Augmenting treatment arms with external data through propensity-score weighted power-priors with an application in expanded access 195 9♥ and after integrating out θ, the marginal posterior of δ is given by Equation 11 With the assumed prior distributions, the conditional posterior of given δ also has a closed-form expression, namely Equation 12 which greatly simplifies posterior sampling. Algorithm The sampling algorithm for the ProPP can be specified as follows: 1. Obtain the propensity scores as the fitted probabilities from a logistic regression for the allocation between current and external data, based on Equation 1. 2. Based on the population of interest and regulatory and statistical properties, choose a suitable weighting scheme from Table 1 to rescale the probabilities obtained in Step 1. 3. Draw a sample of δ from a uniform U(0,1) distribution and accept the values in that sample with probability given by Equation 11; other values in the sample are removed. 4. Draw a sample of from the conditional distribution in Equation 12, using the accepted values of δ from Step 3. In Step 3, we use a sample of size 10,000, which should suffice because the rejection sampling method used in this step generates a random (independent) sample. This sampling algorithm is easy to program, and the code for the analyses in this paper can be downloaded from the GitHub of the first author.14 Simulation Study Setup We implement a simulation design to investigate the performance of our proposed method. The aim of this simulation study is to evaluate our proposed method and compare it with traditional 14 https://GitHub.com/TobiasPolak
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