# Error Models for the Kinetic Evaluation of Chemical Degradation Data

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## Abstract

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## Simple Summary

## Abstract

## 1. Introduction

## 2. Theory and Methods

#### 2.1. Constant Variance

#### 2.2. Variance by Variable

#### 2.3. Two-Component Error Model

#### 2.4. Likelihood Functions

#### 2.5. Algorithms for Parameter Estimation

#### 2.6. Result Comparison Criteria

#### 2.7. Example Datasets

## 3. Results

#### 3.1. Algorithms

#### 3.2. Parent Only Data

#### 3.3. Coupled Fits

`k_A1`for metabolite A1 was nearly the same for all three fits, an interesting observation could be made for the rate constant

`k2`for the second phase of the biphasic decline of the parent compound. While in the OLS fit, it had a very low value of 1.2e-5 per day and would not be deemed significantly different from zero, it took on an appreciable value of about 0.007 per day when the variance by variable error model was used. While the p-value was still far from the usual significance trigger of 0.05 using variance by variable, it dropped to 0.04 if the two-component error model was used, and the estimate for

`k2`rose to around 0.008 per day.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Comparison of bi-exponential (DFOP) kinetic fits assuming constant variance (top panels) and the two-component error model (bottom panels) for Soil Dataset 8. The fitted error models are shown in the plots of the squared residuals to the right as dashed lines.

**Table 1.**Error model parameters obtained for simple first order (SFO) fits to the FOCUS datasets (see text).

Dataset | OLS | Two-Component | |||||
---|---|---|---|---|---|---|---|

$\mathit{\sigma}$ | ${\mathit{\chi}}^{2}$ (%) | AIC | ${\mathit{\sigma}}_{\mathbf{low}}$ | ${\mathbf{rsd}}_{\mathbf{high}}$ | ${\mathit{\chi}}^{2}$ (%) | AIC | |

FOCUS A | 5.27 | 8.39 | 55.3 | 0.00 | 0.09 | 13.77 | 37.5 |

FOCUS B | 1.96 | 4.46 | 39.5 | 0.32 | 0.04 | 4.89 | 31.0 |

FOCUS C | 4.67 | 15.85 | 59.3 | 4.67 | 0.00 | 15.85 | 61.3 |

FOCUS D | 3.40 | 6.45 | 101.1 | 0.00 | 0.07 | 6.59 | 46.8 |

**Table 2.**Overview of AIC values (lowest in bold) obtained for simple first order (SFO) and dual first order in parallel (DFOP) (bi-exponential decline) fits to simple soil datasets.

Dataset | SFO | DFOP | ||
---|---|---|---|---|

OLS | Two-Component | OLS | Two-Component | |

Soil 1 | 131.2 | 133.2 | 107.9 | 109.9 |

Soil 2 | 137.9 | 138.1 | 80.3 | 82.3 |

Soil 3 | 78.8 | 80.8 | 67.0 | 65.9 |

Soil 4 | 82.0 | 84.0 | 75.0 | 77.0 |

Soil 5 | 172.4 | 174.4 | 176.0 | 178.0 |

Soil 6 | 128.3 | 122.9 | 79.0 | 81.0 |

Soil 7 | 127.3 | 129.3 | 89.8 | 88.7 |

Soil 8 | 93.1 | 93.2 | 64.8 | 55.7 |

Soil 9 | 137.4 | 134.9 | 109.0 | 111.0 |

Soil 10 | 111.3 | 110.7 | 76.6 | 75.3 |

Soil 11 | 125.5 | 127.5 | 129.5 | 131.5 |

Soil 12 | 82.0 | 82.0 | 65.9 | 66.9 |

**Table 3.**Overview of AIC values (lowest in bold) obtained for the coupled fits to soil datasets with metabolites.

Dataset | OLS | Variance by Variable | Two-Component |
---|---|---|---|

Soil 1 | 252.0 | 242.9 | 231.5 |

Soil 2 | 207.5 | 210.9 | 209.5 |

Soil 3 | 110.2 | 108.0 | 104.0 |

Soil 4 | 125.7 | 124.4 | 125.4 |

Soil 5 | 266.9 | 201.3 | 210.2 |

Soil 6 | 138.6 | 139.8 | 139.0 |

Soil 7 | 156.7 | 155.6 | 150.5 |

Soil 8 | 106.8 | 105.5 | 99.6 |

Soil 9 | 181.5 | 165.8 | 178.7 |

Soil 10 | 130.5 | 129.0 | 123.4 |

Soil 11 | 268.4 | 216.7 | 238.5 |

Parameter | OLS | Variance by Variable | Two-Component | |||
---|---|---|---|---|---|---|

Estimate | Pr(>t) | Estimate | Pr(>t) | Estimate | Pr(>t) | |

parent_0 | 100 | 5.3 × 10${}^{-37}$ | 100 | 2.9 × 10${}^{-33}$ | 101 | 5.04 × 10${}^{-30}$ |

k1 | 0.0218 | 8.65 × 10${}^{-8}$ | 0.0261 | 0.00358 | 0.0294 | 0.00115 |

k2 | 1.22 × 10${}^{-5}$ | 0.499 | 0.00707 | 0.199 | 0.00826 | 0.0407 |

g | 0.901 | 2.8 × 10${}^{-7}$ | 0.697 | 0.0477 | 0.615 | 0.0162 |

f_parent_to_A1 | 0.299 | 3.16 × 10${}^{-10}$ | 0.298 | 2.45 × 10${}^{-12}$ | 0.288 | 2.2 × 10${}^{-12}$ |

k_A1 | 0.0204 | 2.98 × 10${}^{-7}$ | 0.0204 | 2.44e × 10${}^{-9}$ | 0.0195 | 1.01 × 10${}^{-9}$ |

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**MDPI and ACS Style**

Ranke, J.; Meinecke, S.
Error Models for the Kinetic Evaluation of Chemical Degradation Data. *Environments* **2019**, *6*, 124.
https://doi.org/10.3390/environments6120124

**AMA Style**

Ranke J, Meinecke S.
Error Models for the Kinetic Evaluation of Chemical Degradation Data. *Environments*. 2019; 6(12):124.
https://doi.org/10.3390/environments6120124

**Chicago/Turabian Style**

Ranke, Johannes, and Stefan Meinecke.
2019. "Error Models for the Kinetic Evaluation of Chemical Degradation Data" *Environments* 6, no. 12: 124.
https://doi.org/10.3390/environments6120124