Chapter 9 192 Because our method combines propensity scores with dynamic borrowing based on the MPP, the effective weight for patient i is δ×w i. In our approach neither δ nor w i are allowed to take values greater than 1, so that the proposed method is always more conservative (i.e., provides additional protections against prior-data conflict) than the modified power prior. We additionally assess the causal and practical implications of the choice of weighting schemes. Causal interpretations The weights w i can be chosen in a variety of ways. In applications of propensity score weighting for the estimation of treatment effects in observational studies, the weights are typically allowed to vary between patients within each group and depend on the estimand of interest. Before we choose w i, we here want to provide an explicit causal interpretation of different modeling choices, or different choices of weights w i (see Figure 2). The value of Y can be different in Z = 1 and Z = 0 either due to random error or due to confounding (or selection bias) between Z and Y. An example of such a confounder would be any cause of the outcome that is not equally distributed in the internal and external data (e.g., C1 in Figure 2). Dynamic borrowing methods based on differences in the outcome aim to balance the risk of pooling data with systematic differences (i.e., due to confounding) with the benefit of pooling data with differences only due to random error which increases precision. In this way, dynamic borrowing can never eliminate bias due to a variable such as C1 but it can attenuate the bias by reducing the degree of pooling. Ideally, differences in Y due to variables such as C1 would be removed before dynamic borrowing determines the degree of pooling. Doing so would improve the bias and precision of our estimate. First, it would remove the bias due to C1. Second, if C1 increases the differences in Y across levels of Z, removing the effect of C1 would reduce this difference and would therefore increase the degree of pooling while not sacrificing validity. The goal of the propensity score weights is precisely this: they re-weight the external data in such a way that the distribution of the variables used to compute the propensity score is the same across Z. In the re-weighted population there is no longer any relationship between the variables in the propensity score and Z. In Figure 2, weights constructed from the propensity score estimated using C1 would in essence remove the edge between C1 and Z. Of course, the propensity score-based weights only balance variables used to construct the propensity score. Any confounders which are not included in the propensity score will remain unbalanced across Z even in the re-weighted data set. As was the case before considering propensity score-based weighting, dynamic borrowing can balance the risk of unmeasured confounding with the benefit of increased precision, but with the additional benefit that some of the systematic difference in Y across levels of Z due to confounding has been removed through the propensity score-based weighting process.
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