A measure of asymmetry for ordinal square contingency tables with an application to modified LANZA score data

ABSTRACT

In clinical researches, various clinical scales with ordered categories are used to evaluate the efficacy and safety/toxicity of treatments. Such the clinical scales are sometimes summarized on the transition between the baseline and the study end point as a square contingency table. Also, clinical scales may be reclassified into three groups. However, the cutpoints can be varied depending on clinical researches or clinicians. Hence, this paper proposes a measure which is expressed by using same weights for collapsed tables and which can see the directionality for two kinds of asymmetries. Also, this paper shows an application of the proposed measure to clinical data, and that the proposed measure is a useful statistical method for analyzing ordered categorical data.

1. Introduction

In clinical researches to evaluate the efficacy and safety/toxicity of treatments, various clinical scales having ordered categories are often employed. As an example, the modified LANZA score (MLS), which is defined as an ordered score with 5 stages (from 0 to $+4$), is a popular evaluation scale used for clinical evaluation of gastroduodenal mucosal lesion. Consider the data in Table 1 taken from [5]. These are shift analysis data of MLS after 24 weeks' treatment with esomeprazole 20 mg once daily or placebo. For analyzing such a square contingency table that has the same ordinal row and column classifications, Bowker [1] proposed the symmetry model. If the symmetry model holds in Table 1, it can be interpreted that the tendency to improve from baseline is the same degree of the tendency to worsen from baseline. When the symmetry model does not hold, we are interested in the tendency to improve or worsen. Also, we are interested in whether the drug group has more therapeutic effect than the placebo group, by comparing several tables. For these interests, measures that represent the degree of departure from symmetry were proposed by Tomizawa [7], Tomizawa et al. [8], and so on. When we use the measure proposed by Tahata et al. [6], we can distinguish two kinds of complete asymmetries, namely, we can see one group has more therapeutic effect than the other groups.

Table 1. MLS for Japanese patients (from [5]).

	Baseline
Study end	0	+1	+2	+3	+4
(a) Esomeprazole 20 mg once daily
0	78	9	26	3	1
$+1$	1	5	6	4	0
$+2$	9	1	10	3	1
$+3$	1	0	1	0	0
$+4$	3	0	1	1	2
(b) Placebo
0	41	2	19	0	0
$+1$	8	0	4	0	0
$+2$	12	4	14	3	0
$+3$	0	1	1	3	0
$+4$	29	7	11	6	0

For the MLS in some clinical researches, Kanbayashi and Konishi [2] reclassified the 5 original categories into 3 groups which are 0, $+1$ to $+3$, and $+4$ (combined from $+1$ to $+3$), Kim et al. [3] combined from $+2$ to $+4$ in the original scale, and so on. For the simplicity of interpretation in clinical researches, clinical scales may be combined into 3 categories. When there are various ways of combining a scale, it is reasonable to consider evaluating various collapsed patterns comprehensively. So, Yamamoto et al. [9] proposed the measure which represents the degree of departure from symmetry by using collapsed $3\times 3$ tables. However, this measure cannot distinguish two kinds of complete asymmetries.

Based on these points, the present paper proposes a measure which can distinguish two kinds of complete asymmetries in addition to using collapsed $3\times 3$ tables. The proposed measure is expressed by using the same weights for submeasures that represent the degree of asymmetry for collapsed $3\times 3$ tables.

2. Measure for collapsed $3\times 3$ tables

Consider an $r\times r$ square contingency table with the same row and column classifications. Let $p_{ij}$ denote the probability that an observation will fall in the ith and jth column of the table ($i=1,\dots ,r;j=1,\dots ,r$). The symmetry model is defined by

$$p_{ij}=p_{ji}(i=1,\dots ,r;j=1,\dots ,r;i\ne j).$$

We consider the $(r-1)(r-2)/2$ (being $(\genfrac{}{}{0ex}{}{r-1}{2})$) ways of collapsing the $r\times r$ original table with ordered categories into a $3\times 3$ table by choosing cutpoints after the sth and tth rows and after the sth and tth columns for $1\le s<t\le r-1$. We shall refer to each collapsed $3\times 3$ table as the $T_{st}$ table ($1\le s<t\le r-1$). In the collapsed $T_{st}$ table, let $G_{kl}^{(s,t)}$ denote the corresponding probability for row value $k(k=1,2,3)$ and column value $l(l=1,2,3);$ that is,

