FIGURE 7.4 Bayesian hierarchical framework

To summarize, let 9 be the parameters Rq, Wq, a, and 3 and n be a probability density function. The distribution n describes the variability of the model parameters. The objective of a Bayesian hierarchical model is to generate the distributions of these parameters, based on all available information. 

The hierarchical model can be described in 4 levels. For convenience, the model in Equation (7.3) is expressed in short form as a function of unknown coefficients and the dose concentrations /(0j, C j ). The unknown coefficients 

Intuitively, we assume that individual data points are generated by a normal distribution. A common variance is assumed for model error as in a conventional regression analysis. 

At the second level, coefficients 0jj are modeled as random variables from species-specific distributions  

The species-level coefficients are further modeled as random variables from a common hyperdistribution at the 3rd level  

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A Bayesian hierarchical modeling framework was used to evaluate the effects data for each species and test endpoint (Figure 7.4). Hierarchical models reduce the effect of incomplete data sets, small numbers of tests, inconsistent information on effects among species, and other issues that lend uncertainty to the risk characterization results. 

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