### Abstract

Multivariate probability distributions, such as may be used for mixture dose-response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose-response biomarker and genetic information. In this article, a new two-stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises agradient Markov chain Monte Carlo(GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn-in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termedpredictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose-response function (DRF). Results are shown for the five-parameter common-mode and seven-parameter dissimilar-mode models, based on published data for eight benzene-toluene dose pairs. The common mode conditional DRF is obtained with a 21-fold reduction in data requirement versus MCMC. Example common-mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126-PCB 153 mixture. Applicability is analyzed and discussed. Matlab ^{®} computer programs are provided.

Original language | English |
---|---|

Pages (from-to) | 345-359 |

Number of pages | 15 |

Journal | Risk Analysis |

Volume | 32 |

Issue number | 2 |

DOIs | |

State | Published - Feb 1 2012 |

### Fingerprint

### Keywords

- Dose-response
- Gradient
- Markov chain Monte Carlo
- Mixture
- Predictive

### ASJC Scopus subject areas

- Physiology (medical)
- Safety, Risk, Reliability and Quality

### Cite this

**A Gradient Markov Chain Monte Carlo Algorithm for Computing Multivariate Maximum Likelihood Estimates and Posterior Distributions : Mixture Dose-Response Assessment.** / Li, Ruochen; Englehardt, James Douglas; Li, Xiaoguang.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - A Gradient Markov Chain Monte Carlo Algorithm for Computing Multivariate Maximum Likelihood Estimates and Posterior Distributions

T2 - Mixture Dose-Response Assessment

AU - Li, Ruochen

AU - Englehardt, James Douglas

AU - Li, Xiaoguang

PY - 2012/2/1

Y1 - 2012/2/1

N2 - Multivariate probability distributions, such as may be used for mixture dose-response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose-response biomarker and genetic information. In this article, a new two-stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises agradient Markov chain Monte Carlo(GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn-in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termedpredictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose-response function (DRF). Results are shown for the five-parameter common-mode and seven-parameter dissimilar-mode models, based on published data for eight benzene-toluene dose pairs. The common mode conditional DRF is obtained with a 21-fold reduction in data requirement versus MCMC. Example common-mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126-PCB 153 mixture. Applicability is analyzed and discussed. Matlab ® computer programs are provided.

AB - Multivariate probability distributions, such as may be used for mixture dose-response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose-response biomarker and genetic information. In this article, a new two-stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises agradient Markov chain Monte Carlo(GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn-in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termedpredictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose-response function (DRF). Results are shown for the five-parameter common-mode and seven-parameter dissimilar-mode models, based on published data for eight benzene-toluene dose pairs. The common mode conditional DRF is obtained with a 21-fold reduction in data requirement versus MCMC. Example common-mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126-PCB 153 mixture. Applicability is analyzed and discussed. Matlab ® computer programs are provided.

KW - Dose-response

KW - Gradient

KW - Markov chain Monte Carlo

KW - Mixture

KW - Predictive

UR - http://www.scopus.com/inward/record.url?scp=84857116072&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857116072&partnerID=8YFLogxK

U2 - 10.1111/j.1539-6924.2011.01672.x

DO - 10.1111/j.1539-6924.2011.01672.x

M3 - Article

C2 - 21906114

AN - SCOPUS:84857116072

VL - 32

SP - 345

EP - 359

JO - Risk Analysis

JF - Risk Analysis

SN - 0272-4332

IS - 2

ER -