# Imputation of Below Detection Limit Missing Data in Chemical Mixture Analysis with Bayesian Group Index Regression

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## Abstract

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## 1. Introduction

## 2. Materials & Methods

#### 2.1. Bayesian Grouped Index Regression

#### 2.2. Imputation Methods

#### 2.2.1. Multiple Imputation by Chained Equations (MICE)

#### 2.2.2. Prior Imputation

#### 2.2.3. Pseudo-Gibbs Imputation

#### 2.2.4. Sequential Full Bayes Imputation (SFB)

#### 2.3. Simulation Study Design

#### 2.4. Data Analysis

## 3. Results

#### 3.1. Simulation Study

#### 3.2. Application of Pseudo-Gibbs imputation to house dust chemicals in the CCLS

## 4. Discussion and Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Estimated odds ratio (OR) and power values for Bayesian group index regression using four different imputation methods.

Parameter | Prior Imputation | Sequential Full Bayes | Pseudo-Gibbs | MICE | ||||
---|---|---|---|---|---|---|---|---|

10% BDL | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power |

exp(β_{1}) = 1.00 | 1 | 0.07 | 0.999 | 0.06 | 0.999 | 0.05 | 1 | 0.06 |

exp(β_{2}) = 0.80 | 0.818 | 0.43 | 0.818 | 0.43 | 0.818 | 0.43 | 0.818 | 0.43 |

exp(β_{3}) = 1.25 | 1.251 | 0.43 | 1.251 | 0.42 | 1.251 | 0.44 | 1.251 | 0.43 |

exp(β_{1}) = 1.00 | 0.994 | 0.05 | 0.9934 | 0.04 | 0.993 | 0.04 | 0.994 | 0.05 |

exp(β_{2}) = 0.67 | 0.658 | 0.9 | 0.658 | 0.9 | 0.658 | 0.9 | 0.658 | 0.9 |

exp(β_{3}) = 1.50 | 1.553 | 0.91 | 1.553 | 0.92 | 1.553 | 0.92 | 1.554 | 0.92 |

30% BDL | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power |

exp(β_{1}) = 1.00 | 1.004 | 0.08 | 1.001 | 0.08 | 1.001 | 0.08 | 1 | 0.06 |

exp(β_{2}) = 0.80 | 0.816 | 0.43 | 0.814 | 0.43 | 0.814 | 0.43 | 0.819 | 0.41 |

exp(β_{3}) = 1.25 | 1.246 | 0.4 | 1.254 | 0.43 | 1.253 | 0.43 | 1.247 | 0.42 |

exp(β_{1}) = 1.00 | 0.996 | 0.05 | 0.999 | 0.07 | 0.996 | 0.05 | 0.994 | 0.05 |

exp(β_{2}) = 0.67 | 0.662 | 0.92 | 0.655 | 0.92 | 0.655 | 0.93 | 0.664 | 0.93 |

exp(β_{3}) = 1.50 | 1.539 | 0.9 | 1.552 | 0.89 | 1.556 | 0.9 | 1.535 | 0.89 |

50% BDL | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power |

exp(β_{1}) = 1.00 | 1.002 | 0.05 | 1.004 | 0.07 | 1.003 | 0.07 | 1.002 | 0.07 |

exp(β_{2}) = 0.80 | 0.824 | 0.37 | 0.828 | 0.34 | 0.812 | 0.4 | 0.823 | 0.38 |

exp(β_{3}) = 1.25 | 1.241 | 0.39 | 1.236 | 0.35 | 1.253 | 0.37 | 1.234 | 0.34 |

exp(β_{1}) = 1.00 | 0.995 | 0.04 | 0.995 | 0.03 | 0.994 | 0.05 | 0.991 | 0.06 |

exp(β_{2}) = 0.67 | 0.667 | 0.88 | 0.664 | 0.88 | 0.651 | 0.89 | 0.681 | 0.88 |

exp(β_{3}) = 1.50 | 1.521 | 0.87 | 1.551 | 0.88 | 1.557 | 0.87 | 1.498 | 0.86 |

70% BDL | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power | Estimated OR | Power |

