Bioequivalence and Bioavailability Forum

Dear ALL!

From time to time the question of comparing Reference versus Reference in replicate designs pops up at me. We had discussed this already to some extent in this thread.

Helmut has pointed out a method via the EMA suggested ANOVA (throwing away the data for the Test formulation) recoding the treatments (or periods) to the occasion (replicate) the Reference was applied and comparing occasion 1 to occasion 2. His model must neglect period if he was evaluating the EMA data set I from the EMA Q&A, a 2-sequence-4-periods design.

I was for some reasons interested in applying this idea to the partial replicate design (sequences TRR/RTR/RRT). My test data was the EMA set II. I recoded the treatments to T1, R1 and R2. The numbers after the code are the replicate numbers.

With the usual ANOVA (recommended in the Q&A also for replicate studies) with effects tmt2, period, sequence and subject within sequence I got for the pair R1-R2:
        ----- log data ---------------------------   --- back transf. ------- 
data      se     diff R1-R2        90% CI            point est.  90% CI
R only  0.098703  0.005718  -0.164125 ... 0.175561   1.0057    0.8486 - 1.1919
All     0.046180  0.019498  -0.058095 ... 0.097091   1.0197    0.9436 - 1.1020
(se is the standard error of the difference)

For me that's a big difference and very astonishing because the intra-subject CV's of both analyses don't differ so much (CV=11.4% or 12.0% respectively). Quite the contrary the 90% CI is much broader for the lower CV.

To shed some light on the question "Which analysis is the correct one?" I have made an additional evaluation via the intra-subject contrasts R1-R2 analysed via an ANOVA with sequence as solely effect in the model (this is the so-called robust analysis of Patterson/Jones aka Senn's basic estimator approach).
The naive application of the intra-subject contrasts R1-R2 suffers however from a flaw concerning the period effects (named p1, p2, p3) for the partial replicate design:
sequ. Contrast Expectation
TRR    P2-P3   (R-R')+p2-p3
RTR    P1-P3   (R-R')+p1-p3
RRT    P1-P2   (R-R')+p1-p2
(P1, P2 and P3 term the values of the log-transf. PK metric in the corresponding period)
That is the mean over the sequence groups is contaminated by 2*p1-2*p3 and only an unbiased estimator of (R-R') if there are no period effects. But this flaw can easily corrected within that design if one takes P3-P1 in case of the sequence RTR as a measure of R-R'.
I obtained:
        ----- log data ---------------------------   --- back transf. ------- 
Method    se     diff R1-R2        90% CI            point est.  90% CI
naive   0.032901 -0.021780  -0.078394 ... 0.034835   0.9785    0.9246 - 1.0355
corr.   0.032901  0.001906  -0.054708 ... 0.058520   1.0019    0.9468 - 1.0603

Different results of both methods even if no significant period effect is observed within this data set.
Different results to above ! But more in line with the ANOVA results using all data.
What makes me very curious is that the results of ANOVA using R data only coincide with the evaluation via intra-subject contrasts (corr. for period effects) if the first are divided by 3 (sic). Try it.

Cough .
Any body out there with an idea what's going on here?
Bonus question: Which result do you trust?

BTW: The partial replicate design seems the only one between the common used replicate designs which can be corrected for period effects regarding an estimate of R vs. R'. For others a model without period effects must be employed, if I'm not wrong.