Conditional power computation with non-constant effect size
Arguments
- theta
A vector of length two, which specifies the natural parameter for treatment effect. The first element of
thetais the treatment effect of an interim analysis i. The second element ofthetais the treatment effect of a future analysis j.- info
A vector of two, which specifies the statistical information under the treatment effect
theta.- a
Interim z-value at analysis i (scalar).
- b
Future target z-value at analysis j (scalar).
Details
We assume \(Z_1\) and \(Z_2\) are the z-values at an interim analysis and later analysis, respectively. We assume further \(Z_1\) and \(Z_2\) are bivariate normal with standard group sequential assumptions on independent increments where for \(i=1,2\) $$E(Z_i) = \theta_i\sqrt{I_i}$$ $$Var(Z_i) = 1/I_i$$ $$Cov(Z_1, Z_2) = t \equiv I_1/I_2$$ where \(\theta_1, \theta_2\) are real values and \(0<I_1<I_2\). See https://merck.github.io/gsDesign2/articles/story-npe-background.html for assumption details. Returned value is $$P(Z_2 > b \mid Z_1 = a) = 1 - \Phi\left(\frac{b - \sqrt{t}a - \sqrt{I_2}(\theta_2 - \theta_1\sqrt{t})}{\sqrt{1 - t}}\right)$$
Examples
library(gsDesign2)
library(dplyr)
# Calculate conditional power under arbitrary theta and info
# In practice, the value of theta and info commonly comes from a design.
# More examples are available at the pkgdown vignettes.
gs_cp_npe(theta = c(.1, .2),
info = c(15, 35),
a = 1.5, b = 1.96)
#> Warning: NaNs produced
#> [1] NaN