Group sequential bound computation with non-constant effect

gs_power_npe() derives group sequential bounds and boundary crossing probabilities for a design. It allows a non-constant treatment effect over time, but also can be applied for the usual homogeneous effect size designs. It requires treatment effect and statistical information at each analysis as well as a method of deriving bounds, such as spending. The routine enables two things not available in the gsDesign package: 1) non-constant effect, 2) more flexibility in boundary selection. For many applications, the non-proportional-hazards design function gs_design_nph() will be used; it calls this function. Initial bound types supported are 1) spending bounds, 2) fixed bounds, and 3) Haybittle-Peto-like bounds. The requirement is to have a boundary update method that can each bound without knowledge of future bounds. As an example, bounds based on conditional power that require knowledge of all future bounds are not supported by this routine; a more limited conditional power method will be demonstrated. Boundary family designs Wang-Tsiatis designs including the original (non-spending-function-based) O'Brien-Fleming and Pocock designs are not supported by gs_power_npe().

Usage

gs_power_npe(
  theta = 0.1,
  theta0 = NULL,
  theta1 = NULL,
  info = 1,
  info0 = NULL,
  info1 = NULL,
  info_scale = c(0, 1, 2),
  upper = gs_b,
  upar = qnorm(0.975),
  lower = gs_b,
  lpar = -Inf,
  test_upper = TRUE,
  test_lower = TRUE,
  binding = FALSE,
  r = 18,
  tol = 1e-06
)

Arguments

theta: natural parameter for group sequential design representing expected incremental drift at all analyses; used for power calculation
theta0: natural parameter for null hypothesis, if needed for upper bound computation
theta1: natural parameter for alternate hypothesis, if needed for lower bound computation
info: statistical information at all analyses for input theta
info0: statistical information under null hypothesis, if different than info; impacts null hypothesis bound calculation
info1: statistical information under hypothesis used for futility bound calculation if different from info; impacts futility hypothesis bound calculation
info_scale: the information scale for calculation, default is 2, other options are 0 or 1.
upper: function to compute upper bound
upar: parameter to pass to upper
lower: function to compare lower bound
lpar: parameter to pass to lower
test_upper: indicator of which analyses should include an upper (efficacy) bound; single value of TRUE (default) indicates all analyses; otherwise, a logical vector of the same length as info should indicate which analyses will have an efficacy bound
test_lower: indicator of which analyses should include a lower bound; single value of TRUE (default) indicates all analyses; single value FALSE indicated no lower bound; otherwise, a logical vector of the same length as info should indicate which analyses will have a lower bound
binding: indicator of whether futility bound is binding; default of FALSE is recommended
r: Integer, at least 2; default of 18 recommended by Jennison and Turnbull
tol: Tolerance parameter for boundary convergence (on Z-scale)

Specification

The contents of this section are shown in PDF user manual only.

Author

Keaven Anderson keaven_anderson@merck.com

Examples

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

# Default (single analysis; Type I error controlled)
gs_power_npe(theta = 0) %>% filter(Bound == "Upper")
#> # A tibble: 1 × 10
#>   Analysis Bound     Z Probability theta theta1    IF  info info0 info1
#>      <int> <chr> <dbl>       <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1        1 Upper  1.96      0.0250     0      0     1     1     1     1

# Fixed bound
gs_power_npe(
  theta = c(.1, .2, .3), 
  info = (1:3) * 40, 
  upper = gs_b, 
  upar = gsDesign::gsDesign(k = 3,sfu = gsDesign::sfLDOF)$upper$bound,
  lower = gs_b, 
  lpar =  c(-1, 0, 0))
#> # A tibble: 6 × 10
#>   Analysis Bound     Z Probability theta theta1    IF  info info0 info1
#>      <int> <chr> <dbl>       <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1        1 Upper  3.71     0.00104   0.1    0.1 0.333    40    40    40
#> 2        2 Upper  2.51     0.235     0.2    0.2 0.667    80    80    80
#> 3        3 Upper  1.99     0.869     0.3    0.3 1       120   120   120
#> 4        1 Lower -1        0.0513    0.1    0.1 0.333    40    40    40
#> 5        2 Lower  0        0.0715    0.2    0.2 0.667    80    80    80
#> 6        3 Lower  0        0.0715    0.3    0.3 1       120   120   120

