BDP2 {BDP2}

Operating characteristics of a single-arm trial with a binary endpoint

Description

Determines the operating characteristics of a single-arm trial with a binary endpoint (response, success) and interim efficacy and futility analyses. Declaration of efficacy and futility (including possibly early stopping) is based on the posterior probability that the true response rate is at least pE , pF respectively.

Usage

BDP2(n, interim.at, ptrue, 
      eff.stop = FALSE, 
      pF, cF, pE = NULL, cE = NULL, 
      type="PostProb", alpha=0.05,
      shape1F, shape2F, shape1E = NULL, shape2E = NULL, 
      simulate = FALSE, nsim = 10000)

Arguments

n

sample size at the final analysis

interim.at

vector of sample sizes at the interim analyses

ptrue

true (assumed) response rate used for analytical evaluations or simulating the trial

eff.stop

FALSE: No evaluation of efficacy. "call": no stop for efficacy; in this case the program merely calculates the probability that the efficicacy criterion is satisfied (possibly triggering a notification of the DMC and the start of the planning of a subsequent trial). "stop": the study ends if the efficacy criterion is reached at an interim analysis.

pF

response rate used for the futility criterion (may be identical to pE)

cF

critical level of posterior probabilities used for declaring futility

pE

response rate used for the efficacy criterion

cE

critical level of posterior probabilities used for declaring efficacy

type

"PostProb" for decisions based on posterior probabilities (default) or "PredictivePower" for decisions based on predictive power (currently only implemented for simulate==TRUE)

alpha

significance level for final test (only for simulate==TRUE & type=="PredictivePower")

shape1F

first parameter of the Beta prior for futility analysis

shape2F

second parameter of the Beta prior for futility analysis

shape1E

first parameter of the Beta prior for efficacy analysis

shape2E

second parameter of the Beta prior for efficacy analysis

simulate

FALSE for analytical evaluation and TRUE for simulation

nsim

number of simulation runs (only used if simulate==TRUE)

Details

Assumptions: Endpoint (response/no response) data available for all study patients. Beta-binomial model. Prior distribution = Beta(shape1, shape2).

Decisions based on posterior probabilities

The posterior distribution at interim analysis with n.int patients and k.int successes is Beta(k.int + shape1F, n.int + shape2F - k.int) and Beta(k.int + shape1E, n.int + shape2E - k.int), respectively. Efficacy is declared if the posterior probability P(true response rate > pE) is >= cE. Futility is declared if the posterior probability P(true success rate > pF) is < cF. cF, cE translate into futility/efficacy boundaries (maximum number of responses leading to early termination for futility/ minimum number of responses leading to declaring of, or early termination for, efficacy).

Decisions based on predictive power

Given the results of the interim analysis, the predictive power at the final analysis (n patients, critical number of successes k.crit) is P(X >= k.crit - k.int), where X follows a beta-binomial distribution with parameters n'= n - n.int, a = k.int + shape1, and b = n.int - k.int + shape2.

Efficacy is declared if the predictive power is >= cE (cE must be high, e.g. 0.70). Futility is declared if the predictive power is < cF (cF must be small, e.g. 0.10). cE, cF translate into futility/efficacy boundaries (maximum number of responses leading to early termination for futility/ minimum number of responses leading to declaring of, or early termination for, efficacy).

References

Kopp-Schneider, A., Wiesenfarth, M., Witt, R., Edelmann, D., Witt, O. and Abel, U. (2018).
Monitoring futility and efficacy in phase II trials with Bayesian posterior distributions - a calibration approach. Biometrical Journal, to appear.

Examples

# Operating characteristics with calling for efficacy
BDP2(n=20, interim.at = c(3,9,13,18), ptrue = 0.3, 
      eff.stop = "call",
      pF=0.3, cF=0.01, pE=0.12, cE = 0.9, 
      type="PostProb", 
      shape1F=0.3, shape2F=0.7, shape1E=0.12, shape2E=0.88)

# Operating characteristics with stopping for efficacy
BDP2(n=20, interim.at = c(3,9,13,18), ptrue = 0.3, 
      eff.stop = "stop",
      pF=0.3, cF=0.01, pE=0.12, cE = 0.9, 
      type="PostProb", 
      shape1F=0.3, shape2F=0.7, shape1E=0.12, shape2E=0.88)