Figure 1:

Bias (π_SAT − π_HC)and deviations from a targeted type I error rate of 10%. Bias is due to different patient populations in the single arm trial (SAT) and in the historical study. A single binary characteristic (X=1 or X=0) correlates with the binary outcome Y, and the experimental treatment has no therapeutic effect Pr(Y|X, A = 1) = Pr(Y|X, A = 0). The characteristic X=1 was present in 50% of the patients in the historical control arm, P_HC(X = 1) = 0.5. Panel (A) shows the difference (π_SAT − π_HC) for a range of probabilities P_SAT(X = 1). We consider four levels of association between X and Y; (Pr(Y|X = 1, A = a) and P(Y|X = 0, A = a) equal either to (0.3, 0.9), (0.4, 0.8), (0.5, 0.7) or (0.6, 0.6). Panel (B) indicates, for a SAT (with standard z-test for proportions, H₀: π_SAT ≤ π_HC) how the false positive rate (y-axis) of the design deviates without adjustments from the targeted type I error rate of 10% when the prevalence P_SAT(X = 1) = 0.3, 0.5, 0.6 or 0.8. We consider different sample sizes (x-axis) of the SAT. In panel (B) we assume to know the parameter π_HC.