# Missing Data Calculation Using the Antioxidant Activity in Selected Herbs

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}test was introduced by Karl Pearson [14]. It is used to determine whether the analyzed data is identically distributed across different populations/sub-groups of the same population [15,16]. Two significant modifications to Pearson’s test were introduced by Ronald Aylmer Fisher [17]. First, the degree of freedom was decreased by one unit when applied to a contingency table. The other considered the number of unknown parameters associated with the theoretical distribution. The parameters are estimated from central moments [18]. The Chi-square test has been applied in all research areas, its main uses being goodness-of-fit [19], association/independence [20], homogeneity [21] or classification [22,23].

## 2. Materials and Methods

#### 2.1. Measurement Methods

^{•+}) must be generated by enzymes or chemical reactions [33]. The FRAP method is based on the reduction of a ferroin analog. Total polyphenol content was measured using Folin-Ciocalteu colorimetric method [34]. The results were corrected for dilution and expressed in μM Trolox per 100 g dry weight.

#### 2.2. Primary Data

#### 2.3. Evaluation Methods

^{2}) test was investigated.

^{2}method requires normally distributed data. This method is preferred because it is a natural transformation that also appears in the environment (e.g., pH determination). The nature of the plants, the compounds structure and the antioxidant activity are influencing each other.

- Check that the linearity between antioxidant activity and phenol content is true of the experimental data in the analysis.
- Three alternatives were taken into consideration. The experimental values were introduced in the algorithm in the first step.
- Obtaining the coefficients using linear regression analysis (Equations (1) and (2)); using these to make estimations in the first cycle.
- Fill in the missing places with estimated values.
- Repeat:
- Obtain (new) expected values (Equation (3))
- Calculate χ
^{2}using observed and expected values - Insert in the missing places the (new) expected values
- Until the value of χ
^{2}is not significantly changed (e.g., convergence)

_{i}and y

_{i}; n is the number of points taken into consideration, {(x

_{i}, y

_{i}), i = 1, …, n}.

_{i,j}= mean of the observed value for (i,j) pair of factors; E

_{i,j}= mean of the expected value for (i,j) pair of factors; X

^{2}= the value of Chi-square statistic; m = number of rows; n = number of columns ; 1 ≤ i ≤ m = indices of observations associated to the first factor; 1 ≤ j ≤ n = indices of observations associated to the second factor.

_{i,j}observations are the result of multiplying two factors (repeated observations approximate better the effect of multiplication) [40].

_{i,j}− E

_{i,j})

^{2}) of observation. The measurement is affected by chance errors:

- on a scale with values (X
^{2}, Equation (4)) between absolute and relative errors (step 4); - absolute values (S
^{2}, Equation (5)); - relative values (Cv
^{2}, Equation (5));

^{2}statistic) (Equation (4)) was used:

^{2}) obtained between the model and the observation; and minimization of the squared coefficient of variation (Cv

^{2}):

_{i,j}; 1 ≤ j ≤ c = contribution of the second factor to the expected value E

_{i,j};

^{2}calculations. Different colors indicate changes in data in missing places; these reach their final values in the Cycle n, after X

^{2}is not significantly changed compared to Cycle n − 1.

_{Prs}= the Pearson correlation coefficient; n = number of samples; datasets {y

_{1}… y

_{n}} containing n values and another dataset {$\hat{\mathrm{y}}$

_{1}… $\hat{\mathrm{y}}$

_{n}} containing n values.

_{Spm}= the Spearman rank correlation coefficient; n- number of samples; Rk(Y), Rk($\hat{\mathrm{Y}}$) = the rank of the datasets {y

_{1}… y

_{n}} containing n values and another dataset {$\hat{\mathrm{y}}$

_{1}… $\hat{\mathrm{y}}$

_{n}} containing n values.

_{Sq}= semi-quantitative correlation coefficient.

## 3. Results and Discussion

^{2}were outliers. The experimental data from the plant Acorus calamus was excluded due to its outlier χ

^{2}results (Table 1 see below). The four phenolic acids relationship with the antioxidant capacity in the remained plants were analyzed.

**in bold**), and the expected values (exp.) calculated from the regression. The values are the results of the analysis following the logarithmic transformation of the data.

^{2}as a function of iteration was also investigated after running the algorithm. This showed, as the statistical analysis, that the variables are close to each other. The evolution of X

^{2}as a function of iteration is presented in the next Figure 3.

^{2}fast converged to a minimum after different numbers of cycles. The minimum was reached within a few iterations.

^{2}values led to the stop of the algorithm. 12 iteration in case of ABTS, 10 iterations in case of DPPH, and 11 iterations in case of FRAP values were observed.

^{2}operates on the variables. With the Monte Carlo methods, the table is filled several times, modifying the original variables.

## 4. Conclusions

^{2}statistic is minimized. The results indicate that phenolic compounds influence each other. A linear relationship between all analyzed datasets can be observed. All the three methods (ABTS, DPPH, FRAP) which were used to determine the antioxidant capacity gave comparable results. This is expected based on the experimental data and literature reviews. The estimated values for missing places fit into experimental data. The correlation between the numbers is clearly visible.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The algorithm used to fill the Table S1 (see the Supplementary Material) censored data.

Acorus Calamus | χ^{2} (Outlier) Value | χ^{2} (Average) Value |
---|---|---|

ABTS | 4.6788 | 0.1057 |

DPPH | 4.8017 | 0.1045 |

FRAP | 4.8000 | 0.0989 |

Pearson’s quantitative correlation and significance levels from Student’s t | ABTS | DPPH | FRAP | |

ABTS | - | 0.7746 | 0.7587 | |

DPPH | 4.8812∙10^{−26} | - | 0.6696 | |

FRAP | 1.8376∙10^{−24} | 1.8583∙10^{−17} | - | |

Spearman’s qualitative correlation and significance levels from Student’s t | ABTS | DPPH | FRAP | |

ABTS | - | 0.7743 | 0.7544 | |

DPPH | 3.3332∙10^{−26} | - | 0.6684 | |

FRAP | 3.0492∙10^{−24} | 1.6377∙10^{−17} | - | |

Semi-quantitative correlation and significance levels from Student’s t | ABTS | DPPH | FRAP | |

ABTS | - | 0.7745 | 0.7566 | |

DPPH | 4.0330∙10^{−26} | - | 0.669 | |

FRAP | 2.3699∙10^{−24} | 1.7441∙10^{−17} | - |

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Bálint, D.; Jäntschi, L.
Missing Data Calculation Using the Antioxidant Activity in Selected Herbs. *Symmetry* **2019**, *11*, 779.
https://doi.org/10.3390/sym11060779

**AMA Style**

Bálint D, Jäntschi L.
Missing Data Calculation Using the Antioxidant Activity in Selected Herbs. *Symmetry*. 2019; 11(6):779.
https://doi.org/10.3390/sym11060779

**Chicago/Turabian Style**

Bálint, Donatella, and Lorentz Jäntschi.
2019. "Missing Data Calculation Using the Antioxidant Activity in Selected Herbs" *Symmetry* 11, no. 6: 779.
https://doi.org/10.3390/sym11060779