Defaults

Note that contrary to the other functions of the package a one-sided t-test (instead of TOST) is employed. Hence, \(\small{\alpha}\) defaults to 0.025.
Defaults depending on the argument logscale:

Arguments targetpower, margin, theta0, and CV have to be given as fractions, not in percent.
The CV is generally the within- (intra-) subject coefficient of variation. Only for design = "parallel" it is the total (a.k.a. pooled) CV.

Designs with one (parallel), two (conventional crossover and paired), and three or four periods (replicates) are supported.

#      design                        name   df
#  "parallel"           2 parallel groups  n-2
#       "2x2"               2x2 crossover  n-2
#     "2x2x2"             2x2x2 crossover  n-2
#     "2x2x3"   2x2x3 replicate crossover 2n-3
#     "2x2x4"   2x2x4 replicate crossover 3n-4
#     "2x4x4"   2x4x4 replicate crossover 3n-4
#     "2x3x3"   partial replicate (2x3x3) 2n-3
#     "2x4x2"            Balaam’s (2x4x2)  n-2
#    "2x2x2r" Liu’s 2x2x2 repeated x-over 3n-2
#    "paired"                paired means  n-1

The terminology of the design argument follows this pattern: treatments x sequences x periods. The conventional TR|RT (a.k.a. AB|BA) design can be abbreviated as "2x2". Some call the "parallel" design a ‘one-sequence’ design. The design "paired" has two periods but no sequences, e.g., in studying linear pharmacokinetics a single dose is followed by multiple doses. A profile in steady state (T) is compared to the one after the single dose (R). Note that the underlying model assumes no period effects.

With sampleN.noninf(..., details = FALSE, print = FALSE) results are provided as a data frame ¹ with eight columns Design, alpha, CV, theta0, Margin, Sample size, Achieved power, and Target power. To access e.g., the sample size use either sampleN.noninf[1, 6] or sampleN.noninf[["Sample size"]]. We suggest to use the latter in scripts for clarity.

The estimated sample size gives always the total number of subjects (not subject/sequence in crossovers or subjects/group in parallel designs – like in some other software packages).

Non-Inferiority

If the supplied margin is < 1 (logscale = TRUE) or < 0 (logscale = FALSE), then it is assumed that higher response values are better. The hypotheses are with

• logscale = TRUE \[\small{H_0:\theta_0 \leq \log({margin})\:vs\:H_1:\theta_0>\log({margin})}\] where \(\small{\theta_0=\mu_\textrm{T}/\mu_\textrm{R}}\)

• logscale = FALSE \[\small{H_0:\theta_0 \leq {margin}\:vs\:H_1:\theta_0>{margin}}\]

  where \(\small{\theta_0=\mu_T-\mu_R}\)

sampleN.noninf(CV = 0.25)
# 
# ++++++++++++ Non-inferiority test +++++++++++++
#             Sample size estimation
# -----------------------------------------------
# Study design: 2x2 crossover 
# log-transformed data (multiplicative model)
# 
# alpha = 0.025, target power = 0.8
# Non-inf. margin = 0.8
# True ratio = 0.95,  CV = 0.25
# 
# Sample size (total)
#  n     power
# 36   0.820330

sampleN.noninf(CV = 0.25, details = FALSE, print = FALSE)[["Sample size"]]
# [1] 36

power.noninf(CV = 0.25, n = 35)
# Unbalanced design. n(i)=18/17 assumed.
# [1] 0.8085908

Non-Superiority

If the supplied margin is > 1 (logscale = TRUE) or > 0 (logscale = FALSE), then it is assumed that lower response values are better. The hypotheses are with

• logscale = TRUE \[\small{H_{0}:\theta_0 \geq \log({margin})\:vs\:H_{1}:\theta_0<\log({margin})}\] where \(\small{\theta_0=\mu_\textrm{T}/\mu_\textrm{R}}\)

• logscale = FALSE \[\small{H_{0}:\theta_0 \geq {margin}\:vs\:H_{1}:\theta_0<{margin}}\]

  where \(\small{\theta_0=\mu_T-\mu_R}\)

sampleN.noninf(CV = 0.25, margin = 1.25, theta0 = 1/0.95)
# 
# ++++++++++++ Non-superiority test +++++++++++++
#             Sample size estimation
# -----------------------------------------------
# Study design: 2x2 crossover 
# log-transformed data (multiplicative model)
# 
# alpha = 0.025, target power = 0.8
# Non-inf. margin = 1.25
# True ratio = 1.052632,  CV = 0.25
# 
# Sample size (total)
#  n     power
# 36   0.820330

