power.tsd.inR Documentation

Power calculation of adaptive 2-stage BE studies with 2x2 crossover design, based on the Inverse Normal method

Description

The function calculates the ‘empirical’ power of 2-stage BE studies based on the Inverse-Normal combination method. The main design scheme is according to Maurer et al. (Maximum Combination Test), but it may also be used for other/modified designs, e.g. standard combination method, usage of the observed treatment difference after stage 1 in the sample size re-estimation step or different futility criteria.

Usage

power.tsd.in(alpha, weight, max.comb.test = TRUE, n1, CV, targetpower = 0.8,
             theta0, theta1, theta2, GMR, usePE = FALSE, min.n2 = 4, max.n = Inf,
             fCpower = targetpower, fCrit = "CI", fClower, fCupper, fCNmax,
             ssr.conditional = c("error_power", "error", "no"),
             pmethod = c("nct", "exact", "shifted"), npct = c(0.05, 0.5, 0.95),
             nsims, setseed = TRUE, details = FALSE)

Arguments

alpha

If one element is given, the overall one-sided significance level. In this case the adjusted alpha levels will be calculated internally. If two elements are given, the argument refers to the two adjusted one-sided alpha levels for stage 1 and stage 2, respectively.
If missing, defaults to 0.05.

weight

Pre-defined weight(s) of stage 1, see 'Details' for more information. Note that using the notation from Maurer et al, weight corresponds to information fraction, other literature may refer to sqrt(weight) as being the weight. weight must either contain one element (in case of max.comb.test = FALSE) or two elements (in case of max.comb.test = TRUE).
If missing, defaults to 0.5 for max.comb.test = FALSE and to c(0.5, 0.25) for max.comb.test = TRUE.

max.comb.test

Logical; if TRUE (default) the maximum combination test will be used, otherwise the standard combination test.

n1

Sample size of stage 1.

CV

Coefficient of variation of the intra-subject variability (use e.g., 0.3 for 30%).

targetpower

Desired (overall) target power to declare BE at the end of the trial.

theta0

Assumed ratio of geometric means (T/R) for simulations. If missing, defaults to GMR.

theta1

Lower bioequivalence limit. Defaults to 0.8.

theta2

Upper bioequivalence limit. Defaults to 1.25.

GMR

Assumed ratio of geometric means (T/R) to be used in power calculation for stage 1 and sample size re-estimation for stage 2. If missing, defaults to 0.95.

usePE

If TRUE the sample size re-estimation is done with the observed point estimate (PE) of the treatment difference in stage 1.
Defaults to FALSE.
Note: The power of stage 1 used for the futility inspection and calculation of the estimated conditional target power is always calculated with the the planning value GMR.

min.n2

Minimum sample size of stage 2. Defaults to 4.
If the sample size re-estimation step gives a sample size for stage 2 less than min.n2, then min.n2 will be used for stage 2.

max.n

Maximum overall sample size stage 1 + stage 2.
This is not a futility criterion regarding the maximum sample size! If max.n is set to a finite value and the sample size re-estimation gives a sample size for stage 2 (n2) such that n1 + n2 > max.n, then the sample size for stage 2 will be set to n2 = max.n - n1.
Defaults to Inf, i.e., no constraint on the re-estimated sample size.

fCpower

Threshold for power monitoring step to decide on futility for cases where BE has not been achieved after stage 1: If BE has not been achieved after stage 1 and the power for stage 1 is greater than or equal to fCpower, then the study will be considered a failure.

See ‘Details’ for more information on the choice of fCpower.

fCrit

Futility criterion to use: "No" (no futility criterion regarding observed point estimate, confidence interval and maximum sample size), "PE" (observed point estimate of the geometric mean ratio from stage 1), "CI" (90% confidence interval of the geometric mean ratio from stage 1), "Nmax" (overall maximum sample size); or a combination thereof (concatenate abbreviations; see ‘Examples’). Defaults to "CI".

fClower

Lower futility limit for the PE or CI of stage 1.
If the PE or CI is completely outside of fClower ... fCupper the study is stopped due to futility (not BE).
May be missing. If "PE" or "CI" is specified within fCrit, the default will be set to 0.8 for fCrit = "PE" or 0.95 for fCrit = "CI". If neither "PE" nor "CI" is specified within fCrit, there will be no futility constraint regarding point estimate or confidence interval from stage 1 (regardless of any specification of fClower and/or fCupper).

fCupper

Upper futility limit for the PE or CI of stage 1.
Analogous to fClower: Will be set to 1/fClower if missing.

fCNmax

Futility criterion regarding maximum sample size. If the determined sample size for stage 2 (n2) is such that n1 + n2 > fCNmax, the study will not continue to stage 2 and stopped due to futility (not BE).
If "Nmax" is specified within fCrit and argument fCNmax is missing, the value will be set to fCNmax = 4*n1. If "Nmax" is not specified within fCrit, then there will be no futility constraint regarding maximum sample size (regardless of any specification of fCNmax).

ssr.conditional

Method for sample size re-estimation step: "no" does not use conditional error rates nor the estimated conditional target power for the second stage, "error" uses conditional error rates for the second stage, and "error_power" uses both conditional error rates and the estimated conditional target power for the second stage.
Defaults to "error_power".

