10 Group sequential design boundary with MaxCombo test
In this chapter, we calculate boundaries based on the error spending approach following Gordon Lan and DeMets (1983).
We continue use the same example scenario in the last chapter.
enrollRates <- tibble::tibble(Stratum = "All", duration = 12, rate = 500 / 12)
failRates <- tibble::tibble(
Stratum = "All",
duration = c(4, 100),
failRate = log(2) / 15, # Median survival is 15 months
hr = c(1, .6),
dropoutRate = 0.001
)
# Randomization Ratio is 1:1
ratio <- 1
# Type I error (one-sided)
alpha <- 0.025
# Power (1 - beta)
beta <- 0.2
power <- 1 - beta
# Interim analysis time
analysisTimes <- c(12, 24, 36)- Example 1 in the previous chapter
fh_test <- rbind(
data.frame(
rho = 0, gamma = 0, tau = -1,
test = 1,
Analysis = 1:3,
analysisTimes = c(12, 24, 36)
),
data.frame(
rho = c(0, 0.5), gamma = 0.5, tau = -1,
test = 2:3,
Analysis = 3, analysisTimes = 36
)
)
fh_test#> rho gamma tau test Analysis analysisTimes
#> 1 0.0 0.0 -1 1 1 12
#> 2 0.0 0.0 -1 1 2 24
#> 3 0.0 0.0 -1 1 3 36
#> 4 0.0 0.5 -1 2 3 36
#> 5 0.5 0.5 -1 3 3 3610.1 Sample size calculation based on spending function
- Implementation in
gsdmvn
gs_design_combo(enrollRates,
failRates,
fh_test,
alpha = 0.025,
beta = 0.2,
ratio = 1,
binding = FALSE, # test.type = 4 non-binding futility bound
upper = gs_spending_combo,
upar = list(sf = gsDesign::sfLDOF, total_spend = 0.025), # alpha spending
lower = gs_spending_combo,
lpar = list(sf = gsDesign::sfLDOF, total_spend = 0.2), # beta spending
) %>%
mutate(`Probability_Null (%)` = Probability_Null * 100) %>%
select(-Probability_Null) %>%
mutate_if(is.numeric, round, digits = 2)#> Analysis Bound Time N Events Z Probability Probability_Null (%)
#> 1 1 Upper 12 301.31 64.72 6.18 0.00 0.00
#> 3 2 Upper 24 301.31 148.41 2.80 0.22 0.26
#> 5 3 Upper 36 301.31 199.64 2.10 0.80 2.49
#> 2 1 Lower 12 301.31 64.72 -2.72 0.00 NA
#> 4 2 Lower 24 301.31 148.41 0.65 0.08 NA
#> 6 3 Lower 36 301.31 199.64 2.10 0.20 NA- Simulation results based on 10,000 replications.
#> n d analysisTimes lower upper
#> 7 302 64.78 12 0.00 0.00
#> 8 302 148.75 24 0.08 0.22
#> 9 302 200.18 36 0.20 0.80- Compared with \(FH(0, 0)\) using boundary based on
test.type = 4
x <- gsdmvn::gs_design_wlr(
enrollRates = enrollRates, failRates = failRates,
ratio = ratio, alpha = alpha, beta = beta,
weight = function(x, arm0, arm1) {
gsdmvn::wlr_weight_fh(x, arm0, arm1, rho = 0, gamma = 0)
},
upper = gs_spending_bound,
lower = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = alpha),
lpar = list(sf = gsDesign::sfLDOF, total_spend = beta),
analysisTimes = analysisTimes
)$bounds %>%
mutate_if(is.numeric, round, digits = 2)
x#> # A tibble: 6 × 11
#> Analysis Bound Time N Events Z Probability AHR theta info info0
#> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 12 377. 81 3.79 0 0.84 0.17 20.2 20.3
#> 2 2 Upper 24 377. 186. 2.36 0.46 0.72 0.33 46.3 46.8
#> 3 3 Upper 36 377. 250. 2.01 0.8 0.68 0.38 61.8 63.3
#> 4 1 Lower 12 377. 81 -1.18 0.03 0.84 0.17 20.2 20.3
#> 5 2 Lower 24 377. 186. 1.15 0.14 0.72 0.33 46.3 46.8
#> 6 3 Lower 36 377. 250. 2.01 0.2 0.68 0.38 61.8 63.3- Compared with \(FH(0.5, 0.5)\) using boundary based on
test.type = 4
x <- gsdmvn::gs_design_wlr(
enrollRates = enrollRates, failRates = failRates,
ratio = ratio, alpha = alpha, beta = beta,
weight = function(x, arm0, arm1) {
gsdmvn::wlr_weight_fh(x, arm0, arm1, rho = 0.5, gamma = 0.5)
},
upper = gs_spending_bound,
lower = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = alpha),
lpar = list(sf = gsDesign::sfLDOF, total_spend = beta),
analysisTimes = analysisTimes
)$bounds %>%
mutate_if(is.numeric, round, digits = 2)
x#> # A tibble: 6 × 11
#> Analysis Bound Time N Events Z Probability AHR theta info info0
#> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 12 304. 65.3 5.05 0 0.79 0.68 1.76 1.77
#> 2 2 Upper 24 304. 150. 2.51 0.42 0.68 0.93 6.17 6.28
#> 3 3 Upper 36 304. 201. 1.99 0.8 0.65 0.97 9.16 9.44
#> 4 1 Lower 12 304. 65.3 -1.8 0 0.79 0.68 1.76 1.77
#> 5 2 Lower 24 304. 150. 1.12 0.12 0.68 0.93 6.17 6.28
#> 6 3 Lower 36 304. 201. 1.99 0.2 0.65 0.97 9.16 9.44- Compared with \(FH(0, 0.5)\) using boundary based on
test.type = 4
x <- gsdmvn::gs_design_wlr(
enrollRates = enrollRates, failRates = failRates,
ratio = ratio, alpha = alpha, beta = beta,
weight = function(x, arm0, arm1) {
gsdmvn::wlr_weight_fh(x, arm0, arm1, rho = 0, gamma = 0.5)
},
upper = gs_spending_bound,
lower = gs_spending_bound,
upar = list(sf = gsDesign::sfLDOF, total_spend = alpha),
lpar = list(sf = gsDesign::sfLDOF, total_spend = beta),
analysisTimes = analysisTimes
)$bounds %>%
mutate_if(is.numeric, round, digits = 2)
x#> # A tibble: 6 × 11
#> Analysis Bound Time N Events Z Probability AHR theta info info0
#> <dbl> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 1 Upper 12 288. 61.9 6.18 0 0.78 0.63 2.08 2.09
#> 2 2 Upper 24 288. 142 2.8 0.3 0.67 0.77 8.86 9.07
#> 3 3 Upper 36 288. 191. 1.97 0.8 0.64 0.73 15.7 16.4
#> 4 1 Lower 12 288. 61.9 -2.43 0 0.78 0.63 2.08 2.09
#> 5 2 Lower 24 288. 142 0.93 0.09 0.67 0.77 8.86 9.07
#> 6 3 Lower 36 288. 191. 1.97 0.2 0.64 0.73 15.7 16.410.2 Information fraction
Under the same design assumption, information fraction is different from different weight parameters in WLR.
There are two potential strategies to calculate the information fraction:
- Option 1: Use minimal information fraction of all candidate tests (implemented in
gsdmvn). - Option 2: Use the weighted average of information fraction.
