The statistical properties of RCTs and a proposal for shrinkage.

We abstract the concept of a randomized controlled trial as a triple ( β , b , s ) , where β is the primary efficacy parameter, b the estimate, and s the standard error ( s > 0 ). If the parameter β is either a difference of means, a log odds ratio or a log hazard ratio, then it is reasonable to assume that b is unbiased and normally distributed. This then allows us to estimate the joint distribution of the z-value z = b / s and the signal-to-noise ratio SNR = β / s from a sample of pairs ( b i , s i ) . We have collected 23 551 such pairs from the Cochrane database. We note that there are many statistical quantities that depend on ( β , b , s ) only through the pair ( z , SNR ) . We start by determining the estimated distribution of the achieved power. In particular, we estimate the median achieved power to be only 13%. We also consider the exaggeration ratio which is the factor by which the magnitude of β is overestimated. We find that if the estimate is just significant at the 5% level, we would expect it to overestimate the true effect by a factor of 1.7. This exaggeration is sometimes referred to as the winner's curse and it is undoubtedly to a considerable extent responsible for disappointing replication results. For this reason, we believe it is important to shrink the unbiased estimator, and we propose a method for doing so. We show that our shrinkage estimator successfully addresses the exaggeration. As an example, we re-analyze the ANDROMEDA-SHOCK trial.