Group sequential bound computation with non-constant effect
Source:R/gs_power_npe.r
gs_power_npe.Rdgs_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,
theta1 = NULL,
info = 1,
info1 = NULL,
info0 = NULL,
binding = FALSE,
upper = gs_b,
lower = gs_b,
upar = qnorm(0.975),
lpar = -Inf,
test_upper = TRUE,
test_lower = TRUE,
r = 18,
tol = 1e-06
)Arguments
- theta
natural parameter for group sequential design representing expected incremental drift at all analyses; used for power calculation
- theta1
natural parameter for alternate hypothesis, if needed for lower bound computation
- info
statistical information at all analyses for input
theta- info1
statistical information under hypothesis used for futility bound calculation if different from
info; impacts futility hypothesis bound calculation- info0
statistical information under null hypothesis, if different than
info; impacts null hypothesis bound calculation- binding
indicator of whether futility bound is binding; default of FALSE is recommended
- upper
function to compute upper bound
- lower
function to compare lower bound
- upar
parameter to pass to upper
- 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
infoshould 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
infoshould indicate which analyses will have a lower bound- r
Integer, at least 2; default of 18 recommended by Jennison and Turnbull
- tol
Tolerance parameter for boundary convergence (on Z-scale)
Author
Keaven Anderson keaven_anderson@merck.com
Examples
library(gsDesign)
library(dplyr)
# Default (single analysis; Type I error controlled)
gs_power_npe(theta=0) %>% filter(Bound=="Upper")
#> # A tibble: 1 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 1.96 0.0250 0 0 1 1 1
# Fixed bound
gs_power_npe(theta = c(.1, .2, .3), info = (1:3) * 40, info0 = (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 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 3.71 0.00104 0.1 0.1 40 40 40
#> 2 2 Upper 2.51 0.235 0.2 0.2 80 80 80
#> 3 3 Upper 1.99 0.869 0.3 0.3 120 120 120
#> 4 1 Lower -1 0.0513 0.1 0.1 40 40 40
#> 5 2 Lower 0 0.0715 0.2 0.2 80 80 80
#> 6 3 Lower 0 0.0715 0.3 0.3 120 120 120
# 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 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 3.71 0.000104 0 0 40 40 40
#> 2 2 Upper 2.51 0.00605 0 0 80 80 80
#> 3 3 Upper 1.99 0.0250 0 0 120 120 120
# 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,
upar = c(Inf, 3, 2),
lpar = c(qnorm(.1), -Inf, -Inf))
#> # A tibble: 6 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper Inf 0 0.1 0.1 40 40 40
#> 2 2 Upper 3 0.113 0.2 0.2 80 80 80
#> 3 3 Upper 2 0.887 0.3 0.3 120 120 120
#> 4 1 Lower -1.28 0.0278 0.1 0.1 40 40 40
#> 5 2 Lower -Inf 0.0278 0.2 0.2 80 80 80
#> 6 3 Lower -Inf 0.0278 0.3 0.3 120 120 120
# 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 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 3.71 0.00104 0.1 0.1 40 40 40
#> 2 2 Upper 2.51 0.235 0.2 0.2 80 80 80
#> 3 3 Upper 1.99 0.883 0.3 0.3 120 120 120
#> 4 1 Lower -1.36 0.0230 0.1 0.1 40 40 40
#> 5 2 Lower 0.0726 0.0552 0.2 0.2 80 80 80
#> 6 3 Lower 1.86 0.100 0.3 0.3 120 120 120
# Same bounds, but power under different theta
gs_power_npe(theta = c(.15, .25, .35),
theta1 = 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 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 3.71 0.00288 0.15 0.1 40 40 40
#> 2 2 Upper 2.51 0.392 0.25 0.2 80 80 80
#> 3 3 Upper 1.99 0.956 0.35 0.3 120 120 120
#> 4 1 Lower -1.36 0.0104 0.15 0.1 40 40 40
#> 5 2 Lower 0.0726 0.0221 0.25 0.2 80 80 80
#> 6 3 Lower 1.86 0.0370 0.35 0.3 120 120 120
# Two-sided symmetric spend, O'Brien-Fleming spending
# Typically, 2-sided bounds are binding
xx <- gs_power_npe(theta = rep(0, 3),
theta1 = rep(0, 3),
info = (1:3) * 40,
upper = gs_spending_bound,
binding = TRUE,
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))
xx
#> # A tibble: 6 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 3.71 0.000104 0 0 40 40 40
#> 2 2 Upper 2.51 0.00605 0 0 80 80 80
#> 3 3 Upper 1.99 0.0250 0 0 120 120 120
#> 4 1 Lower -3.71 0.000104 0 0 40 40 40
#> 5 2 Lower -2.51 0.00605 0 0 80 80 80
#> 6 3 Lower -1.99 0.0250 0 0 120 120 120
# Re-use these bounds under alternate hypothesis
# Always use binding = TRUE for power calculations
upar <- (xx %>% filter(Bound == "Upper"))$Z
gs_power_npe(theta = c(.1, .2, .3),
info = (1:3) * 40,
binding = TRUE,
upar = upar,
lpar = -upar)
#> # A tibble: 6 × 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 3.71 0.00104 0.1 0.1 40 40 40
#> 2 2 Upper 2.51 0.235 0.2 0.2 80 80 80
#> 3 3 Upper 1.99 0.902 0.3 0.3 120 120 120
#> 4 1 Lower -3.71 0.00000704 0.1 0.1 40 40 40
#> 5 2 Lower -2.51 0.0000151 0.2 0.2 80 80 80
#> 6 3 Lower -1.99 0.0000151 0.3 0.3 120 120 120