Add footnotes of type I error of non-binding design

[LittleBeannie]

In non-binding design (see example below), it is possible that the type I error is smaller than 0.025. A good strategy is to add a footnote for this probability.

library(gsDesign2)
library(dplyr)

ratio <- 1
alpha <- 0.025
beta <- 0.2
enrollRates <- tibble::tibble(Stratum = "All", 
                              duration = 12, 
                              rate = 500/12)

failRates <- tibble::tibble(Stratum = "All",
                            duration = c(4, 100),
                            failRate = log(2) / 15,  
                            hr = c(1, .6),
                            dropoutRate = 0.001)

fh_test <- rbind(data.frame(rho = 0, gamma = 0, tau = -1, test = 1, Analysis = 1:3,analysisTimes = c(12, 24, 36)),
                 data.frame(rho = c(0, 0.5), gamma = 0.5, tau = -1, test = 2:3, Analysis = 3, analysisTimes = 36))

#### Boundary derived by spending function testing
gs_design_combo(enrollRates, failRates, fh_test,
                alpha = 0.025, beta = 0.2, ratio = 1, binding = FALSE,                 # test.type = 4 non-binding futility bound
                upper = gsDesign2::gs_spending_combo,
                upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025),   # alpha spending
                lower = gsDesign2::gs_spending_combo,
                lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2),     # beta spending
                ) %>% summary() %>% as_gt()

[LittleBeannie]

Templates form Keaven: Cumulative alpha for final analysis (0.0237) is less than the full alpha (0.025) when the futility bound is non-binding. The smaller value subtracts the probability of crossing a futility bound before crossing an efficacy bound at a later analysis (0.025 – 0.0013 = 0.0237) under the null hypothesis.

[LittleBeannie]

See the updates below