$$\begin{array}{rl}{G}_{11}^{(s,t)}& =\sum _{i=1}^s\sum _{j=1}^s{p}_{ij},G_{12}^{(s,t)}=\sum _{i=1}^s\sum _{j=s+1}^t{p}_{ij},G_{13}^{(s,t)}=\sum _{i=1}^s\sum _{j=t+1}^r{p}_{ij},\\ G_{21}^{(s,t)}& =\sum _{i=s+1}^t\sum _{j=1}^s{p}_{ij},G_{22}^{(s,t)}=\sum _{i=s+1}^t\sum _{j=s+1}^t{p}_{ij},G_{23}^{(s,t)}=\sum _{i=s+1}^t\sum _{j=t+1}^r{p}_{ij},\\ G_{31}^{(s,t)}& =\sum _{i=t+1}^r\sum _{j=1}^s{p}_{ij},G_{32}^{(s,t)}=\sum _{i=t+1}^r\sum _{j=s+1}^t{p}_{ij},G_{33}^{(s,t)}=\sum _{i=t+1}^r\sum _{j=t+1}^r{p}_{ij}.\end{array}$$

Then the symmetry model is expressed as

$$G_{kl}^{(s,t)}=G_{lk}^{(s,t)}(k=1,2,3;l=1,2,3;k\ne l),$$

for all s and t ($1\le s<t\le r-1$)  [10].

Assume that $G_{kl}^{(s,t)}+G_{lk}^{(s,t)}\ne 0$ for $1\le s<t\le r-1$ and $1\le k<l\le 3$. Let for $1\le s<t\le r-1$,

$${\mathrm{\Delta }}_{st}=\sum _{k=1}^3\sum _{\begin{array}{c}l=1\\ l\ne k\end{array}}^3{G}_{kl}^{(s,t)}$$

and

$$G_{kl}^{\ast (s,t)}=\frac{G_{kl}^{(s,t)}}{{\mathrm{\Delta }}_{st}},(k=1,2,3;l=1,2,3;k\ne l).$$

We shall consider a measure defined by

$$\mathrm{\Psi }=\frac{1}{(\genfrac{}{}{0ex}{}{r-1}{2})}\sum _{s=1}^{r-2}\sum _{t=s+1}^{r-1}{\mathrm{\Psi }}_{st},$$

where

$$\begin{array}{rl}{\mathrm{\Psi }}_{st}& =\frac{4}{\pi }\sum _{k=1}^2\sum _{l=k+1}^3(G_{kl}^{\ast (s,t)}+G_{lk}^{\ast (s,t)})({\mathrm{\Theta }}_{kl}^{(s,t)}-\frac{\pi }{4}),\\ {\mathrm{\Theta }}_{kl}^{(s,t)}& ={\cos }^{-1}\left(\frac{G_{kl}^{(s,t)}}{\sqrt{{\left(G_{kl}^{(s,t)}\right)}^2+{\left(G_{lk}^{(s,t)}\right)}^2}}\right).\end{array}$$

The range of ${\mathrm{\Theta }}_{kl}^{(s,t)}$ is $0\le {\mathrm{\Theta }}_{kl}^{(s,t)}\le \pi /2$. Thus, the measure Ψ lies between $-1$ and 1. The measure Ψ has characteristics that: from assumption {$G_{kl}^{(s,t)}+G_{lk}^{(s,t)}\ne 0$},

1. $\mathrm{\Psi }=-1$ if and only if $G_{lk}^{(s,t)}=0$ and $G_{kl}^{(s,t)}>0$ for all $1\le s<t\le r-1$ and $1\le k<l\le 3$, i.e. $p_{ji}=0$ for all $1\le i<j\le r$, say, complete-upper-asymmetry;

2. $\mathrm{\Psi }=1$ if and only if $G_{kl}^{(s,t)}=0$ and $G_{lk}^{(s,t)}>0$ for all $1\le s<t\le r-1$ and $1\le k<l\le 3$, i.e. $p_{ij}=0$ for all $1\le i<j\le r$, say, complete-lower-asymmetry.

When $\mathrm{\Psi }=0$, we shall refer to this structure as the collapsed average symmetry. We note that if the symmetry holds then the collapsed average symmetry holds, but the converse does not hold.

3. Relationship between proposed measure and conditional symmetry

We next consider the relationship between the measure Ψ and the conditional symmetry model proposed by McCullagh [4]. The conditional symmetry model is defined by

$$p_{ij}=\mathrm{\Gamma }{p}_{ji}(1\le i<j\le r).$$

The conditional symmetry model indicates that the probability that an observation will fall in cell (i,j) for i<j is Γ times higher than the probability that the observation falls in cell (j,i). A special case of this model obtained by putting $\mathrm{\Gamma }=1$ is the symmetry model.

