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Group sequential bound computation with non-constant effect

Source: R/gs_power_npe.R
gs_power_npe.Rd

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("h0_h1_info", "h0_info", "h1_info"),
  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

Information scale for calculation. Options are:

  • "h0_h1_info" (default): variance under both null and alternative hypotheses is used.

  • "h0_info": variance under null hypothesis is used.

  • "h1_info": variance under alternative hypothesis is used.

upper

Function to compute upper bound.

upar

Parameters passed to upper.

lower

Function to compare lower bound.

lpar

parameters passed 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 of 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 value controlling grid for numerical integration as in Jennison and Turnbull (2000); default is 18, range is 1 to 80. Larger values provide larger number of grid points and greater accuracy. Normally, r will not be changed by the user.

tol

Tolerance parameter for boundary convergence (on Z-scale).

Value

A tibble with columns as analysis index, bounds, z, crossing probability, theta (standardized treatment effect), theta1 (standardized treatment effect under alternative hypothesis), information fraction, and statistical information.

Specification

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

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 info_frac  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 info_frac  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 info_frac  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 info_frac  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 info_frac  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 info_frac  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 info_frac  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

# Different values of `r` and `tol` lead to different numerical accuracy
# Larger `r` and smaller `tol` give better accuracy, but leads to slow computation
n_analysis <- 5
gs_power_npe(
  theta = rep(0.1, n_analysis),
  theta0 = NULL,
  theta1 = NULL,
  info = 1:n_analysis,
  info0 = 1:n_analysis,
  info1 = NULL,
  info_scale = "h0_info",
  upper = gs_spending_bound,
  upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
  lower = gs_b,
  lpar = -rep(Inf, n_analysis),
  test_upper = TRUE,
  test_lower = FALSE,
  binding = FALSE,
  # Try different combinations of (r, tol) with
  # r in 6, 18, 24, 30, 35, 40, 50, 60, 70, 80, 90, 100
  # tol in 1e-6, 1e-12
  r = 6,
  tol = 1e-6
)
#> # A tibble: 10 × 10
#>    analysis bound       z probability theta theta1 info_frac  info info0 info1
#>       <int> <chr>   <dbl>       <dbl> <dbl>  <dbl>     <dbl> <int> <int> <int>
#>  1        1 upper    4.88 0.000000890   0.1    0.1       0.2     1     1     1
#>  2        2 upper    3.36 0.000650      0.1    0.1       0.4     2     2     2
#>  3        3 upper    2.68 0.00627       0.1    0.1       0.6     3     3     3
#>  4        4 upper    2.29 0.0200        0.1    0.1       0.8     4     4     4
#>  5        5 upper    2.03 0.0408        0.1    0.1       1       5     5     5
#>  6        1 lower -Inf    0             0.1    0.1       0.2     1     1     1
#>  7        2 lower -Inf    0             0.1    0.1       0.4     2     2     2
#>  8        3 lower -Inf    0             0.1    0.1       0.6     3     3     3
#>  9        4 lower -Inf    0             0.1    0.1       0.8     4     4     4
#> 10        5 lower -Inf    0             0.1    0.1       1       5     5     5