japan Abstract. For R×R square contingency tables with the same ordinal classifications for rows and columns, this study investigates models in which the relationship between the row and column variables is symmetric or asymmetric with respect to the anti-diagonal, rather than the main diag- onal. The recently proposed anti-diagonals-parameter symmetry model includes R−1 asymmetric parameters and is capable of representing complex asymmetric structures. However, this model is saturated in the following cells: the cell in the first row and first column, the cell in the Rth row and Rth column, and all anti-diagonal cells. Since observed frequencies in square contingency tables tend to concentrate along the main diagonal, a model saturated only in the anti-diagonal cells may be more appropriate. We propose the anti-linear diagonals-parameter symmetry model, which captures asymmetry with respect to the anti-diagonal. The proposed model is saturated solely in the anti-diagonal cells and, like the anti-diagonals-parameter symmetry model, expresses how the degree of asymmetry varies according to the distance from the anti-diagonal. Furthermore, we demonstrate a decomposition of the anti-symmetry model using the proposed model and derive an orthogonal decomposition of the test statistic for the anti-symmetry model. 2020 Mathematics Subject Classifications: 62H17 Key Words and Phrases: Anti-diagonal, Asymmetry, Decomposition, Symmetry, Test statistic. 1. Introduction For R×R square contingency tables with the same ordinal classifications for rows and columns, we denote by πij , for (i, j) ∈ A, the probability that an observed frequency falls in the (i, j)th cell of the table, where A = {(i, j) | i, j = 1, 2, . . . , R}. Assume that πij , for all (i, j) ∈ A, are positive. For the analysis of square contingency tables, we often use a class of models defined by the following equation: πij = δijπji for (i, j) ∈ D, DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6439 Email address: shuji.ando@rs.tus.ac.jp (S. Ando) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 2 of 13 where δij , for all (i, j) ∈ D, are unknown parameters, andD = {(i, j) | i, j = 1, 2, . . . , R; i < j}. When no restrictions are imposed on δij for (i, j) ∈ D, the model above corresponds to the saturated model. Conversely: (i) When the restrictions δij = 1 for (i, j) ∈ D are imposed, the model corresponds to the symmetry (S) model [1]. (ii) When the restrictions δij = δ for (i, j) ∈ D are imposed, the model corresponds to the conditional symmetry (CS) model [2]. (iii) When the restrictions δij = δj−i for (i, j) ∈ D are imposed, the model corresponds to the diagonals-parameter symmetry (DPS) model [3]. (iv) When the restrictions δij = δj−i for (i, j) ∈ D are imposed, the model corresponds to the linear DPS (LDPS) model [4]. The cells on the main diagonal of the table represent cases where the difference between the row variable X and the column variable Y is zero, which corresponds to the mean of X − Y . The S model exhibits a symmetric structure in the cell probabilities with respect to the main diagonal of the table. Conversely, the CS, DPS, and LDPS models exhibit asymmetric structures in the cell probabilities with respect to the main diagonal. Next, we consider a class of models defined by the following equation: πij = δijπj∗i∗ for (i, j) ∈ E, where δij , for all (i, j) ∈ E, are unknown parameters, i∗ = R+ 1− i, j∗ = R+ 1− j, and E = {(i, j) | i, j = 1, 2, . . . , R; i+j < R+1}. The model above exhibits either a symmetric or asymmetric structure in the cell probabilities with respect to the anti-diagonal of the table. The cells on the anti-diagonal correspond to cases where the sum of