Bioequivalence and Bioavailability Forum

Dear HS,

❝ Yes; see this post for TRT/RTR (75% of TR/RT) and TRRT/RTTR (50% of TR/RT).

❝ No idea about TRR/RTT...

as I have understood until now all 3 period and all 4 period replicate designs, regardless of which sequences are used, have the same variance of the treatment effect namely
   3 period : 3/8*s2/N
   4 period : 1/8*s2/N

if no carry-over term is incorporated in the model.

Compared to the common 2x2 design with variance
   1/2*s2/N
you get your mentioned 75% or 50% respectively.

See for that
Jones and Kenward
Design and Analsis of Cross-over Trials
Chapter 3

Only if a carry-over term (as simple carry over in different flavors) or other interaction terms are introduced in the model different variances arose for the different designs.

Regarding the carry-over (and the model for simple carry-over) I adopt SENN's position: With an adequate wash-out period an standardization of the study it is futile in BE studies.
Interesting enough there is no teensy-weensy attempt of incorporation of carry-over in the FDA code for analysing average BE in replicate designs.

Thus IMHO we have always the same sample size estimation formula for 3 or 4 period replicate designs.

Dear Vidula,

❝ Can we calculate the sample size for a 4-way crossover design? We are designing a study where each person will get 1 cycle each of 4 treatment regimes ABCD.

This is not a replicate crossover design (as suggested in the subject line of your post). It is a crossover design for more then 2 treatments. Search the forum for “Williams' design“ to find some discussions about such designs, but mainly about crossover designs for 3-treatments-3-periods.
See this thread/post with literature hints, also for sample size estimation.

❝ I could calculate sample size for a two-way crossover design from an online calculator.

(emphasis by me)
Fine. But how are you sure that the online calculator gives you reasonable results? Which methods, approximations did it use? Is it really for an equivalence test? Does it calculate with a multiplicative model (log-transformed) or in the untransformed domain?
Be so kind to post the URL. I myself hadn't seen an online calculator for the sample size for BE studies up to know.

❝ How is the sample size of a 4-way crossover related to 2-way crossover study?

The formulas for calculation of the power of a TOST of a specific comparison, let's say A versus B, are nearly the same as for the classical 2x2 crossover design. With the exception that the degrees of freedom for the ANOVA error mean square (MSE) used are different. But this makes seldom a great difference in terms of the sample size (see below). It mainly affects the attained power.

To be on the safe side I (knowing the author since many years ) would suggest you the R project add-on package PowerTOST within which you can calculate power/sample size for a variety of designs used in BE studies, among them your 4x4 crossover.

The best of this solution is: It's free of any charge.
The second best is: It's source code is open, you and any other of the community can inspect it and assure it self it is calculating what he/she need the correct way.
(Change the rating of the bests to your preference)

Here some results of PowerTOST function sampleN.TOST() for both the classical 2x2 crossover and the 4x4 crossover, calculated with an assumed true ratio of 0.95, BE margins as usual 80 - 125% and target power 80%:
CV    "2x2"  "4x4"
10%      8      8
15%     12     12
20%     20     20
25%     28     28
30%     40     40

Depending on the aim of your study you should eventually consider some alpha adjustment.
See Chapter 7 of
Hauschke, Steinijans and Pigeot
Bioequivalence Studies in Drug Development
Wiley, Chichester 2007

❝ As you may have guessed, I am a beginner in sample size calculation problems in crossover designs.

Welcome to the club.