# Same fixed efficacy bounds, no futility bound (i.e., non-binding bound), null hypothesis
gs_power_npe(
  theta = rep(0, 3), 
  info = (1:3) * 40,
  upar = gsDesign::gsDesign(k = 3,sfu = gsDesign::sfLDOF)$upper$bound,
  lpar = rep(-Inf, 3)) %>% 
  filter(Bound == "Upper")
#> # A tibble: 3 × 10
#>   Analysis Bound     Z Probability theta theta1    IF  info info0 info1
#>      <int> <chr> <dbl>       <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1        1 Upper  3.71    0.000104     0      0 0.333    40    40    40
#> 2        2 Upper  2.51    0.00605      0      0 0.667    80    80    80
#> 3        3 Upper  1.99    0.0250       0      0 1       120   120   120

# Fixed bound with futility only at analysis 1; efficacy only at analyses 2, 3
gs_power_npe(
  theta = c(.1, .2, .3), 
  info = (1:3) * 40,
  upper = gs_b,
  upar = c(Inf, 3, 2), 
  lower = gs_b,
  lpar = c(qnorm(.1), -Inf, -Inf))
#> # A tibble: 6 × 10
#>   Analysis Bound       Z Probability theta theta1    IF  info info0 info1
#>      <int> <chr>   <dbl>       <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1        1 Upper  Inf         0        0.1    0.1 0.333    40    40    40
#> 2        2 Upper    3         0.113    0.2    0.2 0.667    80    80    80
#> 3        3 Upper    2         0.887    0.3    0.3 1       120   120   120
#> 4        1 Lower   -1.28      0.0278   0.1    0.1 0.333    40    40    40
#> 5        2 Lower -Inf         0.0278   0.2    0.2 0.667    80    80    80
#> 6        3 Lower -Inf         0.0278   0.3    0.3 1       120   120   120

# Spending function bounds
# Lower spending based on non-zero effect
gs_power_npe(
  theta = c(.1, .2, .3), 
  info = (1:3) * 40,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfHSD, total_spend = 0.1, param = -1, timing = NULL))
#> # A tibble: 6 × 10
#>   Analysis Bound       Z Probability theta theta1    IF  info info0 info1
#>      <int> <chr>   <dbl>       <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1        1 Upper  3.71       0.00104   0.1    0.1 0.333    40    40    40
#> 2        2 Upper  2.51       0.235     0.2    0.2 0.667    80    80    80
#> 3        3 Upper  1.99       0.883     0.3    0.3 1       120   120   120
#> 4        1 Lower -1.36       0.0230    0.1    0.1 0.333    40    40    40
#> 5        2 Lower  0.0726     0.0552    0.2    0.2 0.667    80    80    80
#> 6        3 Lower  1.86       0.100     0.3    0.3 1       120   120   120

# Same bounds, but power under different theta
gs_power_npe(
  theta = c(.15, .25, .35), 
  info = (1:3) * 40,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfHSD, total_spend = 0.1, param = -1, timing = NULL))
#> # A tibble: 6 × 10
#>   Analysis Bound      Z Probability theta theta1    IF  info info0 info1
#>      <int> <chr>  <dbl>       <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1        1 Upper  3.71      0.00288  0.15   0.15 0.333    40    40    40
#> 2        2 Upper  2.51      0.391    0.25   0.25 0.667    80    80    80
#> 3        3 Upper  1.99      0.931    0.35   0.35 1       120   120   120
#> 4        1 Lower -1.05      0.0230   0.15   0.15 0.333    40    40    40
#> 5        2 Lower  0.520     0.0552   0.25   0.25 0.667    80    80    80
#> 6        3 Lower  2.41      0.100    0.35   0.35 1       120   120   120

# Two-sided symmetric spend, O'Brien-Fleming spending
# Typically, 2-sided bounds are binding
x <- gs_power_npe(
  theta = rep(0, 3), 
  info = (1:3) * 40,
  binding = TRUE,
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_spending_bound,
  lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL))

# Re-use these bounds under alternate hypothesis
# Always use binding = TRUE for power calculations
gs_power_npe(
  theta = c(.1, .2, .3),
  info = (1:3) * 40,
  binding = TRUE,
  upar = (x %>% filter(Bound == "Upper"))$Z,
  lpar = -(x %>% filter(Bound == "Upper"))$Z)
#> # A tibble: 6 × 10
#>   Analysis Bound     Z Probability theta theta1    IF  info info0 info1
#>      <int> <chr> <dbl>       <dbl> <dbl>  <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1        1 Upper  3.71  0.00104      0.1    0.1 0.333    40    40    40
#> 2        2 Upper  2.51  0.235        0.2    0.2 0.667    80    80    80
#> 3        3 Upper  1.99  0.902        0.3    0.3 1       120   120   120
#> 4        1 Lower -3.71  0.00000704   0.1    0.1 0.333    40    40    40
#> 5        2 Lower -2.51  0.0000151    0.2    0.2 0.667    80    80    80
#> 6        3 Lower -1.99  0.0000151    0.3    0.3 1       120   120   120