X and Y equals R+ 1, which represents the mean of X + Y . When no restrictions are imposed on δij for (i, j) ∈ E, the model above corresponds to the saturated model. Conversely: (i) When the restrictions δij = 1 for (i, j) ∈ E are imposed, the model corresponds to the anti-symmetry (AS) model [5]. (ii) When the restrictions δij = δ for (i, j) ∈ E are imposed, the model corresponds to the anti-conditional symmetry (ACS) model [6]. (iii) When the restrictions δij = δR+1−(i+j) for (i, j) ∈ E are imposed, the model corre- sponds to the anti-diagonals-parameter symmetry (ADPS) model [7]. Given that the ADPS model includesR−1 asymmetric parameters (i.e., δ1, δ2, . . . , δR−1), while the ACS model includes only one (i.e., δ), the ADPS model offers greater flexibility than the ACS model in representing complex asymmetric structures. However, the ADPS model is saturated in the following cells: the cell in the first row and first column, the cell in the Rth row and Rth column, and all anti-diagonal cells. Observed frequencies in square S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 3 of 13 contingency tables tend to concentrate along the main diagonal. In light of this tendency, it may be advantageous to consider a model that, like the ACS model, is saturated only on the anti-diagonal, but that offers greater flexibility in capturing asymmetric structures. To address this issue, the present paper proposes a new model that is saturated only in the anti-diagonal cells of the table. Like the ADPS model, the proposed model allows the degree of asymmetry to vary depending on the distance from the anti-diagonal. The remainder of this paper is organized as follows. In Section 2, we propose a new model. Section 2 also presents the decomposition of the AS model using the proposed model, while Section 3 describes the decomposition of the test statistic for the AS model. Section 4 demonstrates the advantages of the proposed model through an application to a real dataset. Finally, Section 5 provides concluding remarks. 2. Proposed model This study proposes the anti-linear diagonals-parameter symmetry (ALDPS) model, which is defined by the following equation: πij = δR+1−(i+j)πj∗i∗ for (i, j) ∈ E, where δ is unknown parameter. Note that the ALDPS model is saturated only in the anti-diagonal cells of the table. Similar to the ADPS model, the ALDPS model allows the degree of asymmetry to vary depending on the distance from the anti-diagonal (i.e., R+1− (i+ j)). However, while the ADPS model includes R− 1 asymmetric parameters, the ALDPS model includes only a single asymmetric parameter. As a special case of the ADPS model, the ADPS model with δR+1−(i+j) = δR+1−(i+j) for (i, j) ∈ E coincides with the ALDPS model. Therefore, the ADPS model offers greater flexibility than the ALDPS model in representing complex asymmetric structures. Nevertheless, the ADPS model is saturated in the following cells: the cell in the first row and first column, the cell in the Rth row and Rth column, and all anti-diagonal cells. Given that observed frequencies in square contingency tables tend to concentrate along the main diagonal, the ALDPS model may be considered more advantageous than the ADPS model in practice. As a special case of the ALDPS model, the ALDPS model with δR+1−(i+j) = 1 for (i, j) ∈ E coincides with the AS model. When the AS model holds, the ALDPS model necessarily holds as well, although the converse is not always true. In Section 3, we identify a counterpart to the ALDPS model for the purpose of decomposing the AS model. We discuss the relationship between the ALDPS model and the bivariate normal distri- bution. Let V andW be random variables that follow a joint bivariate