Integer designs

Yujie Zhao

library(gsDesign)
library(gsDesign2)
library(tibble)
library(dplyr)
library(gt)

Unstratified design

Binary outcome

x <- gs_design_rd(
  p_c = tibble(stratum = "All", rate = 0.2),
  p_e = tibble(stratum = "All", rate = 0.15),
  info_frac = c(0.5, 0.8, 1),
  rd0 = 0,
  alpha = 0.025,
  beta = 0.1,
  ratio = 1,
  stratum_prev = NULL,
  weight = "unstratified",
  upper = gs_spending_bound,
  lower = gs_b,
  upar = list(sf = gsDesign::sfLDOF, timing = c(0.5, 0.8, 1), total_spend = 0.025, param = NULL),
  lpar = rep(-Inf, 3)
)

xi <- x %>% to_integer()

Note that in the original design, the sample size is 1243.307021, 1989.2912336, 2486.614042, and in the integer design, the sample size is updated to 1243, 1989, 2488. For the 2 interim analysis, we floor to the closet multiplier of 2, since the randomization ratio is 1. At the final analysis, we ceiling the sample size from 2486.614042 to 2488 and also make sure the integer sample size is a multiplier of 2.

Please also note that, since the sample size is rounded, the power of the new design also changes a little bit, that is, from 0.9 to 0.9001336.

tibble(
  Design = rep(c("Original design", "Integer design"), each = 3),
  `Sample size` = c(x$analysis$n, xi$analysis$n),
  Z = c(
    (x$bound %>% filter(bound == "upper"))$z,
    (xi$bound %>% filter(bound == "upper"))$z
  ),
  `Information fraction` = c(x$analysis$info_frac, xi$analysis$info_frac),
  Power = c(
    (x$bound %>% filter(bound == "upper"))$probability,
    (xi$bound %>% filter(bound == "upper"))$probability
  )
) %>%
  group_by(Design) %>%
  gt() %>%
  tab_header(
    title = "Comparison between the original/integer design",
    subtitle = "on binary endpoints (unstratified design)"
  ) %>%
  fmt_number(columns = 2:5, decimals = 4)

Comparison between the original/integer design
on binary endpoints (unstratified design)
Sample size	Z	Information fraction	Power
Original design
1,243.3070	2.9626	0.5000	0.2598
1,989.2912	2.2662	0.8000	0.7501
2,486.6140	2.0278	1.0000	0.9000
Integer design
1,243.0000	2.9626	0.4996	0.2597
1,989.0000	2.2662	0.7994	0.7500
2,488.0000	2.0280	1.0000	0.9001

Survival outcome

x <- gs_design_ahr(
  analysis_time = c(12, 24, 36),
  upper = gs_spending_bound,
  lower = gs_b,
  upar = list(sf = gsDesign::sfLDOF, timing = 1:3 / 3, total_spend = 0.025, param = NULL),
  lpar = rep(-Inf, 3)
)
xi <- x %>% to_integer()

Notice that with the integer design, the (i) number of events, (ii) sample size, (iii) power, (iv) information fraction will be different.

tibble(
  Design = rep(c("Original design", "Integer design"), each = 3),
  Events = c(x$analysis$event, xi$analysis$event),
  `Sample size` = c(x$analysis$n, xi$analysis$n),
  Z = c(
    (x$bound %>% filter(bound == "upper"))$z,
    (xi$bound %>% filter(bound == "upper"))$z
  ),
  `Information fraction` = c(x$analysis$info_frac, xi$analysis$info_frac),
  Power = c(
    (x$bound %>% filter(bound == "upper"))$probability,
    (xi$bound %>% filter(bound == "upper"))$probability
  )
) %>%
  group_by(Design) %>%
  gt() %>%
  tab_header(
    title = "Comparison between the original/integer design",
    subtitle = "on survival endpoints (unstratified design)"
  ) %>%
  fmt_number(columns = 2:5, decimals = 4)

Comparison between the original/integer design
on survival endpoints (unstratified design)
Events	Sample size	Z	Information fraction	Power
Original design
91.9812	405.7098	3.7103	0.3091	0.003627376
221.2005	486.8518	2.5122	0.7376	0.481734802
298.6001	486.8518	1.9828	1.0000	0.900000000
Integer design
92.0000	406.0000	3.7103	0.3088	0.003620194
221.0000	488.0000	2.5122	0.7360	0.479605447
299.0000	488.0000	1.9830	1.0000	0.900134317

Stratified design

x <- gs_design_rd(
  p_c = tibble(
    stratum = c("biomarker positive", "biomarker negative"),
    rate = c(0.2, 0.25)
  ),
  p_e = tibble(
    stratum = c("biomarker positive", "biomarker negative"),
    rate = c(0.15, 0.22)
  ),
  info_frac = c(0.7, 1),
  rd0 = 0,
  alpha = 0.025,
  beta = 0.1,
  ratio = 1,
  stratum_prev = tibble(
    stratum = c("biomarker positive", "biomarker negative"),
    prevalence = c(0.4, 0.6)
  ),
  weight = "ss",
  upper = gs_spending_bound,
  lower = gs_b,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = c(0.7, 1)),
  lpar = rep(-Inf, 2)
)

xi <- x %>% to_integer()

Note that in the original design, the sample size is 3426.1318268, 4894.4740382, and in the integer design, the sample size is updated to 3426, 4896. For the 2 interim analysis, we floor to the closet multiplier of 2, since the randomization ratio is 1. At the final analysis, we ceiling the sample size from 4894.4740382 to 4896 and also make sure the integer sample size is a multiplier of 2.

tibble(
  Design = rep(c("Original design", "Integer design"), each = 2),
  `Sample size` = c(x$analysis$n, xi$analysis$n),
  Z = c(
    (x$bound %>% filter(bound == "upper"))$z,
    (xi$bound %>% filter(bound == "upper"))$z
  ),
  `Information fraction` = c(x$analysis$info_frac, xi$analysis$info_frac),
  Power = c(
    (x$bound %>% filter(bound == "upper"))$probability,
    (xi$bound %>% filter(bound == "upper"))$probability
  )
) %>%
  group_by(Design) %>%
  gt() %>%
  tab_header(
    title = "Comparison between the original/integer design",
    subtitle = "on binary endpoints (unstratified design)"
  ) %>%
  fmt_number(columns = 2:5, decimals = 4)

Comparison between the original/integer design
on binary endpoints (unstratified design)
Sample size	Z	Information fraction	Power
Original design
3,426.1318	2.4380	0.7000	0.6161
4,894.4740	1.9999	1.0000	0.9000
Integer design
3,426.0000	2.4380	0.6998	0.6160
4,896.0000	2.0000	1.0000	0.9001