DesignWithSpending.RmdThis vignette covers how to implement designs for trials with spending assuming non-proportional hazards. We are primarily concerned with practical issues of implementation rather than design strategies, but we will not ignore design strategy.
Here we set up enrollment, failure and dropout rates along with assumptions for enrollment duration and times of analyses.
We derive statistical information at targeted analysis times.
library(gsDesign2)
xx <- gsDesign2::AHR(enrollRates = enrollRates, failRates = failRates, totalDuration = analysisTimes)
Events <- ceiling(xx$Events)
yy <- gs_info_ahr(enrollRates = enrollRates, failRates = failRates, events = Events)Now we can examine power using gs_power_npe():
zz <- gs_power_npe(theta = yy$theta, info = yy$info, info0 = yy$info0,
upper = gs_spending_bound, lower = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
)
zz
#> # A tibble: 8 x 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 2.67 0.105 0.301 0.301 22.3 22.7 22.3
#> 2 2 Upper 2.36 0.307 0.346 0.346 28.4 29.0 28.4
#> 3 3 Upper 2.18 0.493 0.370 0.370 33.4 34.0 33.4
#> 4 4 Upper 2.07 0.627 0.385 0.385 37.5 38.0 37.5
#> 5 1 Lower -1.26 0.00365 0.301 0.301 22.3 22.7 22.3
#> 6 2 Lower -0.523 0.0101 0.346 0.346 28.4 29.0 28.4
#> 7 3 Lower -0.0449 0.0176 0.370 0.370 33.4 34.0 33.4
#> 8 4 Lower 0.286 0.0250 0.385 0.385 37.5 38.0 37.5If we were using a fixed design, we would approximate the sample size as follows:
K <- 4
minx <- ((qnorm(.025) / sqrt(zz$info0[K]) + qnorm(.1) / sqrt(zz$info[K])) / zz$theta[K])^2
minx
#> [1] 1.875516If we inflate the enrollment rates by minx and use a fixed design, we will see this achieves the targeted power.
gs_power_npe(theta = yy$theta[K], info = yy$info[K] * minx, info0 = yy$info0[K] * minx, upar = qnorm(.975), lpar = -Inf) %>%
filter(Bound == "Upper")
#> # A tibble: 1 x 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 1.96 0.898 0.385 0.385 70.3 71.3 70.3The power for a group sequential design with the same final sample size is a bit lower:
zz <- gs_power_npe(theta = yy$theta, info = yy$info * minx, info0 = yy$info0 * minx,
upper = gs_spending_bound, lower = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
)
zz
#> # A tibble: 8 x 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 2.67 0.233 0.301 0.301 41.8 42.7 41.8
#> 2 2 Upper 2.36 0.569 0.346 0.346 53.3 54.4 53.3
#> 3 3 Upper 2.18 0.778 0.370 0.370 62.7 63.8 62.7
#> 4 4 Upper 2.07 0.881 0.385 0.385 70.3 71.3 70.3
#> 5 1 Lower -0.739 0.00365 0.301 0.301 41.8 42.7 41.8
#> 6 2 Lower 0.159 0.0101 0.346 0.346 53.3 54.4 53.3
#> 7 3 Lower 0.746 0.0176 0.370 0.370 62.7 63.8 62.7
#> 8 4 Lower 1.16 0.0250 0.385 0.385 70.3 71.3 70.3If we inflate this a bit we will be overpowered.
zz <- gs_power_npe(theta = yy$theta, info = yy$info * minx * 1.2, info0 = yy$info0 * minx * 1.2,
upper = gs_spending_bound, lower = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
)
zz
#> # A tibble: 8 x 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 2.67 0.294 0.301 0.301 50.2 51.2 50.2
#> 2 2 Upper 2.36 0.660 0.346 0.346 64.0 65.3 64.0
#> 3 3 Upper 2.18 0.851 0.370 0.370 75.3 76.5 75.3
#> 4 4 Upper 2.07 0.931 0.385 0.385 84.4 85.5 84.4
#> 5 1 Lower -0.554 0.00365 0.301 0.301 50.2 51.2 50.2
#> 6 2 Lower 0.400 0.0101 0.346 0.346 64.0 65.3 64.0
#> 7 3 Lower 1.03 0.0176 0.370 0.370 75.3 76.5 75.3
#> 8 4 Lower 1.47 0.0250 0.385 0.385 84.4 85.5 84.4Now we use gs_design_npe() to inflate the information proportionately to power the trial.
theta <- yy$theta
info <- yy$info
info0 <- yy$info0
upper = gs_spending_bound
lower = gs_spending_bound
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
alpha = .025
beta = .1
binding = FALSE
test_upper = TRUE
test_lower = TRUE
r = 18
tol = 1e-06
zz <- gs_design_npe(theta = yy$theta, info = yy$info, info0 = yy$info0,
upper = gs_spending_bound, lower = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL)
)
zz
#> # A tibble: 8 x 9
#> Analysis Bound Z Probability theta theta1 info info0 info1
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 2.67 0.252 0.301 0.301 44.5 45.4 44.5
#> 2 2 Upper 2.36 0.600 0.346 0.346 56.8 57.9 56.8
#> 3 3 Upper 2.18 0.804 0.370 0.370 66.8 67.9 66.8
#> 4 4 Upper 2.07 0.900 0.385 0.385 74.9 75.8 74.9
#> 5 1 Lower -0.678 0.00365 0.301 0.301 44.5 45.4 44.5
#> 6 2 Lower 0.238 0.0101 0.346 0.346 56.8 57.9 56.8
#> 7 3 Lower 0.839 0.0176 0.370 0.370 66.8 67.9 66.8
#> 8 4 Lower 1.26 0.0250 0.385 0.385 74.9 75.8 74.9