Bioequivalence and Bioavailability Forum

Hi Martin,

❝ Body weight stratum 1: x mg for T1, T2 and R for subjects < z kg

❝ Body weight stratum 2: y mg for T1, T2 and R for subjects > z kg

❝ Study design: Williams’ design (3-treatment, 3-period, 6-sequence) with stratified randomization according to body weight stratum (<z kg and > z kg) to ensure balance regarding this strata.

Modifying this code:

Randomization table          created: 2018-06-08 11:06:57
(seed: 8503789 blocksize: 6 )

 subject stratum seqno sequence period 1 period 2 period 3
       1  low BW     1  T2-T1-R       T2       T1        R
       2  low BW     5  R-T2-T1        R       T2       T1
       3 high BW     3  T1-T2-R       T1       T2        R
       4 high BW     2  T2-R-T1       T2        R       T1
       5  low BW     6  R-T1-T2        R       T1       T2
       6  low BW     4  T1-R-T2       T1        R       T2
       7  low BW     3  T1-T2-R       T1       T2        R
       8 high BW     6  R-T1-T2        R       T1       T2
       9  low BW     2  T2-R-T1       T2        R       T1
      10 high BW     5  R-T2-T1        R       T2       T1
      11 high BW     4  T1-R-T2       T1        R       T2
      12 high BW     1  T2-T1-R       T2       T1        R
      13  low BW     1  T2-T1-R       T2       T1        R
      14  low BW     6  R-T1-T2        R       T1       T2
      15  low BW     4  T1-R-T2       T1        R       T2
      16 high BW     2  T2-R-T1       T2        R       T1
      17 high BW     3  T1-T2-R       T1       T2        R
      18  low BW     5  R-T2-T1        R       T2       T1
      19 high BW     5  R-T2-T1        R       T2       T1
      20  low BW     3  T1-T2-R       T1       T2        R
      21 high BW     1  T2-T1-R       T2       T1        R
      22 high BW     4  T1-R-T2       T1        R       T2
      23 high BW     6  R-T1-T2        R       T1       T2
      24  low BW     2  T2-R-T1       T2        R       T1


Summary of randomisation

24 subjects randomized into 6 sequence groups.
Number of subjects in sequence groups:
R-T1-T2 R-T2-T1 T1-R-T2 T1-T2-R T2-R-T1 T2-T1-R
      4       4       4       4       4       4
Runs test of randomness: p.value=0.4038

R-T1-T2,     low BW: 2
R-T1-T2,    high BW: 2
R-T2-T1,     low BW: 2
R-T2-T1,    high BW: 2
T1-R-T2,     low BW: 2
T1-R-T2,    high BW: 2
T1-T2-R,     low BW: 2
T1-T2-R,    high BW: 2
T2-R-T1,     low BW: 2
T2-R-T1,    high BW: 2
T2-T1-R,     low BW: 2
T2-T1-R,    high BW: 2

Trivial: Your sample size has to be a multiple of 12.

❝ 1) Can the BE assessment be based on all data combined or does this require a separate analysis per body weight stratum

Strictly speaking BE is a comparison of at the same molar dose. Hence, separate?

❝ 2) If a combined assessment is the way to go how should the model look like

❝ a) Ignoring body weight strata and use classical model (FDA): fixed effects for period, sequence, treatment and random effect for subject nested in sequence.

❝ b) Including body weight strata as mentioned in ICH E9 (i.e. factors on which randomization has been stratified should be accounted for later in the analysis): fixed effects for period, sequence, treatment, body weight strata and random effect for subject nested in sequence.

If you go with a) what is the purpose of the stratification?
In b) the stratum is a between-group term anyhow and I guess that the residual error will be the same as in a).

Dear Martin,

❝ ...

❝ Questions:

❝

❝ 1) Can the BE assessment be based on all data combined or does this require a separate analysis per body weight stratum

❝

❝ 2) If a combined assessment is the way to go how should the model look like

❝

❝ a) Ignoring body weight strata and use classical model (FDA): fixed effects for period, sequence, treatment and random effect for subject nested in sequence.

❝

❝ b) Including body weight strata as mentioned in ICH E9 (i.e. factors on which randomization has been stratified should be accounted for later in the analysis): fixed effects for period, sequence, treatment, body weight strata and random effect for subject nested in sequence.

Additionally to what Helmut wrote:

If you will use a Proc GLM or lm() for the combined assessment you will face a confounding between subject effects and body-weight-strata effects and the analysis code will show you the finger. Something like df=0 will happen, at least for type III tests of effects.

You have to include at least a treatment by body-weight-strata interaction to get meaningfull results.
I'm not really sure if this confounding is also an issue if you plan to use Proc MIXED or lme()/lmer(). Make a example data set and try it.

I suggest that you modify the FDA code for logistic groups (see f.i. this post) accordingly (change group to body-weight-strata and drop Period(nested within Group)).
And read Helmut's lectures about "Multi-Group Studies in BE. To pool or not to pool?". All the criticism regarding the group effects apply also to body-weight-strata effects. The treatment effect (diff in the log domain, ratio on the original scale) in x-over studies is determined intra-subject. Thus all subject characteristics constant over the study can not influence it really.

If you are interested in having a look at BE assesment (ratio & CI) for the different body-weight-strata you have to go with 1) anyway.

BTW: "factors on which randomization has been stratified should be accounted for" should read the other way round: randomize stratified for factors planned to account for in the analysis.

Dear Martin,
additionally to what Helmut and Detlew wrote:

❝ 1) Can the BE assessment be based on all data combined or does this require a separate analysis per body weight stratum

Maybe easier way would be to separate the analysis as 2 studies:
- for subjects with weight <z kg,
- for subjects with weight >z kg.

Pro: standard evaluation (2x)
Con: approx. double number of subjects required (than in option 2))

❝ 2) If a combined assessment is the way to go how should the model look like ...

I would be inspired by FDA model with Groups (as Detlew pointed). Then it is an interesting question which factors should be included in the model:
Maybe:

Period                                  (1 or 2 or 3)
Sequence                                (six options acc. to Williams’ design)
BWS                                     (Body Weight Strata, values 1 or 2)
Subject (nested within BWS*Sequence)    (sequence and BWS are between-subject factors)
BWS*Sequence
Treatment (nested within BWS)           (same situation as Period (nested within Group) in FDA suggested model for studies in multiple groups ... periods differ for different groups, here treatments differ for different BWS)

- The factor Treatment (nested within BWS) is THE END. It is not possible (for me) to evaluate the treatment difference as it is not the main effect but it is nested. When changed to main effect (i.e. just Treatment instead of Treatment(BWS) in the model) then we ignore the fact that treatment differs for BWS. (Wenn ich mich nicht irre.)

Possible option would be to use dose correction and then evaluation using the standard model.
Though I realized that there is the same dose correction for T1,T2,R in the half of the subjects and similarly in the second half ... so the ratios will stay unchanged regardless the correction.

Pro: Less subjects required in total (if you are not interested in separate analysis)
Con: model , varied questions may be arised

Best regards,
zizou