If there is a structure of conditional symmetry in the $r\times r$ table with ordinal categories, then the measure Ψ is identical to the measure ϕ proposed by Tahata et al. [6]. The measure to represent the degree of departure from average symmetry proposed by Tahata et al. [6] is given as follows: assuming that {$p_{ij}+p_{ji}\ne 0$},

$$\phi =\frac{4}{\pi }\sum _{i=1}^{r-1}\sum _{j=i+1}^r(p_{ij}^{\ast }+p_{ji}^{\ast })({\theta }_{ij}-\frac{\pi }{4}),$$

where

$$\begin{array}{rl}\delta & =\sum _{i=1}^r\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^r{p}_{ij},\\ p_{ij}^{\ast }& =\frac{p_{ij}}{\delta },\\ {\theta }_{ij}& ={\cos }^{-1}\left(\frac{p_{ij}}{\sqrt{p_{ij}^2+p_{ji}^2}}\right).\end{array}$$

Under the conditional symmetry, the measure ϕ can be expressed as

$$\phi =\frac{4}{\pi }{\cos }^{-1}\left(\frac{\mathrm{\Gamma }}{\sqrt{{\mathrm{\Gamma }}^2+1}}\right)-1,$$

since

$$\begin{array}{rl}& {\theta }_{ij}={\cos }^{-1}\left(\frac{p_{ij}}{\sqrt{p_{ij}^2+p_{ji}^2}}\right)={\cos }^{-1}\left(\frac{\mathrm{\Gamma }{p}_{ji}}{\sqrt{{\mathrm{\Gamma }}^2{p}_{ji}^2+p_{ji}^2}}\right)={\cos }^{-1}\left(\frac{\mathrm{\Gamma }}{\sqrt{{\mathrm{\Gamma }}^2+1}}\right),\\ & \sum _{i=1}^{r-1}\sum _{j=i+1}^r(p_{ij}^{\ast }+p_{ji}^{\ast })=\sum _{i=1}^{r-1}\sum _{j=i+1}^r\frac{p_{ij}+p_{ji}}{\delta }=\sum _{i=1}^r\sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^r\frac{p_{ij}}{\delta }=1.\end{array}$$

In a similar manner, under the conditional symmetry, the measure Ψ can also be expressed as

$$\mathrm{\Psi }=\frac{4}{\pi }{\cos }^{-1}\left(\frac{\mathrm{\Gamma }}{\sqrt{{\mathrm{\Gamma }}^2+1}}\right)-1.$$

This is because the submeasure ${\mathrm{\Psi }}_{st}$ can be expressed as

$${\mathrm{\Psi }}_{st}=\frac{4}{\pi }{\cos }^{-1}\left(\frac{\mathrm{\Gamma }}{\sqrt{{\mathrm{\Gamma }}^2+1}}\right)-1.$$

Therefore, if there is a structure of conditional symmetry in the ordinal $r\times r$ table, the measure Ψ is identical to the measure ϕ. In addition, $\mathrm{\Psi }=0$ if and only if $\mathrm{\Gamma }=1$, i.e. the symmetry model holds. As Γ becomes larger than 1, the measure Ψ approaches $-1$. Also, the measure Ψ approaches 1 as Γ becomes smaller than 1. Thus, the measure Ψ would be adequate for representing the degree of departure from symmetry toward the complete-upper-asymmetry and toward the complete-lower-asymmetry.

4. Approximate confidence interval for measure

Let $n_{ij}$ denote the observed frequency in the ith row and jth column of the table ($i=1,\dots ,r;j=1,\dots ,r$). The sample version of Ψ, that is, $\hat{\mathrm{\Psi }}$, is given by Ψ with {$p_{ij}$} replaced by {${\hat{p}}_{ij}$}, where ${\hat{p}}_{ij}=n_{ij}/n$ and $n=\sum \sum n_{ij}$. Assuming that {$n_{ij}$} result from full multinomial sampling, we shall consider an approximate standard error for $\hat{\mathrm{\Psi }}$ and a large-sample confidence interval for Ψ. Using the delta method, $\sqrt{n}(\hat{\mathrm{\Psi }}-\mathrm{\Psi })$ has asymptotically (as $n\to \mathrm{\infty }$) a normal distribution with mean zero and variance ${\sigma }^2[\mathrm{\Psi }]$. See Appendix for the details of ${\sigma }^2[\mathrm{\Psi }]$.

