In phase II single-arm studies, the response rate of the experimental treatment is typically compared with a fixed target value that should ideally represent the true response rate for the standard of care therapy. Generally, this target value is estimated through previous data, but the inherent variability in the historical response rate is not taken into account. In this paper, we present a Bayesian procedure to construct single-arm two-stage designs that allows to incorporate uncertainty in the response rate of the standard treatment. In both stages, the sample size determination criterion is based on the concepts of conditional and predictive Bayesian power functions. Different kinds of prior distributions, which play different roles in the designs, are introduced, and some guidelines for their elicitation are described. Finally, some numerical results about the performance of the designs are provided and a real data example is illustrated. Copyright © 2016 John Wiley & Sons, Ltd.
In basketball and hockey, state-of-the-art player value statistics are often variants of Adjusted Plus-Minus (APM). But APM hasn't had the same impact in soccer, since soccer games are low scoring with a low number of substitutions. In soccer, perhaps the most comprehensive player value statistics come from video games, and in particular FIFA. FIFA ratings combine the subjective evaluations of over 9000 scouts, coaches, and season-ticket holders into ratings for over 18,000 players. This paper combines FIFA ratings and APM into a single metric, which we call Augmented APM. The key idea is recasting APM into a Bayesian framework, and incorporating FIFA ratings into the prior distribution. We show that Augmented APM predicts better than both standard APM and a model using only FIFA ratings. We also show that Augmented APM decorrelates players that are highly collinear.
Adjusted plus-minus (APM) can sometimes lack common sense. This happens, for instance, when mediocre players move into the top ten, and superstars fall out of the top 100. These occasional outliers hurt the credibility of APM, and mask the benefits, such as increased prediction accuracy. We address this problem with a new method, called Augmented APM. Augmented APM incorporates external player ratings into APM methodology. The purpose of the external rating system is to capture common sense player value. Augmented APM maintains the benefits of APM, and improves credibility by leveraging external ratings that pass the eye test. The key technical idea is recasting APM into a Bayesian framework and using external ratings in the prior distribution. This paper instantiates the Augmented APM method by applying it to soccer. APM methods have not had a substantial impact on soccer, because soccer matches are low scoring, with a low number of substitutions. For external ratings, we use the video game FIFA, which provides subjective evaluations from thousands of scouts, coaches, and season ticket holders. Our paper shows that Augmented APM predicts match outcomes better than (1) APM, and (2) FIFA ratings. We also show that Augmented APM de-correlates players on the same team, which helps for players that play most of their minutes together. Although our results are specific to soccer and FIFA ratings, Augmented APM is a principled method to combine subjective and objective ratings into a single system.
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