utility <- gsdmvn:::gs_utility_combo(enrollRates, failRates, fh_test = fh_test, ratio = 1)
utility$info_all %>% mutate_if(is.numeric, round, digits = 3)#> test Analysis Time N Events AHR delta sigma2 theta info info0
#> 1 1 1 12 500 107.394 0.842 -0.009 0.054 0.172 26.841 26.899
#> 2 1 2 24 500 246.283 0.716 -0.041 0.123 0.333 61.352 62.087
#> 3 1 3 36 500 331.291 0.683 -0.062 0.164 0.381 81.918 83.944
#> 4 2 1 12 500 107.394 0.781 -0.005 0.007 0.626 3.605 3.624
#> 5 2 2 24 500 246.283 0.666 -0.024 0.031 0.765 15.371 15.737
#> 6 2 3 36 500 331.291 0.639 -0.040 0.054 0.732 27.205 28.484
#> 7 3 1 12 500 107.394 0.790 -0.004 0.006 0.676 2.898 2.910
#> 8 3 2 24 500 246.283 0.675 -0.019 0.020 0.927 10.155 10.335
#> 9 3 3 36 500 331.291 0.650 -0.029 0.030 0.973 15.070 15.526
info_frac <- tapply(utility$info_all$info0, utility$info_all$test, function(x) x / max(x))
info_frac#> $`1`
#> [1] 0.3204395 0.7396239 1.0000000
#>
#> $`2`
#> [1] 0.1272245 0.5525079 1.0000000
#>
#> $`3`
#> [1] 0.1874599 0.6656420 1.0000000#> [1] 0.1272245 0.5525079 1.000000010.3 Spending function
-
gsDesignbound can not be directly used for MaxCombo test:- Multiple test statistics are considered in interim analysis or final analysis.
-
Design example:
- \(\alpha = 0.025\)
- \(\beta = 0.2\)
- \(K=3\) total analysis.
-
test.type=4: A two-sided, asymmetric, beta-spending with non-binding lower bound.
Lan-DeMets spending function to approximate an O’Brien-Fleming bound Gordon Lan and DeMets (1983). (
gsDesign::sfLDOF)\(t\) is information fraction in the formula below.
\[f(t; \alpha)=2-2\Phi\left(\Phi^{-1}\left(\frac{1-\alpha/2}{t^{\rho/2}}\right)\right)\]
Other spending functions are discussed in the gsDesign technical manual and implemented in
gsDesign::sf*()functions.-
Upper bound:
- Non-binding lower bound: lower bound are all
-Inf.
- Non-binding lower bound: lower bound are all
alpha_spend <- gsDesign::sfLDOF(alpha = 0.025, t = min_info_frac)$spend
alpha_spend#> [1] 3.300225e-10 2.566068e-03 2.500000e-02- Lower bound:
beta_spend <- gsDesign::sfLDOF(alpha = 0.2, t = min_info_frac)$spend
beta_spend#> [1] 0.0003269604 0.0846866352 0.200000000010.4 Technical details
This section describes the details of calculating the sample size and events required for WLR under group sequential design. It can be skipped for the first read of this training material.
10.5 Upper bound in group sequential design
10.5.1 One test in each interim analysis
- First interim analysis
\[\alpha_1 = \text{Pr}(Z_1 > b_1 \mid H_0)\]
qnorm(1 - alpha_spend[1])#> [1] 6.17538- General formula (non-binding futility bound)
\[\alpha_k = \text{Pr}(\cap_{i=1}^{i=k-1} Z_i < b_i, Z_k > b_k \mid H_0)\]
10.5.2 MaxCombo upper bound
- First interim analysis upper bound
\[\alpha_1 = \text{Pr}(G_1 > b_1 \mid H_0) = 1 - \text{Pr}(\cap_i Z_{i1} < b_1 \mid H_0)\]
- General formula (non-binding futility bound)
\[\alpha_k = \text{Pr}(\cap_{i=1}^{i=k-1} G_i < b_i, G_k > b_k \mid H_0)\] \[ = \text{Pr}(\cap_{i=1}^{i=k-1} G_i < b_i \mid H_0) - \text{Pr}(\cap_{i=1}^{i=k} G_i < b_i \mid H_0)\]
