# Global and Geographically Weighted Quantile Regression for Modeling the Incident Rate of Children’s Lead Poisoning in Syracuse, NY, USA

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Quantile Regression

**cdf**) $\mathrm{F}\left(\mathrm{y}\right)=\mathrm{Pr}\left(\mathrm{Y}\le \mathrm{y}\right)$, the

**τ**th quantile of Y is defined as the inverse of the

**cdf**at τ, that is the value of Y such that $\mathrm{F}\left(\mathrm{Y}\right)=\mathrm{Pr}\left(\mathrm{Y}\le \mathsf{\xi}\right)=\mathsf{\tau}$, where 0 < τ < 1. Thus, the proportion of the population with the response variable below $\mathsf{\xi}\left(\mathsf{\tau}\right)$ is τ. For example, the 90th quantile of the standard normal distribution is $\mathsf{\xi}\left(\mathsf{\tau}=0.9\right)$ = 1.284 (i.e., the Z-score value). The inverse function $\mathrm{Q}\left(\mathsf{\tau}\right)={\mathrm{F}}^{-1}\left(\mathsf{\tau}\right)=\mathrm{inf}\left(\mathrm{y}:\mathrm{F}\left(\mathrm{Y}\right)\ge \mathsf{\tau}\right)$ is called the quantile function of F(Y). The general τ-th sample quantile $\mathsf{\xi}\left(\mathsf{\tau}\right)$, which is the analogue of $\mathrm{Q}\left(\mathsf{\tau}\right)$, can be obtained by minimizing:

**τ**to positive residuals ${\mathrm{Y}}_{\mathrm{i}}-\mathsf{\xi}\left(\mathsf{\tau}\right)$ and a weight of (1 − τ) to negative residuals [45].

**ε**(τ) is unspecified and is only assumed that

**ε**(τ) satisfies the quantile restriction $\mathrm{Q}\left(\mathsf{\epsilon}\left(\mathsf{\tau}\right)|\mathrm{X}\right)=0$ [38]. The quantile regression coefficients can be obtained by solving for any quantile 0 < τ < 1:

_{1}regression estimator.

#### 2.2. Geographically Weighted Quantile Regression (GWQR)

_{i}, v

_{i}) for each observation i (i = 1, 2,…, n). The local information leads to estimate the localized regression coefficients of the relationship of interest. Geographically weighted quantile regression (GWQR) developed by [44] is expressed as follows:

_{i}is the response variable, X

_{k}is a set of p predictor variables (k = 1, 2, …, p), ε

_{i}is the error term of the conditional τ-th quantile function, and ${\mathsf{\beta}}_{0}\left(\mathsf{\tau}\right)\left({\mathrm{u}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{i}}\right),{\mathsf{\beta}}_{1}\left(\mathsf{\tau}\right)\left({\mathrm{u}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{i}}\right),\dots ,{\mathsf{\beta}}_{\mathrm{p}}\left(\mathsf{\tau}\right)\left({\mathrm{u}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{i}}\right)$ are the local quantile regression coefficients for the τ-th quantile at the i-th location. The model coefficient ${\mathsf{\beta}}_{\mathrm{k}}\left(\mathsf{\tau}\right)\left({\mathrm{u}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{i}}\right)$ measures the change in a specified quantile τ of the response variable corresponding to one unit change in the predictor variable. The comparison between percentiles or quantiles of the response variable may reveal where a specific percentile or quantile is more affected by the set of predictor variables than others.

_{ij}between the location j and center i, and (iii) estimate the model coefficients using linear programming optimization [44]. Thus, for a given regression point (u

_{0}, v

_{0}), the solution of the GWQR coefficients for the τ-th quantile in Equation (3) can be obtained by minimizing the geographically weighted loss function using the data within the kernel window:

_{0}is the spatial weights defined by a kernel function K (d

_{0i}/h) where h is the bandwidth and d

_{0i}is distance between each neighboring location i and the regression point (u

_{0}, v

_{0}). Note: there is no explicit form available for the solution of the model coefficient vector in Equation (4). Instead, it can be equivalently formulated as a linear programming optimization problem [45,46].

_{i}denotes the vector of observed predictor variables at the i-th location $\left({\mathrm{u}}_{\mathrm{i}},{\mathrm{v}}_{\mathrm{i}}\right)$. More details on the GWQR coefficient estimates, standard errors of regression coefficients, kernel function and bandwidth selection, and the assessment of spatial non-stationarity can be viewed in [44].

