Calculate optimal design parameters for a two-stage gold-standard design
Source:R/optimization_methods.R
optimize_design_twostage.RdCalculate optimal design parameters for a two-stage gold-standard design
Usage
optimize_design_twostage(
cT2 = NULL,
cP1 = NULL,
cP2 = NULL,
cC1 = NULL,
cC2 = NULL,
bTP1f = NULL,
bTP1e = NULL,
bTC1f = NULL,
bTC1e = NULL,
alpha = 0.025,
beta = 0.2,
alternative_TP = 0.4,
alternative_TC = 0,
Delta = 0.2,
varT = 1,
varP = 1,
varC = 1,
binding_futility = FALSE,
always_both_futility_tests = TRUE,
round_n = TRUE,
lambda = 1,
kappa = 0,
eta = 0,
objective = quote(sqrt(lambda)^2 * ASN[["H11"]] + (1 - sqrt(lambda)) * sqrt(lambda) *
ASN[["H10"]] + (1 - sqrt(lambda)) * sqrt(lambda) * ASN[["H01"]] + (1 -
sqrt(lambda))^2 * ASN[["H00"]] + kappa * (sqrt(lambda)^2 * ASNP[["H11"]] + (1 -
sqrt(lambda)) * sqrt(lambda) * ASNP[["H10"]] + (1 - sqrt(lambda)) * sqrt(lambda) *
ASNP[["H01"]] + (1 - sqrt(lambda))^2 * ASNP[["H00"]] + eta * cumn[[2]][["P"]]) + eta
* (cumn[[2]][["T"]] + cumn[[2]][["P"]] + cumn[[2]][["C"]])),
inner_tol_objective = .Machine$double.eps^0.25,
mvnorm_algorithm = mvtnorm::Miwa(steps = 128, checkCorr = FALSE, maxval = 1000),
nloptr_x0 = NULL,
nloptr_lb = NULL,
nloptr_ub = NULL,
nloptr_opts = list(algorithm = "NLOPT_LN_SBPLX", ftol_rel = 1e-04, xtol_abs = 0.001,
xtol_rel = 0.01, maxeval = 1000, print_level = 0),
print_progress = TRUE,
...
)Arguments
- cT2
(numeric) allocation ratio nT2 / nT1. Parameter to be optimized if left unspecified.
- cP1
(numeric) allocation ratio nP1 / nT1. Parameter to be optimized if left unspecified.
- cP2
(numeric) allocation ratio nP2 / nT1. Parameter to be optimized if left unspecified.
- cC1
(numeric) allocation ratio nC1 / nT1. Parameter to be optimized if left unspecified.
- cC2
(numeric) allocation ratio nC2 / nT1. Parameter to be optimized if left unspecified.
- bTP1f
(numeric) first stage futility boundary for the T vs. P testing problem. Parameter to be optimized if left unspecified.
- bTP1e
(numeric) first stage critical value for the T vs. P testing problem. Parameter to be optimized if left unspecified.
- bTC1f
(numeric) first stage futility boundary for the T vs. C testing problem. Parameter to be optimized if left unspecified.
- bTC1e
(numeric) first stage critical value for the T vs. C testing problem. Parameter to be optimized if left unspecified.
- alpha
type I error rate.
- beta
type II error rate.
- alternative_TP
assumed difference between T and P under H1. Positive values favor T.
- alternative_TC
assumed difference between T and C under H1. Positive values favor T.
- Delta
non-inferiority margin for the test \(X_T - X_C \leq - \Delta\) vs. \(X_T - X_C > - \Delta\).
- varT
variance of treatment group.
- varP
variance of placebo group.
- varC
variance of control group.
- binding_futility
(logical) controls if futility boundaries are binding.
- always_both_futility_tests
(logical) if true, both futility tests are performed after the first stage. If false, a 'completely sequential' testing procedure is employed (see Appendix of (Meis et al. 2023) ).
- round_n
(logical) if TRUE, a design with integer valued sample sizes is returned.
- lambda
(numeric) weight of the alternative hypothesis in the default objective function. 1-lambda is the weight of the null hypothesis.
