The limited scope of Randomized Controlled Trials (RCT) is increasingly under scrutiny, in particular when samples are unrepresentative. Indeed, some RCTs over- or under- sample individuals with certain characteristics compared to the target population, for which one want to draw conclusions on treatment effectiveness. Over the last decades, several methods have been proposed to account for this sampling bias, relying on the RCT data, and an external data set representative of the target population of interest. We will review the different methods available and the current state-of-the-art. One of the method consists in re-weighting the trial individuals to match the target population, which is usually called Inverse Propensity of Sampling Weighting (IPSW). Such procedures require an estimation of the ratio of two densities (trial and target distributions). In this presentation, we establish the exact expressions of the bias and variance of such reweighting procedures in presence of categorical covariates for any sample size. These results show how the performance (bias, variance and quadratic risk) of IPSW estimates depends on the two sample sizes (RCT and target population). In addition, we study how including covariates that are unnecessary to a proper estimation of the weights may impact the asymptotic variance, for the best or the worse. We illustrate all the takeaways twice: in a toy and didactic example, and on a semi-synthetic simulation inspired from critical care medicine.

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Physically
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