Bioequivalence and Bioavailability Forum

Dear Helmut, dear All,

Meanwhile I have experimented a little bit with the ANOVA described in the Liu paper. I have taken Example 4.4 from the book

Patterson, Jones
"BIOEQUIVALENCE and STATISTICS in CLINICAL PHARMACOLOGY"
Chapman & Hall / CRC 2006

a 4-period study with the sequences TRTR and RTRT as the dataset (to be found here).
[edit: /CRC WEB PAGE/chapter4/exam44.dat (Helmut)]

The subjects with missing data were left out from the analyses.

It took me some headache (and some beer ) to figure out how to formulate a Proc GLM ANOVA with the effects sequence, subject (within sequence), formulation, period and subject-by-formulation interaction. "Simple" attempts catched me often in the Type III sum-of-squares trap.

Here comes the code that worked without curiosities:
Proc GLM data=four;
  class sequence subject period formula;
  model ln_AUC = sequence subject(sequence) formula period
                 subject(sequence*formula) / CLparm alpha=0.1;
  random subject(sequence) subject(sequence*formula) /Test;
  Estimate 'T-R' formula -1 1;
*without subjects with missings;
 where subject not in(3 27);
quit;

The random statement is only for letting SAS figure out the "appropriate" F-tests for the effects, because we all know meanwhile that Proc GLM is fitting the model as if all effects are fixed and the random statement only affects the F-tests, which in an all-fixed model always have MS(error) as denominator.

Results:
...
                             The GLM Procedure

Dependent Variable: ln_auc

                                   Sum of
 Source                  DF       Squares   Mean Square  F Value  Pr > F

 Model                  105   55.42179511    0.52782662     5.33  <.0001
 Error                  102   10.10810776    0.09909910
 Corrected Total        207   65.52990287

           R-Square     Coeff Var      Root MSE    ln_auc Mean
           0.845748      5.349792      0.314800       5.884343

*-----------------------------------------------------------------------
the all effects fixed part;
 Source                    DF     Type I SS   Mean Square  F Value  Pr > F

 sequence                   1    0.17364001    0.17364001     1.75  0.1886
 subject(sequence)         50   49.37630741    0.98752615     9.97  <.0001
 formula                    1    0.56857602    0.56857602     5.74  0.0184
 period                     3    0.30656009    0.10218670     1.03  0.3821
 subje(sequen*formul)      50    4.99671158    0.09993423     1.01  0.4752


 Source                    DF   Type III SS   Mean Square  F Value  Pr > F

 sequence                   1    0.17364001    0.17364001     1.75  0.1886
 subject(sequence)         50   49.37630741    0.98752615     9.97  <.0001
 formula                    1    0.56857602    0.56857602     5.74  0.0184
 period                     2    0.27974027    0.13987013     1.41  0.2485
 subje(sequen*formul)      50    4.99671158    0.09993423     1.01  0.4752

*-----------------------------------------------------------------------
expected mean square from the MIXED model;
Source                Type III Expected Mean Square

sequence              Var(Error) + 2 Var(subje(sequen*formul)) +
                      4 Var(subject(sequence)) + Q(sequence)

subject(sequence)     Var(Error) + 2 Var(subje(sequen*formul)) +
                      4 Var(subject(sequence))

formula               Var(Error) + 2 Var(subje(sequen*formul)) + Q(formula)

period                Var(Error) + Q(period)

subje(sequen*formul)  Var(Error) + 2 Var(subje(sequen*formul))

*----------------------------------------------------------------------
appropriate (for the MIXED model) F-tests;
         Tests of Hypotheses for Mixed Model Analysis of Variance

Dependent Variable: ln_auc

 Source                    DF   Type III SS   Mean Square  F Value  Pr > F

 sequence                   1      0.173640      0.173640     0.18  0.6768
 Error                     50     49.376307      0.987526
 Error: MS(subject(sequence))

 subject(sequence)         50     49.376307      0.987526     9.88  <.0001
 formula                    1      0.568576      0.568576     5.69  0.0209
 Error                     50      4.996712      0.099934
 Error: MS(subje(sequen*formul))

 period                     2      0.279740      0.139870     1.41  0.2485
 subje(sequen*formul)      50      4.996712      0.099934     1.01  0.4752
 Error: MS(Error)         102     10.108108      0.099099

*--------------------------------------------------------------
estimated treatment effect, 90% CIs in log domain
Parameter  Estimate       Error    t Value    Pr > |t| 90% Confidence Limits

T-R      0.10456651    0.04365492   2.40      0.0184   0.03210240   0.17703061
...

Some points to consider:

• The "appropriate" F-tests will only be obtained if the model is formulated with some random effects.
  
• If there is no subject-by-formulation interaction the model reduces to the classical GLM-ANOVA used for the 2x2 crossover (the Bear way, but there estimation with "real" mixed model software lme()).
  
• The appropriate error term for the formula (treatment) effect is the MS(subject-by-formulation) if mixed model is used! This is the meaning of s_Int not known in this thread in connection with SABE in replicate design.
  
• The estimate statement is not affected by the random statement and uses the MS(error) as the error term. This can be show if the random statement is deleted (results not shown here).
  Thus it treats all effects as fixed. (EM: Now I understand you, from now on I will call it also bogus! )
  
• This is in most cases anti-conservative. You can verify this in looking at the expected mean square for MS(SxF) and noticing that it is the sum of the error variance + variance from subject-by-treatment interaction. Thus you will get wider CIs for the treatment diffs in the mixed model, even more if you look at the associated degrees of freedom.
  But this anti-conservative method is what is required by the new EMA guidance (or do we err?)! BTW: This is different to the classical 2x2 design where the correct error term is due to missing SxF interaction always MS(error)(subject as random effect or not).
  
• The different intra-individual variabilities are not part of the Proc GLM output and have to calculate in a additional step (using method of moments?).
  
• Somehow curious for me is the period effect df. Don't know if this is a SAS curiosity. It is different from the Liu paper, but the Liu design is also different (2 periods with 2 replicates within periods).

BTW: Excuse this very long post, but I had no idea how to shorten.
BTW2: The fixed effects story gets more and more curious for me.
BTW3: If this ANOVA would be sufficient for the EMA we had also in SAS to follow Helmut's SOP with the adaption "delete Mixed model" .