# Bayesian Weighted Sums: A Flexible Approach to Estimate Summed Mixture Effects

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Rationale for Bayesian Weighted Sums

_{i}represents the estimated effect of each exposure, X

_{i}. Researchers can choose from a variety of approaches, including penalized estimators, to improve estimation and identify exposure effects that are more strongly related to the outcome [9]. Alternatively, because the exposures often occur in tandem and an overall effect of the exposures may be of interest, we propose a model in which each exposure is weighted and the effect of the weighted summed exposures is estimated. This model is of the form:

_{1}represents the estimated effect of the weighted sum of exposures X

_{1}–X

_{5}, and w

_{1}–w

_{5}are the estimated weights for each exposure of interest and are constrained to sum to 1. This is a straightforward reparameterization of model (1), but directly addresses a different research question [9]. That is, rather than estimating independent effects (β

_{i}in expression (1)), we estimate a single summed mixture effect ($\theta $

_{1}). The weights represent a percent contribution of each exposure to that mixture effect. This approach is a natural extension of toxic equivalency factors estimated in toxicology [5]; but rather than using a single exposure as a reference, weights are estimated for each exposure in the model.

_{1}…w

_{k}, to 1. Second, the values for the weights must be positive real numbers, thus ensuring that no exposures receive negative weights. We note here that while the weights are constrained to have positive values, the summed effect estimate, $\theta $

_{1}, has no such constraints, and can take any value. The Dirichlet prior is specified so that: ${w}_{1}\dots {w}_{5}~Dirichlet\left({\alpha}_{1},\dots ,{\alpha}_{5}\right),$where ${\alpha}_{k}=1.$This weak prior specification implies that, prior to observing any data, we think it most likely that all weights are equal. To complete the Bayesian specification of the model, we assume $\theta $

_{1}~ N(μ = 0, σ

^{2}= 100), a very weakly informative prior on these values. We note that one could use an informative prior on any of these effects, if prior information were available to inform that distribution. We provide the likelihood function and approximate posterior distribution as an eAppendix in Supplementary Materials.

#### 2.2. Simulations

#### 2.3. Association of Summed PBDEs with ASD and SRS in the EARLI Cohort

## 3. Results

#### 3.1. Simulations

_{1}, is approximately unbiased. In addition, these estimates have relatively low MSE and bias, and expected or near expected 95% confidence interval coverage in all scenarios. As would be expected, MSE and average bias decreases as sample size increases, and as the standard error is reduced. Notably, MSE, average bias, and average standard deviation of θ

_{1}decreased when the five exposures were highly correlated vs. low to moderately correlated. For example, with a sample size of 250, θ

_{1}= 1.0 and standard deviation of the linear model = 0.5, the MSE, average bias, and average standard deviation fell from 0.003 to 0.001, −0.013 to 0.001, and 0.053 to 0.032; this decrease in MSE and average bias for the overall effect was offset by a corresponding increase in these quantities for the weights as the correlation increased. Simulations for a negative summed effect were conducted and had similar performance (data not shown).

#### 3.2. EARLI Results

^{2}> 0.75; only PBDE 153 is not moderately to highly correlated with the remaining four PBDEs (28, 47, 99, and 100).

## 4. Discussion

^{3}for air pollutants, users can decide to maintain this exposure scale.

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Correlations of polybrominated diphenyl ethers (PBDEs) measured in maternal biospecimens in the Early Autism Risk Longitudinal Investigation (EARLI) cohort.

**Table 1.**Simulation results. Weights are specified so that w

_{1}= 0.1, w

_{2}= 0.3, w

_{3}= 0.2, w

_{4}= 0.1, w

_{5}= 0.3.

β = 1.0, Standard Error = 0.5 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

N = 250 | N = 500 | |||||||||||

Correlation | Coefficient | Estimate | Average Standard Error | Average Bias | MSE | 95% CI Coverage | Estimate | Average Standard Error | Average Bias | MSE | 95% CI Coverage | |

Low and moderate | θ | 0.99 | 0.053 | −0.013 | 0.003 | 95% | θ | 1.00 | 0.035 | −0.004 | 0.001 | 95% |

w_{1} | 0.10 | 0.036 | −0.002 | 0.001 | 96% | w_{1} | 0.10 | 0.025 | −0.001 | 0.001 | 95% | |

w_{2} | 0.31 | 0.048 | 0.005 | 0.002 | 95% | w_{2} | 0.30 | 0.031 | 0.001 | 0.001 | 94% | |

w_{3} | 0.20 | 0.045 | −0.005 | 0.002 | 94% | w_{3} | 0.21 | 0.031 | 0.002 | 0.001 | 96% | |

w_{4} | 0.10 | 0.043 | 0.005 | 0.002 | 97% | w_{4} | 0.10 | 0.031 | −0.001 | 0.001 | 96% | |

w_{5} | 0.30 | 0.037 | −0.003 | 0.001 | 96% | w_{5} | 0.30 | 0.025 | −0.002 | 0.001 | 95% | |

