8 Scenario VII: binomial distribution, Gaussian prior
8.1 Details
In this scenario, we revisit the case from ScenarioVI, but are not assuming a point prior anymore. We assume \(p_C=0.3\) for the event rate in the control group. The parameter which can be varied is the event rate in the experimental group \(p_E\) and we assume that the rate difference \(p_E-p_C \sim \textbf{1}_{(-0.29,0.69)} \mathcal{N}(0.2,0.2)\).The restriction to the interval \((-0.29,0.69)\) is necessary to ensure that the parameter \(p_E\) does not become smaller than \(0\) or larger than \(1\).
In order to fulfill regulatory considerations, the type one error rate is still protected under the point prior \(\delta=0\) at the level of significance \(\alpha=0.025\).
Since effect sizes less than a minimal clinically relevant effect do not show evidence against the null hypothesis, we assume a clinically relevant effect size \(\delta =0.0\) and condition the prior on values \(\delta > 0\) in order to compute the expected power. We assume a minimal expected power of at least \(0.8\).
# data distribution and priors
datadist <- Binomial(rate_control = 0.3, two_armed = TRUE)
H_0 <- PointMassPrior(.0, 1)
prior <- ContinuousPrior(function(x) 1 /
(pnorm(0.69, 0.2, 0.2) - pnorm(-0.29, 0.2, 0.2)) * dnorm(x, 0.2, 0.2),
support = c(-0.29, 0.69),
tighten_support = TRUE)
# define constraints on type one error rate and expected power
alpha <- 0.025
min_epower <- 0.8
toer_cnstr <- Power(datadist, H_0) <= alpha
epow_cnstr <- Power(datadist, condition(prior, c(0.0, 0.69))) >= min_epower8.2 Variant VII-1: Minimizing Expected Sample Size under Continuous Prior
8.2.1 Objective
Expected sample size under the full prior is minimized, i.e., \(\boldsymbol{E}\big[n(\mathscr{D})\big]\).
8.2.3 Initial Designs
For this example, the optimal one-stage, group-sequential, and generic two-stage designs are computed. The initial design for the one-stage case is determined heuristically and both the group sequential and the generic two-stage designs are optimized starting from the corresponding group-sequential design as computed by the rpact package.
order <- 7L
# data frame of initial designs
tbl_designs <- tibble(
type = c("one-stage", "group-sequential", "two-stage"),
initial = list(
OneStageDesign(200, 2.0),
rpact_design(datadist, 0.2, 0.025, 0.8, TRUE, order),
TwoStageDesign(get_initial_design(0.2, 0.025, 0.2, "two-stage", dist = datadist) )))The order of integration is set to 7.
8.2.5 Test Cases
To avoid improper solutions, it is first verified that the maximum number of iterations was not exceeded in any of the three cases.
tbl_designs %>%
transmute(
type,
iterations = purrr::map_int(tbl_designs$optimal,
~.$nloptr_return$iterations) ) %>%
{print(.); .} %>%
{testthat::expect_true(all(.$iterations < opts$maxeval))}## # A tibble: 3 × 2
## type iterations
## <chr> <int>
## 1 one-stage 21
## 2 group-sequential 642
## 3 two-stage 22550Furthermore, the type one error rate constraint needs to be tested.
tbl_designs %>%
transmute(
type,
toer = purrr::map(tbl_designs$optimal,
~sim_pr_reject(.[[1]], .0, datadist)$prob) ) %>%
unnest(., cols = c(toer)) %>%
{print(.); .} %>% {
testthat::expect_true(all(.$toer <= alpha * (1 + tol))) }## # A tibble: 3 × 2
## type toer
## <chr> <dbl>
## 1 one-stage 0.0251
## 2 group-sequential 0.0250
## 3 two-stage 0.0249The optimal two-stage design is more flexible than the other two designs, so the expected sample sizes under the prior should be ordered upwards as follows: optimal two-stage design < group sequential < one-stage. We additionally compare the simulated and theoretical outcomes of the sample size under the null.
essh0 <- ExpectedSampleSize(datadist, H_0)
tbl_designs %>%
mutate(
ess = map_dbl(optimal,
~evaluate(ess, .$design) ),
essh0 = map_dbl(optimal,
~evaluate(essh0, .$design) ),
essh0_sim = map_dbl(optimal,
~sim_n(.$design, .0, datadist)$n ) ) %>%
{print(.); .} %>% {
# sim/evaluate same under null?
testthat::expect_equal(.$essh0, .$essh0_sim,
tolerance = tol_n,
scale = 1)
# monotonicity with respect to degrees of freedom
testthat::expect_true(all(diff(.$ess) < 0)) }## # A tibble: 3 × 6
## type initial optimal ess essh0 essh0_sim
## <chr> <list> <list> <dbl> <dbl> <dbl>
## 1 one-stage <OnStgDsg> <adptrOpR [3]> 238 238 238
## 2 group-sequential <GrpSqntD> <adptrOpR [3]> 103. 161. 161.
## 3 two-stage <TwStgDsg> <adptrOpR [3]> 99.2 169. 169.8.3 Variant VII-2: Conditional Power Constraint {variantVII_2}
8.3.1 Objective
As in VariantVII-1, the expected sample size under the full prior is minimized.
8.3.2 Constraints
In addition to the constraints on type one error rate and expected power, a constraint on conditional power to be larger than \(0.7\) is included.
8.3.5 Test Cases
As always, we start checking whether the maximum number of iterations was reached or not.
testthat::expect_true(opt_cp$nloptr_return$iterations < opts$maxeval)
print(opt_cp$nloptr_return$iterations)## [1] 17551The type one error rate is tested via simulation and compared to the value obtained by evaluate().
tbl_toer <- tibble(
toer = evaluate(Power(datadist, H_0), opt_cp$design),
toer_sim = sim_pr_reject(opt_cp$design, .0, datadist)$prob
)
print(tbl_toer)## # A tibble: 1 × 2
## toer toer_sim
## <dbl> <dbl>
## 1 0.0250 0.0249testthat::expect_true(tbl_toer$toer <= alpha * (1 + tol))
testthat::expect_true(tbl_toer$toer_sim <= alpha * (1 + tol))The conditional power is evaluated via numerical integration on several points inside the continuation region and it is tested whether the constraint is fulfilled on all these points.
tibble(
x1 = seq(opt_cp$design@c1f, opt_cp$design@c1e, length.out = 25),
cp = map_dbl(x1, ~evaluate(cp, opt_cp$design, .)) ) %>%
{print(.); .} %>% {
testthat::expect_true(all(.$cp >= 0.7 * (1 - tol))) }## # A tibble: 25 × 2
## x1 cp
## <dbl> <dbl>
## 1 0.190 0.699
## 2 0.288 0.700
## 3 0.387 0.700
## 4 0.486 0.700
## 5 0.585 0.700
## 6 0.684 0.698
## 7 0.783 0.697
## 8 0.882 0.699
## 9 0.981 0.709
## 10 1.08 0.725
## # ℹ 15 more rowsDue to the additional constraint, this variant should show a larger expected sample size.