Group sequential design using average hazard ratio under non-proportional hazards

Usage

gs_update_ahr(
  x = NULL,
  alpha = NULL,
  ia_alpha_spending = c("actual_info_frac", "min_of_planned_and_actual_info_frac"),
  fa_alpha_spending = c("full_alpha", "info_frac"),
  observed_data = NULL
)

Arguments

x: A original design created by either gs_design_ahr or gs_power_ahr.
alpha: Alpha for the updated design.
ia_alpha_spending: Alpha spending strategy for interim analyses, either "actual_info_frac" (default), or "min_of_planned_and_actual_info_frac".
fa_alpha_spending: Alpha spending strategy for final analysis, either "info_frac" or "full_alpha" (default).
observed_data: a list of observed datasets by analyses.

Value

A list with input parameters, enrollment rate, analysis, and bound.

Examples

library(gsDesign)
library(gsDesign2)
library(dplyr)

alpha <- 0.025
beta <- 0.1
ratio <- 1

# Enrollment
enroll_rate <- define_enroll_rate(
  duration = c(2, 2, 10),
  rate = (1:3) / 3)

# Failure and dropout
fail_rate <- define_fail_rate(
  duration = c(3, Inf), fail_rate = log(2) / 9,
  hr = c(1, 0.6), dropout_rate = .0001)

# IA and FA analysis time
analysis_time <- c(20, 36)

# Randomization ratio
ratio <- 1

# Example A: one-sided design (efficacy only) ----

# Original design
upper <- gs_spending_bound
upar <- list(sf = sfLDOF, total_spend = alpha)
x <- gs_design_ahr(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  alpha = alpha, beta = beta, ratio = ratio,
  info_scale = "h0_info",
  info_frac = NULL,
  analysis_time = c(20, 36),
  upper = gs_spending_bound, upar = upar,
  lower = gs_b, lpar = rep(-Inf, 2),
  test_upper = TRUE, test_lower = FALSE) |> to_integer()

# Observed dataset at IA and FA
set.seed(123)

observed_data <- simtrial::sim_pw_surv(
  n = x$analysis$n[x$analysis$analysis == 2],
  stratum = data.frame(stratum = "All", p = 1),
  block = c(rep("control", 2), rep("experimental", 2)),
  enroll_rate = x$enroll_rate,
  fail_rate = (fail_rate |> simtrial::to_sim_pw_surv())$fail_rate,
  dropout_rate = (fail_rate |> simtrial::to_sim_pw_surv())$dropout_rate)

observed_data_ia <- observed_data |> simtrial::cut_data_by_date(x$analysis$time[1])
observed_data_fa <- observed_data |> simtrial::cut_data_by_date(x$analysis$time[2])

