Group sequential design computation with non-constant effect and information.
Source:R/gs_design_npe.R, R/gs_power_npe.R
gs_power_design_npe.RdThe following two functions allow a non-constant treatment effect over time, but also can be applied for the usual homogeneous effect size designs. They require treatment effect and statistical information at each analysis as well as a method of deriving bounds, such as spending. Initial bound types supported are spending bounds and fixed bounds. These routines enables two things not available in the gsDesign package: 1) non-constant effect, 2) more flexibility in boundary selection.
gs_power_npe() derives group sequential bounds and boundary crossing probabilities
for a design, given treatment effect and information at each analysis and the
method of deriving bounds, such as spending.
gs_design_npe() derives group sequential design size,
bounds and boundary crossing probabilities based on proportionate
information and effect size at analyses, as well as the
method of deriving bounds, such as spending.
The only differences in arguments between the two functions are the alpha and beta
parameters used in the gs_design_npe().
Usage
gs_design_npe(
theta = 0.1,
theta0 = 0,
theta1 = theta,
info = 1,
info0 = NULL,
info1 = NULL,
info_scale = c("h0_h1_info", "h0_info", "h1_info"),
alpha = 0.025,
beta = 0.1,
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
)
gs_power_npe(
theta = 0.1,
theta0 = 0,
theta1 = theta,
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 cumulative drift at all analyses; used for power calculation. It can be a scalar (constant treatment effect) or a vector (non-constant treatment effect). The user must provide a value for
theta.- theta0
Natural parameter for null hypothesis. It can be a scalar (constant treatment effect) or a vector (non-constant treatment effect). The default is 0. If a value other than 0 is provided, it affects upper bound computation.
- theta1
Natural parameter for alternate hypothesis, if needed for lower bound computation. It can be a scalar (constant treatment effect) or a vector (non-constant treatment effect). The default is the same as
theta, which yields the usual beta-spending. If set to 0, spending is 2-sided under the null hypothesis.- info
Statistical information at all analyses for input
theta. It is a vector of positive numbers in increasing order. The user must provide values corresponding totheta.- info0
Statistical information under null hypothesis. It is a vector of all positive numbers with increasing order. Default is set to be the same as
info. Ifinfo0is different thaninfo, it impacts null hypothesis bound calculation.- info1
Statistical information under hypothesis used for futility bound calculation. It is a vector of all positive numbers with increasing order. Default is set to be the same as
info. Ifinfo1is different frominfo, it impacts futility 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. This is often used for testing methods that use local alternatives, such as the Schoenfeld method."h1_info": variance under alternative hypothesis is used.
- alpha
One-sided Type I error.
- beta
Type II error.
- 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 asinfoshould 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 ofFALSEindicated no lower bound; otherwise, a logical vector of the same length asinfoshould indicate which analyses will have a lower bound.- binding
Indicator of whether futility bound is binding; default of
FALSEis 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,
rwill not be changed by the user.- tol
Tolerance parameter for boundary convergence (on Z-scale).
Value
A tibble with columns of
analysis: analysis index.bound: either of value"upper"or"lower", indicating the upper and lower bound.z: the Z-score bounds.probability: cumulative probability of crossing the bound at or before the analysis.theta: same as the input.theta1: same as the input.info: statistical information at each analysis.If it is returned by
gs_power_npe, theinfo,info0,info1are same as the input.If it is returned by
gs_design_npe, theinfo,info0,info1are changed by a constant scale factor. factor to ensure the design has power1 - beta.
info0: statistical information under the null at each analysis.info1: statistical information under the alternative at each analysis.info_frac: information fraction at each analysis, i.e.,info / max(info).
Details
The bound specifications (upper, lower, upar, lpar) of gs_design_npe()
will be used to ensure Type I error and other boundary properties are as specified.
See the help file of gs_spending_bound() for details on spending function.
Specification
The contents of this section are shown in PDF user manual only.
The contents of this section are shown in PDF user manual only.
