# Partial Asymmetry Measures for Square Contingency Tables

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## Abstract

**:**

## 1. Introduction

## 2. Review of Previous Research

## 3. The Proposed Measure

## 4. Numerical Examples

## 5. Example

## 6. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Estimate of measure ${\varphi}_{ij}$ and approximate 95% confidence interval for ${\varphi}_{ij}$ applied to Table 1.

(a) | (1) | (2) | (3) | (4) | (5) |

(1) | 37 | 544 | 12 | 7 | 8 |

(2) | 102 | 26 | 15 | 15 | 12 |

(3) | 9 | 8 | 29 | 10 | 11 |

(4) | 9 | 9 | 12 | 40 | 12 |

(5) | 14 | 9 | 10 | 11 | 29 |

(b) | (1) | (2) | (3) | (4) | (5) |

(1) | 47 | 11 | 37 | 44 | 48 |

(2) | 3 | 38 | 34 | 37 | 49 |

(3) | 44 | 44 | 52 | 56 | 48 |

(4) | 38 | 25 | 55 | 45 | 47 |

(5) | 35 | 25 | 51 | 43 | 44 |

(c) | (1) | (2) | (3) | (4) | (5) |

(1) | 33 | 316 | 13 | 18 | 18 |

(2) | 321 | 37 | 20 | 18 | 20 |

(3) | 7 | 6 | 26 | 16 | 14 |

(4) | 5 | 5 | 1 | 30 | 19 |

(5) | 5 | 10 | 5 | 2 | 35 |

(d) | (1) | (2) | (3) | (4) | (5) |

(1) | 39 | 4 | 70 | 42 | 50 |

(2) | 5 | 34 | 45 | 110 | 84 |

(3) | 17 | 12 | 54 | 103 | 63 |

(4) | 31 | 14 | 20 | 39 | 48 |

(5) | 9 | 29 | 26 | 6 | 46 |

(e) | (1) | (2) | (3) | (4) | (5) |

(1) | 7 | 103 | 1 | 1 | 4 |

(2) | 19 | 10 | 2 | 2 | 4 |

(3) | 2 | 1 | 6 | 2 | 2 |

(4) | 3 | 4 | 4 | 5 | 2 |

(5) | 1 | 2 | 4 | 1 | 8 |

(a) | (1) | (2) | (3) | (4) | (5) |

(1) | 0.030 | 0.570 | 0.010 | 0.010 | 0.010 |

(2) | 0.010 | 0.030 | 0.010 | 0.010 | 0.010 |

(3) | 0.010 | 0.010 | 0.030 | 0.010 | 0.010 |

(4) | 0.010 | 0.010 | 0.010 | 0.030 | 0.010 |

(5) | 0.010 | 0.010 | 0.010 | 0.010 | 0.030 |

(1) | 0.040 | 0.008 | 0.040 | 0.040 | 0.040 |

(2) | 0.004 | 0.040 | 0.040 | 0.040 | 0.040 |

(3) | 0.040 | 0.040 | 0.050 | 0.050 | 0.050 |

(4) | 0.040 | 0.040 | 0.050 | 0.040 | 0.049 |

(5) | 0.040 | 0.040 | 0.050 | 0.049 | 0.040 |

(c) | (1) | (2) | (3) | (4) | (5) |

(1) | 0.030 | 0.320 | 0.015 | 0.015 | 0.015 |

(2) | 0.320 | 0.030 | 0.021 | 0.019 | 0.024 |

(3) | 0.005 | 0.007 | 0.030 | 0.020 | 0.016 |

(4) | 0.005 | 0.006 | 0.005 | 0.030 | 0.018 |

(5) | 0.005 | 0.007 | 0.004 | 0.003 | 0.030 |

(d) | (1) | (2) | (3) | (4) | (5) |