# Example A1 ----
# IA spending = observed events / final planned events
gs_update_ahr(
  x = x,
  ia_alpha_spending = "actual_info_frac",
  fa_alpha_spending = "full_alpha",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.2
#> 2 All            2  20.3
#> 3 All           10  30.5
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0       z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>   <dbl>          <dbl>       <dbl>
#> 1        1 upper       0.315      0.00484    2.59          0.686     0.00484
#> 2        2 upper       0.901      0.0250     1.99          0.794     0.0235 
#> 3        1 lower       0          0       -Inf           Inf         1      
#> 4        2 lower       0          0       -Inf           Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.86657 366   188 0.7356221 0.3070388 47.00 47.00 0.6329966
#> 2        2 35.81007 366   295 0.6832088 0.3809547 73.75 73.75 1.0000000
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example A2 ----
# IA, FA spending = observed events / final planned events
gs_update_ahr(
  x = x,
  ia_alpha_spending = "actual_info_frac",
  fa_alpha_spending = "info_frac",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.2
#> 2 All            2  20.3
#> 3 All           10  30.5
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0       z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>   <dbl>          <dbl>       <dbl>
#> 1        1 upper       0.315      0.00484    2.59          0.686     0.00484
#> 2        2 upper       0.899      0.0245     2.00          0.793     0.0230 
#> 3        1 lower       0          0       -Inf           Inf         1      
#> 4        2 lower       0          0       -Inf           Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.86657 366   188 0.7356221 0.3070388 47.00 47.00 0.6329966
#> 2        2 35.81007 366   295 0.6832088 0.3809547 73.75 73.75 0.9932660
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example A3 ----
# IA spending = min(observed events, planned events) / final planned events
gs_update_ahr(
  x = x,
  ia_alpha_spending = "min_of_planned_and_actual_info_frac",
  fa_alpha_spending = "full_alpha",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.2
#> 2 All            2  20.3
#> 3 All           10  30.5
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0       z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>   <dbl>          <dbl>       <dbl>
#> 1        1 upper       0.315      0.00484    2.59          0.686     0.00484
#> 2        2 upper       0.901      0.0250     1.99          0.794     0.0235 
#> 3        1 lower       0          0       -Inf           Inf         1      
#> 4        2 lower       0          0       -Inf           Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.86657 366   188 0.7356221 0.3070388 47.00 47.00 0.6329966
#> 2        2 35.81007 366   295 0.6832088 0.3809547 73.75 73.75 1.0000000
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example A4 ----
# IA spending = min(observed events, planned events) / final planned events
gs_update_ahr(
  x = x,
  ia_alpha_spending = "min_of_planned_and_actual_info_frac",
  fa_alpha_spending = "info_frac",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.2
#> 2 All            2  20.3
#> 3 All           10  30.5
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0       z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>   <dbl>          <dbl>       <dbl>
#> 1        1 upper       0.315      0.00484    2.59          0.686     0.00484
#> 2        2 upper       0.899      0.0245     2.00          0.793     0.0230 
#> 3        1 lower       0          0       -Inf           Inf         1      
#> 4        2 lower       0          0       -Inf           Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.86657 366   188 0.7356221 0.3070388 47.00 47.00 0.6329966
#> 2        2 35.81007 366   295 0.6832088 0.3809547 73.75 73.75 0.9932660
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example A5 ----
# IA spending = min(observed events, planned events) / final planned events
# alpha is upadted to 0.05
gs_update_ahr(
  x = x,
  alpha = 0.05,
  ia_alpha_spending = "min_of_planned_and_actual_info_frac",
  fa_alpha_spending = "info_frac",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.2
#> 2 All            2  20.3
#> 3 All           10  30.5
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0       z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>   <dbl>          <dbl>       <dbl>
#> 1        1 upper       0.461       0.0138    2.20          0.725      0.0138
#> 2        2 upper       0.943       0.0492    1.69          0.821      0.0451
#> 3        1 lower       0           0      -Inf           Inf          1     
#> 4        2 lower       0           0      -Inf           Inf          1     
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.86657 366   188 0.7356221 0.3070388 47.00 47.00 0.6329966
#> 2        2 35.81007 366   295 0.6832088 0.3809547 73.75 73.75 0.9932660
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example B: Two-sided asymmetric design,
# beta-spending with non-binding lower bound ----

# Original design
x <- gs_design_ahr(
  enroll_rate = enroll_rate, fail_rate = fail_rate,
  alpha = alpha, beta = beta, ratio = ratio,
  info_scale = "h0_info",
  info_frac = NULL, analysis_time = c(20, 36),
  upper = gs_spending_bound,
  upar = list(sf = sfLDOF, total_spend = alpha, param = NULL),
  test_upper = TRUE,
  lower = gs_spending_bound,
  lpar = list(sf = sfLDOF, total_spend = beta, param = NULL),
  test_lower = c(TRUE, FALSE),
  binding = FALSE) |> to_integer()

