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

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## Abstract

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.

## Keywords

Confidence region Double symmetry Index vector Visualization## Mathematics Subject Classification

62H17## 1 Introduction

Consider an \(r \times r\) square contingency table \({\varvec{P}}=(p_{ij})\) with the same row and column nominal classifications. Here \(p_{ij}\) denotes the probability that an observation will fall in the *i*th row and *j*th column of the table (\(i=1,\dots ,r;j=1,\dots ,r\)). 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).

*i*and column category

*j*is equal to the probability that the observation falls in row category

*j*and column category

*i*\((i\ne j)\). Wall and Lienert (1976) proposed the point symmetry model defined by

*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.

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. 226)

Forecast 1956 | Actual 1959 | |||
---|---|---|---|---|

Higher | No change | Lower | Total | |

(a) For prices | ||||

Higher | 209 | 169 | 6 | 384 |

No change | 190 | 3073 | 184 | 3447 |

Lower | 3 | 62 | 81 | 146 |

Total | 402 | 3304 | 271 | 3977 |

(b) For production | ||||

Higher | 532 | 394 | 69 | 995 |

No change | 447 | 1727 | 334 | 2508 |

Lower | 39 | 230 | 231 | 500 |

Total | 1018 | 2351 | 634 | 4003 |

In case of unknown probabilities \(p_{ij}\) these are estimated from an observed \(r \times r\) contingency table \({\varvec{N}}=(x_{ij})\) 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 model. For evaluating goodness-of-fit of the model, test statistics (e.g., Pearson’s Chi-squared statistic or likelihood ratio statistic) are used. When the model does not hold for several tables, we may be interested in comparing the degrees of departure from the corresponding model. However, test statistics would not be useful for comparing the degrees of departure from the model in several tables because test statistics depend on the dimension *r* and sample size. Thus, for comparing the degrees of departure from the model in several tables, we are interested in an index that does not depend on the dimension *r* and sample size.

### 1.1 Index \(\varPhi _{S}\) for departure from symmetry

*r*. Assume that \(p_{ij}+p_{ji}>0\) for all \(i\ne j\). The index \(\varPhi _{S}\) is expressed as follows:

### 1.2 Index \(\varPhi _{PS}\) for departure from point symmetry

*r*. Let

*i*,

*j*) outside of the center of the table. Assume that \(p_{ij}+p_{r+1-i,r+1-j}>0\) for all \((i, j)\in D\). The index \(\varPhi _{PS}\) is expressed as follows:

### 1.3 Index \(\varPhi _{DS}\) for departure from double symmetry

*i*,

*j*) on the diagonal or on the secondary diagonal (without center point if any), while \(E_2\) includes the remaining cells. Assume that \(p_{ij}+p_{ji}+p_{r+1-i,r+1-j}+p_{r+1-j,r+1-i}>0\) for all \((i, j)\in E_{1}\cup E_{2}\). Let for \(t=1,2,\)

## 2 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.

### 2.1 Definition of the index vector

*r*is even) or \(i=1, \dots , (r-1)/2\) (when

*r*is odd). Note that the definition of the maximum degree of departure from double symmetry for the proposed index vector \({\varvec{\varPsi }}\) is different from that for the index \(\varPhi _{DS}\).

### 2.2 A confidence region for the index vector

*n*and probability vector \({\varvec{p}}\). Then \(\sqrt{n}(\hat{{\varvec{p}}}-{\varvec{p}})\) has asymptotically a normal distribution with zero mean and covariance matrix \(\mathbf{Diag}({\varvec{p}})-{\varvec{pp}}^{\prime }\), where \(\hat{{\varvec{p}}}={\varvec{x}}/n\) and \(\mathbf{Diag}({\varvec{p}})\) is diagonal matrix with the elements of \({\varvec{p}}\) on the main diagonal (see, e.g., Agresti 2013, p. 590). In order to estimate the indexes, \(\hat{\varPhi }_{S}\) and \(\hat{\varPhi }_{PS}\) are given by \(\varPhi _{S}\) and \(\varPhi _{PS}\) with \(\{p_{ij}\}\) replaced by \(\{\hat{p}_{ij}\}\), respectively. Therefore, the sample version of \({\varvec{\varPsi }}\), i.e., \(\widehat{{\varvec{\varPsi }}}\), is given by \({\varvec{\varPsi }}\) with \(\varPhi _{S}\) and \(\varPhi _{PS}\) replaced by \(\hat{\varPhi }_{S}\) and \(\hat{\varPhi }_{PS}\), respectively. Let \((\partial {\varvec{\varPsi }}/\partial {\varvec{p}}^{\prime })\) denote the \(2\times r^{2}\) matrix for which the entry in row

