## 3 treatments with two-at-a-time approach [Power / Sample Size]

Hi Chris,

❝ First, thank you to the authors of PowerTOST and the maintainers of this wonderful forum.

Welcome!

❝ […] how the sample size is calculated for 1 test (A) and 2 references (B and C) based on 3x6x3 design. I am interested in showing the BE based on both A/C and B/C.

Not uncommon.

❝ If I were to use the two-at-a-time analysis approach, …

Good idea!

❝ … can I use the following function to get the sample size?

sampleN.TOST(CV = 0.2, design = "2x2")

Sample size (total)

 n     power

20   0.834680

❝ Based on this output, I am to assign 20/6 ~ 4 participants in each of the 6 sequences. I am hoping my reasoning is correct

Almost.

I suggest the section ‘Two at a Time’ as a starter.
Scroll down a bit and click (sorry, JavaScript required). Additionally to PowerTOST you need the package randomizeBE. You are interested in the -script make.ibds(). It has the same defaults like sampleN.TOST() and performs randomization of Williams’ designs for an arbitrary number of treatments in such a way that the extracted IBDs are balanced. Since for three treatments the default codes are A, B, C and the default comparisons are A/C and B/C, it’s easy.

library(PowerTOST) library(randomizeBE) make.ibds(CV = 0.2)

Gives:

 subject seqno sequence IBD 1 IBD 2        1     3      BAC   •AC   B•C        2     4      BCA   •CA   BC•        3     6      CBA   C•A   CB•        4     5      CAB   CA•   C•B        5     1      ABC   A•C   •BC        6     2      ACB   AC•   •CB        7     4      BCA   •CA   BC•        8     1      ABC   A•C   •BC        9     5      CAB   CA•   C•B       10     6      CBA   C•A   CB•       11     2      ACB   AC•   •CB       12     3      BAC   •AC   B•C       13     6      CBA   C•A   CB•       14     1      ABC   A•C   •BC       15     2      ACB   AC•   •CB       16     5      CAB   CA•   C•B       17     3      BAC   •AC   B•C       18     6      CBA   C•A   CB•       19     3      BAC   •AC   B•C       20     4      BCA   •CA   BC• Reference             : C Tests                 : A, B Sequences             : ABC, ACB, BAC, BCA, CAB, CBA Subjects per sequence : 3 | 3 | 4 | 3 | 3 | 4 (imbalanced) Estimated sample size : 20 Achieved power        : 0.8347 Randomized            : 2022-08-01 13:37:56 CEST Seed                  : 9540602 Runtime               : 20 ms (2 calls for balance of IBDs)

Of course, the sample size agrees with yours. However, there is no guarantee that you get balanced IBDs from multiple sequences of a given Williams’ design. Therefore, the script calls the randomization repeatedly until balance is achieved. Unpredictable, how many calls are needed. Sometimes just one, sometimes many more.

calls <- numeric(0) for (call in 1:250) {   calls[call] <- make.ibds(CV = 0.2, print = FALSE, details = FALSE)\$calls } summary(calls, digits = 1) Min. 1st Qu.  Median    Mean 3rd Qu.    Max.    1      2        3       4       6      29

If you want that the study (not only the IBDs) are balanced, use make.ibds(CV = 0.2, bal = TRUE). Even if IBDs are balanced, they are not necessarily balanced for period effects. In the example above: A is administered 6× in period 1 and 7× in periods 2 and 3. C is administered 6× in period 2 and 7× in periods 1 and 3. Too lazy to repeat this game for B/C.
Period-balance is a desirable property and I strongly recommend to aim at it. You would have a hard time to convince regulators that a true period effect does not exist. No, the p-value of an ANOVA is not relevant. However, there’s a downside: The sample size will be rounded up to a multiple of the number of sequences of the respective Williams’ design, i.e., in this example to 24.
In most cases you will increase the estimated sample size taking the anticipated dropout-rate into account anyhow. That’s especially important for Higher-Order crossovers and replicate designs as well. Let’s assume a conservative dropout-rate of 10%.

make.ibds(CV = 0.2, do.rate = 0.1)

Gives:

 subject seqno sequence IBD 1 IBD 2        1     4      BCA   •CA   BC•        2     5      CAB   CA•   C•B        3     1      ABC   A•C   •BC        4     2      ACB   AC•   •CB        5     5      CAB   CA•   C•B        6     6      CBA   C•A   CB•        7     3      BAC   •AC   B•C        8     4      BCA   •CA   BC•        9     6      CBA   C•A   CB•       10     3      BAC   •AC   B•C       11     1      ABC   A•C   •BC       12     2      ACB   AC•   •CB       13     6      CBA   C•A   CB•       14     1      ABC   A•C   •BC       15     3      BAC   •AC   B•C       16     5      CAB   CA•   C•B       17     5      CAB   CA•   C•B       18     6      CBA   C•A   CB•       19     4      BCA   •CA   BC•       20     2      ACB   AC•   •CB       21     4      BCA   •CA   BC•       22     3      BAC   •AC   B•C       23     1      ABC   A•C   •BC       24     2      ACB   AC•   •CB Reference                : C Tests                    : A, B Sequences                : ABC, ACB, BAC, BCA, CAB, CBA Subjects per sequence    : 4 | 4 | 4 | 4 | 4 | 4 (balanced) Estimated sample size    : 20 Achieved power           : 0.8347  Anticipated dropout-rate: 10%  Adjusted sample size    : 24  Achieved power          : 0.8960 Randomized               : 2022-08-01 13:38:47 CEST Seed                     : 8582810 Runtime                  : 20 ms (1 call for balance of IBDs)

• the IBDs are balanced, and
• all IBDs are period-balanced.
If you want to use another randomization (SAS, whatsoever), you must check whether the extracted IBDs are balanced.

❝ Also, I was unable to find any reference for the sample size calculation and wondered if anyone can kindly point me to the right direction.

<nitpick>

Estimation, if you don’t mind.

</nitpick>

I know only two. However, this one1 is for the ‘All at Once’ approach – which I don’t recommend – and that one2 pretty technical.

1. Yuan J, Tong T, Tang M-L. Sample Size Calculation for Bioequivalence Studies Assessing Drug Effect and Food Effect at the Same Time With a 3-Treatment Williams Design. Regul Sci. 2013; 47(2): 242–7. doi:10.1177/2168479012474273.
2. Zheng C, Wang J, Zhao L. Testing bioequivalence for multiple formulations with power and sample size calculations. Pharm Stat. 2012; 11(4): 334–41. doi:10.1002/pst.1522.

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Helmut Schütz

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