(1) | 0.040 | 0.003 | 0.080 | 0.050 | 0.050 |

(2) | 0.003 | 0.040 | 0.050 | 0.100 | 0.080 |

(3) | 0.020 | 0.010 | 0.050 | 0.100 | 0.064 |

(4) | 0.030 | 0.020 | 0.020 | 0.040 | 0.050 |

(5) | 0.010 | 0.020 | 0.020 | 0.020 | 0.040 |

**Table 3.**Estimate of measure ${\varphi}_{ij}$, estimated approximate variance for ${\varphi}_{ij}$, approximate 95% confidence interval for ${\varphi}_{ij}$, weights for measures ${\mathrm{\Phi}}_{S}$ and ${\mathrm{\Phi}}_{T}$, and estimates of measures ${\mathrm{\Phi}}_{S}$ and ${\mathrm{\Phi}}_{T}$, applied to Table 1(a)–(d).

Applied Data | Cells | ${\widehat{\mathit{\varphi}}}_{\mathbf{ij}}$ | ${\widehat{\mathit{\sigma}}}_{\mathbf{ij}}^{2}$ | Confidence Interval for ${\mathit{\varphi}}_{\mathbf{ij}}$ | ${\widehat{w}}_{\mathbf{ij}}^{*}$ | ${\widehat{w}}_{\mathbf{ij}}$ | ${\widehat{\mathbf{\Phi}}}_{\mathit{S}}$ | ${\widehat{\mathbf{\Phi}}}_{\mathit{T}}$ |
---|---|---|---|---|---|---|---|---|

Table 1(a) | (1,2), (2,1) | 0.37075 | 0.0012005 | (0.303, 0.439) | 0.057987 | 0.769964 | ||

(1,3), (3,1) | 0.01477 | 0.0020088 | (−0.073, 0.103) | 0.034653 | 0.025030 | |||

(1,4), (4,1) | 0.01130 | 0.0020219 | (−0.077, 0.099) | 0.034428 | 0.019070 | |||

(1,5), (5,1) | 0.05434 | 0.0068561 | (−0.108, 0.217) | 0.010153 | 0.026222 | |||

(2,3), (3,2) | 0.06789 | 0.0081116 | (−0.109, 0.244) | 0.008582 | 0.027414 | 0.02624 | 0.29124 | |

(2,4), (4,2) | 0.04557 | 0.0053039 | (−0.097, 0.188) | 0.013124 | 0.028605 | |||

(2,5), (5,2) | 0.01477 | 0.0020088 | (−0.073, 0.103) | 0.034653 | 0.025030 | |||

(3,4), (4,3) | 0.00597 | 0.0007797 | (−0.049, 0.061) | 0.089277 | 0.026222 | |||

(3,5), (5,3) | 0.00164 | 0.0002246 | (−0.028, 0.031) | 0.309966 | 0.025030 | |||

(4,5), (5,4) | 0.00136 | 0.0001710 | (−0.024, 0.027) | 0.407178 | 0.027414 | |||

Table 1(b) | (1,2), (2,1) | 0.25041 | 0.0422558 | (−0.152, 0.653) | 0.000032 | 0.018088 | ||

(1,3), (3,1) | 0.00539 | 0.0001914 | (−0.022, 0.033) | 0.006979 | 0.104651 | |||

(1,4), (4,1) | 0.00387 | 0.0001357 | (−0.019, 0.027) | 0.009848 | 0.105943 | |||

(1,5), (5,1) | 0.01777 | 0.0006101 | (−0.031, 0.066) | 0.002190 | 0.107235 | |||

(2,3), (3,2) | 0.01189 | 0.0004362 | (−0.029, 0.053) | 0.003063 | 0.100775 | 0.00036 | 0.01843 | |

