Calculates the derivative of nu for a given conditional error and conditional power.
Usage
getNuPrime(alpha, conditionalPower)
Value
Value for nu prime.
Arguments
alpha
The (conditional) type I error rate of the design. Must be a numeric vector with values between 0 and 1.
conditionalPower
The target conditional power \(CP\) of the design. Must be a numeric value.
Details
The function \(\nu'\) is defined as
$$\nu'(p_1) = -2 \cdot (\Phi^{-1}(1-\alpha_2(p_1)) + \Phi^{-1}(CP))/\phi(\Phi^{-1}(1-\alpha_2(p_1))).$$
Note that in this implementation, the the factor -2 is used instead of -4, which is used in by Brannath & Bauer (2004), who explicitly investigate the setting of a balanced two-group trial.
The argument conditionalPower is either the fixed target conditional power or the value of the conditional power function at the corresponding first-stage p-value.
References
Brannath, W. & Bauer, P. (2004). Optimal conditional error functions for the control of conditional power. Biometrics. https://www.jstor.org/stable/3695393