normal distribution with means E(V ) = µ1 and E(W ) = µ2, variances Var(V ) = Var(W ) = σ2, and correla- tion Corr(V,W ) = ρ. The joint bivariate normal density f(v, w) is given by: f(v, w) = 1 2πσ2 √ 1− ρ2 exp [ − 1 4σ2(1− ρ) {(v − w)− (µ1 − µ2)}2 − 1 4σ2(1 + ρ) {(v + w)− (µ1 + µ2)}2 ] . S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 4 of 13 Therefore, the density f(v, w) satisfies the following identity: f(v, w) f(w∗, v∗) = exp [ 1 σ2(1 + ρ) {η − (µ1 + µ2)}{η − (v + w)} ] for v < w, where v∗ = η − v and w∗ = η − w for any η. This identity resembles the functional form of the ALDPS model. Therefore, when we aim to capture the linear structure in the log odds log{f(v, w)/f(w∗, v∗)}, the ALDPS model provides an appropriate framework. In terms of simulation studies, Tables 1a and b, taken from Tomizawa, Miyamoto and Ashihara [8], give the 4×4 tables of sample size 10,000 formed by using cut points for each variable at µ1, µ1 ± 0.6σ, for underlying bivariate normal distribution with the conditions σ21 = σ22 = σ2, µ2 = µ1+0.4, and the correlations ρ = 0 (Table 1a) and ρ = 0.3 (Table 1b). Table 1: The 4× 4 tables of sample size 10,000, formed by using cut points for each variable at µ1, µ1 ± 0.6σ, from an underlying bivariate normal distribution with the conditions µ2 = µ1+0.4, σ2 1 = σ2 2 = σ2 and ρ = 0, 0.3 (a) For ρ = 0 (b) For ρ = 0.3 428 526 671 1174 696 666 678 785 358 416 561 951 384 436 587 836 374 405 544 875 269 388 554 1008 405 509 658 1145 216 366 615 1516 Source: Tomizawa, Miyamoto and Ashihara [8] The ALDPS model fits these data well yielding the likelihood ratio chi-squared values G2 = 5.89 (for Table 1a) and G2 = 4.35 (for Table 1b) with both 5 degrees of freedom. Therefore, the ALDPS model may be appropriate for a square table if it is reasonable to assume an underlying bivariate normal distribution with equal marginal variances. 3. Separation of the anti-symmetry model via the anti-linear diagonals-parameter symmetry model In this section, we present a decomposition of the AS model using the ALDPS model. The objective is to identify a counterpart to the ALDPS model that enables the decom- position of the AS model. As such a counterpart, we introduce the anti-marginal equity (AME) model [9], which is defined by the following equation: E(X) = E(R+ 1− Y ), where E(X) = ∑ (i,j)∈A iπij and E(R+ 1− Y ) = ∑ (i,j)∈A (R+ 1− j)πij = ∑ (i,j)∈A j∗πij . The decomposition of the AS model is derived using the ALDPS and AME models, as stated in the following theorem. S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 5 of 13 Theorem 1. The AS model holds if and only if both the ALDPS and AME models hold simultaneously. Proof. We first show the necessity. Assume that the AS model holds, i.e., πij = πj∗i∗ for (i, j) ∈ E. Then, the ALDPS model clearly holds. Consider the following equation: E(X − (R+ 1− Y )) = ∑ i+jR+1 (i− j∗)πij . (1) In equation (1), the second term on the right-hand side equals zero since i− j∗ = 0 when i+ j = R+ 1. Therefore, under the AS model, equation (1) becomes: E(X − (R+ 1− Y )) = ∑ i+jR+1 (i− j∗)πij = − ∑ i+jR+1 (i− j∗)πij = − ∑ i+j>R+1 (i− j∗)πj∗i∗ + ∑ i+j>R+1 (i− j∗)πij = − ∑ i+j>R+1 (i− j∗)πij + ∑ i+j>R+1 (i− j∗)πij = 0. Hence, we obtain E(X) = E(R + 1 − Y ), which implies that the AME model holds. Therefore, the necessary condition is satisfied. We now show the sufficiency. Assume that both the ALDPS and AME models hold. Then we have: E(X − (R+ 1− Y )) = − ∑ i+j>R+1 (i− j∗)πijδ R+1−(i+j) + ∑ i+j>R+1 (i− j∗)πij = ∑ i+j>R+1 (i− j∗)πij(1− δR+1−(i+j)) = 0. Since πij > 0 for (i, j) ∈ A, it follows that δ = 1. Therefore, the ALDPS model reduces to the AS model, and the sufficiency is established. When the AS model does