Model Averaging Using Fractional Polynomials to Estimate a Safe Level of Exposure

Abstract

Quantitative risk assessment involves the determination of a safe level of exposure. Recent techniques use the estimated dose-response curve to estimate such a safe dose level. Although such methods have attractive features, a low-dose extrapolation is highly dependent on the model choice. Fractional polynomials, basically being a set of (generalized) linear models, are a nice extension of classical polynomials, providing the necessary flexibility to estimate the dose-response curve. Typically, one selects the best-fitting model in this set of polynomials and proceeds as if no model selection were carried out. We show that model averaging using a set of fractional polynomials reduces bias and has better precision in estimating a safe level of exposure (say, the benchmark dose), as compared to an estimator from the selected best model. To estimate a lower limit of this benchmark dose, an approximation of the variance of the model-averaged estimator, as proposed by Burnham and Anderson, can be used. However, this is a conservative method, often resulting in unrealistically low safe doses. Therefore, a bootstrap-based method to more accurately estimate the variance of the model averaged parameter is proposed.

Accounting for Model Selection Uncertainty: Model Averaging of Prevalence and Force of Infection Using Fractional Polynomials

“…Also for the force of infection, model averaging using the full set of fractional polynomials shows consistently superior results as compared with the estimation based on one single best fitting model. These results are comparable to the findings reported by Faes et al (2007) in the framework of the estimation of a safe level of exposure with a continuous response. Table 3: Simulation results for 500 runs.…”

“…After ranking the models the second step is scaling them to obtain the relative plausibility of each fitted model by a weight of evidence (w i ) relative to the selected best model. The AIC is an asymptotically unbiased estimator of the relative, expected Kullback-Leibler (K-L) distance (Faes et al 2007). As a result, the weight w i is the weight of evidence in favor of model i for being the K-L best model, given the design and sample size.…”

“…The latter is also called the conditional sampling variance, var(η|model i ). Faes et al (2007) derived the unconditional variance of the model-averaged estimate as follows. When η is estimated from a specific logistic fractional polynomial model with parameters β, then the variance ofη is estimated as…”

“…This approach assumes that the data has been generated from the selected fractional polynomial model, ignoring model selection uncertainty which leads to over-confident inferences (Hoeting et al 1999). Instead of focusing on one single model, one can treat each model in the set of fractional polynomial models as a possible model of interest (Faes et al 2007). …”

“…Faes, Aerts, Geys & Molenberghs (2007) explained that normally it is not clear what models should be used for model averaging and suggested that fractional polynomials are a natural and flexible family of parametric models, lending itself nicely as the set of models to be used in model averaging. They also showed the application of fractional polynomials for the estimation of a safe dose level of exposure in the framework of model averaging.…”