Bracketing Approach

Compare a new modified release formulation (regimen once a day) with an intermediate release formulation (twice a day).² C_min is the target metric for efficacy (non-inferiority) and C_max for safety (non-superiority). Margins are 0.80 for C_min and 1.25 for C_max. CVs are 0.35 for C_min and 0.20 for C_max; \(\small{\theta_{0}}\) 0.95 for C_min and 1.05 for C_max. Full replicate design due to the high variability of C_min.
Which PK metric leads the sample size?

res <- data.frame(design = "2x2x4", metric = c("Cmin", "Cmax"),
                  margin = c(0.80, 1.25), CV = c(0.35, 0.20),
                  theta0 = c(0.95, 1.05), n = NA, power = NA,
                  stringsAsFactors = FALSE) # this line for R <4.0.0)
for (i in 1:2) {
  res[i, 6:7] <- sampleN.noninf(design = res$design[i],
                                margin = res$margin[i],
                                theta0 = res$theta0[i],
                                CV = res$CV[i],
                                details = FALSE,
                                print = FALSE)[6:7]
}
print(res, row.names = FALSE)
#  design metric margin   CV theta0  n     power
#   2x2x4   Cmin   0.80 0.35   0.95 32 0.8077926
#   2x2x4   Cmax   1.25 0.20   1.05 12 0.8406410

The sample size depends on C_min. Hence, the study is ‘overpowered’ for C_max.

power.noninf(design = "2x2x4", margin = 1.25, CV = 0.20,
             theta0 = 1.05, n = 32)
# [1] 0.9984996

Therefore, that gives us some ‘safety margin’ for C_max.

power.noninf(design = "2x2x4", margin = 1.25, CV = 0.25,
             theta0 = 1.10, n = 32) # higher CV, worse theta0
# [1] 0.8279726

The bracketing approach does not necessarily give lower sample sizes than tests for equivalence. In this example we could aim at reference-scaling for the highly variable C_min and at conventional ABE for C_max.

res <- data.frame(design = "2x2x4", intended = c("ABEL", "ABE"),
                  metric = c("Cmin", "Cmax"), CV = c(0.35, 0.20),
                  theta0 = c(0.90, 1.05), n = NA, power = NA,
                  stringsAsFactors = FALSE) # this line for R <4.0.0
res[1, 6:7] <- sampleN.scABEL(CV = res$CV[1], theta0 = res$theta0[1],
                              design = res$design[1], print = FALSE,
                              details = FALSE)[8:9]
res[2, 6:7] <- sampleN.TOST(CV = res$CV[2], theta0 = res$theta0[2],
                            design = res$design[2], print = FALSE,
                            details = FALSE)[7:8]
print(res, row.names = FALSE)
#  design intended metric   CV theta0  n     power
#   2x2x4     ABEL   Cmin 0.35   0.90 34 0.8118400
#   2x2x4      ABE   Cmax 0.20   1.05 10 0.8517596

Which method is optimal is a case-to-case decision. Although in this example the bracketing approach seems to be the ‘winner’ (32 subjects instead of 34), we might fail if the CV of C_min is larger than assumed, whereas in reference-scaling we might still pass due to the expanded limits.

n <- sampleN.scABEL(CV = 0.35, theta0 = 0.90, design = "2x2x4",
                    print = FALSE, details = FALSE)[["Sample size"]]
# CV and theta0 of both metrics worse than assumed
res <- data.frame(design = "2x2x4", intended = c("ABEL", "ABE"),
                  metric = c("Cmin", "Cmax"), CV = c(0.50, 0.25),
                  theta0 = c(0.88, 1.12), n = n, power = NA,
                  stringsAsFactors = FALSE) # this line for R <4.0.0
res[1, 7] <- power.scABEL(CV = res$CV[1], theta0 = res$theta0[1],
                          design = res$design[1], n = n)
res[2, 7] <- power.TOST(CV = res$CV[2], theta0 = res$theta0[2],
                        design = res$design[2], n = n)
print(res, row.names = FALSE)
#  design intended metric   CV theta0  n     power
#   2x2x4     ABEL   Cmin 0.50   0.88 34 0.8183300
#   2x2x4      ABE   Cmax 0.25   1.12 34 0.8258111

See also the vignettes RSABE, ABE, and PA.

Author

Detlew Labes

License

GPL-3 2024-03-18 Helmut Schütz