# Example B1 ----
# IA spending = observed events / final planned events
# the remaining alpha will be allocated to FA.
gs_update_ahr(
  x = x,
  ia_alpha_spending = "actual_info_frac",
  fa_alpha_spending = "full_alpha",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.6
#> 2 All            2  21.2
#> 3 All           10  31.8
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0        z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>    <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.290       0.00394    2.66           0.679     0.00394
#> 2        2 upper      0.894       0.0248     1.98           0.794     0.0239 
#> 3        1 lower      0.0344      0.612      0.285          0.959     0.388  
#> 4        2 lower      0.0344      0.612   -Inf            Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.91897 382   188 0.7356221 0.3070388 47.00 47.00 0.6045016
#> 2        2 36.06513 382   295 0.6832088 0.3809547 73.75 73.75 1.0000000
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example B2 ----
# IA, FA spending = observed events / final planned events
gs_update_ahr(
  x = x,
  ia_alpha_spending = "actual_info_frac",
  fa_alpha_spending = "info_frac",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.6
#> 2 All            2  21.2
#> 3 All           10  31.8
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0        z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>    <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.290       0.00394    2.66           0.679     0.00394
#> 2        2 upper      0.882       0.0213     2.05           0.788     0.0201 
#> 3        1 lower      0.0344      0.612      0.285          0.959     0.388  
#> 4        2 lower      0.0344      0.612   -Inf            Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.91897 382   188 0.7356221 0.3070388 47.00 47.00 0.6045016
#> 2        2 36.06513 382   295 0.6832088 0.3809547 73.75 73.75 0.9485530
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example B3 ----
# IA spending = min(observed events, planned events) / final planned events
gs_update_ahr(
  x = x,
  ia_alpha_spending = "min_of_planned_and_actual_info_frac",
  fa_alpha_spending = "full_alpha",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.6
#> 2 All            2  21.2
#> 3 All           10  31.8
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0        z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>    <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.290       0.00394    2.66           0.679     0.00394
#> 2        2 upper      0.894       0.0248     1.98           0.794     0.0239 
#> 3        1 lower      0.0344      0.612      0.285          0.959     0.388  
#> 4        2 lower      0.0344      0.612   -Inf            Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.91897 382   188 0.7356221 0.3070388 47.00 47.00 0.6045016
#> 2        2 36.06513 382   295 0.6832088 0.3809547 73.75 73.75 1.0000000
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example B4 ----
# IA spending = min(observed events, planned events) / final planned events
gs_update_ahr(
  x = x,
  ia_alpha_spending = "min_of_planned_and_actual_info_frac",
  fa_alpha_spending = "info_frac",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.6
#> 2 All            2  21.2
#> 3 All           10  31.8
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0        z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>    <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.290       0.00394    2.66           0.679     0.00394
#> 2        2 upper      0.882       0.0213     2.05           0.788     0.0201 
#> 3        1 lower      0.0344      0.612      0.285          0.959     0.388  
#> 4        2 lower      0.0344      0.612   -Inf            Inf         1      
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.91897 382   188 0.7356221 0.3070388 47.00 47.00 0.6045016
#> 2        2 36.06513 382   295 0.6832088 0.3809547 73.75 73.75 0.9485530
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"

# Example A5 ----
# IA spending = min(observed events, planned events) / final planned events
# alpha is upadted to 0.05
gs_update_ahr(
  x = x,
  alpha = 0.05,
  ia_alpha_spending = "min_of_planned_and_actual_info_frac",
  fa_alpha_spending = "info_frac",
  observed_data = list(observed_data_ia, observed_data_fa))
#> Joining with `by = join_by(analysis, bound, z)`
#> $enroll_rate
#> # A tibble: 3 × 3
#>   stratum duration  rate
#>   <chr>      <dbl> <dbl>
#> 1 All            2  10.6
#> 2 All            2  21.2
#> 3 All           10  31.8
#> 
#> $fail_rate
#> # A tibble: 2 × 5
#>   stratum duration fail_rate dropout_rate    hr
#>   <chr>      <dbl>     <dbl>        <dbl> <dbl>
#> 1 All            3    0.0770       0.0001   1  
#> 2 All          Inf    0.0770       0.0001   0.6
#> 
#> $bound
#> # A tibble: 4 × 7
#>   analysis bound probability probability0        z `~hr at bound` `nominal p`
#>      <int> <chr>       <dbl>        <dbl>    <dbl>          <dbl>       <dbl>
#> 1        1 upper      0.436        0.0117    2.27           0.718      0.0117
#> 2        2 upper      0.924        0.0436    1.74           0.816      0.0406
#> 3        1 lower      0.0344       0.612     0.285          0.959      0.388 
#> 4        2 lower      0.0344       0.612  -Inf            Inf          1     
#> 
#> $analysis
#>   analysis     time   n event       ahr     theta  info info0 info_frac
#> 1        1 19.91897 382   188 0.7356221 0.3070388 47.00 47.00 0.6045016
#> 2        2 36.06513 382   295 0.6832088 0.3809547 73.75 73.75 0.9485530
#> 
#> attr(,"class")
#> [1] "non_binding"    "ahr"            "gs_design"      "list"          
#> [5] "updated_design"