- kappa
(numeric) penalty factor for placebo patients in the default objective function.
- eta
(numeric) penalty factor for the maximum sample size in the default objective function.
- objective
(expression) objective criterion.
- inner_tol_objective
(numeric) used to determine the tolerances for integrals and nuisance optimization problems inside the objective function.
- mvnorm_algorithm
algorithm for multivariate integration passed to
pmvnorm.- nloptr_x0
(numeric vector) starting point for optimization.
- nloptr_lb
(numeric vector) lower bound box for box constrained optimization.
- nloptr_ub
(numeric vector) upper bound box for box constrained optimization.
- nloptr_opts
(list) nloptr options. See
nloptr.- print_progress
(logical) controls whether optimization progress should be visualized during the calculation.
- ...
additional arguments passed along.
Details
This function calculates optimal design parameters for a two-stage three-arm gold-standard
non-inferiority trial. Run vignette("Introduction", package = "OptimalGoldstandardDesigns")
to see some examples related to the associated paper (Meis et al. 2023)
.
Parameters which can be optimized are the allocation ratios for all groups and stages and the futility and efficacy boundaries of the first stage. The allocation ratios are cT2 = nT2 / nT1, cP1 = nP1 / nT1, cP2 = nP2 / nT1, cC1 = nC1 / nT1 and cC2 = nC2 / nT1. Here, nT1 denotes the sample size of the treatment group in the first stage, nP2 the sample size of the placebo group in the second stage, etc. The first stage efficacy boundaries are bTP1e for the treatment vs placebo testing problem, and bTC1e for the treatment vs control non-inferiority testing problem. The futility boundaries are denoted by bTP1f and bTC1f.
If these parameters are left unspecified or set to NULL, they will be included into the
optimization process, otherwise they will be considered boundary constraints.
You may also supply quoted expressions as arguments for these
parameters to solve a constrained optimization problem. For example, you can supply
cT2 = 1, cP2 = quote(cP1), cC2 = quote(cC1) to ensure that the first and second
stage allocation ratios are equal.
The design is optimized with respect to the objective criterion given by the parameter
objective. The default objective function is described in the
Subsection Optimizing group sequential gold-standard designs in Section 2
of (Meis et al. 2023)
. Additionally,
this objective includes a term to penalize the maximum sample size of a trial,
which can be controlled by the parameter eta (default is eta=0).
Designs are calculated to fulfill the following constraints: the family-wise type I error
rate is controlled at alpha under any combination of the two null hypotheses
muT - muP = 0 and muT - muC + Delta = 0.
The power to reject both hypothesis given both alternative
hypotheses muT - muP = alternative_TP and muT - muC + Delta = alternative_TC + Delta
is at least 1 - beta. Variances are assumed to be given by varT, varP and varC.
If binding_futility is TRUE, type I error recycling is used.
If always_both_futility_tests is TRUE, it is assumed that futility tests for both
hypotheses are performed at interim, regardless of whether the treatment vs placebo null hypothesis
was successfully rejected. If always_both_futility_tests is FALSE, the futility
test for the treatment vs. control testing problem only needs to be done if the null for the
treatment vs. placebo testing problem was rejected in the first stage.
References
Meis J, Pilz M, Herrmann C, Bokelmann B, Rauch G, Kieser M (2023). “Optimization of the two-stage group sequential three-arm gold-standard design for non-inferiority trials.” Statistics in Medicine, 42(4), 536-558. doi:10.1002/sim.9630 .
