Conditional power computation with non-constant effect size for non-/crossing an upper boundary at analysis j given observed Z value at analysis i

Usage

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

Arguments

x: An object of type gsDesign2.
theta: Optional numeric vector with length $j-i+1$, which specifies the natural parameter for treatment effect of interim analysis $i$ through analysis $j$. The default is NULL.
i: Index of current analysis, with default of 1.
zi: Numeric scalar z-value observed at analysis $i$.

Value

A list with the following elements:

prob_alpha: A numeric vector of $(\alpha_{i,i+1}, \ldots, \alpha_{i,j-1}, \alpha_{i,j})$, where $\alpha_{i,j} = P(\{Z_j \geq b_j\} \& \{\cap_{m=i+1}^{j-1} a_m \leq Z_m < b_m\} \mid Z_i = z_i)$.
prob_alpha_plus: A numeric vector of $(\alpha^+_{i,i+1}, \ldots, \alpha^+_{i,j-1}, \alpha^+_{i,j})$, where $\alpha^+_{i,j} = P(\{Z_j \geq b_j\} \& \{\cap_{m=i+1}^{j-1} Z_m < b_m\} \mid Z_i = z_i)$.
prob_beta: A numeric vector of $(\beta_{i,i+1}, \ldots, \beta_{i,j-1}, \beta_{i,j})$, where $\beta_{i,j} = P(\{Z_j \leq b_j\} \& \{\cap_{m=i+1}^{j-1} a_m \leq Z_m < b_m\} \mid Z_i = z_i)$.

Details

We assume $Z_i, i = 1, ..., K$ are the z-statistics at an interim analysis i, respectively. We assume further $Z_i, i = 1, ..., K$ follows multivariate normal distribution $$E(Z_i) = \theta_i\sqrt{I_i}$$ $$Cov(Z_i, Z_j) = I_i/I_j$$. See https://merck.github.io/gsDesign2/articles/story-npe-background.html for assumption details.

The returned value is list of $$P(\{Z_j \geq b_j\} \& \{\cap_{m=i+1}^{j-1} a_m \leq Z_m < b_m\} \mid Z_i = z_i).$$ $$P(\{Z_j \geq b_j\} \& \{\cap_{m=i+1}^{j-1} Z_m < b_m\} \mid Z_i = z_i).$$ $$P(\{Z_j \leq b_j\} \& \{\cap_{m=i+1}^{j-1} a_m \leq Z_m < b_m\} \mid Z_i = z_i).$$

Examples

library(gsDesign2)
library(gsDesign)
#> 
#> Attaching package: ‘gsDesign’
#> The following objects are masked from ‘package:gsDesign2’:
#> 
#>     as_gt, as_rtf
library(dplyr)
#> 
#> Attaching package: ‘dplyr’
#> The following objects are masked from ‘package:stats’:
#> 
#>     filter, lag
#> The following objects are masked from ‘package:base’:
#> 
#>     intersect, setdiff, setequal, union
library(mvtnorm)
# Example 1
# original design ----
enroll_rate <- define_enroll_rate(duration = c(2, 2, 2, 18),
                                  rate = c(1, 2, 3, 4))
fail_rate <- define_fail_rate(duration = c(3, Inf),
                              fail_rate = log(2) / 10,
                              dropout_rate = 0.001,
                              hr = c(1, 0.7))
x <- gs_design_ahr(enroll_rate = enroll_rate, fail_rate = fail_rate,
                   alpha = 0.025, beta = 0.1, ratio = 1,
                   info_frac = c(0.4, 0.6, 0.8, 1), analysis_time = 30,
                   binding = FALSE,
                   upper = gs_spending_bound,
                   upar = list(sf = sfLDOF, total_spend = 0.025, param = NULL),
                   lower = gs_spending_bound,
                   lpar = list(sf = sfLDOF, total_spend = 0.1),
                   h1_spending = TRUE,
                   test_lower = TRUE,
                   info_scale = "h0_h1_info") |> to_integer()

# calculate conditional power
# case 1: currently at IA1, compute conditional power at IA2, IA3 and FA, with default theta = NULL
gs_cp(x = x, i = 1, zi = -gsDesign::hrn2z(hr = 0.8, n = 150+180, ratio = 1))
#> $prob_alpha
#> [1] 0.4092021 0.4521861 0.1219315
#> 
#> $prob_alpha_plus
#> [1] 0.4092021 0.4521865 0.1232153
#> 
#> $prob_beta
#> [1] 0.5907979 0.0000000 0.0000000
#> 

# case 2: currently at IA1, compute conditional power at IA2, IA3 and FA, with user-input theta
gs_cp(x = x, theta = c(0.15, 0.2, 0.25, 0.3), i = 1, zi = -gsDesign::hrn2z(hr = 0.8, n = 150+180, ratio = 1))
#> $prob_alpha
#> [1] 0.53387369 0.43584552 0.02968923
#> 
#> $prob_alpha_plus
#> [1] 0.53387369 0.43584635 0.02989685
#> 
#> $prob_beta
#> [1] 0.4661263 0.0000000 0.0000000
#>