Augmenting treatment arms with external data through propensity-score weighted power-priors with an application in expanded access 193 9♥ Some caution is required when considering which variables to include in the propensity score model.Variables that are a cause of the outcome but unrelated to Z (C2, Figure 2), are not necessary to include but they may help increase precision. Variables related to Z but not directly related to Y (C3, Figure 2) should not be included and may in fact amplify any bias due to uncontrolled confounding between Z and Y. Figure 2: Directed acyclic graph to explore the causal implication of combining propensity scores and dynamic borrowing methods The choice of weights In dynamic borrowing, the usual goal is to improve the estimate of the outcomes from the trial. We should choose w i according to a weighting scheme that corresponds with our estimand of interest, i.e., the average causal effect among those in the trial should be our target estimand (see Table 1). Therefore, we use a weighting scheme based that targets that estimand. However, we slightly adapt the above weighting scheme to make sure that no subject in the external data obtains a weight larger than 1. Weights larger than 1 would be undesirable for two reasons. First, this would amount to an inflation of the sample size in a Bayesian analysis, which in turn would lead to an overestimation in the precision of the estimates. Second, weighing a non-trial participant higher than current trial participants may cause regulatory concerns. Therefore, we propose to maximize the weight of the patients in the external data set at 1 (meaning that this patient is equally likely to have come from the trial). We set the weight wi of all trial patients to 1, and of all external patients to:
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