exp(β_{1}) = 1.00 | 0.997 | 0.06 | 0.992 | 0.01 | 0.997 | 0.06 | 0.994 | 0.03 |

exp(β_{2}) = 0.80 | 0.857 | 0.2 | 0.843 | 0.2 | 0.81 | 0.29 | 0.857 | 0.18 |

exp(β_{3}) = 1.25 | 1.209 | 0.26 | 1.25 | 0.28 | 1.256 | 0.26 | 1.184 | 0.22 |

exp(β_{1}) = 1.00 | 0.993 | 0.02 | 0.979 | 0.04 | 0.987 | 0.05 | 0.984 | 0.01 |

exp(β_{2}) = 0.67 | 0.724 | 0.68 | 0.693 | 0.66 | 0.655 | 0.81 | 0.753 | 0.6 |

exp(β_{3}) = 1.50 | 1.425 | 0.69 | 1.53 | 0.74 | 1.542 | 0.75 | 1.356 | 0.59 |

**Table 2.**MSE and bias of index effects from Bayesian group index regression using different imputation methods.

Parameter | Prior Imputation | Sequential Full Bayes | Pseudo-Gibbs | MICE | ||||
---|---|---|---|---|---|---|---|---|

10% BDL | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias |

exp(β_{1}) = 1.00 | 0.012 | −0.006 | 0.012 | −0.007 | 0.011 | −0.007 | 0.012 | −0.006 |

exp(β_{2}) = 0.80 | 0.017 | 0.014 | 0.017 | 0.014 | 0.017 | 0.014 | 0.017 | 0.014 |

exp(β_{3}) = 1.25 | 0.014 | −0.007 | 0.014 | −0.007 | 0.014 | −0.006 | 0.014 | −0.006 |

exp(β_{1}) = 1.00 | 0.012 | −0.012 | 0.012 | −0.012 | 0.012 | −0.013 | 0.012 | −0.012 |

exp(β_{2}) = 0.67 | 0.015 | −0.026 | 0.015 | −0.025 | 0.015 | −0.025 | 0.015 | −0.026 |

exp(β_{3}) = 1.50 | 0.017 | 0.027 | 0.017 | 0.027 | 0.016 | 0.027 | 0.017 | 0.028 |

30% BDL | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias |

exp(β_{1}) = 1.00 | 0.012 | −0.002 | 0.013 | −0.005 | 0.013 | −0.005 | 0.012 | −0.006 |

exp(β_{2}) = 0.80 | 0.017 | 0.012 | 0.018 | 0.009 | 0.017 | 0.008 | 0.016 | 0.015 |

exp(β_{3}) = 1.25 | 0.014 | −0.010 | 0.015 | −0.004 | 0.014 | −0.005 | 0.014 | −0.009 |

exp(β_{1}) = 1.00 | 0.012 | −0.010 | 0.013 | −0.008 | 0.012 | −0.010 | 0.012 | −0.012 |

exp(β_{2}) = 0.67 | 0.014 | −0.019 | 0.015 | -0.03 | 0.015 | −0.031 | 0.013 | −0.015 |

exp(β_{3}) = 1.50 | 0.017 | 0.018 | 0.018 | 0.025 | 0.018 | 0.028 | 0.016 | 0.015 |

50% BDL | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias |

exp(β_{1}) = 1.00 | 0.014 | −0.005 | 0.015 | −0.003 | 0.015 | −0.004 | 0.013 | −0.004 |

exp(β_{2}) = 0.80 | 0.018 | 0.021 | 0.021 | 0.024 | 0.02 | 0.006 | 0.017 | 0.021 |

exp(β_{3}) = 1.25 | 0.014 | −0.014 | 0.015 | −0.019 | 0.015 | −0.005 | 0.013 | −0.020 |

exp(β_{1}) = 1.00 | 0.013 | −0.012 | 0.013 | −0.012 | 0.014 | −0.013 | 0.012 | −0.015 |

exp(β_{2}) = 0.67 | 0.015 | −0.011 | 0.015 | −0.017 | 0.017 | −0.036 | 0.013 | 0.009 |

exp(β_{3}) = 1.50 | 0.018 | 0.005 | 0.021 | 0.024 | 0.02 | 0.028 | 0.017 | −0.010 |