Let ${\hat{\sigma }}^2[\mathrm{\Psi }]$ denote ${\sigma }^2[\mathrm{\Psi }]$ with {$p_{ij}$} replaced by {${\hat{p}}_{ij}$}. Then $\hat{\sigma }[\mathrm{\Psi }]/\sqrt{n}$ is an estimated approximate standard error for $\hat{\mathrm{\Psi }}$, and $\hat{\mathrm{\Psi }}\pm z_{p/2}\hat{\sigma }[\mathrm{\Psi }]/\sqrt{n}$ is an approximate $100(1-p)$ percent confidence interval for Ψ, where $z_{p/2}$ is the $100(1-p/2)$th percentile of the standard normal distribution.

5. Data analysis: application to clinical trial data

Consider the data in Table 1 (a and b) again. For these data, Sugano et al. [5] just described the proportion of improvement or deterioration for esomeprazole group (say, drug group) and placebo group. In our paper, we apply the proposed measure Ψ to these data so that we can compare the treatment effects for the drug and placebo statistically.

We see from Table 2 that for the data in Table 1 (a), the estimated value of measure Ψ is $-0.5041$ and all values in the confidence interval for Ψ are negative. Therefore, the structure of collapsed average symmetry departs toward the complete-upper-asymmetry. As a result, we can statistically estimate that the drug group tends to be getting better. While for the data in Table 1 (b), the estimated measure of Ψ is 0.5591 and all values in the confidence interval for Ψ are positive. Therefore, the structure of collapsed average symmetry departs toward the complete-lower-asymmetry. So, we can statistically estimate that the placebo group tends to be getting worse. In addition, for comparing the confidence intervals in Table 2, we can see that the drug is more effective than placebo, because the confidence intervals do not overlap each other.

Table 2. The estimates of measure Ψ, approximate standard error for $\hat{\mathrm{\Psi }}$, and approximate 95% confidence intervals for Ψ applied to the data in Table 1.

Applied data	Estimated measure	Standard error	Confidence interval
Table 1 (a)	−0.5041	0.1194	(−0.7382, −0.2700)
Table 1 (b)	0.5591	0.0861	(0.3903, 0.7278)

However, the existing measure ϕ cannot be estimated, because the sum of symmetric cells is zero (i.e. $n_{25}+n_{52}=0$ in Table 1 (a), $n_{14}+n_{41}=0$ in Table 1 (b)).

In  [5], to evaluate the efficacy of the drug in preventing NSAID induced peptic ulcers in Japanese at-risk patients, the primary end point was the Kaplan–Meier estimated proportion of ulcer-free patients. As the result of the Kaplan–Meier survival analysis, they concluded that the ulcer-free proportion in the drug group was significantly higher than that in the placebo group. In addition to that, by using our proposed measure Ψ, it is shown that the drug also has a more therapeutic effect than placebo.

6. Discussion

In Section 5, we introduced the case that the proposed measure can be estimated but the existing measure cannot be estimated. Since the existing measure is defined under the assumption $\{p_{ij}+p_{ji}\ne 0\}$, we cannot estimate the measure when a sum of the observed frequencies for one (or more) pair of symmetric cells is zero. On the other hand, the assumption for the proposed measure is more relaxed than that for the existing measure, because the proposed measure is defined for the collapsed contingency tables. This point (i.e. sampling zeros problem) may be advantageous for the proposed measure.

6.1. Comparison of two measures on artificial data

We compare two measures using artificial data in Table 3 (a–c). We set some sums of symmetric cells in these contingency tables to be zero.

Table 3. There are artificial data which have some sums of symmetric cells zero (i.e. $n_{13}+n_{31}=0$ and $n_{24}+n_{42}=0$ in (a), $n_{13}+n_{31}=0$ and $n_{34}+n_{43}=0$ in (b), and $n_{13}+n_{31}=0$ and $n_{25}+n_{52}=0$ in (c)).