not fit the observed data well, Theorem 1 provides insight into the possible reasons for the poor fit. 4. Separation of test statistic for the anti-symmetry model Let N denote the sample size, i.e., N = ∑∑ (i,j)∈A nij , where nij for (i, j) ∈ A is the observed frequency in the (i, j)th cell of the table. We assume that the observed S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 6 of 13 frequencies nij for (i, j) ∈ A follow a multinomial distribution with probability vector π, where π = (π11, π12, . . . , π1R, π21, π22, . . . , π2R, . . . , πR1, πR2, . . . , πRR) ⊤. The symbol “⊤” denotes the transpose of a vector or matrix. Let êij for (i, j) ∈ A denote the maximum likelihood estimator (MLE) of the expected frequencies eij under the model of interest. The MLE êij can be obtained by maximizing the log-likelihood function subject to the constraints imposed by the model. Specifically, under the ALDPS model, êij for (i, j) ∈ A can be obtained by maximizing the following Lagrangian with respect to {πij}, ϕ, {ψij}, and δ, using the Newton–Raphson method: L = ∑ (i,j)∈A nij log πij − ϕ  ∑ (i,j)∈A πij − 1 − ∑ (i,j)∈E ψij ( πij − δR+1−(i+j)πj∗i∗ ) . The likelihood ratio statistic for testing the goodness-of-fit of model M is given by G2(M) = 2 ∑ (i,j)∈A nij log ( nij êij ) , where êij is the MLE of the expected frequency eij under model M. Table 2 shows the degrees of freedom for testing the goodness of fit for each model. The number of degrees of freedom for the ALDPS model is given by (R + 1)(R − 2)/2, which is R− 2 fewer than that for the ADPS model. Note that the number of degrees of freedom for the AS model equals the sum of that for the ALDPS and AME models. Table 2: The number of degrees of freedom for testing goodness-of-fit for each model Models Degrees of freedom AS R(R− 1)/2 ACS (R+ 1)(R− 2)/2 ADPS (R− 1)(R− 2)/2 ALDPS (R+ 1)(R− 2)/2 AME 1 Assume that model M3 holds if and only if both models M1 and M2 hold, where the number of degrees of freedom for the model M3 equals the sum of that for the models M1 and M2. Darroch and Silvey [10] described that (i) when the following asymptotic equivalence holds: G2(M3) ≃ G2(M1) +G2(M2), (2) if both models M1 and M2 are accepted (at the α significance level) with high probability, then the model M3 would be accepted; however, (ii) when the equation (2) does not hold, such an incompatible situation that both models M1 and M2 are accepted with high probability but the model M3 is rejected with high probability is quite possible. In S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 7 of 13 fact, Darroch and Silvey [10], Tahata, Ando and Tomizawa [11] showed such interesting examples. We now demonstrate that Theorem 1 satisfies the asymptotic equivalence given in equation (2). Accordingly, the following theorem is obtained. Theorem 2. For R × R square contingency tables, the following asymptotic equivalence holds: G2(AS) ≃ G2(ALDPS) +G2(AME). Proof. The ALDPS model may be expressed as log πij = {R+ 1− (i+ j)}β1 + ϕij i = 1, . . . , r; j = 1, . . . , r, (3) where ϕij = ϕj∗i∗ . Let β = (β1,β2) ⊤, where β2 = (ϕ11, ϕ12, . . . , ϕ1r, ϕ21, ϕ22, . . . , ϕ2,R−1, . . . , ϕRR), is the 1×R(R+1)/2 vector of ϕij for i+ j ≤ R+1. Then, the ALDPS model is expressed as logπ = Xβ = (X1,X2)β, where X is the R2 ×K matrix with K = (R2 +R+ 2)/2 and X1 = (R+ 1)1R2 − (1R ⊗ JR + JR ⊗ 1R); the R2 × 1vector, and X2 is the R 2×R(R+1)/2 matrix of 1 or 0 elements, determined from the equation (3), 1s is the s×1 vector of 1 elements and JR = (1, 2, . . . , R)⊤, and the symbol “⊗” represents the Kronecker product. Note that X21R(R+1)/2 = 1R2 holds. Note that the matrix X is full column rank which is K. In a similar manner to Haber [12], and