Chapter 9 194 Table 1: Propensity score weighting schemes under different populations of interest. Implementation for dichotomous outcomes Here we illustrate the implementation of our method for data with a Bernoulli-distributed dichotomous outcome measure, with mean and likelihood function (before applying propensity score weights) given by L(θ)y i ) = θy i. (1 = ) 1-y i. Filling in this expression in Equation 5gives the propensity score weighted likelihood function: Equation 7 Combining this propensity score weighted likelihood function with the posterior of the modified power prior in Equation 4 gives the joint posterior of the power parameter δ and the mean δ as Equation 8 With a uniform U(0,1) prior for the mean parameter the integral in the scaling constant can be solved analytically as Equation 9 If we further assume a Beta(αδ,βδ) prior for δ, the joint posterior becomes Equation 10
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