(2,4), (4,2) | 0.02719 | 0.0012416 | (−0.042, 0.096) | 0.001076 | 0.080103 | |||

(2,5), (5,2) | 0.07727 | 0.0028494 | (−0.027, 0.182) | 0.000469 | 0.095607 | |||

(3,4), (4,3) | 0.00006 | 0.0000015 | (−0.002, 0.002) | 0.877858 | 0.143411 | |||

(3,5), (5,3) | 0.00066 | 0.0000193 | (−0.008, 0.009) | 0.069221 | 0.127907 | |||

(4,5), (5,4) | 0.00143 | 0.0000457 | (−0.012, 0.015) | 0.029264 | 0.116279 | |||

Table 1(c) | (1,2), (2,1) | 0.00004 | 0.0000002 | (−0.001, 0.001) | 0.999900 | 0.759237 | ||

(1,3), (3,1) | 0.06593 | 0.0090727 | (−0.121, 0.253) | 0.000022 | 0.023838 | |||

(1,4), (4,1) | 0.24463 | 0.0252616 | (−0.067, 0.556) | 0.000008 | 0.027414 | |||

(1,5), (5,1) | 0.24463 | 0.0252616 | (−0.067, 0.556) | 0.000008 | 0.027414 | |||

(2,3), (3,2) | 0.22065 | 0.0205989 | (−0.061, 0.502) | 0.000010 | 0.030989 | 0.00006 | 0.06270 | |

(2,4), (4,2) | 0.24463 | 0.0252616 | (−0.067, 0.556) | 0.000008 | 0.027414 | |||

(2,5), (5,2) | 0.08170 | 0.0074074 | (−0.087, 0.250) | 0.000027 | 0.035757 | |||

(3,4), (4,3) | 0.67724 | 0.0521067 | (0.230, 1.125) | 0.000004 | 0.020262 | |||

(3,5), (5,3) | 0.16853 | 0.0225185 | (−0.126, 0.463) | 0.000009 | 0.022646 | |||

(4,5), (5,4) | 0.54628 | 0.0432851 | (0.139, 0.954) | 0.000005 | 0.025030 | |||

Table 1(d) | (1,2), (2,1) | 0.00892 | 0.0028433 | (−0.096, 0.113) | 0.113903 | 0.011421 | ||

(1,3), (3,1) | 0.28736 | 0.0075340 | (0.117, 0.457) | 0.042986 | 0.110406 | |||

(1,4), (4,1) | 0.01644 | 0.0006424 | (−0.033, 0.066) | 0.504107 | 0.092640 | |||

(1,5), (5,1) | 0.38383 | 0.0134101 | (0.157, 0.611) | 0.024150 | 0.074873 | |||

(2,3), (3,2) | 0.25751 | 0.0106028 | (0.056, 0.459) | 0.030545 | 0.072335 | 0.11518 | 0.28830 | |

(2,4), (4,2) | 0.49139 | 0.0071440 | (0.326, 0.657) | 0.045333 | 0.157360 | |||

(2,5), (5,2) | 0.17837 | 0.0039745 | (0.055, 0.302) | 0.081484 | 0.143401 | |||

(3,4), (4,3) | 0.35950 | 0.0061895 | (0.205, 0.514) | 0.052323 | 0.156091 | |||

(3,5), (5,3) | 0.12854 | 0.0037881 | (0.008, 0.249) | 0.085494 | 0.112944 | |||

(4,5), (5,4) | 0.49674 | 0.0164609 | (0.245, 0.748) | 0.019674 | 0.068528 | |||

Table 1(e) | (1,2), (2,1) | 0.37599 | 0.0064089 | (0.219, 0.533) | 0.486330 | 0.743902 | ||

(1,3), (3,1) | 0.08170 | 0.0740741 | (−0.452, 0.615) | 0.042077 | 0.018293 | |||

(1,4), (4,1) | 0.18872 | 0.1177550 | (−0.484, 0.861) | 0.026469 | 0.024390 | |||

(1,5), (5,1) | 0.27807 | 0.1280000 | (−0.423, 0.979) | 0.024350 | 0.030488 | |||

(2,3), (3,2) | 0.08170 | 0.0740741 | (−0.452, 0.615) | 0.042077 | 0.018293 | 0.23244 | 0.30922 | |

