# Applying Grey Relational Analysis in the Evaluation of the Balance of Children with Intellectual Disability

## Abstract

**:**

## 1. Introduction

## 2. Grey Relational Analysis

- I.
- ${x}_{0}\text{}$is a standard sequence, whereas ${x}_{i}\text{}$are comparison sequences.
- II.
- ${\Delta}_{0i}=\left|\right|{x}_{0}(k)-{x}_{i}(k)\left|\right|\text{}$ is the norm between$\text{}{x}_{0}\text{}$and$\text{}{x}_{i}\text{}$, whereas ${\overline{\Delta}}_{0i}$ is the mean of ${\Delta}_{0i}$.
- III.
- ${\Delta}_{\mathrm{min}.}=\underset{j\in i}{\overset{\mathrm{min}.\mathrm{min}.}{\forall}}\forall k\left|\right|{x}_{0}(k)-{x}_{j}(k)\left|\right|$ ${\Delta}_{\mathrm{max}.}=\underset{j\in i}{\overset{\mathrm{max}.\mathrm{max}.}{\forall}}\forall k\left|\right|{x}_{0}(k)-{x}_{j}(k)\left|\right|$.

## 3. Method

#### 3.1. Participants

#### 3.2. Apparatus

#### 3.3. Procedure

#### 3.4. Measurements

_{s}), L (L

_{s}), and T (T

_{s}) were identified immediately after the action cue when the difference between the GRA at that timepoint and the previous value was not equal to zero. T

_{p}and L

_{p}occurred when the ground reaction force for Curves T and L, respectively, were maximal. Seat off was defined as the time at which the thighs lost contact with the chair, when the instant of Curve B was at zero (B

_{0}). The GRF oscillated following seat off, and T

_{min}represents the minimum body force during the ascending phase [36]. T

_{pw}and L

_{pw}are the maximum forces of Curves T and L, respectively.

## 4. Calculation Results

#### 4.1. Time Factors

_{max}= 1622.923, Δ

_{min}= 01538.

#### 4.2. Weight Factors

_{max}= 0.4479, Δ

_{min}= 0.0015.

## 5. Discussion

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Variable | Children Without Intellectual Disability (n = 13) | Children with Mild Intellectual Disability (n = 12) | Children with Moderate Intellectual Disability (n = 8) | |||
---|---|---|---|---|---|---|

M | SD | M | SD | M | SD | |

Age (years) | 9.6 | 1.4 | 9.8 | 0.9 | 8.9 | 0.8 |

Weight (kg) | 33.1 | 6.4 | 37.7 | 12.6 | 25.9 | 8.5 |

Hight (cm) | 139.2 | 10.0 | 138.4 | 10.8 | 130.2 | 10.2 |

BMI | 16.9 | 1.6 | 19.3 | 4.5 | 15.3 | 3.8 |

Time Factors (Time) | |||||
---|---|---|---|---|---|

No. | Factors | Explanation | No. | Factors | Explanation |

01 | B_{s}–T_{p} | duration between B_{s} and T_{p} | 05 | T_{p}–L_{p} | duration between T_{p} and L_{p} |

02 | B_{s}–L_{p} | duration between B_{s} and L_{p} | 06 | T_{p}–seat off | duration between T_{p} and seat off |

03 | B_{s}–seat off | duration between B_{s} and seat off | 07 | L_{p}–seat off | duration between L_{p} and seatoff |

04 | B_{s}–end | duration between B_{s} and end |

Weight Factors (Weight) | ||
---|---|---|

No. | Factors | Explanation |

01 | T_{pw} | The maximum ground reaction force of Curve T |

02 | L_{pw} | The maximum ground reaction force of Curve L |

03 | T_{minw} | The minimum body force during the ascending phase |

04 | T_{pw}–T_{minw} | The force difference between T_{pw} and T_{minw} |

05 | L_{pw}–T_{minw} | The force difference between L_{pw} and T_{minw} |

Time Factors (ms) | Weight Factors | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

B_{s}–T_{p} | B_{s}–L_{p} | B_{s}–Seat off | B_{s}–End | T_{p}–L_{p} | T_{p}–Seat off | L_{p}–Seat off | T_{pw} | L_{pw} | T_{minw} | T_{pw}–T_{minw} | L_{pw}–T_{minw} | |

1 | 433 | 499 | 632 | 1256 | 66 | 199 | 133 | 1.30 | 1.20 | 0.77 | 0.52 | 0.43 |

2 | 726 | 781 | 900 | 1739 | 55 | 174 | 119 | 1.47 | 1.31 | 0.70 | 0.78 | 0.61 |

3 | 725 | 745 | 821 | 2034 | 20 | 96 | 76 | 1.43 | 1.41 | 0.88 | 0.55 | 0.52 |

4 | 1074 | 1173 | 1245 | 2135 | 99 | 171 | 72 | 1.27 | 1.12 | 0.86 | 0.40 | 0.26 |