Lang and Agresti [13], we denote the linear space spanned by the columns of the matrix X by S(X) with the dimension K. Let U be an R2 × d1 matrix, where d1 = R2 −K = (R + 1)(R − 2)/2, full column rank matrix such that the linear space spanned by the columns of U , i.e., S(U), is the orthogonal complement of the space S(X). Thus, U⊤X = Od1,K where Od1,K is the d1 × K zero matrix. Therefore, the ALDPS model is expressed as h1(π) = 0d1 , where 0d1 is the d1 × 1 zero vector and h1(π) = U⊤ logπ. The AME model may be expressed as h2(π) = 0, S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 8 of 13 where h2(π) = Wπ, with W = ((R+ 1)1R2 − (1R ⊗ JR + JR ⊗ 1R)) ⊤; the 1×R2vector. Namely, W⊤ = X1. Thus W⊤ belongs to the space S(X), i.e., S(W⊤) ⊂ S(X). Hence WU = 0⊤d1 . From Theorem 1, the AS model may be expressed as h3(π) = 0d3 , where d3 = d1 + d2 = R(R− 1)/2 with d2 = 1, h3 = (h1, h2) ⊤. Note that hs(π), s = 1, 2, 3, are the vectors of order ds × 1, and ds, s = 1, 2, 3, are the numbers of degrees of freedom for testing goodness-of-fit of the ALDPS, AME and AS models, respectively. Let Hs(π), s = 1, 2, 3, denote the ds × R2 matrix of partial derivatives of hs(π) with respect to π, i.e., Hs(π) = ∂hs(π)/∂π ⊤. Let Σ(π) = diag(π) − ππ⊤, where diag(π) denotes a diagonal matrix with ith component of π as ith diagonal component. Let π̂ denote π with {πij} replaced by {π̂ij}, where π̂ij = nij/N . Then √ N(π̂ −π) has asymp- totically a normal distribution with mean 0R2 and covariance matrix Σ(π). Using the delta method, √ N(h3(π̂)− h3(π)) is asymptotically distributed to a normal distribution with mean 0d3 and covariance matrix H3(π)Σ(π)H3(π) ⊤ = [ H1(π)Σ(π)H1(π) ⊤ H1(π)Σ(π)H2(π) ⊤ H2(π)Σ(π)H1(π) ⊤ H2(π)Σ(π)H2(π) ⊤ ] . We see that H1(π) = U⊤1R2 = 0d1 since 1R2 ⊂ S(X), H1(π)diag(π) = U⊤ and H2(π) = W . Therefore we obtain H1(π)Σ(π)H2(π) ⊤ = U⊤W⊤ = 0d1 . Thus we obtain ∆3(π̂) = ∆1(π̂) + ∆2(π̂), where ∆s(π̂) = hs(π̂) ⊤ [ Hs(π̂)Σ(π̂)Hs(π̂) ⊤ ]−1 hs(π̂). (4) Under each hs(p) = 0ds (s = 1, 2, 3), the Wald statistic N∆s(π̂) has asymptotically a chi-squared distribution with ds degrees of freedom. From the equation (4), we see that N∆3(π̂) = N∆1(π̂) + N∆2(π̂). From the asymptotic equivalence of the Wald statistic and likelihood ratio statistic [10], we obtain Theorem 2. The proof is completed. S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 9 of 13 Table 3: The grip strength test dataset for men between the ages of 15 and 69 Left hand Right hand (1) (2) (3) (4) (5) Total Highest (1) 215 124 46 14 2 401 (2) 37 143 165 74 16 435 (3) 7 45 156 166 51 425 (4) 2 20 62 226 210 520 Lowest (5) 1 2 16 61 495 575 Total 262 334 445 541 774 2356 Source: Yamamoto, Aizawa and Tomizawa [14] 5. Real data analysis We examine the dataset presented in Table 3, which is taken directly from Yamamoto, Aizawa, and Tomizawa [14]. This dataset summarizes the results of grip strength mea- surements conducted on a cohort of 2,356 men aged 15 to 69 years, as part of the National Health and Nutrition Examination Survey (NHANES) conducted in 2011–2012. Grip strength for the right and left hands is categorized according to the NHANES manual. In the case of grip strength data, such as that presented in Table 3, symmetry or asym- metry between right and left hand grip strength may be of less concern. This is because the dominant hand typically exhibits greater grip strength than the non-dominant hand, with the majority of individuals being right-handed, as reported by an Intage Group self- administered survey (https://gallery.intage.co.jp/smartphone-operation/). Fur- thermore, Iki [5] and Ando [7, 15, 16] analyzed grip strength data using models in which the relationship between the row and column