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| 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 |
| CVb | CVb |
Coefficient of variation of the between-subject variability | – |
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 CVb 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 1 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).
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)
#
# ++++ Dose proportionality study, power model ++++
# Sample size estimation
# -------------------------------------------------
# Study design: crossover (3x3 Latin square)
# alpha = 0.05, target power = 0.8
# Equivalence margins of R(dnm) = 0.8 ... 1.25
# Doses = 10 20 33.3
# True slope = 1.02, CV = 0.2
# Slope acceptance range = 0.81451 ... 1.1855
#
# Sample size (total)
# n power
# 15 0.808127As 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
#
# ++++ Dose proportionality study, power model ++++
# Sample size estimation
# -------------------------------------------------
# Study design: crossover (4x4 Latin square)
# alpha = 0.05, target power = 0.8
# Equivalence margins of R(dnm) = 0.8 ... 1.25
# Doses = 10 20 33.3 50
# True slope = 1.02, CV = 0.2
# Slope acceptance range = 0.86135 ... 1.1386
#
# Sample size (total)
# n power
# 16 0.867441Note 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$power >= 0.80, ]
print(res, row.names = FALSE)
# n power
# 16 0.86744
# 15 0.82914
# 14 0.81935
# 13 0.81516As usual nothing to worry about.
Rather extreme: Five levels and we desire only three periods. Hence,
we opt for an incomplete block design. The design matrix of a balanced
minimal repeated measurements design is obtained by the function
balmin.RMD() of package crossdes.
levels <- 5
doses <- mod.fibo(lowest, levels)
per <- 3
block <- levels*(levels-1)/(per-1)
dm <- crossdes::balmin.RMD(levels, block, per)
x <- sampleN.dp(CV = 0.20, doses = doses, beta0 = 1.02,
design = "IBD", dm = dm)
#
# ++++ Dose proportionality study, power model ++++
# Sample size estimation
# -------------------------------------------------
# Study design: IBD (5x10x3)
# alpha = 0.05, target power = 0.8
# Equivalence margins of R(dnm) = 0.8 ... 1.25
# Doses = 10 20 33.3 50 66.7
# True slope = 1.02, CV = 0.2, CVb = 0.4
# Slope acceptance range = 0.88241 ... 1.1176
#
# Sample size (total)
# n power
# 30 0.898758The IBD comes with a price since we need at least two blocks.
res <- data.frame(n = seq(x[["Sample size"]], nrow(dm), -1),
power = NA)
for (i in 1:nrow(res)) {
res$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)
# n power
# 30 0.89876
# 29 0.89196
# 28 0.87939
# 27 0.87201
# 26 0.85793
# 25 0.83587
# 24 0.82470
# 23 0.80405Again, 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")
#
# ++++ Dose proportionality study, power model ++++
# Sample size estimation
# -------------------------------------------------
# Study design: crossover (5x5 Latin square)
# alpha = 0.05, target power = 0.8
# Equivalence margins of R(dnm) = 0.8 ... 1.25
# Doses = 1 4 16 64 256
# True slope = 1.02, CV = 0.3
# Slope acceptance range = 0.95976 ... 1.0402
#
# Sample size (total)
# n power
# 70 0.809991In 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)
#
# ++++ Dose proportionality study, power model ++++
# Sample size estimation
# -------------------------------------------------
# Study design: crossover (5x5 Latin square)
# alpha = 0.05, target power = 0.8
# Equivalence margins of R(dnm) = 0.75 ... 1.333333
# Doses = 1 4 16 64 256
# True slope = 1.02, CV = 0.3
# Slope acceptance range = 0.94812 ... 1.0519
#
# Sample size (total)
# n power
# 30 0.828246Hummel et al. 2 proposed even more liberal \(\small{\theta_{1},\theta_{2}}\) of {0.50, 2.0}.
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.
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.↩︎
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