	Y
X	1	2	3	4	5
(a)
1	11	21	0	14	6
2	21	4	23	0	25
3	0	23	33	12	15
4	14	0	12	16	9
5	6	25	15	9	10
(b)
1	2	12	0	5	6
2	0	22	6	16	25
3	0	0	27	0	8
4	0	0	0	44	20
5	0	0	0	0	25
(c)
1	11	0	0	0	0
2	3	4	0	0	0
3	0	9	27	0	0
4	5	42	34	8	0
5	15	0	35	20	55

For Table 3 (a), the estimated measure of Ψ (i.e. $\hat{\mathrm{\Psi }}$) is 0 but the estimated measure of ϕ (i.e. $\hat{\phi }$) cannot be calculated. Also, $\hat{\mathrm{\Psi }}$ is $-1$ for Table 3 (b), and $\hat{\mathrm{\Psi }}$ is 1 for Table 3 (c). However, $\hat{\phi }$ cannot be calculated for Table 3 (b and c). Although $\hat{\phi }$ cannot be calculated because of the assumption {$p_{ij}+p_{ji}\ne 0$}, $\hat{\mathrm{\Psi }}$ can be calculated because of the assumption {$G_{kl}^{(s,t)}+G_{lk}^{(s,t)}\ne 0$}.

6.2. Simulation studies

We compare two measures in simulation studies. Consider random variables $Z_1$ and $Z_2$ having a joint bivariate normal distribution with means $\mathrm{E}(Z_1)={\mu }_1$ and $\mathrm{E}(Z_2)={\mu }_2$, variances $\mathrm{Var}(Z_1)=\mathrm{Var}(Z_2)={\sigma }^2$, and correlation $\mathrm{Corr}(Z_1,Z_2)=\rho$. Suppose that there is an underlying bivariate normal distribution and suppose that a $4\times 4$ table is formed using cutpoints for each variable at ${\mu }_1,{\mu }_1\pm 0.6\sigma$; a $6\times 6$ table is formed using cutpoints for each variable at ${\mu }_1,{\mu }_1\pm 0.6\sigma ,{\mu }_1\pm 0.8\sigma$; a $10\times 10$ table is formed using cutpoints for each variable at ${\mu }_1,{\mu }_1\pm 0.2\sigma ,{\mu }_1\pm 0.4\sigma ,{\mu }_1\pm 0.6\sigma ,{\mu }_1\pm 0.8\sigma$. Then, we consider the several simulation scenarios; r=4,6,10 (r is the number of categories); $\rho =0.3,0.6$; n=50,80,100,200,300. The simulation studies are performed based on 100,000 trials per scenario.

We summarize the proportions of the successfully calculated $\hat{\mathrm{\Psi }}$ and $\hat{\phi }$ for each simulation scenarios in Table 4. We shall denominate this proportion as ‘calculable proportion’. That is, we count up when the estimated measure can be calculated, and divide the counts by the number of trials. When r=4, there are not much difference between these measures. It may be because the proportions for $\hat{\phi }$ are high enough, namely, there is almost no chance that the sum of symmetric cells is zero. On the other hand, when r=6,10, we can confirm big difference between these measures. We can see that $\hat{\mathrm{\Psi }}$ can be calculated even if the proportions for $\hat{\phi }$ are zero by some scenarios. Moreover, the difference of the calculable proportions between $\hat{\mathrm{\Psi }}$ and $\hat{\phi }$ are more than $50\mathrm{\%}$ in some scenarios. Therefore, even if the sample size n is equal to or less than the number of cells, i.e. $r^2$, the proposed measure Ψ can be estimated.

Table 4. Results of simulation studies comparing the measures Ψ and ϕ.

		Calculable proportion ($\mathrm{\%}$)
		$\rho =0.3$		$\rho =0.6$
r	n	$\hat{\mathrm{\Psi }}$	$\hat{\phi }$	$\hat{\mathrm{\Psi }}$	$\hat{\phi }$
4
	50	98.8	97.7	80.2	76.7
	80	99.9	99.9	92.6	92.3
	100	100.0	100.0	96.3	96.2
	200	100.0	100.0	99.9	99.9
	300	100.0	100.0	100.0	100.0
6
	50	57.5	1.5	30.9	0.2
	80	84.6	12.9	54.8	2.3
	100	92.4	24.7	65.5	5.7
	200	99.7	68.8	89.6	34.3
	300	100.0	85.8	96.8	60.4
10
	50	34.7	0.0	19.3	0.0
	80	77.4	0.0	51.3	0.0
	100	89.6	0.0	64.1	0.0
	200	99.7	1.7	89.7	0.9
	300	100.0	25.2	96.7	18.3

Note: Proportions of the successfully calculated $\hat{\mathrm{\Psi }}$ and $\hat{\phi }$ are summarized based on 100,000 simulation replications per scenario.

