12 Scenario XI: F-Distribution

12.1 Details

In this scenario, we compute the necessary sample size for a one-way ANOVA with \(k\) groups. The resulting test statistic is \(F\)-distributed with \(k-1\) and \(nk-k\) degrees of freedom, where \(n\) stands for the sample size per group.

12.2 Variant XI-1: ANOVA

12.2.1 Setup

Under the alternative, we assume a mean of \(0.0\) in the first group, \(0.4\) in the second group and \(0.2\) in the third group.

means <- c(0, 0.4, 0.2)
theta <- get_tau_ANOVA(means)

datadist <- ANOVA(3)
H_0 <- PointMassPrior(0, 1)
prior <- PointMassPrior(theta, 1)

alpha <- 0.05
min_power <- 0.8

toer_cnstr <- Power(datadist, H_0)  <= alpha
pow_cnstr <- Power(datadist, prior) >= min_power

12.2.2 Objective

We want to minimize the expected sample size under the alternative.

ess <- ExpectedSampleSize(datadist, prior)

12.2.3 Initial Design

The initial designs are chosen via the in-built function to create initial designs.

tbl_designs <- tibble(
    type    = c("one-stage", "group-sequential", "two-stage"),
    initial = list(
        get_initial_design(theta, alpha, 1 - min_power, 
                           "one-stage", dist = datadist),
        get_initial_design(theta, alpha, 1 - min_power, 
                           "group-sequential", dist = datadist),
        get_initial_design(theta, alpha, 1 - min_power, 
                           "two-stage", dist = datadist) ))

12.2.4 Optimization

tbl_designs <- tbl_designs %>% 
    mutate(
       optimal = purrr::map(initial, ~minimize(
         
          ess,
          subject_to(
              toer_cnstr,
              pow_cnstr 
          ),
          
          initial_design = ., 
          opts           = opts)) )

12.2.5 Test Cases

We first verify that in none of the three cases, the maximal number of iterations was exceeded.

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                22
## 2 group-sequential       1573
## 3 two-stage              1912

We then check via simulations of the test statistic, that the type I error and power constraints are fulfilled.

tbl_designs %>% 
  transmute(
      type, 
      toer  = purrr::map(tbl_designs$optimal, 
                         ~sim_pr_reject(.[[1]], .0, datadist)$prob), 
      power = purrr::map(tbl_designs$optimal, 
                         ~sim_pr_reject(.[[1]], theta , datadist)$prob) ) %>% 
  unnest(., cols = c(toer, power)) %>% 
  {print(.); .} %>% {
  testthat::expect_true(all(.$toer  <= alpha * (1 + tol)))
  testthat::expect_true(all(.$power >= min_power * (1 - tol))) }

## # A tibble: 3 × 3
##   type               toer power
##   <chr>             <dbl> <dbl>
## 1 one-stage        0.0503 0.799
## 2 group-sequential 0.0502 0.798
## 3 two-stage        0.0500 0.799

The sample size function \(n_2\) should be monotonously decreasing.

testthat::expect_true(
    all(diff(
        # get optimal two-stage design n2 pivots
        tbl_designs %>% filter(type == "two-stage") %>%
           {.[["optimal"]][[1]]$design@n2_pivots} 
        ) < 0) )

Since the degrees of freedom of the three design classes are ordered as ‘two-stage’ > ‘group-sequential’ > ‘one-stage’, the expected sample sizes (under the alternative) should be ordered in reverse (‘two-stage’ smallest). Additionally, expected sample sizes under both null and alternative are computed both via evaluate() and simulation-based.

ess0 <- ExpectedSampleSize(datadist, H_0)

tbl_designs %>% 
    mutate(
        ess      = map_dbl(optimal,
                           ~evaluate(ess, .$design) ),
        ess_sim  = map_dbl(optimal,
                           ~sim_n(.$design, theta, datadist)$n ),
        ess0     = map_dbl(optimal,
                           ~evaluate(ess0, .$design) ),
        ess0_sim = map_dbl(optimal,
                           ~sim_n(.$design, .0, datadist)$n ) ) %>% 
    {print(.); .} %>% {
    # sim/evaluate same under alternative?
    testthat::expect_equal(.$ess, .$ess_sim, 
                           tolerance = tol_n,
                           scale = 1)
    # sim/evaluate same under null?
    testthat::expect_equal(.$ess0, .$ess0_sim, 
                           tolerance = tol_n,
                           scale = 1)
    # monotonicity with respect to degrees of freedom
    testthat::expect_true(all(diff(.$ess) < 0)) }

## # A tibble: 3 × 7
##   type             initial    optimal          ess ess_sim  ess0 ess0_sim
##   <chr>            <list>     <list>         <dbl>   <dbl> <dbl>    <dbl>
## 1 one-stage        <OnStgDsg> <adptrOpR [3]>  121     121   121      121 
## 2 group-sequential <GrpSqntD> <adptrOpR [3]>  111.    111.  103.     103.
## 3 two-stage        <TwStgDsg> <adptrOpR [3]>  111.    111.  104.     105.

The expected sample size under the alternative of the optimized designs should be lower than the sample size of the initial designs.

testthat::expect_lte(
  evaluate(ess, 
             tbl_designs %>% 
                dplyr::pull(optimal) %>% 
                .[[1]]  %>%
                .$design ),
    evaluate(ess, 
             tbl_designs %>% 
                dplyr::pull(initial) %>% 
                .[[1]] ) )