High | Θ | 1.00 | 0.032 | 0.001 | 0.001 | 95% | θ | 1.00 | 0.024 | −0.001 | 0.001 | 95% |

w_{1} | 0.11 | 0.068 | 0.024 | 0.003 | 97% | w_{1} | 0.10 | 0.052 | 0.008 | 0.002 | 98% | |

w_{2} | 0.28 | 0.084 | −0.024 | 0.006 | 95% | w_{2} | 0.29 | 0.064 | −0.006 | 0.004 | 96% | |

w_{3} | 0.18 | 0.079 | −0.008 | 0.005 | 97% | w_{3} | 0.19 | 0.061 | −0.013 | 0.003 | 95% | |

w_{4} | 0.11 | 0.071 | 0.028 | 0.003 | 98% | w_{4} | 0.10 | 0.053 | 0.013 | 0.002 | 97% | |

w_{5} | 0.28 | 0.083 | −0.021 | 0.006 | 95% | w_{5} | 0.30 | 0.061 | −0.002 | 0.004 | 95% | |

β = 0.2 Standard Error = 0.1 | ||||||||||||

N = 250 | N = 500 | |||||||||||

Correlations | Coefficient | Estimate | Average Standard Deviation | Average Bias | MSE | 95% CI Coverage | Estimate | Average Standard Deviation | Average Bias | MSE | 95% CI Coverage | |

Low and moderate | θ | 0.20 | 0.011 | −0.003 | 0.000 | 95% | θ | 0.20 | 0.007 | −0.001 | 0.000 | 95% |

w_{1} | 0.10 | 0.036 | −0.002 | 0.001 | 97% | w_{1} | 0.10 | 0.025 | −0.001 | 0.001 | 95% | |

w_{2} | 0.31 | 0.049 | 0.005 | 0.002 | 95% | w_{2} | 0.30 | 0.031 | 0.001 | 0.001 | 94% | |

w_{3} | 0.20 | 0.046 | −0.005 | 0.002 | 94% | w_{3} | 0.21 | 0.031 | 0.002 | 0.001 | 96% | |

w_{4} | 0.10 | 0.043 | 0.005 | 0.002 | 96% | w_{4} | 0.10 | 0.031 | −0.001 | 0.001 | 96% | |

w_{5} | 0.30 | 0.038 | −0.003 | 0.001 | 96% | w_{5} | 0.30 | 0.025 | −0.002 | 0.001 | 95% | |

High | θ | 0.20 | 0.006 | 0.000 | 0.000 | 95% | θ | 0.20 | 0.005 | 0.000 | 0.000 | 95% |

w_{1} | 0.11 | 0.068 | 0.025 | 0.003 | 97% | w_{
1
} | 0.10 | 0.052 | 0.008 | 0.002 | 98% | |

w_{2} | 0.28 | 0.084 | −0.024 | 0.006 | 96% | w_{2} | 0.29 | 0.064 | −0.006 | 0.004 | 96% | |

w_{3} | 0.18 | 0.079 | −0.008 | 0.005 | 97% | w_{3} | 0.19 | 0.061 | −0.013 | 0.003 | 95% | |

w_{4} | 0.11 | 0.071 | 0.028 | 0.003 | 98% | w_{4} | 0.10 | 0.053 | 0.013 | 0.002 | 97% | |

w_{5} | 0.28 | 0.084 | −0.021 | 0.006 | 95% | w_{5} | 0.30 | 0.061 | −0.002 | 0.004 | 94% |

Autism Spectrum Disorder (ASD) (N = 42) | No ASD (N = 124) | |
---|---|---|

Maternal Age | 33 (5.4) | 34 (4.9) |

Gestational age | 38 (2.8) | 39 (2.0) |

Social Responsiveness Scores (SRS) t-score | 62 (14) | 49 (8) |

Site | ||

Drexel | 6 (14%) | 34 (27%) |

Johns Hopkins | 9 (21%) | 32 (26%) |

Kaiser Permanente | 16 (38%) | 33 (27%) |

UC Davis | 11 (26%) | 25 (20%) |

Family income | ||

<$10,000 | 4 (9.5%) | 4 (3.2%) |

$10,000 to <$20,000 | 0 (0.0%) | 3 (2.4%) |

$20,000 to <$30,000 | 3 (7.1%) | 4 (3.2%) |

$30,000 to <$50,000 | 2 (4.8%) | 22 (18%) |

$50,000 to <$75,000 | 8 (19%) | 17 (14%) |

$75,000 to <$100,000 | 8 (19%) | 19 (15%) |

$100,000 to <$200,000 | 11 (26%) | 40 (32%) |

$200,000 or more | 0 (0.0%) | 12 (9.7%) |

missing | 6 (14%) | 3 (2.4%) |

Maternal Race-Ethnicity | ||

White, non-Hispanic | 20 (48%) | 72 (58%) |

White, Hispanic | 4 (9.5%) | 9 (7.3%) |

Black | 6 (14%) | 12 (9.7%) |

Asian | 5 (12%) | 16 (13%) |

Other | 7 (17%) | 15 (12%) |

Male birth sex | 34 (81%) | 59 (48%) |

Scheme | ASD Crude (Top) and Adjusted (Bottom) | ||||||
---|---|---|---|---|---|---|---|

