# Using Statistical and Probabilistic Methods to Evaluate Health Risk Assessment: A Case Study

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Health Risk Assessment (HRA)

- I = Intake rate (mg/kg·day)
- C = Concentration at exposure point (e.g., mg/L)
- CF = Conversion factor = 10
^{−6}(kg/mg) - SA = Skin surface area available (cm
^{2}/event) - PC = Chemical-specific dermal permeability constant (cm/h)
- ET = Length of exposure time (h/day)
- EF = Exposure frequency (days/year)
- ED = Exposure duration (years)
- Abs = Absorption into bloodstream (decimal fraction)
- BW = Body weight (kg)
- AT = Average time (days)
- HI
_{j}= Hazard index of one contaminant - RfD = Reference dose (mg/kg·day)

#### 2.2. Statistical Methods for Censored Data

**and Z**

_{i}_{i}was developed by using only above-limit data. Z

_{i}can be obtained by (q

_{i}is the ranked original data):

## 3. Case Study

## 4. Data Preparation

#### 4.1. Comparison of Concentrations of Five Heavy Metals in Different Sites

- H
_{0}: the medians of each site are identical; - H
_{1}: at least the median of one site is different from the others.

Site | N | Median | Ave Rank | Z-value |
---|---|---|---|---|

Site 1 | 13 | 0.17 | 40 | −1.91 |

Site 2 | 14 | 0.3803 | 64.8 | 1.1 |

Site 3 | 14 | 0.35 | 63.8 | 0.97 |

Site 4 | 14 | 0.3 | 55.7 | −0.04 |

Site 5 | 14 | 0.3 | 58.7 | 0.33 |

Site 6 | 14 | 0.2 | 46.6 | −1.16 |

Site 7 | 14 | 0.215 | 45.8 | −1.27 |

Site 8 | 14 | 0.845 | 71.5 | 1.92 |

Overall | 111 | 56 | ||

H = 11.04 | DF = 7 | p = 0.137 | ||

H = 11.06 | DF = 7 | p = 0.136 | (adjusted for ties) |

#### 4.2. Uncertainties in Concentration

Logqi1 = −1097 + 0.916 × Zi1 | |||||

S = 0.291865 | R-Sq = 83.9% | R-Sq(adj) = 81.9% | |||

Analysis of variance | |||||

Source | DF | SS | MS | F | p |

Regression | 1 | 3.5642 | 3.5642 | 41.84 | <0.001 |

Error | 8 | 0.68148 | 0.08519 | ||

Total | 9 | 4.24568 |

^{2}in (Table 2) is 81.9%, indicating the good performance of the regression model. The p-value is smaller than 0.001, indicating the significance of the regression model. Therefore, the predicted equation could be used to estimate the data value below the detection limit. After the above procedures, the data below the detection limit were estimated and combined with the above-limit data as treated concentration data. Similarly, U and Zn, which contained censored data, were also treated by this robust method. The values of adj-R

^{2}for U and Zn are 96.3% and 88.1%, respectively, and both p-values for regression models are smaller than 0.001. Due to the significance of the regression models, all the estimated concentrations of U and Zn along with real concentrations were used for further tests.

#### 4.3. Data Generalization

#### 4.4. Other Parameters

^{−6}kg/mg [18,25]. According to a survey, children mainly play in the water at Site 4. Due to the climate, there are just 3 months suitable for playing in water, and children cannot play for a very long time every day. Children would play there for about 15 years before they become adults [19]. Therefore, considered as a practical case, some other parameters can be assumed as follows: the length of exposure time as 2 h/day, the exposure frequency as 50 days/year, the exposure duration as 15 years, and the average time as 365 days/year × 15 years = 5475 days. The rest of the parameters can be viewed as uncertain parameters, and it is better to use probability distributions to evaluate. Table 3 shows some suggested distributions with typical values for important parameters in Equation (1).

Definition | units | Infant | Children | Adult |
---|---|---|---|---|

Age | year | 0–1 | 5–19 | 20–70 |

Body weight (BW) | kg | LN(6.79, 1.27) | LN(36.24, 1.05) | LN(59.78, 1.07) |

Dermal surface exposure (SA) | cm^{2} | LN(719, 1.19) | LN(2196, 1.08) | LN(3067, 1.06) |

#### 4.5. Reference Dose

**Table 4.**Reference dose of exposure for five contaminants [27]. RfD, reference dose.

Contaminants | Arsenic | Manganese | Molybdenum | Uranium | Zinc | |
---|---|---|---|---|---|---|

CAS No. | 7440-38-2 | 7439-96-5 | 7439-98-7 | 7440-61-1 | 7440-66-6 | |

Carcinogenicity | Car. and Non-Car. | Non-car. | Non-car. | Non-car. | Non-car. | |

RfD (mg/kg·day) | 3.0 × 10^{−4} | 1.4 × 10^{−1} | 5.0 × 10^{−3} | 3 × 10^{−3} | 0.3 |

#### 4.6. Monte Carlo Simulation

**Figure 5.**Probability plot of log(hazard index (HI)) for five heavy metals and the corresponding parameters.

## 5. Results and Analysis

_{As}) (7.989) among five heavy metals is greater than 3.59. Therefore, all of the probabilities of HI exceeding one for the five heavy metals are definitely less than 0.0001. It can be concluded that the concentration level of the five heavy metals in Site 4 is quite safe for dermal contact exposure for children. The total hazard index (THI) Equation (10) is shown as follows [28]:

- THI = Total hazard index
- RfD = Reference dose for chemical
- p = Number of pathways
- n = Number of chemicals

_{As}+ HI

_{Mn}+ HI

_{Mo}+ HI

_{U}+ HI

_{Zn}⇒

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wu, H.; Chen, B.
Using Statistical and Probabilistic Methods to Evaluate Health Risk Assessment: A Case Study. *Toxics* **2014**, *2*, 291-306.
https://doi.org/10.3390/toxics2020291

**AMA Style**

Wu H, Chen B.
Using Statistical and Probabilistic Methods to Evaluate Health Risk Assessment: A Case Study. *Toxics*. 2014; 2(2):291-306.
https://doi.org/10.3390/toxics2020291

**Chicago/Turabian Style**

Wu, Hongjing, and Bing Chen.
2014. "Using Statistical and Probabilistic Methods to Evaluate Health Risk Assessment: A Case Study" *Toxics* 2, no. 2: 291-306.
https://doi.org/10.3390/toxics2020291