Lee and Liu's criterion function for determining the trial decision cutoffs based on the predictive probability.
1  PredP(x, n, nmax, a, b, p0, theta_t)

x 
the number of responses among n patients treated by the experimental drug at a certain stage of the trial. 
n 
the number of patients treated by the experimental drug at a certain stage of the trial. 
nmax 
the maximum number of patients treated by the experimental drug. 
a 
the hyperparameter (shape1) of the Beta prior for the experimental drug. 
b 
the hyperparameter (shape2) of the Beta prior for the experimental drug. 
p0 
the the response rate for the standard drug. 
theta_t 
the cutoff probability for efficacy including future patients; typically, θ_T = [0.85, 0.95]. Set 0.9 by default. 
prob 
the predictive probability: PP = ∑\limits_{y=0}^{n_{max}n} Pr(Y=y  x) I(\Pr(p > p_0  Y=y, x) ≥q θ_T) 
Lee, J. J., Liu, D. D. (2008). A predictive probability design for phase II cancer clinical trials. Clinical Trials 5: 93106.
Yin, G. (2012). Clinical Trial Design: Bayesian and Frequentist Adaptive Methods. New York: Wiley.
1 2  # Using vague prior Uniform(0,1), i.e. Beta(1,1)
PredP(16, 23, 40, 1, 1, 0.5, 0.9)

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