5 | 1043 | 1077 | 1233 | 2204 | 34 | 190 | 156 | 1.34 | 1.30 | 0.82 | 0.51 | 0.48 |

6 | 732 | 794 | 927 | 1675 | 62 | 195 | 133 | 1.31 | 1.25 | 0.84 | 0.47 | 0.41 |

7 | 753 | 767 | 811 | 1691 | 14 | 58 | 44 | 1.52 | 1.50 | 0.76 | 0.76 | 0.74 |

8 | 703 | 717 | 752 | 1375 | 14 | 49 | 35 | 1.44 | 1.42 | 0.66 | 0.78 | 0.76 |

9 | 484 | 629 | 717 | 1481 | 145 | 233 | 88 | 1.29 | 1.21 | 0.97 | 0.32 | 0.24 |

10 | 904 | 999 | 1183 | 1621 | 95 | 279 | 184 | 1.34 | 1.25 | 0.74 | 0.60 | 0.50 |

11 | 744 | 777 | 927 | 1784 | 33 | 183 | 150 | 1.52 | 1.41 | 0.93 | 0.59 | 0.48 |

12 | 623 | 699 | 798 | 1213 | 76 | 175 | 99 | 1.28 | 1.20 | 0.78 | 0.50 | 0.42 |

13 | 820 | 874 | 896 | 1607 | 54 | 76 | 22 | 1.23 | 1.21 | 0.77 | 0.46 | 0.44 |

Time Factors (ms) | Weight Factors | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

B_{s}–T_{p} | B_{s}–L_{p} | B_{s}–Seat off | B_{s}–End | T_{p}–L_{p} | T_{p}–Seat off | L_{p}–Seat off | T_{pw} | L_{pw} | T_{minw} | T_{pw}–T_{minw} | L_{pw}–T_{minw} | |

14 | 1196 | 1359 | 1495 | 2656 | 163 | 299 | 136 | 1.19 | 1.13 | 0.88 | 0.31 | 0.25 |

15 | 918 | 980 | 1070 | 2189 | 62 | 152 | 90 | 1.23 | 1.21 | 0.89 | 0.34 | 0.32 |

16 | 1255 | 1315 | 1448 | 2388 | 60 | 193 | 133 | 1.20 | 1.16 | 0.93 | 0.27 | 0.23 |

17 | 885 | 903 | 1046 | 1921 | 18 | 161 | 143 | 1.33 | 1.30 | 0.72 | 0.61 | 0.58 |

18 | 1057 | 1089 | 1193 | 2421 | 32 | 136 | 104 | 1.18 | 1.15 | 0.81 | 0.36 | 0.33 |

19 | 697 | 817 | 879 | 1485 | 120 | 182 | 62 | 1.24 | 1.19 | 0.74 | 0.50 | 0.45 |

20 | 841 | 873 | 952 | 1638 | 32 | 111 | 79 | 1.10 | 1.08 | 0.89 | 0.21 | 0.19 |

21 | 1033 | 1096 | 1210 | 2289 | 63 | 177 | 114 | 1.17 | 1.11 | 0.83 | 0.35 | 0.29 |

22 | 885 | 929 | 1045 | 1914 | 44 | 160 | 116 | 1.26 | 1.23 | 0.83 | 0.43 | 0.40 |

23 | 946 | 1127 | 1217 | 1713 | 181 | 271 | 90 | 1.22 | 1.13 | 0.93 | 0.28 | 0.20 |

24 | 861 | 885 | 977 | 2189 | 24 | 116 | 92 | 1.20 | 1.19 | 0.96 | 0.24 | 0.23 |

25 | 902 | 978 | 1081 | 1948 | 76 | 179 | 103 | 1.28 | 1.16 | 0.95 | 0.34 | 0.22 |

Time Factors (ms) | Weight Factors | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

B_{s}–T_{p} | B_{s}–L_{p} | B_{s}–Seat off | B_{s}–End | T_{p}–L_{p} | T_{p}–Seat off | L_{p}–Seat off | T_{pw} | L_{pw} | T_{minw} | T_{pw}–T_{minw} | L_{pw}–T_{minw} | |

26 | 497 | 777 | 814 | 3085 | 280 | 317 | 37 | 1.06 | 1.03 | 0.90 | 0.16 | 0.13 |

27 | 1041 | 1124 | 1269 | 2754 | 83 | 228 | 145 | 1.12 | 1.09 | 0.94 | 0.18 | 0.14 |