Examples
# Should take about 15 seconds.
# \donttest{
optimize_design_twostage(
cT2 = 1,
cP2 = quote(cP1),
cC2 = quote(cC1),
bTP1f = -Inf,
bTC1f = -Inf,
beta = 0.2,
alternative_TP = 0.4,
alternative_TC = 0,
Delta = 0.2,
binding_futility = TRUE,
lambda = .9,
kappa = 1,
nloptr_opts = list(algorithm = "NLOPT_LN_SBPLX", ftol_rel = 1e-01)
)
#>
iteration: 1/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.250000, 1.000000, 2.105099, 2.270933) f(x) = 915.8373
iteration: 2/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.250000, 1.000000, 2.105099, 2.270933) f(x) = 915.8373
iteration: 3/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.250000, 1.000000, 2.105099, 2.270933) f(x) = 915.8373
iteration: 4/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.428125, 1.000000, 2.105099, 2.270933) f(x) = 917.2013
iteration: 5/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.250000, 1.712500, 2.105099, 2.270933) f(x) = 1022.588
iteration: 6/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.428125, 0.287500, 2.105099, 2.270933) f(x) = 1412.46
iteration: 7/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.2945313, 1.3562500, 2.1050990, 2.2709330) f(x) = 935.5908
iteration: 8/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3835937, 0.6437500, 2.1050990, 2.2709330) f(x) = 963.2407
iteration: 9/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3167969, 1.1781250, 2.1050990, 2.2709330) f(x) = 911.3385
iteration: 10/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.1386719, 1.1781250, 2.1050990, 2.2709330) f(x) = 1108.736
iteration: 11/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1050990, 2.2709330) f(x) = 904.044
iteration: 12/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.2139503, 2.2709330) f(x) = 908.8018
iteration: 13/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1050990, 2.5041598) f(x) = 907.0289
iteration: 14/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 1.9962477, 2.5041598) f(x) = 918.6257
iteration: 15/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1595246, 2.3292397) f(x) = 903.9803
iteration: 16/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1595246, 2.0960129) f(x) = 935.7317
iteration: 17/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1187054, 2.4021231) f(x) = 903.2534
iteration: 18/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1731310, 2.4604297) f(x) = 906.1748
iteration: 19/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1221070, 2.3183072) f(x) = 902.9637
iteration: 20/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.0812878, 2.3911905) f(x) = 903.0437
iteration: 21/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.0846894, 2.3073747) f(x) = 902.7042
iteration: 22/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.0676814, 2.2600005) f(x) = 904.3596
iteration: 23/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1255086, 2.2344913) f(x) = 906.9245
iteration: 24/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.0923430, 2.3520157) f(x) = 902.4161
iteration: 25/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.0549254, 2.3410832) f(x) = 903.1108
iteration: 26/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1053116, 2.3240012) f(x) = 902.5799
iteration: 27/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1129652, 2.3686423) f(x) = 902.6578
iteration: 28/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1058963, 2.3533254) f(x) = 902.4788
iteration: 29/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.0929276, 2.3813399) f(x) = 902.7331
iteration: 30/100 cP1 <- x[1L] cC1 <- x[2L] bTP1e <- x[3L] bTC1e <- x[4L] x = c(0.3557617, 1.0445312, 2.1022156, 2.3383359) f(x) = 902.4439
#> Optimization finished. Calculating final design with greater accuracy...
#> Sample sizes (stage 1): T: 209, P: 75, C: 219
#> Sample sizes (stage 2): T: 209, P: 75, C: 219
#> Efficacy boundaries (stage 1): Z_TP_e: 2.09234, Z_TC_e: 2.35202
#> Futility boundaries (stage 1): Z_TP_f: -Inf, Z_TC_f: -Inf
#> Efficacy boundaries (stage 2): Z_TP_e: 2.29395, Z_TC_e: 2.06436
#> Inverse normal combination test weights (TP): w1: 0.70711, w2: 0.70711
#> Inverse normal combination test weights (TC): w1: 0.70711, w2: 0.70711
#> Maximum overall sample size: 1006
#> Expected sample size (H1): 795.7
#> Expected sample size (H0): 1004.2
#> Expected placebo group sample size (H1): 89.4
#> Expected placebo group sample size (H0): 148.6
#> Objective function value: 905.7
#> (local) type I error for TP testing: 2.50%
#> (local) type I error for TC testing: 2.50%
#> Probability of futility stop (H1): 0.00%
#> Probability of futility stop (H0): 0.00%
#> Minimum conditional power: 0.00%
#> Power: 80.03%
#> Futility boundaries: binding
#> Futility testing method: always both futility tests
# }