## 3. Materials and Methods

#### 3.1. Data

_{1}), (2) natural logarithm of town taxable values (in thousand dollars) (X

_{2}), and (3) soil lead concentration (ppm) (X

_{3}). The descriptive statistics of the response variable and three predictor variables were listed in Table 1. The histogram of the response variable is illustrated in Figure 1. The distribution of the response variable was summarized and showed in Table 2. For example, the 50th quantile (median) was 0.04, the 75th quantile 0.18, and the 90th quantile 0.40. Thus, $\mathsf{\xi}\left(\mathsf{\tau}=0.75\right)$ = 0.18 indicated that 75% of these 1393 census blocks had less than 18% children’s BLL ≥ 5 µg/dL among all tested children in the census block, while 25% of these 1393 census blocks (or 348 blocks) had 18% or more children’s BLL ≥ 5 µg/dL among all tested children in the census block. Those (348) census blocks may be considered “high risk” areas of children lead poisoning because about 20% of children’s BLL were over the elevated BLL, which should be identified for lead control treatment.

#### 3.2. Methods

#### 3.2.1. Regression Model

_{i}) of children’s BLL ≥ 5 µg/dL and three environment risk factors at the four quantiles of Y

_{i}(τ = 0.25, 0.50, 0.75, and 0.90), respectively:

#### 3.2.2. Bandwidth Selection for GWQR

#### 3.2.3. Assessment of Spatial Nonstationary

## 4. Results

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The histogram of the observed incident rate of children’s BLL ≥ 5 µg/dL in Syracuse, NY, USA.

**Figure 2.**Geographical map of the observed incident rate of children’s BLL ≥ 5 µg/dL in the city of Syracuse, NY, USA, (the blanks are non-residential areas).

**Figure 3.**Geographical map of (

**a**) the building year of residence; (

**b**) the town taxable value of residence; and (

**c**) the soil lead concentration in the city of Syracuse, NY, USA, (the blanks are non-residential areas).

**Figure 4.**The model coefficient estimates of the global quantile regression models at different quantiles.

**Figure 6.**Geographical map of the significant coefficients (±1.96) of GWQR for Building Year: (

**a**) 25th; (

**b**) 50th; (

**c**) 75th; (

**d**) 90th.

**Figure 7.**Geographical map of the significant coefficients (±1.96) of GWQR for LogTax: (

**a**) 25th; (

**b**) 50th; (

**c**) 75th; (

**d**) 90th.

**Figure 8.**Geographical map of the significant coefficients (±1.96) of GWQR for Soil_Lead: (

**a**) 25th; (

**b**) 50th; (

**c**) 75th; (

**d**) 90th.

Variable | Mean | Std Dev | Minimum | Maximum |
---|---|---|---|---|

Rate of BLL ≥ 5 µg/dL | 0.1375 | 0.2075 | 0.0 | 1.0 |

Building Year | 1923 | 17.6 | 1860 | 1978 |

Town Taxable Values (K$) | 58.445 | 23.815 | 14.000 | 230.106 |

Soil Lead (ppm) | 185.8 | 112.5 | 10.03 | 840.8 |

**Table 2.**Distribution summary of the response variable (the incident rate of children’s BLL ≥ 5 µg/dL).

τ | 0.25 | 0.50 | 0.60 | 0.70 | 0.75 | 0.80 | 0.90 | 0.95 | 0.99 | 1.0 |

ξ(τ) | 0.0 | 0.05 | 0.08 | 0.14 | 0.18 | 0.25 | 0.40 | 0.57 | 1.0 | 1.0 |

Coefficient | Estimate | 95% Confidence Limits | p-Value | STB ^{†} | |
---|---|---|---|---|---|

τ = 0.25 | |||||

Intercept | 0.5707 | −0.0234 | 1.1647 | 0.0597 | |

Building Year | −0.0002 | −0.0005 | 0.0001 | 0.1119 | −0.0201 |

Log Tax Values | −0.0288 | −0.0393 | −0.0183 | <0.0001 | −0.0502 |

Soil Lead | 0.0001 | 0.0001 | 0.0002 | <0.0001 | 0.0688 |

τ = 0.50 | |||||

Intercept | 2.4739 | 1.3337 | 3.6141 | <0.001 | |

Building Year | −0.0011 | −0.0017 | −0.0005 | 0.0002 | −0.0914 |

Log Tax Values | −0.0905 | −0.1120 | −0.0689 | <0.001 | −0.1576 |

Soil Lead | 0.0003 | 0.0002 | 0.0004 | <0.001 | 0.1411 |

τ = 0.75 | |||||

Intercept | 4.9176 | 3.6101 | 6.2251 | <0.0001 | |

Building Year | −0.0021 | −0.0028 | −0.0015 | <0.0001 | −0.1813 |

Log Tax Values | −0.1734 | −0.1954 | −0.1515 | <0.0001 | −0.3012 |

Soil Lead | 0.0005 | 0.0004 | 0.0006 | <0.0001 | 0.2629 |

τ = 0.90 | |||||

Intercept | 7.4102 | 4.5054 | 10.3150 | <0.0001 | |

Building Year | −0.0033 | −0.0047 | −0.0018 | <0.0001 | −0.2764 |

Log Tax Values | −0.2303 | −0.2761 | −0.1844 | <0.0001 | −0.4011 |

Soil Lead | 0.0007 | 0.0003 | 0.0011 | 0.0007 | 0.3715 |

^{†}Note: STB—standardized model coefficients.