variables is symmetric or asymmetric with respect to the anti-diagonal of the table, rather than the main diagonal. Accordingly, we are interested in applying the AS, ACS, ADPS, and ALDPS models to the dataset in Table 3. When models M1 and M2 are nested (i.e., model M1 is more parsimonious than model M2), the likelihood ratio statistic for testing whether model M1 holds under the assumption that model M2 is true is given by G2(M1 | M2) = G2(M1)−G2(M2). Under the null hypothesis, the statistic G2(M1 | M2) asymptotically follows a chi-squared distribution with the number of degrees of freedom equal to the difference in the number of degrees of freedom between models M1 and M2; see, for example, Agresti [17, Sec. 3.4.4]. When models M1 and M2 are not nested, the statistic G2(M1 | M2) cannot be used for model comparison. In such cases, alternative criteria are required. Among the most widely used are the Akaike Information Criterion (AIC) [18] and the Bayesian Information Criterion (BIC) [19], both of which are used to identify the best-fitting model among competing candidates. The model with the minimum AIC or BIC is considered the best- fitting model. Since only the difference between AIC (or BIC) values is required for comparison, the constant part can be omitted. Therefore, we define the modified AIC and S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 10 of 13 BIC as follows: AIC+ = G2 − 2× (number of degrees of freedom), BIC+ = G2 − logN × (number of degrees of freedom), where N denotes the sample size. The model with the smallest AIC+ or BIC+ is selected as the best-fitting model among the models under consideration. Table 4 provides the values of G2 and AIC+ for the AS, ACS, ADPS, ALDPS, and AME models applied to the dataset in Table 3. From Table 4, we observe that the goodness-of-fit for both the ADPS and ALDPS models is well and shows a substantial improvement over the ACS model. The value of the test statistic G2(ALDPS | ADPS) is 2.30. Therefore, the ALDPS model is preferable to the ADPS model. Furthermore, in terms of AIC+, the ALDPS model is identified as the best-fitting model among those applied to the dataset in Table 3. Table 4 provided the values of G2 and AIC+ for each AS, ACS, ADPS, ALDPS, and AME model applied to the dataset in Table 3. From Table 4, we see that the goodness-of- fit of the ADPS model and ALDPS modell are well and dramatically improves compared to the ACS model. The value of test statistics G2(ALDPS | ADPS) is 2.30. Therefore, the ALDPS model is preferable to the ADPS model. Moreover, in terms of AIC, the ALDPS model is the best-fitting model among the models applied to the dataset in Table 3. Table 4: Results of goodness-of-fit test applied each model to the dataset of Table 3 Applied models Degrees of freedom G2 p-value AIC+ AS 10 167.37∗ < 0.01 147.37 ACS 9 43.92∗ < 0.01 25.92 ADPS 6 5.35 0.50 −7.35 ALDPS 9 7.65 0.57 −10.35 AME 1 160.07∗ < 0.01 158.07 The symbol ∗ represents significance at the 0.05 level. Table 5 presents the estimated expected frequencies êij under the ALDPS and ADPS models. The maximum likelihood estimate of δ in the ALDPS model is 0.82. These results reveal a clear discrepancy in the ratio of men classified as having high versus low grip strength. It can therefore be concluded that the current criteria used to determine grip strength levels may be inadequate. In the future, it is anticipated that the classification criteria for grip strength will be revised. Given that observations in square contingency tables tend to concentrate in the main diagonal cells, it may be desirable to consider a model that is saturated only on the anti-diagonal cells. In fact, the ratio of (n11 + n55) to N in