28 | 972 | 1039 | 1160 | 1439 | 67 | 188 | 121 | 1.22 | 1.16 | 0.65 | 0.58 | 0.51 |

29 | 574 | 803 | 1079 | 1775 | 229 | 505 | 276 | 1.06 | 0.94 | 0.91 | 0.16 | 0.04 |

30 | 1851 | 2017 | 2218 | 3301 | 166 | 367 | 201 | 1.10 | 0.90 | 0.84 | 0.26 | 0.06 |

31 | 1154 | 1290 | 1375 | 3095 | 136 | 221 | 85 | 1.11 | 1.04 | 0.81 | 0.30 | 0.23 |

32 | 577 | 604 | 763 | 1082 | 27 | 186 | 159 | 1.21 | 1.19 | 0.56 | 0.65 | 0.63 |

33 | 1517 | 1602 | 1747 | 2666 | 85 | 230 | 145 | 1.15 | 1.10 | 0.84 | 0.30 | 0.26 |

No. | Grey Relational Grade | No. | Grey Relational Grade | No. | Grey Relational Grade |
---|---|---|---|---|---|

x_{1} | 0.876104 | x_{15} | 0.909550 | x_{29} | 0.899785 |

x_{2} | 0.985806 | x_{16} | 0.795669 | x_{30} | 0.502607 |

x_{3} | 0.941580 | x_{17} | 0.939427 | x_{31} | 0.743228 |

x_{4} | 0.863041 | x_{18} | 0.853622 | x_{32} | 0.890893 |

x_{5} | 0.866535 | x_{19} | 0.964177 | x_{33} | 0.690029 |

x_{6} | 0.989125 | x_{20} | 0.970895 | ||

x_{7} | 0.968267 | x_{21} | 0.866975 | ||

x_{8} | 0.927481 | x_{22} | 0.942596 | ||

x_{9} | 0.911080 | x_{23} | 0.903536 | ||

x_{10} | 0.920046 | x_{24} | 0.925307 | ||

x_{11} | 0.977166 | x_{25} | 0.929919 | ||

x_{12} | 0.925165 | x_{26} | 0.803517 | ||

x_{13} | 0.966067 | x_{27} | 0.808698 | ||

x_{14} | 0.750596 | x_{28} | 0.912563 |

No. | Grey Relational Grade | No. | Grey Relational Grade | No. | Grey Relational Grade |
---|---|---|---|---|---|

x_{1} | 0.8753 | x_{15} | 0.6995 | x_{29} | 0.2879 |

x_{2} | 0.7420 | x_{16} | 0.5694 | x_{30} | 0.3733 |

x_{3} | 0.8676 | x_{17} | 0.8785 | x_{31} | 0.5442 |

x_{4} | 0.6888 | x_{18} | 0.6964 | x_{32} | 0.6681 |

x_{5} | 0.9560 | x_{19} | 0.8309 | x_{33} | 0.5903 |

x_{6} | 0.8778 | x_{20} | 0.4673 | ||

x_{7} | 0.6153 | x_{21} | 0.6455 | ||

x_{8} | 0.6258 | x_{22} | 0.8226 | ||

x_{9} | 0.6493 | x_{23} | 0.5610 | ||

x_{10} | 0.9163 | x_{24} | 0.5637 | ||

x_{11} | 0.8071 | x_{25} | 0.6274 | ||

x_{12} | 0.8537 | x_{26} | 0.3680 | ||

x_{13} | 0.8235 | x_{27} | 0.4199 | ||

x_{14} | 0.6051 | x_{28} | 0.7889 |

Children Without Intellectual Disability | Children with Mild Intellectual Disability | Children with Moderate Intellectual Disability | |
---|---|---|---|

I | 100.0% | 50.0% | 25.0% |

II | 0% | 8.3% | 0% |

III | 0% | 16.7% | 62.5% |

IV | 0% | 25.0% | 12.5% |

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**MDPI and ACS Style**

Chang, C.-S. Applying Grey Relational Analysis in the Evaluation of the Balance of Children with Intellectual Disability. *Axioms* **2021**, *10*, 341.
https://doi.org/10.3390/axioms10040341

**AMA Style**

Chang C-S. Applying Grey Relational Analysis in the Evaluation of the Balance of Children with Intellectual Disability. *Axioms*. 2021; 10(4):341.
https://doi.org/10.3390/axioms10040341

**Chicago/Turabian Style**

Chang, Chih-Sheng. 2021. "Applying Grey Relational Analysis in the Evaluation of the Balance of Children with Intellectual Disability" *Axioms* 10, no. 4: 341.
https://doi.org/10.3390/axioms10040341