Author
Keaven Anderson keaven_anderson@merck.com
Examples
library(gsDesign)
# Example 1 ----
# gs_design_npe with single analysis
# Lachin book p 71 difference of proportions example
pc <- .28 # Control response rate
pe <- .40 # Experimental response rate
p0 <- (pc + pe) / 2 # Ave response rate under H0
# Information per increment of 1 in sample size
info0 <- 1 / (p0 * (1 - p0) * 4)
info <- 1 / (pc * (1 - pc) * 2 + pe * (1 - pe) * 2)
# Result should round up to next even number = 652
# Divide information needed under H1 by information per patient added
gs_design_npe(theta = pe - pc, info = info, info0 = info0)
#> # A tibble: 1 × 10
#> analysis bound z probability probability0 theta info info0 info1
#> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 upper 1.96 0.9 0.025 0.12 737. 725. 737.
#> # ℹ 1 more variable: info_frac <dbl>
# Example 2 ----
# gs_design_npe with with fixed bound
x <- gs_design_npe(
alpha = 0.0125,
theta = c(.1, .2, .3),
info = (1:3) * 80,
info0 = (1:3) * 80,
upper = gs_b,
upar = gsDesign::gsDesign(k = 3, sfu = gsDesign::sfLDOF, alpha = 0.0125)$upper$bound,
lower = gs_b,
lpar = c(-1, 0, 0)
)
x
#> # A tibble: 6 × 10
#> analysis bound z probability probability0 theta info_frac info info0
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 upper 4.17 0.000278 0.0000152 0.1 0.333 51.6 51.6
#> 2 1 lower -1 0.0429 0.159 0.1 0.333 51.6 51.6
#> 3 2 upper 2.85 0.208 0.00222 0.2 0.667 103. 103.
#> 4 2 lower 0 0.0537 0.513 0.2 0.667 103. 103.
#> 5 3 upper 2.26 0.900 0.0125 0.3 1 155. 155.
#> 6 3 lower 0 0.0537 0.606 0.3 1 155. 155.
#> # ℹ 1 more variable: info1 <dbl>
# Same upper bound; this represents non-binding Type I error and will total 0.025
gs_power_npe(
theta = rep(0, 3),
info = (x |> dplyr::filter(bound == "upper"))$info,
upper = gs_b,
upar = (x |> dplyr::filter(bound == "upper"))$z,
lower = gs_b,
lpar = rep(-Inf, 3)
)
#> # 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 4.17 0.0000152 0 0 0.333 51.6 51.6 51.6
#> 2 2 upper 2.85 0.00222 0 0 0.667 103. 103. 103.
#> 3 3 upper 2.26 0.0125 0 0 1 155. 155. 155.
#> 4 1 lower -Inf 0 0 0 0.333 51.6 51.6 51.6
#> 5 2 lower -Inf 0 0 0 0.667 103. 103. 103.
#> 6 3 lower -Inf 0 0 0 1 155. 155. 155.
# Example 3 ----
# gs_design_npe with spending bound
# Design with futility only at analysis 1; efficacy only at analyses 2, 3
# Spending bound for efficacy; fixed bound for futility
# NOTE: test_upper and test_lower DO NOT WORK with gs_b; must explicitly make bounds infinite
# test_upper and test_lower DO WORK with gs_spending_bound
gs_design_npe(
theta = c(.1, .2, .3),
info = (1:3) * 40,
info0 = (1:3) * 40,
upper = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
lower = gs_b,
lpar = c(-1, -Inf, -Inf),
test_upper = c(FALSE, TRUE, TRUE)
)
#> # A tibble: 6 × 10
#> analysis bound z probability probability0 theta info_frac info info0
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 upper Inf 0 0 0.1 0.333 44.6 44.6
#> 2 1 lower -1 0.0477 0.159 0.1 0.333 44.6 44.6
#> 3 2 upper 2.51 0.267 0.00605 0.2 0.667 89.1 89.1
#> 4 2 lower -Inf 0.0477 0.159 0.2 0.667 89.1 89.1
#> 5 3 upper 1.99 0.900 0.0249 0.3 1 134. 134.