*k*and column

*l*is \(\partial \varPsi _{k}({\varvec{p}})/\partial p_{l}\), where \(\varPsi _{1}\) and \(\varPsi _{2}\) denote \(\varPhi _{S}\) and \(\varPhi _{PS}\), respectively, and \(p_{l}\) denotes the

*l*th element of \({\varvec{p}}\). For

*n*approaching infinity, the estimated index vector can be approximated by

## 3 Examples

Consider the data in Table 1 taken from Andersen (1997, p. 226). 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) (symmetry), 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) (point symmetry). Thus, for these data, we are interested in the structure of symmetry and point symmetry (namely, double symmetry).

Estimates of \(\varPhi _{S}\), \(\varPhi _{PS}\) and \(\varPhi _{DS}\), approximate standard errors for \(\hat{\varPhi }_{S}\), \(\hat{\varPhi }_{PS}\) and \(\hat{\varPhi }_{DS}\), and approximate 95% confidence intervals for \(\varPhi _{S}\), \(\varPhi _{PS}\) and \(\varPhi _{DS}\), for the data of Table 1a, b

Estimated index | Standard error | Confidence interval | |
---|---|---|---|

(a) For Table 1a (prices) | |||

\(\varPhi _{S}\) | 0.0770 | 0.0181 | (0.0415, 0.1125) |

\(\varPhi _{PS}\) | 0.0887 | 0.0161 | (0.0571, 0.1202) |

\(\varPhi _{DS}\) | 0.0817 | 0.0134 | (0.0554, 0.1080) |

(b) For Table 1b (production) | |||

\(\varPhi _{S}\) | 0.0148 | 0.0053 | (0.0045, 0.0251) |

\(\varPhi _{PS}\) | 0.0604 | 0.0085 | (0.0437, 0.0771) |

\(\varPhi _{DS}\) | 0.0536 | 0.0075 | (0.0389, 0.0684) |

Also, we shall compare the degrees of departure from double symmetry in Table 1a, b using the confidence interval of \(\varPhi _{DS}\). 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.

## 4 Discussion

Three artificial contingency tables with \(n = 4020\) each

(a) | |||

100 | 10 | 100 | 100 |

2000 | 100 | 100 | 100 |

100 | 100 | 100 | 800 |

100 | 100 | 10 | 100 |

(b) | |||

100 | 10 | 100 | 100 |

10 | 100 | 100 | 100 |

100 | 100 | 100 | 800 |

100 | 100 | 2000 | 100 |

(c) | |||

100 | 10 | 100 | 100 |

2000 | 100 | 100 | 100 |

100 | 100 | 100 | 10 |

100 | 100 | 800 | 100 |

Estimates of \(\varPhi _{S}\), \(\varPhi _{PS}\) and \(\varPhi _{DS}\), approximate standard errors for \(\hat{\varPhi }_{S}\), \(\hat{\varPhi }_{PS}\) and \(\hat{\varPhi }_{DS}\), and approximate 95% confidence intervals for \(\varPhi _{S}\), \(\varPhi _{PS}\) and \(\varPhi _{DS}\), for the data of Table 3a–c

Estimated index | Standard error | Confidence interval | |
---|---|---|---|

(a) For Table 3a | |||

\(\varPhi _{S}\) | 0.7324 | 0.0108 | (0.7113, 0.7536) |

\(\varPhi _{PS}\) | 0.0953 | 0.0079 | (0.0798, 0.1109) |

\(\varPhi _{DS}\) | 0.3771 | 0.0068 | (0.3637, 0.3905) |

(b) For Table 3b | |||

\(\varPhi _{S}\) | 0.1059 | 0.0088 | (0.0887, 0.1231) |

\(\varPhi _{PS}\) | 0.6595 | 0.0103 | (0.6393, 0.6798) |

\(\varPhi _{DS}\) | 0.3771 | 0.0068 | (0.3637, 0.3905) |

(c) For Table 3c | |||

\(\varPhi _{S}\) | 0.7324 | 0.0108 | (0.7113, 0.7536) |

\(\varPhi _{PS}\) | 0.6595 | 0.0103 | (0.6393, 0.6798) |

\(\varPhi _{DS}\) | 0.3771 | 0.0068 | (0.3637, 0.3905) |

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 from point symmetry is large, (3) for Table 3c, both the degree of departure from symmetry and the degree of departure from point symmetry are large.

## 5 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 \(\varPhi _{S}\) and \(\varPhi _{PS}\) 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 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.

## Notes

### Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their comments and suggestions to improve this paper.

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