70% BDL | MSE | Bias | MSE | Bias | MSE | Bias | MSE | Bias |

exp(β_{1}) = 1.00 | 0.02 | −0.013 | 0.019 | −0.018 | 0.022 | −0.014 | 0.012 | −0.012 |

exp(β_{2}) = 0.80 | 0.024 | 0.058 | 0.024 | 0.041 | 0.026 | 0 | 0.017 | 0.062 |

exp(β_{3}) = 1.25 | 0.018 | −0.042 | 0.021 | −0.011 | 0.019 | −0.005 | 0.016 | −0.060 |

exp(β_{1}) = 1.00 | 0.016 | −0.015 | 0.025 | −0.032 | 0.022 | −0.024 | 0.014 | −0.023 |

exp(β_{2}) = 0.67 | 0.024 | 0.069 | 0.023 | 0.023 | 0.024 | −0.034 | 0.023 | 0.112 |

exp(β_{3}) = 1.50 | 0.025 | −0.062 | 0.031 | 0.005 | 0.028 | 0.014 | 0.028 | −0.109 |

**Table 3.**Sensitivity and specificity for Bayesian group index regression using different imputation methods.

Parameter | Prior Imputation | Sequential Full Bayes | Pseudo-Gibbs | MICE | ||||
---|---|---|---|---|---|---|---|---|

10% BDL | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity |

exp(β_{1}) = 1.00 | 0.34 | 0.573 | 0.33 | 0.58 | 0.31 | 0.575 | 0.31 | 0.568 |

exp(β_{2}) = 0.80 | 0.91 | 0.797 | 0.89 | 0.803 | 0.9 | 0.8 | 0.9 | 0.8 |

exp(β_{3}) = 1.25 | 0.82 | 0.738 | 0.85 | 0.753 | 0.82 | 0.733 | 0.84 | 0.748 |

exp(β_{1}) = 1.00 | 0.39 | 0.615 | 0.38 | 0.6 | 0.42 | 0.623 | 0.41 | 0.615 |

exp(β_{2}) = 0.67 | 0.98 | 0.943 | 0.98 | 0.94 | 0.98 | 0.94 | 0.98 | 0.94 |

exp(β_{3}) = 1.50 | 0.99 | 0.918 | 1 | 0.918 | 1 | 0.918 | 0.99 | 0.92 |

30% BDL | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity |

exp(β_{1}) = 1.00 | 0.28 | 0.573 | 0.32 | 0.56 | 0.32 | 0.55 | 0.29 | 0.568 |

exp(β_{2}) = 0.80 | 0.87 | 0.797 | 0.89 | 0.8 | 0.9 | 0.8 | 0.89 | 0.793 |

exp(β_{3}) = 1.25 | 0.82 | 0.705 | 0.86 | 0.723 | 0.84 | 0.713 | 0.85 | 0.703 |

exp(β_{1}) = 1.00 | 0.38 | 0.58 | 0.36 | 0.593 | 0.36 | 0.6 | 0.4 | 0.613 |

exp(β_{2}) = 0.67 | 0.98 | 0.92 | 0.97 | 0.92 | 0.98 | 0.927 | 0.98 | 0.92 |

exp(β_{3}) = 1.50 | 0.99 | 0.893 | 0.99 | 0.903 | 0.99 | 0.9 | 0.99 | 0.903 |

50% BDL | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity |

exp(β_{1}) = 1.00 | 0.38 | 0.593 | 0.33 | 0.593 | 0.35 | 0.585 | 0.37 | 0.603 |

exp(β_{2}) = 0.80 | 0.85 | 0.76 | 0.81 | 0.787 | 0.83 | 0.8 | 0.81 | 0.783 |

exp(β_{3}) = 1.25 | 0.83 | 0.705 | 0.86 | 0.7 | 0.83 | 0.715 | 0.81 | 0.703 |

exp(β_{1}) = 1.00 | 0.38 | 0.578 | 0.41 | 0.605 | 0.4 | 0.598 | 0.41 | 0.603 |

exp(β_{2}) = 0.67 | 0.96 | 0.89 | 0.98 | 0.903 | 0.98 | 0.903 | 0.98 | 0.89 |

exp(β_{3}) = 1.50 | 0.98 | 0.87 | 0.98 | 0.875 | 0.99 | 0.885 | 0.99 | 0.873 |

70% BDL | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity | Sensitivity | Specificity |

