## Tricky question, lengthy answer [Power / Sample Size]

Unfortunately I am not sure to completely understand the following :

» Although you plan for a 3×6×3 Williams’ design, the two parts will be evaluated as incomplete block designs (IBD), having the same degrees of freedom as the conventional 2×2×2 crossover. Hence, in sampleN.TOST() use the argument design="2x2x2" and not design="3x6x3". You will get a small reward:

library(PowerTOST)» sampleN.TOST(CV=0.3, design="2x2x2", targetpower=0.9, »              print=FALSE)[["Sample size"]]» [1] 52» sampleN.TOST(CV=0.3, design="3x6x3", targetpower=0.9, »              print=FALSE)[["Sample size"]]» [1] 54

If my understanding is correct,
sampleN.TOST(CV=0.3, design="3x6x3", targetpower=0.9, print=FALSE)[["Sample size"]] would give me the sample size necessary for a power of 90% to detect BE for at least one given comparison (either A vs B or C vs B) if I evaluated all data at the same time, while sampleN.TOST(CV=0.3, design="2x2x2", targetpower=0.9, print=FALSE)[["Sample size"]] would give me the sample size for a given comparison if I evaluated the data two at a time (keeping in mind that I should potentially increase the result so that the sample size is a multiple of 6).

However, what should I do if I want the sample size for a power of 90% to detect BE for both comparisons ? If I assume that there is not correlation between both comparisons, and be n = sampleN.TOST(CV=0.3, design="2x2x2", targetpower=0.9, print=FALSE)[["Sample size"]], then:
power.TOST(CV=0.3, design="2x2x2", n=n) * power.TOST(CV=0.3, design="2x2x2", n=n) would evaluate to less than 0.9.
Is it correct to calculate the sample size only based on one comparison?

Furthermore, while the two parts of the trial will be evaluated as incomplete block designs, it seems to me that the original sequences and periods are preserved (e.g. an observation from period 3 is still coded as period 3), so that the degree of freedom would not be the same as for the conventional 2x2x2 crossover, no?

Lastly, when you said no multiplicity adjustment procedure was needed, was it implied that I should specify that BE between the clinical and marketed formulation will be tested first, before the food effect (i.e. hierarchical testing)? It is just to be in accordance with EMA and FDA.