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

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,(1) 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,(2) 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.