exp(β_{1}) = 1.00 | 0.32 | 0.605 | 0.41 | 0.62 | 0.37 | 0.595 | 0.37 | 0.573 |

exp(β_{2}) = 0.80 | 0.64 | 0.67 | 0.72 | 0.69 | 0.75 | 0.693 | 0.71 | 0.673 |

exp(β_{3}) = 1.25 | 0.63 | 0.675 | 0.68 | 0.675 | 0.74 | 0.67 | 0.62 | 0.66 |

exp(β_{1}) = 1.00 | 0.39 | 0.625 | 0.41 | 0.62 | 0.38 | 0.585 | 0.4 | 0.58 |

exp(β_{2}) = 0.67 | 0.88 | 0.767 | 0.87 | 0.817 | 0.95 | 0.79 | 0.92 | 0.737 |

exp(β_{3}) = 1.50 | 0.89 | 0.775 | 0.88 | 0.778 | 0.89 | 0.8 | 0.87 | 0.743 |

**Table 4.**Model fit statistics and computation time for Bayesian group index regression using different imputation methods.

Scenario 1 | Prior Imputation | Sequential Full Bayes | Pseudo-Gibbs | MICE |
---|---|---|---|---|

10% BDL | ||||

DIC | 585.04 | 585.51 | 585.53 | 585.64 |

pD | 5.04 | 5.03 | 5.25 | 5.21 |

Runtime (min) | 7.32 | 679.71 | 538.2 | 7.78 |

30% BDL | ||||

DIC | 585.49 | 585.58 | 585.77 | 585.52 |

pD | 5.39 | 5.68 | 5.46 | 5.1 |

Runtime (min) | 7.31 | 1567.51 | 1333.01 | 7.93 |

50% BDL | ||||

DIC | 585.58 | 585.56 | 585.77 | 586.32 |

pD | 5.15 | 5.83 | 6.21 | 5.52 |

Runtime (min) | 7.03 | 2375.42 | 2108.65 | 8.31 |

70% BDL | ||||

DIC | 587.56 | 586.25 | 586.57 | 588.56 |

pD | 5.05 | 8.69 | 9.3 | 5.59 |

Runtime (min) | 6.33 | 3557.38 | 2686.91 | 9.67 |

Scenario 2 | Prior Imputation | Sequential Full Bayes | Pseudo-Gibbs | MICE |

10% BDL | ||||

DIC | 577.71 | 577.66 | 577.33 | 577.57 |

pD | 5.98 | 6.05 | 5.7 | 5.79 |

Runtime (min) | 7.19 | 683.38 | 565.97 | 7.89 |

30% BDL | ||||

DIC | 578.83 | 578.36 | 579.46 | 578.89 |

pD | 6.07 | 7.08 | 7.26 | 5.86 |

Runtime (min) | 7.22 | 1573.61 | 1304.99 | 7.97 |

50% BDL | ||||

DIC | 581.55 | 580.27 | 579.18 | 582.49 |

pD | 6.53 | 8.01 | 8.06 | 6.35 |

Runtime (min) | 6.9 | 2407.21 | 2067.7 | 8.16 |

70% BDL | ||||

DIC | 589.33 | 586.2 | 586.11 | 591.42 |

pD | 5.53 | 13.42 | 15.91 | 6.4 |

Runtime (min) | 6.29 | 3487.45 | 2711.79 | 8.51 |

**Table 5.**Odds ratio estimates for chemical groups and demographic covariates from the Bayesian group index model (n = 583).