-
gsdmvnimplementation for MaxCombo test
gsdmvn:::gs_bound(alpha_spend, beta_spend,
analysis = utility$info$Analysis, # Analysis indicator
theta = rep(0, nrow(fh_test)), # Under the null hypothesis
corr = utility$corr # Correlation
)$upper#> [1] 6.175397 2.798651 2.097603- Compared with upper bound calculated from
gsDesign
x <- gsDesign::gsSurv(
k = 3, test.type = 4, alpha = 0.025,
beta = 0.2, astar = 0, timing = c(1),
sfu = sfLDOF, sfupar = c(0), sfl = sfLDOF,
sflpar = c(0), lambdaC = c(0.1),
hr = 0.6, hr0 = 1, eta = 0.01,
gamma = c(10),
R = c(12), S = NULL,
T = 36, minfup = 24, ratio = 1
)
x$upper$bound#> [1] 3.710303 2.511407 1.99297010.6 Lower bound in group sequential design
10.6.1 One test in each interim analysis
- First interim analysis
\[\beta_1 = \text{Pr}(Z_1 < a_1 \mid H_1)\]
#> [1] -2.610665#> [1] -2.516528- General formula \[\beta_k = \text{Pr}(\cap_{i=1}^{i=k-1} a_i < Z_i < b_i, Z_k < a_k \mid H_1)\]
10.6.2 MaxCombo lower bound
- First interim analysis upper bound
\[\beta_1 = \text{Pr}(G_1 < a_1 \mid H_1) = \text{Pr}(\cap_i Z_{i1} < a_1 \mid H_0)\]
- General formula (non-binding futility bound)
\[\beta_k = \text{Pr}(\cap_{i=1}^{i=k-1} a_i <G_i < b_i, G_k < a_k \mid H_0)\]
-
gsdmvnimplementation for MaxCombo test
n <- 400
utility <- gsdmvn:::gs_utility_combo(enrollRates, failRates, fh_test = fh_test, ratio = 1)
bound <- gsdmvn:::gs_bound(alpha_spend, beta_spend,
analysis = utility$info$Analysis, # Analysis indicator
theta = utility$theta * sqrt(n), # Under the alternative hypothesis
corr = utility$corr # Correlation
)
bound$lower#> [1] -2.6106678 0.9619422 2.0968098- Compared with lower bound calculated from
gsDesign
x$lower$bound#> [1] -0.2361874 1.1703638 1.992970210.7 Sample size calculation
- Sample size and boundaries are set simultaneously using an iterative algorithm.
10.7.1 Initiate the calculation from lower bound derived at \(N = 400\)
bound#> upper lower
#> 1 6.175397 -2.6106678
#> 2 2.798651 0.9619422
#> 3 2.096810 2.0968098
gs_design_combo(enrollRates,
failRates,
fh_test,
alpha = 0.025,
beta = 0.2,
ratio = 1,
binding = FALSE,
upar = bound$upper,
lpar = bound$lower,
) %>% mutate_if(is.numeric, round, digits = 2)#> Analysis Bound Time N Events Z Probability Probability_Null
#> 1 1 Upper 12 324.88 69.78 6.18 0.00 0.00
#> 3 2 Upper 24 324.88 160.03 2.80 0.24 0.00
#> 5 3 Upper 36 324.88 215.26 2.10 0.80 0.03
#> 2 1 Lower 12 324.88 69.78 -2.61 0.00 NA
#> 4 2 Lower 24 324.88 160.03 0.96 0.13 NA
#> 6 3 Lower 36 324.88 215.26 2.10 0.20 NA10.7.2 Update bound based on newly calculated sample size
n <- 355
utility <- gsdmvn:::gs_utility_combo(enrollRates, failRates, fh_test = fh_test, ratio = 1)
bound <- gsdmvn:::gs_bound(alpha_spend, beta_spend,
analysis = utility$info$Analysis, # Analysis indicator
theta = utility$theta * sqrt(n), # Under the alternative hypothesis
corr = utility$corr # Correlation
)
bound#> upper lower
#> 1 6.175397 -2.6568642
#> 2 2.798651 0.8266125
#> 3 2.097617 2.0976166- Repeat the procedure above until the sample size and lower bound converge.