Median | Mean | 95% HPD | Median | Mean | 95% HPD | ||

θ | 0.25 | 0.25 | (0.05, 0.45) | θ | 1.28 | 1.29 | (0.83, 1.99) |

w(PBDE28) | 0.19 | 0.22 | (0.00, 0.56) | w(PBDE28) | 0.21 | 0.25 | (0.00, 0.61) |

w(PBDE47) | 0.21 | 0.24 | (0.00, 0.59) | w(PBDE47) | 0.15 | 0.19 | (0.00, 0.51) |

w(PBDE99) | 0.17 | 0.21 | (0.00, 0.54) | w(PBDE99) | 0.15 | 0.19 | (0.00, 0.50) |

w(PBDE100) | 0.15 | 0.19 | (0.00, 0.52) | w(PBDE100) | 0.17 | 0.21 | (0.00, 0.55) |

w(PBDE153) | 0.10 | 0.13 | (0.00, 0.34) | w(PBDE153) | 0.12 | 0.16 | (0.00, 0.45) |

Median | Mean | 95% HPD | Median | Mean | 95% HPD | ||

θ | 0.15 | 0.15 | (−0.08, 0.38) | θ | 1.41 | 1.41 | (0.82, 2.50) |

w(PBDE28) | 0.19 | 0.23 | (0.00, 0.57) | w(PBDE28) | 0.18 | 0.22 | (0.00, 0.56) |

w(PBDE47) | 0.19 | 0.23 | (0.00, 0.58) | w(PBDE47) | 0.17 | 0.21 | (0.00, 0.55) |

w(PBDE99) | 0.15 | 0.19 | (0.00, 0.50) | w(PBDE99) | 0.14 | 0.18 | (0.00, 0.48) |

w(PBDE100) | 0.15 | 0.19 | (0.00, 0.51) | w(PBDE100) | 0.18 | 0.22 | (0.00, 0.57) |

w(PBDE153) | 0.13 | 0.16 | (0.00, 0.44) | w(PBDE153) | 0.13 | 0.17 | (0.00, 0.46) |

**Table 4.**Independent and summed effects of PBDEs estimated with traditional regression techniques. Summed effects are per 1 unit increase in natural logged exposure values.

Models Include Each Exposure Individually | Single Model Includes All Exposures | |||||
---|---|---|---|---|---|---|

SRS | ASD | ASD | ||||

Mean Difference | 95% CI | Odds Ratio | 95% CI | Odds Ratio | 95% CI | |

PBDE28 | 0.20 | (−0.05, 0.45) | 1.53 | (0.81, 2.96) | 1.09 | (0.25, 4.70) |

PBDE47 | 0.16 | (−0.01, 0.33) | 1.38 | (0.89, 2.17) | 3.19 | (0.53, 24.32) |

PBDE99 | 0.12 | (−0.04, 0.28) | 1.14 | (0.76, 1.71) | 0.25 | (0.06, 0.91) |

PBDE100 | 0.13 | (−0.05, 0.31) | 1.42 | (0.89, 2.28) | 2.20 | (0.61, 8.11) |

PBDE153 | 0.01 | (−0.15, 0.17) | 0.93 | (0.59, 1.44) | 0.75 | (0.38, 1.42) |

summed PBDEs | 0.10 | (−0.10, 0.30) | 1.30 | (0.78, 2.18) | n/a | n/a |

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**MDPI and ACS Style**

Hamra, G.B.; Maclehose, R.F.; Croen, L.; Kauffman, E.M.; Newschaffer, C. Bayesian Weighted Sums: A Flexible Approach to Estimate Summed Mixture Effects. *Int. J. Environ. Res. Public Health* **2021**, *18*, 1373.
https://doi.org/10.3390/ijerph18041373

**AMA Style**

Hamra GB, Maclehose RF, Croen L, Kauffman EM, Newschaffer C. Bayesian Weighted Sums: A Flexible Approach to Estimate Summed Mixture Effects. *International Journal of Environmental Research and Public Health*. 2021; 18(4):1373.
https://doi.org/10.3390/ijerph18041373

**Chicago/Turabian Style**

Hamra, Ghassan B., Richard F. Maclehose, Lisa Croen, Elizabeth M. Kauffman, and Craig Newschaffer. 2021. "Bayesian Weighted Sums: A Flexible Approach to Estimate Summed Mixture Effects" *International Journal of Environmental Research and Public Health* 18, no. 4: 1373.
https://doi.org/10.3390/ijerph18041373