Variable | Odds Ratio | 2.5% CI | 97.5% CI |
---|---|---|---|

PCBs | 1.19 | 0.96 | 1.51 |

Insecticides | 0.64 | 0.39 | 1.00 |

Herbicides | 1.17 | 0.82 | 1.69 |

Metals | 0.79 | 0.59 | 1.06 |

PAHs | 1.27 | 1.01 | 1.60 |

Tobacco | 0.82 | 0.66 | 1.01 |

PBDEs | 1.21 | 0.79 | 1.83 |

Child’s age | 1.01 | 0.92 | 1.12 |

Female | 0.98 | 0.70 | 1.37 |

Child’s ethnicity | |||

Hispanic | 1.25 | 0.81 | 2.00 |

Non-Hispanic | 1.42 | 0.91 | 2.27 |

Household Income | |||

$15,000–$29,999 | 1.02 | 0.47 | 2.15 |

$30,000–$44,999 | 0.79 | 0.36 | 1.61 |

$45,000–$59,999 | 0.78 | 0.34 | 1.66 |

$60,000–$74,999 | 0.45 | 0.18 | 1.06 |

$75,000 or more | 0.38 | 0.17 | 0.79 |

Income missing | 0.56 | 0.17 | 1.61 |

Mother’s education | |||

High school | 1.25 | 0.63 | 2.81 |

Some college | 1.22 | 0.60 | 2.84 |

Bachelor’s or higher | 1.21 | 0.57 | 2.89 |

Mother’s age | 1.01 | 0.98 | 1.05 |

Residence since birth | 0.66 | 0.44 | 0.96 |

**Table 6.**Odds ratio estimates for chemical groups and demographic covariates from the Bayesian group index model for subjects in highest income bracket (n = 266).

Variable | Odds Ratio | 2.5% CI | 97.5% CI |
---|---|---|---|

PCBs | 1.55 | 1.04 | 2.36 |

Insecticides | 0.51 | 0.19 | 1.12 |

Herbicides | 2.02 | 1.00 | 3.99 |

Metals | 0.42 | 0.25 | 0.69 |

PAHs | 1.19 | 0.83 | 1.75 |

Tobacco | 0.77 | 0.52 | 1.09 |

PBDEs | 1.12 | 0.63 | 2.23 |

Child’s age | 0.98 | 0.83 | 1.15 |

Female | 0.70 | 0.38 | 1.22 |

Child’s ethnicity | |||

Hispanic | 1.14 | 0.47 | 2.83 |

Non-Hispanic | 1.62 | 0.87 | 3.18 |

Mother’s education | |||

High school | 0.49 | 0.00 | 1930.56 |

Some college | 0.20 | 0.00 | 730.17 |

Bachelor’s or higher | 0.36 | 0.00 | 1375.01 |

Mother’s age | 0.99 | 0.93 | 1.05 |

Residence since birth | 0.40 | 0.21 | 0.76 |

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## Share and Cite

**MDPI and ACS Style**

Carli, M.; Ward, M.H.; Metayer, C.; Wheeler, D.C.
Imputation of Below Detection Limit Missing Data in Chemical Mixture Analysis with Bayesian Group Index Regression. *Int. J. Environ. Res. Public Health* **2022**, *19*, 1369.
https://doi.org/10.3390/ijerph19031369

**AMA Style**

Carli M, Ward MH, Metayer C, Wheeler DC.
Imputation of Below Detection Limit Missing Data in Chemical Mixture Analysis with Bayesian Group Index Regression. *International Journal of Environmental Research and Public Health*. 2022; 19(3):1369.
https://doi.org/10.3390/ijerph19031369

**Chicago/Turabian Style**

Carli, Matthew, Mary H. Ward, Catherine Metayer, and David C. Wheeler.
2022. "Imputation of Below Detection Limit Missing Data in Chemical Mixture Analysis with Bayesian Group Index Regression" *International Journal of Environmental Research and Public Health* 19, no. 3: 1369.
https://doi.org/10.3390/ijerph19031369