Chapter 9 190 The power prior The power prior is one of the most prominent methods for dynamically borrowing information from the external data to aid inference of the current trial. The amount of borrowing - and hence the dynamic aspect - is based on how comparable the external data are to the current data. The more alike they are, the more is borrowed. An excellent review of these methods is provided by Viele and others.2 The power prior is a Bayesian methodology that incorporates the external data into an informative prior to facilitate the analysis of the current study. In this informative prior the external data is downweighted by raising its likelihood to a power parameter δ, where the value of δ (with 0 ≤ δ ≤ 1) controls the amount of borrowing: Equation 3 In the above specification, δ = 1 results in a simple pooling of the two data sources, whereas δ= 0 effectively ignores the external data. As it is unclear how δ should be chosen, Duan together with Ibrahim and Chen have proposed to estimate this in a fully Bayesian way,34,35 in the so-called ’modified power prior’ (MPP). This leads to: Equation 4 Where integral of theta (θ) C(δ) = ∫ L θ|D e δπ θ dθ is a scaling constant to ensure Equation 4 abides by the likelihood principle. Reviews of different power-prior specifications and their characteristics can be found in Van Rosmalen et al. or Ibrahim and Chen.7,40
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