Simple conditional power computation with non-constant effect size

Usage

gs_cp_simple(x = NULL, theta = NULL, i = 1, zi = NULL)

Arguments

x: An object of type gsDesign2.
theta: Optional numeric vector of natural parameter for treatment effects from analysis $i$ through final analysis $K$. If NULL, values are taken from x$analysis$theta.
i: Integer index for current analysis. Default is 1.
zi: Numeric scalar z-value observed at analysis $i$.

Value

A numeric vector with the conditional power $P(Z_j > b_j \mid Z_i = z_i)$ for $j = i+1, ..., K$.

Details

Suppose there are $K$ analyses. Let $Z_i$ and $Z_j$ be the Z-statistics at a current analysis $i$ and a future analysis $j$, respectively. Let's denote the statistical information at the $i$-th analysis as $I_i$. We further assume $Z_i$ and $Z_j$ are bivariate normal under standard group sequential assumptions with independent increments, then $$E(Z_i) = \theta_i\sqrt{I_i}$$ $$Var(Z_i) = 1/I_i$$ $$Cov(Z_i, Z_j) = t \equiv I_i/I_j$$ See https://merck.github.io/gsDesign2/articles/story-npe-background.html for assumption details. Returned value is $$P(Z_j > b_j \mid Z_i = z_i) = 1 - \Phi\left(\frac{b_j - \sqrt{t}z_i - \sqrt{I_j}(\theta_j - \theta_i\sqrt{t})}{\sqrt{1 - t}}\right)$$ where $b_j$ is the efficacy bound at analysis $j$, $z_i$ is the observed Z-value at analysis $i$, and $\theta_i$ is the natural parameter for treatment effect at analysis $i$.

Examples

library(gsDesign2)
library(gsDesign)
# Calculate conditional power with optional user-input theta (if NULL, will come from the input design)
alpha <- 0.025
beta <- 0.1
ratio <- 1
enroll_rate <- define_enroll_rate(
  duration = c(2, 2, 10),
  rate = (1:3) / 3)
fail_rate <- define_fail_rate(
  duration = Inf, fail_rate = log(2) / 9,
  hr = 0.6, dropout_rate = .0001)
analysis_time <- c(12, 24, 36)
ratio <- 1
upper <- gs_spending_bound
lower <- gs_b
upar <- list(sf = sfLDOF, total_spend = alpha)
lpar <- rep(-Inf, 3)
x <- gs_design_ahr(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  alpha = alpha, beta = beta, ratio = ratio,
  info_scale = "h0_h1_info",
  info_frac = NULL,
  analysis_time = c(12, 24, 36),
  upper = upper, upar = upar, test_upper = TRUE,
  lower = lower, lpar = lpar, test_lower = FALSE,
) |>
  to_integer()

# theta is NULL case
gs_cp_simple(x = x, theta = NULL, i = 1, zi = 1.5)
#> [1] 0.6914911 0.9129447

# User-input theta case
gs_cp_simple(x = x, theta = c(0.1, 0.2, 0.3), i = 1, zi = 1.5)
#> [1] 0.2562896 0.7194958