**Table 4.**Moran’s I for the observed incident rate of children’s BLL ≥ 5 µg/dL and the residuals of global quantile regression models.

Test | Observed (BLL ≥ 5 µg/dL) | Residuals τ = 0.25 | Residuals τ = 0.50 | Residuals τ = 0.75 | Residuals τ = 0.90 |
---|---|---|---|---|---|

Moran’s Index | 0.0974 | 0.0782 | 0.045 | 0.02 | 0.0276 |

Z-score | 69.10 | 55.56 | 32.20 | 14.61 | 19.91 |

p-value | <0.0001 | <0.0001 | <0.0001 | <0.0001 | <0.0001 |

Coefficient | Mean | Median | Min | Max | IQR | Ste ^{†} | Status |
---|---|---|---|---|---|---|---|

τ = 0.25 | |||||||

Intercept | 0.5409 | 0.5996 | −0.8802 | 1.3253 | 0.7439 | 0.3028 | Nonstationary |

Building Year | −0.00021 | −0.0002 | −0.0007 | 0.0006 | 0.0004 | 0.0001 | Nonstationary |

Log Tax Values | −0.0315 | −0.0279 | −0.0879 | 0.0398 | 0.0453 | 0.0055 | Nonstationary |

Soil Lead | 0.000095 | 0.0001 | −0.0001 | 0.0003 | 0.0000 | 0.0000 | Stationary |

τ = 0.50 | |||||||

Intercept | 2.3455 | 2.4911 | −0.9819 | 5.0047 | 1.7971 | 0.5812 | Nonstationary |

Building Year | −0.001051 | −0.0011 | −0.0024 | 0.0006 | 0.0008 | 0.0003 | Nonstationary |

Log Tax Values | −0.07374 | −0.0615 | −0.2094 | 0.0657 | 0.0959 | 0.0110 | Nonstationary |

Soil Lead | 0.000176 | 0.0002 | −0.0001 | 0.0005 | 0.0001 | 0.0000 | Nonstationary |

τ = 0.75 | |||||||

Intercept | 3.7948 | 4.1557 | −4.4179 | 8.8531 | 3.5624 | 0.6665 | Nonstationary |

Building Year | −0.001703 | −0.0018 | −0.0043 | 0.0022 | 0.0002 | 0.0003 | Nonstationary |

Log Tax Values | −0.1102 | −0.0975 | −0.2993 | 0.0994 | 0.1244 | 0.0112 | Nonstationary |

Soil Lead | 0.000299 | 0.0003 | −0.0003 | 0.0006 | 0.002 | 0.0001 | Nonstationary |

τ = 0.90 | |||||||

Intercept | 4.0228 | 3.6329 | −10.0867 | 12.0888 | 6.2613 | 1.4808 | Nonstationary |

Building Year | −0.001726 | −0.0014 | −0.0059 | 0.03699 | 0.0033 | 0.0008 | Nonstationary |

Log Tax Values | −0.1406 | −0.1413 | −0.3871 | 0.1247 | 0.1589 | 0.0204 | Nonstationary |

Soil Lead | 0.000444 | 0.0004 | −0.0006 | 0.1342 | 0.0003 | 0.0002 | Stationary |

^{†}Note: Standard error (Ste) was estimated from the global quantile regression.

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**MDPI and ACS Style**

Zhen, Z.; Cao, Q.; Shao, L.; Zhang, L.
Global and Geographically Weighted Quantile Regression for Modeling the Incident Rate of Children’s Lead Poisoning in Syracuse, NY, USA. *Int. J. Environ. Res. Public Health* **2018**, *15*, 2300.
https://doi.org/10.3390/ijerph15102300

**AMA Style**

Zhen Z, Cao Q, Shao L, Zhang L.
Global and Geographically Weighted Quantile Regression for Modeling the Incident Rate of Children’s Lead Poisoning in Syracuse, NY, USA. *International Journal of Environmental Research and Public Health*. 2018; 15(10):2300.
https://doi.org/10.3390/ijerph15102300

**Chicago/Turabian Style**

Zhen, Zhen, Qianqian Cao, Liyang Shao, and Lianjun Zhang.
2018. "Global and Geographically Weighted Quantile Regression for Modeling the Incident Rate of Children’s Lead Poisoning in Syracuse, NY, USA" *International Journal of Environmental Research and Public Health* 15, no. 10: 2300.
https://doi.org/10.3390/ijerph15102300