(2,4), (4,2) | 0.08170 | 0.0370370 | (−0.295, 0.459) | 0.084155 | 0.036585 | |||

(2,5), (5,2) | 0.08170 | 0.0370370 | (−0.295, 0.459) | 0.084155 | 0.036585 | |||

(3,4), (4,3) | 0.08170 | 0.0370370 | (−0.295, 0.459) | 0.084155 | 0.036585 | |||

(3,5), (5,3) | 0.08170 | 0.0370370 | (−0.295, 0.459) | 0.084155 | 0.036585 | |||

(4,5), (5,4) | 0.08170 | 0.0740741 | (−0.452, 0.615) | 0.042077 | 0.018293 |

Fathers’ Birth Order | |||||
---|---|---|---|---|---|

Mothers’ Birth Order | First | Second | Third | Fourth or More | Total |

First | 224 | 179 | 53 | 22 | 478 |

Second | 162 | 153 | 35 | 15 | 365 |

Third | 37 | 37 | 18 | 11 | 103 |

Fourth or more | 12 | 7 | 3 | 5 | 27 |

Total | 435 | 376 | 109 | 53 | 973 |

**Table 5.**Estimate of measure ${\varphi}_{ij}$, estimated approximate variance for ${\varphi}_{ij}$, approximate 95% confidence interval for ${\varphi}_{ij}$, estimates of measures of ${\mathrm{\Phi}}_{S}$ and ${\mathrm{\Phi}}_{T}$, and weights for measures of ${\mathrm{\Phi}}_{S}$ and ${\mathrm{\Phi}}_{T}$, applied to Table 4.

Cells | ${\widehat{\mathit{\varphi}}}_{\mathbf{ij}}$ | ${\widehat{\mathit{\sigma}}}_{\mathbf{ij}}^{2}$ | Confidence Interval for ${\mathit{\varphi}}_{\mathbf{ij}}$ | ${\widehat{w}}_{\mathbf{ij}}^{*}$ | ${\widehat{w}}_{\mathbf{ij}}$ | ${\widehat{\mathbf{\Phi}}}_{\mathit{S}}$ | ${\widehat{\mathbf{\Phi}}}_{\mathit{T}}$ |
---|---|---|---|---|---|---|---|

(1,2), (2,1) | 0.0018 | 0.00002 | (−0.006, 0.009) | 0.5864 | 0.5951 | ||

(1,3), (3,1) | 0.0229 | 0.00072 | (−0.030, 0.076) | 0.0123 | 0.1571 | ||

(1,4), (4,1) | 0.0633 | 0.00514 | (−0.077, 0.204) | 0.0017 | 0.0593 | ||

(2,3), (3,2) | 0.0006 | 0.00002 | (−0.009, 0.010) | 0.3986 | 0.1257 | 0.0018 | 0.0184 |

(2,4), (4,2) | 0.0976 | 0.01192 | (−0.116, 0.312) | 0.0007 | 0.0384 | ||

(3,4), (4,3) | 0.2504 | 0.04226 | (−0.152, 0.653) | 0.0002 | 0.0244 |

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**MDPI and ACS Style**

Ishihara, T.; Yamamoto, K.; Tahata, K.; Tomizawa, S.
Partial Asymmetry Measures for Square Contingency Tables. *Symmetry* **2022**, *14*, 1936.
https://doi.org/10.3390/sym14091936

**AMA Style**

Ishihara T, Yamamoto K, Tahata K, Tomizawa S.
Partial Asymmetry Measures for Square Contingency Tables. *Symmetry*. 2022; 14(9):1936.
https://doi.org/10.3390/sym14091936

**Chicago/Turabian Style**

Ishihara, Takuma, Kouji Yamamoto, Kouji Tahata, and Sadao Tomizawa.
2022. "Partial Asymmetry Measures for Square Contingency Tables" *Symmetry* 14, no. 9: 1936.
https://doi.org/10.3390/sym14091936