See also ‘Details’.

pmethod

Power calculation method, also to be used in the sample size estimation for stage 2.
Implemented are "nct" (approximate calculations via non-central t-distribution, "exact" (exact calculations via Owen’s Q), and "shifted" (approximate calculation via shifted central t-distribution like in the paper of Potvin et al.)
Defaults to "nct" as it is a reasonable compromise between speed and accuracy in the sample size estimation step.

npct

Percentiles to be used for the presentation of the distribution of n(total)=n1+n2.
Defaults to c(0.05, 0.5, 0.95) to obtain the 5% and 95% percentiles and the median.

nsims

Number of studies to simulate.
If missing, nsims is set to 1E+05 = 100,000 or to 1E+06 = 1 Mio if estimating the empiric Type I Error ('alpha'), i.e., with theta0 at the border of the acceptance range theta1 ... theta2.

setseed

Simulations are dependent on the starting point of the (pseudo) random number generator. To avoid differences in power for different runs a set.seed(1234567) is issued if setseed=TRUE, the default.
Set this argument to FALSE to view the variation in power between different runs.

details

If set to TRUE the function prints the results of time measurements of the simulation steps. Default to FALSE.

Details

The underlying subject data are assumed to be evaluated after log-transformation. But instead of simulating subject data, the statistics pe1, mse1 and pe2, mse2 are simulated via their associated distributions (Normal and χ2 distribution).

The weight argument always refers to the first weight of a pair of weights. For example, in case of max.comb.test = FALSE the standard combination test requires two weights (w, 1-w) but only the first one, w, is required as input argument here because the second weight is automatically specified once the first is given. Similarly for max.comb.test = TRUE, w and w* need to be specified, which in turn define the two pairs of weights (w, 1-w) and (w*, 1-w*).

If ssr.conditional = "error_power", the design scheme generally calculates the estimated conditional target power of the second stage and uses this value as desired target power in the sample size re-estimation process:
If fCpower > targetpower, then the conditional estimated target power may be negative. This does not seem sensible. Therefore, for such cases the desired target power for the sample size re-calculation will be set to targetpower, i.e. ssr.conditional will be set to "error".

Value

Returns an object of class "pwrtsd" with all the input arguments and results as components. As part of the input arguments a component cval is also presented, containing the critical values for stage 1 and 2 according to the input based on alpha, weight and max.comb.test.
The class "pwrtsd" has an S3 print method.

The results are in the components:

pBE

Fraction of studies found BE.

pBE_s1

Fraction of studies found BE in stage 1.

pct_stop_s1

Percentage of studies stopped after stage 1 (due to BE or due to futility).

pct_stop_fut

Percentage of studies stopped after stage 1 due to futility.

pct_s2

Percentage of studies continuing to stage 2.

nmean

Mean of n(total).

nrange

Range (min, max) of n(total).

nperc

Vector of percentiles of the distribution of n(total).

Author(s)

B. Lang

References

König F, Wolfsegger M, Jaki T, Schütz H, Wassmer G.
Adaptive two-stage bioequivalence trials with early stopping and sample size re-estimation.
Vienna: 2014; 35th Annual Conference of the International Society for Clinical Biostatistics. Poster P1.2.88
\Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.13140/RG.2.1.5190.0967")}.

Patterson SD, Jones B. Bioequivalence and Statistics in Clinical Pharmacology.
Boca Raton: CRC Press; 2nd edition 2017.

Kieser M, Rauch G. Two-stage designs for cross-over bioequivalence trials.
Stat Med. 2015; 34(16): 2403–16. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1002/sim.6487")}.

Maurer W, Jones B, Chen Y. Controlling the type 1 error rate in two-stage sequential designs when testing for average bioequivalence.
Stat Med. 2018; 37(10): 1587–1607. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1002/sim.7614")}.

Wassmer G, Brannath W. Group Sequential and Confirmatory Adaptive Designs in Clinical Trials.
Springer 2016. \Sexpr[results=rd,stage=build]{tools:::Rd_expr_doi("10.1007/978-3-319-32562-0")}.

See Also

interim.tsd.in, final.tsd.in

Examples

# 12 subjects in stage 1, CV of 20%, no futility rule, otherwise all defaults
# except too low number of sims due to CRAN policy for run-time of examples
# This setting resembles values in Table 6.8 and 6.10 in Patterson and Jones
# if nsims=1e5 (default)
power.tsd.in(n1 = 12, CV = 0.2, fCrit = "No", nsims=1e4)

# Note that actual replication would require the following additional
# arguments (extremely long run-time)
## Not run: 
power.tsd.in(n1 = 12, CV = 0.2, fCrit = "No", max.n = 4000,
             pmethod = "exact", nsims = 1E6)
## End(Not run)

# Table 8 in Maurer et al
power.tsd.in(n1 = 36, CV = 0.4, max.n = 4000)

# Same example as before but with additional futility criterion regarding
# maximum sample size (using the default 4*n1 as maximum bound)
power.tsd.in(n1 = 36, CV = 0.4, max.n = 4000, fCrit = c("CI", "Nmax"))