#> 6 3 lower -Inf 0.0477 0.159 0.3 1 134. 134.
#> # ℹ 1 more variable: info1 <dbl>
# one can try `info_scale = "h1_info"` or `info_scale = "h0_info"` here
gs_design_npe(
theta = c(.1, .2, .3),
info = (1:3) * 40,
info0 = (1:3) * 30,
info_scale = "h1_info",
upper = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025, param = NULL, timing = NULL),
lower = gs_b,
lpar = c(-1, -Inf, -Inf),
test_upper = c(FALSE, TRUE, TRUE)
)
#> # A tibble: 6 × 10
#> analysis bound z probability probability0 theta info_frac info info0
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 upper Inf 0 0 0.1 0.333 44.6 44.6
#> 2 1 lower -1 0.0477 0.159 0.1 0.333 44.6 44.6
#> 3 2 upper 2.51 0.267 0.00605 0.2 0.667 89.1 89.1
#> 4 2 lower -Inf 0.0477 0.159 0.2 0.667 89.1 89.1
#> 5 3 upper 1.99 0.900 0.0249 0.3 1 134. 134.
#> 6 3 lower -Inf 0.0477 0.159 0.3 1 134. 134.
#> # ℹ 1 more variable: info1 <dbl>
# Example 4 ----
# gs_design_npe with spending function bounds
# 2-sided asymmetric bounds
# Lower spending based on non-zero effect
gs_design_npe(
theta = c(.1, .2, .3),
info = (1:3) * 40,
info0 = (1:3) * 30,
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 probability0 theta info_frac info info0
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 upper 3.71 0.000145 0.000104 0.1 0.333 43.5 32.7
#> 2 1 lower -1.34 0.0139 0.0909 0.1 0.333 43.5 32.7
#> 3 2 upper 2.51 0.258 0.00605 0.2 0.667 87.1 65.3
#> 4 2 lower 0.150 0.0460 0.562 0.2 0.667 87.1 65.3
#> 5 3 upper 1.99 0.900 0.0249 0.3 1 131. 98.0
#> 6 3 lower 2.00 0.0908 0.976 0.3 1 131. 98.0
#> # ℹ 1 more variable: info1 <dbl>
# Example 5 ----
# gs_design_npe with two-sided symmetric spend, O'Brien-Fleming spending
# Typically, 2-sided bounds are binding
xx <- gs_design_npe(
theta = c(.1, .2, .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)
)
xx
#> # A tibble: 6 × 10
#> analysis bound z probability probability0 theta info_frac info info0
#> <int> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 upper 3.71 0.00104 0.000104 0.1 0.333 39.8 39.8
#> 2 1 lower -3.08 0.000104 0.00104 0.1 0.333 39.8 39.8
#> 3 2 upper 2.51 0.233 0.00605 0.2 0.667 79.5 79.5
#> 4 2 lower -0.728 0.00605 0.233 0.2 0.667 79.5 79.5
#> 5 3 upper 1.99 0.900 0.0250 0.3 1 119. 119.
#> 6 3 lower 1.28 0.0250 0.900 0.3 1 119. 119.
#> # ℹ 1 more variable: info1 <dbl>
# 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,
upper = gs_b,
lower = gs_b,
upar = (xx |> dplyr::filter(bound == "upper"))$z,
lpar = -(xx |> dplyr::filter(bound == "upper"))$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
# Example 6 ----
# Default of gs_power_npe (single analysis; Type I error controlled)
gs_power_npe(theta = 0) |> dplyr::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
# Example 7 ----
# gs_power_npe with 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)
) |>
dplyr::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
# Example 8 ----
# gs_power_npe with fixed bound testing 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
# Example 9 ----
# gs_power_npe with 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
# Example 10 ----
# gs_power_npe with 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 |> dplyr::filter(bound == "upper"))$z,
lpar = -(x |> dplyr::filter(bound == "upper"))$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
# Example 11 ----
# 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 = 0.1,
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