Augmenting treatment arms with external data through propensity-score weighted power-priors with an application in expanded access 191 9♥ METHODS Integrating propensity scores and power prior Recently, various researchers have proposed ‘propensity-score integrated hybrid approaches’, which combine propensity score methodology with dynamic borrowing methods. Methods have been developed that focus on combining propensity score stratification with power priors,21,22 or meta-analytic predictive priors.14 Other methods focus on the inclusion of propensity score matching in dynamic borrowing.23 Finally, a recent review of several of these methods has proposed both propensity score-weighting together with fixed and commensurate priors.25 All these methods focus primarily on augmented control designs, designs in which the control arm of a trial is combined with external data on (historical) control arms. The main rationale for all these methods is the dual safeguard mechanism within the two-stage analysis: observed confounding is addressed by using propensity score methods in the first stage, and remaining unobserved confounding is attenuated via dynamic borrowing methodology in the second stage. We add to this literature by proposing a novel method based on propensity score weighting and the modified power-prior to augment the current treatment arm with external treatment data.The basis of our method, which we refer to as the ProPP, is the modified power prior, which is designed to only address imbalances due to unmeasured confounding (see Equation 4). To also safeguard against the effects of measured imbalances in patient characteristics, we apply propensity score weighting to this likelihood function before it is used in the MPP. The propensity-score weighted likelihood function is given by Equation 5 where w i, the weight used for patient i, is chosen as a function of the propensity score λ i. If we now substitute this likelihood for the external data in Equation 5, we obtain: Equation 6
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