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Parametric estimation of P(X > Y) for normal distributions in the context of probabilistic environmental risk assessment.

Jacobs R, Bekker AA, van der Voet H, Ter Braak CJ - PeerJ (2015)

Bottom Line: We compare maximum likelihood and Bayesian estimators with the non-parametric estimator and study the influence of sample size and risk on the (interval) estimators via simulation.We found that the parametric estimators enable us to estimate and bound the risk for smaller sample sizes and small risks.Also, the Bayesian estimator outperforms the maximum likelihood estimators in terms of coverage and interval lengths and is, therefore, preferred in our motivating case study.

View Article: PubMed Central - HTML - PubMed

Affiliation: Biometris, Wageningen University and Research Centre , Wageningen , The Netherlands.

ABSTRACT
Estimating the risk, P(X > Y), in probabilistic environmental risk assessment of nanoparticles is a problem when confronted by potentially small risks and small sample sizes of the exposure concentration X and/or the effect concentration Y. This is illustrated in the motivating case study of aquatic risk assessment of nano-Ag. A non-parametric estimator based on data alone is not sufficient as it is limited by sample size. In this paper, we investigate the maximum gain possible when making strong parametric assumptions as opposed to making no parametric assumptions at all. We compare maximum likelihood and Bayesian estimators with the non-parametric estimator and study the influence of sample size and risk on the (interval) estimators via simulation. We found that the parametric estimators enable us to estimate and bound the risk for smaller sample sizes and small risks. Also, the Bayesian estimator outperforms the maximum likelihood estimators in terms of coverage and interval lengths and is, therefore, preferred in our motivating case study.

No MeSH data available.


Related in: MedlinePlus

Histograms and normal density curves of exposure (nx = 1,000) and effect (ny = 12) concentration nano-Ag (µg/L).Data taken from Gottschalk, Kost & Nowack (2013).
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fig-1: Histograms and normal density curves of exposure (nx = 1,000) and effect (ny = 12) concentration nano-Ag (µg/L).Data taken from Gottschalk, Kost & Nowack (2013).

Mentions: In the motivating case study of aquatic risk assessment of nano-Ag (Gottschalk, Kost & Nowack, 2013), we are confronted with such a small dataset of effect concentrations. Gottschalk, Kost & Nowack (2013) modeled the exposure of nano-Ag from surface water with a probabilistic material-flow model (Gottschalk, Scholz & Nowack, 2010) to obtain a distribution of exposure concentration values. They collected the effect concentration data from available toxicity studies found in the literature. These effect concentration data consist of toxic endpoints (eq. LC50, EC50, NOEC) for 12 aquatic species. For some of these species there were more than one data point. We averaged these to obtain one value for each species. Histograms and normal density curves of the exposure and effect concentration data are given in Fig. 1.


Parametric estimation of P(X > Y) for normal distributions in the context of probabilistic environmental risk assessment.

Jacobs R, Bekker AA, van der Voet H, Ter Braak CJ - PeerJ (2015)

Histograms and normal density curves of exposure (nx = 1,000) and effect (ny = 12) concentration nano-Ag (µg/L).Data taken from Gottschalk, Kost & Nowack (2013).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4548477&req=5

fig-1: Histograms and normal density curves of exposure (nx = 1,000) and effect (ny = 12) concentration nano-Ag (µg/L).Data taken from Gottschalk, Kost & Nowack (2013).
Mentions: In the motivating case study of aquatic risk assessment of nano-Ag (Gottschalk, Kost & Nowack, 2013), we are confronted with such a small dataset of effect concentrations. Gottschalk, Kost & Nowack (2013) modeled the exposure of nano-Ag from surface water with a probabilistic material-flow model (Gottschalk, Scholz & Nowack, 2010) to obtain a distribution of exposure concentration values. They collected the effect concentration data from available toxicity studies found in the literature. These effect concentration data consist of toxic endpoints (eq. LC50, EC50, NOEC) for 12 aquatic species. For some of these species there were more than one data point. We averaged these to obtain one value for each species. Histograms and normal density curves of the exposure and effect concentration data are given in Fig. 1.

Bottom Line: We compare maximum likelihood and Bayesian estimators with the non-parametric estimator and study the influence of sample size and risk on the (interval) estimators via simulation.We found that the parametric estimators enable us to estimate and bound the risk for smaller sample sizes and small risks.Also, the Bayesian estimator outperforms the maximum likelihood estimators in terms of coverage and interval lengths and is, therefore, preferred in our motivating case study.

View Article: PubMed Central - HTML - PubMed

Affiliation: Biometris, Wageningen University and Research Centre , Wageningen , The Netherlands.

ABSTRACT
Estimating the risk, P(X > Y), in probabilistic environmental risk assessment of nanoparticles is a problem when confronted by potentially small risks and small sample sizes of the exposure concentration X and/or the effect concentration Y. This is illustrated in the motivating case study of aquatic risk assessment of nano-Ag. A non-parametric estimator based on data alone is not sufficient as it is limited by sample size. In this paper, we investigate the maximum gain possible when making strong parametric assumptions as opposed to making no parametric assumptions at all. We compare maximum likelihood and Bayesian estimators with the non-parametric estimator and study the influence of sample size and risk on the (interval) estimators via simulation. We found that the parametric estimators enable us to estimate and bound the risk for smaller sample sizes and small risks. Also, the Bayesian estimator outperforms the maximum likelihood estimators in terms of coverage and interval lengths and is, therefore, preferred in our motivating case study.

No MeSH data available.


Related in: MedlinePlus