Bioequivalence and Bioavailability Forum

Dear Helmut!

Sorry, but I cannot follow your reasoning, mostly.

First: Senns quote's

❝ [...] To quote Senn¹:

Perfect balance with respect to sequences, is not, however, an absolute requirement for an analysis which allows for period effects as well as treatments: it is simply that other things being equal such designs are more efficient.

(my emphasis)

I must confess that this paragraph is a little bit confusing to me (especially his example with 14 subjects in a 3x3 cross-over). But I think he is not talking about balance in the sense of the 3. condition of Williams designs (every treatment follows each other the same number) but rather in terms of balance of number of subjects to sequences (which ever chosen).

A hint that this is so cames from your second quote of Senn.
But in full context this reads:
" Frequently there is no good reason to choose any one Latin square in preference to another and the choice may then be made at random. There is often no reason either why a single Latin square should be used in a given trial (although some analyses are simpler if this is done) rather than a number of squares, apart from convenience in preparing trial material.

❝ It has been my habit in designing cross-over trials to use all six sequences for a three-period three-treatment design and to use a single Latin square for a four-treatment design.

Senn's attitude in his book is always to depict with personal sentences if any thing he is discussing is a personal view/opinion or doesn't matter.

Second: Regarding your reasoning in terms of efficiencies from Jones and Kenward

If I read my Holy bible^[2] of cross-over designs again and again I cannot follow you.
Again I must confess that it is like some white or black magic to me, what they do in chapter 4.
But undoubtedly: The efficiency numbers they have given for various designs (which you cite) are not obtained with no carry-over.
"In the presence of carry-over effects we define efficiency exactly as above but with Var[tau_i-tau_j] now denoting the difference between two treatments adjusted for carry-over effects: We label this efficiency as E_d ... For information we will always include the values of the efficiencies of a design, in the order of E_d, ..." (emphasis by me). Your cited numbers are the E_d efficiencies.

BTW: I think their 100% base is different from yours.

Thus the efficiencies show that in a model with carry-over not all designs are equal as they were without such effects. But you have always to pay for inclusion of such effects (efficiencies <100%).
That is why part of Senn's criticism of the simple carry-over model is ^[2]Chapter 10.3.4 The estimators based on it are inefficient and Chapter 10.3.5 The designs associated with it are not necessarily better than others.

Third:

❝ ❝ ... what are then the benefits of using balance with respect to condition 1.-3., the so called Williams designs?

❝ ❝ And why should regulators insist on it?

❝ Do they?

This question of mine is directly emerging from the statement "The chance of regulatory acceptance of a 3×3 is close to zero." in this post by HS .

Fourth:

❝ [...] Any Williams' design has the advantage that pairwise comparisons may be extracted …

❝ I've seen small studies where due to dropouts the extracted 2x2 sets were extremely imbalanced - or even worse, didn't work any more at all.

Here I can follow you, but it applies only for very imbalanced, small studies, in which we end up with many drop outs or missings.

Therefore it is my habit to choose all six sequences for a three-period three-treatment design also and to use a single Latin square for a 4-treatment design, not necessarily Williams .
For sequence stratified non-parametric evaluation Senn^[1] (p162-163) mentioned among others the 4-period-4-treatment Latin square
  ABCD
  BADC
  CDAB
  DCBA
which is not an Williams’s design. This and similar designs have the properties that for any pair of treatments in a given sequence there is another sequence in which the treatments are reversed in periods. This allows the evaluation stratified to these sequence pairs as described by Duchateau^[3] in the context of 3-period-3-treatment studies.
This evaluation is not possible with Williams design in a 4-period-4-treatment study or with a single 3x3 Latin square in a 3-period-3-treatment study.

But non-parametrics …. You know!

[1] S Senn
    Cross-over Trials in Clinical Research
    John Wiley & Sons, Chichester (2^nd ed. 2002)
[2] B Jones and MG Kenward
    Design and Analysis of Cross-over Trials
    Chapman & Hall/CRC, Boca Raton, Chapter 4 (2^nd ed. 2003)
[3] Duchatau et. al
    Adjusting pairwise nonparametric equivalence hypothesis
    tests and confidence intervals for period effects
    in 3x3 crossover trials
    J. Biopharm. Stat. Vol 12(2), 149-160, 2002