Dose-Proportionality"
In PowerTOST: Power and Sample Size for (Bio)Equivalence Studies

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#"
)

[Main Vignette](vignette.html)
[ABE](ABE.html "Average Bioequivalence")
[RSABE](RSABE.html "Reference-scaled Average Bioequivalence")
[Non-Inferiority](NI.html)
Dose-Proportionality
[Power Analysis](PA.html)

Details and examples of other methods are accessible via the menu bar on top of the page and the [online manual](https://cran.r-project.org/package=PowerTOST/PowerTOST.pdf "PDF") of all functions.

library(PowerTOST) # attach the library

Defaults

| Parameter | Argument | Purpose | Default | |---|---|-------------|----| | $\small{\alpha}$ | alpha | Nominal level of the test | 0.05 | | CV | CV | CV | none | | doses | doses | Vector of doses | see examples | | $\small{\pi}$ | targetpower | Minimum desired power | 0.80 | | $\small{\beta_0}$ | beta0 | ‘True’ or assumed slope of the power model | see below | | $\small{\theta_1}$ | theta1 | Lower limit for the ratio of dose normalized means Rdmn | see below | | $\small{\theta_2}$ | theta2 | Upper limit for the ratio of dose normalized means Rdmn | see below | | design | design | Planned design | "crossover" | | dm | dm | Design matrix | NULL | | CV~b~ | CVb | Coefficient of variation of the between-subject variability | – | | print | print | Show information in the console? | TRUE | | details | details | Show details of the sample size search? | FALSE | | imax | imax | Maximum number of iterations | 100 |

Arguments targetpower, theta1, theta2, 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. The between- (intra-) subject coefficient of variation CV~b~ is only necessary if design = "IBD" (if missing, it will be set to 2*CV).

The ‘true’ or assumed slope of the power model $\small{\beta_0}$ defaults to 1+log(0.95)/log(rd), where rd is the ratio of the highest/lowest dose.

Supported designs are "crossover" (default; Latin Squares), "parallel", and "IBD" (incomplete block design). Note that when "crossover" is chosen, instead of Latin Squares any Williams’ design could be used as well (identical degrees of freedom result in the same sample size).

With sampleN.dp(..., details = FALSE, print = FALSE) results are provided as a data frame ^[R Documentation. Data Frames. 2022-02-08. R-manual.] with ten (design = "crossover") or eleven (design = "parallel" or design = "IBD") columns: Design, alpha, CV, (CVb,) doses, beta0, theta1, theta2, Sample size, Achieved power, and Target power. To access e.g., the sample size use sampleN.dp[["Sample size"]].

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

Examples

Estimate the sample size for a modified Fibonacci dose-escalation study, lowest dose 10, three levels. Assumed CV 0.20 and $\small{\beta_0}$ slightly higher than 1. Defaults employed.

mod.fibo <- function(lowest, levels) {
  # modified Fibonacci dose-escalation
  fib      <- c(2, 1 + 2/3, 1.5, 1 + 1/3)
  doses    <- numeric(levels)
  doses[1] <- lowest
  level    <- 2
  repeat {
    if (level <= 4) {
      doses[level] <- doses[level-1] * fib[level-1]
    } else {  # ratio 1.33 for all higher doses
      doses[level] <- doses[level-1] * fib[4]
    }
    level <- level + 1
    if (level > levels) {
      break
    }
  }
  return(signif(doses, 3))
}
lowest <- 10
levels <- 3
doses  <- mod.fibo(lowest, levels)
sampleN.dp(CV = 0.20, doses = doses, beta0 = 1.02)

As above but with an additional level.

levels <- 4
doses  <- mod.fibo(lowest, levels)
x <- sampleN.dp(CV = 0.20, doses = doses, beta0 = 1.02) # we need the data.frame later

Note that with the wider dose range the acceptance range narrows.

Explore the impact of dropouts.

res <- data.frame(n = seq(x[["Sample size"]], 12, -1), power = NA)
for (i in 1:nrow(res)) {
  res$power[i] <- signif(suppressMessages(
                           power.dp(CV = 0.20, doses = doses,
                                    beta0 = 1.02, n = res$n[i])), 5)
}
res <- res[res$power >= 0.80, ]
print(res, row.names = FALSE)

As usual nothing to worry about.

Rather extreme: Five levels and we desire only three periods. Hence, we opt for an incomplete block design. The design matrix of a balanced minimal repeated measurements design is obtained by the function balmin.RMD() of package crossdes.

levels <- 5
doses  <- mod.fibo(lowest, levels)
per    <- 3
block  <- levels*(levels-1)/(per-1)
dm     <- crossdes::balmin.RMD(levels, block, per)
x      <- sampleN.dp(CV = 0.20, doses = doses, beta0 = 1.02,
                     design = "IBD", dm = dm)

The IBD comes with a price since we need at least two blocks.

res <- data.frame(n = seq(x[["Sample size"]], nrow(dm), -1),
                  power = NA)
for (i in 1:nrow(res)) {
  res$power[i] <- signif(suppressMessages(
                           power.dp(CV = 0.20, doses = doses,
                                    beta0 = 1.02, design = "IBD",
                                    dm = dm, n = res$n[i])), 5)
}
res <- res[res$power >= 0.80, ]
print(res, row.names = FALSE)

Again, we don’t have to worry about dropouts.

For a wide dose range the acceptance range narrows and becomes increasingly difficult to meet.

doses  <- 2^(seq(0, 8, 2))
levels <- length(doses)
sampleN.dp(CV = 0.30, doses = doses, beta0 = 1.02,
           design = "crossover")

In an exploratory setting more liberal limits could be specified (only one has to be specified; the other is calculated as the reciprocal of it).

sampleN.dp(CV = 0.30, doses = doses, beta0 = 1.02,
           design = "crossover", theta1 = 0.75)

Hummel et al. ^[Hummel J, McKendrick S, Brindley C, French R. Exploratory assessment of dose proportionality: review of current approaches and proposal for a practical criterion. Pharm. Stat. 2009; 8(1): 38--49. doi:10.1002/pst.326.] proposed even more liberal $\small{\theta_{1},\theta_{2}}$ of {0.50, 2.0}.

Cave!

There is no guarantee that a desired incomplete block design (for given dose levels and number of periods) can be constructed. If you provide your own design matrix (in the argument dm) it is not assessed for meaningfulness.