Group sequential boundary crossing probabilities
Arguments
- theta
natural parameter for group sequentia design representing expected drift at time of each analysis
- upper
function to compute upper bound
- lower
function to compare lower bound
- upar
parameter to pass to upper
- lpar
parameter to pass to lower
- info
statistical information at each analysis
- r
Integer, at least 2; default of 18 recommended by Jennison and Turnbull
Value
A tibble with a row for each finite bound and analysis containing the following variables:
Analysis analysis number
Bound Upper (efficacy) or Lower (futility)
Z Z-value at bound
Probability probability that this is the first bound crossed under the given input
theta approximate natural parameter value required to cross the bound
Details
Approximation for theta is based on Wald test and assumes the observed information is equal to the expected.
Examples
library(dplyr)
# Asymmetric 2-sided design
gs_prob(theta = 0, upar = rep(2.2, 3), lpar = rep(0, 3),
upper=gs_b, lower=gs_b, info = 1:3)
#> # A tibble: 6 × 6
#> Analysis Bound Z Probability theta info
#> <int> <chr> <dbl> <dbl> <dbl> <int>
#> 1 1 Upper 2.2 0.0139 0 1
#> 2 2 Upper 2.2 0.0236 0 2
#> 3 3 Upper 2.2 0.0305 0 3
#> 4 1 Lower 0 0.5 0 1
#> 5 2 Lower 0 0.625 0 2
#> 6 3 Lower 0 0.687 0 3
# One-sided design
x <- gs_prob(theta = 0, upar = rep(2.2, 3), lpar = rep(-Inf, 3),
upper=gs_b, lower=gs_b, info = 1:3)
# Without filtering, this shows unneeded lower bound
x
#> # A tibble: 6 × 6
#> Analysis Bound Z Probability theta info
#> <int> <chr> <dbl> <dbl> <dbl> <int>
#> 1 1 Upper 2.2 0.0139 0 1
#> 2 2 Upper 2.2 0.0237 0 2
#> 3 3 Upper 2.2 0.0311 0 3
#> 4 1 Lower -Inf 0 0 1
#> 5 2 Lower -Inf 0 0 2
#> 6 3 Lower -Inf 0 0 3
# Filter to just show bounds intended for use
x %>% filter(abs(Z) < Inf)
#> # A tibble: 3 × 6
#> Analysis Bound Z Probability theta info
#> <int> <chr> <dbl> <dbl> <dbl> <int>
#> 1 1 Upper 2.2 0.0139 0 1
#> 2 2 Upper 2.2 0.0237 0 2
#> 3 3 Upper 2.2 0.0311 0 3