Interim analysis of first stage data of 2-stage 2x2 crossover...

Description Usage Arguments Details Value Author(s) References See Also Examples


Following the design scheme according to the function performs the interim analysis of the first stage data.


6, weight, max.comb.test = TRUE, targetpower = 0.8,
               GMR1, n1, CV1, df1 = NULL, SEM1 = NULL, 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("exact", "nct", "shifted"))



If one element is given, the overall one-sided significance level (not the adjusted level for stage 1). In this case the adjusted alpha levels will be calcualted 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.


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.


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


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


Observed ratio of geometric means (T/R) of stage 1 data (use e.g., 0.95 for 95%).


Sample size of stage 1.


Observed coefficient of variation of the intra-subject variability of stage 1 (use e.g., 0.3 for 30%).


Optional; Error degrees of freedom of stage 1 that can be specified in addition to n1.


Optional; Standard error of the difference of means of stage 1 that can be specified in addition to CV1. Must be on additive scale (i.e. usually log-scale).


Lower bioequivalence limit. Defaults to 0.8.


Upper bioequivalence limit. Defaults to 1.25.


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 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 planning value GMR.


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.


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.


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.


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). Defaults to "CI".


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 to be 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).


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


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).


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’.


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.)
In contrast to the default value here is "exact".


The observed values of stage 1 (e.g. GMR1, n1, CV1) may be obtained based on the first stage data via the usual ANOVA approach.

The optional arguments df1 and SEM1 require a somewhat advanced knowledge (provided in the raw output from for example the software SAS, or may be obtained via emmeans::emmeans). However, it has the advantage that if there were missing data the exact degrees of freedom and standard error of the difference can be used, the former possibly being non-integer valued (e.g. if the Kenward-Roger method was used).

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".
Also, if the futility criterion based on the power of stage 1 is met, then the conditional estimated target power will be negative. Thus, no further sample size calculation can be made. To acknowledge that this rule is nonbinding, for the purpose of calculating n2 the argument ssr.conditional is set to "error".


Returns an object of class "evaltsd" 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 "evaltsd" has an S3 print method.

The results are in the components:


Observed p-value for first hypothesis.


Observed p-value for second hypothesis.


z statistic value for first null hypothesis.


z statistic value for second null hypothesis.


Repeated confidence interval for stage 1. Corresponds to the usual CI with level alpha1.


If the study stops, the median unbiased point estimate as estimate for the final adjusted geometric mean ratio after stage 1 (note that the value is identical to GMR1.)


Three dimensional vector with either 0 or 1. The first component represents futility due to Power of first stage > fCpower, the second futility due to CI (or PE) outside of fClower ... fCupper, the third futility due to n1 + n2 > fCNmax.
Note that the futility rules can be applied in a non-binding manner.


90% Confidence interval for observed ratio of geometric means from stage 1. If fCrit != "CI" result will be NULL.

Power Stage 1

Calculated power of stage 1.


Logical, indicating whether to stop after stage 1 (due to BE or due to futility).


Logical, indicating whether study is recommended to be stopped after stage 1 due to futility.


Logical, indicating whether BE could be concluded after stage 1 or not (regardless of any futility criterion).


Required (total) sample size for stage 2 (will be zero if BE has been shown after stage 1).


Only applicable if BE has not been shown after stage 1. Contains alpha values for the two hypotheses required for sample size re-calculation. If ssr.conditional = "no" the result is equal to alpha, otherwise it contains the conditional error rates for the standard combination test (in case of max.comb.test = FALSE) or maximum combination test (in case of max.comb.test = TRUE).


Only applicable if BE has not been shown after stage 1. Contains the geometric mean ratio used for sample size re-calculation (accounts for adaptive planning step).


Only applicable if BE has not been shown after stage 1. Contains the target power used for the sample size re-calculation (see also 'Details').


B. Lang


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
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.

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. doi: 10.1002/sim.7614.

Wassmer G, Brannath W. Group Sequential and Confirmatory Adaptive Designs in Clinical Trials.
Springer 2016. doi: 10.1007/978-3-319-32562-0.

See Also,


# Example from Maurer et al. = 0.95, max.n = 4000,
               GMR1 = exp(0.0424), CV1 = 0.3682, n1 = 20)
# Example 2 from Potvin et al. = 0.95, GMR1 = 1.0876, CV1 = 0.18213, n1 = 12,
               fCrit = "No", ssr.conditional = "no")

Power2Stage documentation built on Nov. 21, 2021, 1:07 a.m.