Augmenting treatment arms with external data through propensity-score weighted power-priors with an application in expanded access 189 9♥ Background of methodology Notation The data consist of a current (internal) trial y and data from an external source ye. In total, we have data on N = N + N e patients. For every patient i, i = 1, ...N in either the current study or the external source, we observe the outcome Y i, a realization of Y i , and the covariate vector x i of length K, which is a realization of the set of covariates . Let Z be an indicator variable, where i = 1 if patient i belongs to the internal study and i = 0 if patient i belongs to the external data source. In our case study, the estimand is the baseline rate in a single-arm study and hence there is no treatment effect. Propensity scores Propensity scores are frequently used to address biases that arise due to confounding in nonrandomized experimental settings,29 by modeling the allocation to treatment (T = 1) or control (T = 0) as a function of the covariates that one wishes to balance across these two groups: Equation 1 Among patients with the same propensity scores e(x), covariates included in the propensity score will be balanced across the treated and untreated groups. Under the assumption that the variables in X are sufficient to make the treatment groups conditionally exchangeable (Y ⊥T| ), the propensity score can be used to estimate the causal effect of treatment. Weighting, matching, and stratification are the main methods in the propensity score toolbox.39 To use the propensity score to compare current trial data with external data, several authors have slightly redefined the propensity score.21–23,36 Instead of modeling assignment to a control or treatment group, the propensity score is now used to model the allocation between current and external data (Z): Equation 2 where λ i is the probability of patient i being in the internal study given the patient characteristics. Now, patients with similar propensity scores are equally likely to have been in the trial or external data conditional on . If the variables in are sufficient to satisfy Y =0⊥Z|T=0, =x, then the internal and external populations are exchangeable.
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