As a conclusion of simulation studies, when an original table is a little larger than the collapsed table, such as $4\times 4$ table, the effect of collapse may be minor. To the contrast, when an original table is larger, such as $6\times 6$ or $10\times 10$ table (we may have sparse table), the effect of collapse may be major. Note that the benefits of collapsing tables are not only the improvement for calculable proportion, but also the simplicity of interpretation as mentioned in Section 1.

7. Concluding remarks

The proposed measure Ψ is useful for representing what degree the departure from collapsed average symmetry is toward two kinds of complete asymmetries (i.e. the complete-upper-asymmetry and the complete-lower-asymmetry). Since $\hat{\mathrm{\Psi }}$ lies between $-1$ and 1 without the dimension and the sample size, $\hat{\mathrm{\Psi }}$ would also be useful for comparing the degrees of departure from collapsed average symmetry in several tables. Especially, if it can be estimated that there is a structure of conditional symmetry in the table, then $\hat{\mathrm{\Psi }}$ would be adequate for representing the degree of departure from symmetry toward two kinds of complete asymmetries.

As described above, it is meaningful to consider collapsed $3\times 3$ tables. If it is difficult to decide uniquely how to choose cutpoints for a clinical scale, it is reasonable to consider equally evaluating the square contingency tables collapsed in various patterns. Thus, we have considered that all ${\mathrm{\Psi }}_{st}$ are combined at the same weights $1/(\genfrac{}{}{0ex}{}{r-1}{2})$. In addition, even if the estimated measure of ϕ cannot be calculated because the sum of symmetric cells is zero in the original table, the estimated measure of Ψ can be calculated in some cases. It seems useful to analyze an original square contingency table by using the measure Ψ. This property plays a great role especially in small-size clinical trials.

Acknowledgements

The authors would like to thank the editor and two anonymous referees for the meaningful comments.

Appendix.

Using the delta method, $\sqrt{n}(\hat{\mathrm{\Psi }}-\mathrm{\Psi })$ has asymptotic variance ${\sigma }^2[\mathrm{\Psi }]$ as follows:

$${\sigma }^2[\mathrm{\Psi }]=\sum _{i=1}^{r-1}\sum _{j=i+1}^r(p_{ij}{D}_{ij}^2+p_{ji}{D}_{ji}^2),$$

where

$$\begin{array}{rl}{D}_{ij}& =\frac{1}{(\genfrac{}{}{0ex}{}{r-1}{2})}\sum _{s=1}^{r-2}\sum _{t=s+1}^{r-1}[I(1\le i\le s,s+1\le j\le t)A_{12}^{(s,t)}\\ & +I(1\le i\le s,t+1\le j\le r)A_{13}^{(s,t)}\\ & +I(s+1\le i\le t,t+1\le j\le r)A_{23}^{(s,t)}],\\ D_{ji}& =\frac{1}{(\genfrac{}{}{0ex}{}{r-1}{2})}\sum _{s=1}^{r-2}\sum _{t=s+1}^{r-1}[I(1\le i\le s,s+1\le j\le t)B_{12}^{(s,t)}\\ & +I(1\le i\le s,t+1\le j\le r)B_{13}^{(s,t)}\\ & +I(s+1\le i\le t,t+1\le j\le r)B_{23}^{(s,t)}],\\ A_{kl}^{(s,t)}& =\frac{4}{\pi {\mathrm{\Delta }}_{st}}({\mathrm{\Theta }}_{kl}^{(s,t)}-\frac{G_{lk}^{(s,t)}(G_{kl}^{(s,t)}+G_{lk}^{(s,t)})}{{\left(G_{kl}^{(s,t)}\right)}^2+{\left(G_{lk}^{(s,t)}\right)}^2})-\frac{{\mathrm{\Psi }}_{st}+1}{{\mathrm{\Delta }}_{st}},\\ B_{kl}^{(s,t)}& =\frac{4}{\pi {\mathrm{\Delta }}_{st}}({\mathrm{\Theta }}_{kl}^{(s,t)}+\frac{G_{kl}^{(s,t)}(G_{kl}^{(s,t)}+G_{lk}^{(s,t)})}{{\left(G_{kl}^{(s,t)}\right)}^2+{\left(G_{lk}^{(s,t)}\right)}^2})-\frac{{\mathrm{\Psi }}_{st}+1}{{\mathrm{\Delta }}_{st}},\end{array}$$

and $I(\cdot )$ is the indicator function.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Satoru Shinoda  http://orcid.org/0000-0003-0651-438X