the dataset in Table 3 is approximately 30%. The ADPS model is saturated in the first row and first column cell, the Rth row and Rth column cell, as well as in the anti-diagonal cells of the table. For the dataset in Table 3, under the ADPS model, ê11 and ê55 are equal to n11 and n55, respectively. In contrast, the ALDPS model is saturated only in the anti-diagonal cells. S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 11 of 13 Table 5: Maximum likelihood estimates of expected frequencies under the anti-diagonals-parameter symmetry (ADPS) model and anti-linear diagonals-parameter symmetry (ALDPS) model applied to the dataset in Ta- ble 3.The parenthesized values in lines 2 and 3 are the maximum likelihood estimates of expected frequencies under the ADPS and ALDPS models, respectively. Left hand Right hand (1) (2) (3) (4) (5) Total Highest (1) 215 124 46 14 2 401 (215) (124.48) (38.88) (14.36) (2) (222.66) (119.31) (39.12) (13.54) (2) (2) 37 143 165 74 16 435 (36.52) (147.90) (158.49) (74) (15.64) (35.01) (148.82) (149.35) (74) (16.46) (3) 7 45 156 166 51 425 (9.22) (51.23) (156) (172.51) (58.12) (9.28) (48.28) (156) (181.65) (57.88) (4) 2 20 62 226 210 520 (1.92) (20) (55.77) (221.10) (209.52) (1.80) (20) (58.72) (220.18) (214.69) Lowest (5) 1 2 16 61 495 575 (1) (2.08) (13.78) (61.48) (495) (1) (2.20) (13.72) (62.99) (487.34) Total 262 334 445 541 774 2356 Accordingly, for the same dataset, under the ALDPS model, ê11 and ê55 are not equal to n11 and n55, respectively. In accordance with Theorem 1, for the dataset in Table 3, the poor goodness-of-fit of the AS model can be attributed to the AMEmodel rather than to the ALDPS model. From Table 4, we observe that G2(AS) = 167.37, and the sum of G2(ALDPS) and G2(AME) is 167.72. According to Theorem 2, since the following asymptotic equivalence holds: G2(AS) ≃ G2(ALDPS) +G2(AME), we conclude that G2(AS) is nearly equal to the sum of G2(ALDPS) and G2(AME) in this dataset. 6. Conclusion This study proposed the ALDPS model, which captures the asymmetric structure of cell probabilities with respect to the anti-diagonal of the table. We also examined the rela- tionship between the ALDPS model and the bivariate normal distribution. Furthermore, we provided a decomposition of the AS model using the ALDPS model (Theorem 1), as well as a decomposition of the test statistic for the AS model (Theorem 2). The ALDPS model was shown to offer substantial advantages through its application to the real dataset in Table 3. The results supported the hypothesis that males with high S. Ando / Eur. J. Pure Appl. Math, 18 (4) (2025), 6439 12 of 13 grip strength levels are less prevalent than those with low grip strength levels. Moreover, the findings suggested that the current criteria used to classify grip strength levels may be inadequate. Readers may be interested in an extended ALDPS model, which would be suitable for a square contingency table under the assumption of an underlying bivariate normal distribution without requiring equality of the marginal variances. Following the approach of Tomizawa [20], extensions of the ALDPS model can also be considered. A detailed discussion of such extensions is left for future research. Acknowledgements The author would like to thank the anonymous reviewers and the editors for their comments and suggestions to improve this paper. References [1] H Bowker. A test for symmetry in contingency tables. Journal of the American Statistical Association, 43(244):572–574, 1948. [2] P McCullagh. A class of parametric models for the analysis of square contingency tables with ordered categories. Biometrika, 65(2):413–418, 1978. [3] A Goodman. 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