Historical controls in clinical trials: a note on linking Pocock's model with the robust mixture priors

Several Bayesian methods have been proposed to borrow information dynamically from historical controls in clinical trials. In this note, we identify key features of the relationship between the first method proposed, the bias-variance method, which is strongly related to the commensurate prior approach, and a more recent and widely used approach called robust mixture priors (RMP). We focus on the two key terms that need to be chosen to define the RMP, namely $w$, the prior probability that the new trial differs systematically from the historical trial, and $s_v^2$, the variance of the vague component of the RMP. The relationship with Pocock's prior reveals that different combinations of these two terms can express similar prior beliefs about the amount of information provided by the historical data. This demonstrates the value of fixing $s_v^2$, e.g., so the vague component is "worth one subject." Prior belief about the relevance of the historical data is then driven by a single value, the prespecified weight $w$.