Bioequivalence and Bioavailability Forum

Hi ElMaestro,

❝ ❝ It should be noted that in rare cases (e.g., extremely unbalanced sequences) the fixed effects model gives no solution and the mixed effects model has to be used.

❝

❝ a realistic linear model will have a single analytical solution unless you make a specification error. Imbalance would not affect that, please describe where/how you came a cross a fit which failed with the lm.

I was right about failing in JMP and Phoenix WinNonlin.
Sorry I can’t disclose the data set. Naïve pooling was performed. Deficiency letter by the MHRA in summer 2016:

The applicant should present estimates and 95% confidence interval for the difference between the Test and the Reference product on a ratio scale from ANOVA model, that reflects the design of the study, with terms for Group, Sequence, Sequence * Group, Subject (Sequence * Group), Period (Group), Treatment as fixed effects.

Note that this is the FDA’s model 2 with fixed effects. Why the 95% CI instead of the 90% CI was required is another story. The data set (subjects fixed) did not converge in JMP. Switched to random and all was good. Was accepted by the MHRA’s assessor.

Phoenix showed me the finger with the fixed effect Subject(Sequence*Group) in Model 1

and execution stopped (no results at all).
In model 2 I got the same warning as above but these results:

Partial Sum of Squares
            Hypothesis        DF          SS         MS     F_stat  P_value
---------------------------------------------------------------------------
                 Group         2   0.0758837  0.0379418   3.48232    0.0381
              Sequence         1   0.0708455  0.0708455   6.50224    0.0138
        Group*Sequence         2   0.145263   0.0726313   6.66614    0.0026
Sequence*Group*Subject        50   7.67886    0.153577   14.0954     0.0000
          Group*Period         3   0.0111135  0.0037045   0.340001   0.7965
             Treatment         1   0.144129   0.144129   13.2283     0.0006
                 Error        52   0.566569   0.0108956

Partial Tests of Model Effects
            Hypothesis  Numer_DF  Denom_DF     F_stat  P_value
--------------------------------------------------------------
                 Group         2        52   3.48232    0.0381
              Sequence         1        52   6.50224    0.0138
        Group*Sequence         2        52   6.66614    0.0026
Sequence*Group*Subject        50        52  14.0954     0.0000
          Group*Period         3        52   0.340001   0.7965
             Treatment         1        52  13.2283     0.0006

End of the story. No LSMs. Hence, no difference, no CI…

No problems in R.
Model 1:

Analysis of Variance Table

Response: log(Cmax)
                       Df   Sum Sq    Mean Sq  F value     Pr(>F)   
group                   2 0.078270 0.03913490  3.54171 0.03643331 * 
sequence                1 0.073106 0.07310604  6.61611 0.01312377 * 
treatment               1 0.141465 0.14146461 12.80257 0.00078035 ***
group:period            3 0.011114 0.00370450  0.33526 0.79988452   
group:sequence          2 0.145263 0.07263128  6.57314 0.00292116 **
group:treatment         2 0.014083 0.00704174  0.63728 0.53296911   
group:sequence:subject 50 7.678856 0.15357712 13.89875 < 2.22e-16 ***
Residuals              50 0.552485 0.01104970

Model 2:

Model 2:Analysis of Variance Table

Response: log(Cmax)
                       Df   Sum Sq    Mean Sq  F value     Pr(>F)   
group                   2 0.078270 0.03913490  3.59182 0.03458136 * 
sequence                1 0.073106 0.07310604  6.70971 0.01241215 * 
treatment               1 0.141465 0.14146461 12.98370 0.00070289 ***
group:period            3 0.011114 0.00370450  0.34000 0.79647341   
group:sequence          2 0.145263 0.07263128  6.66614 0.00264706 **
group:sequence:subject 50 7.678856 0.15357712 14.09540 < 2.22e-16 ***
Residuals              52 0.566569 0.01089555

Diving deeper into it. Originally I set up the models in Phoenix WinNonlin’s Bioequivalence module, which sits on top of Linear Mixed Effects. When I send the data directly to Linear Mixed Effects (all fixed) no error, no warning, nada. CI identical to the one from R to 12 significant digits.
Conclusion: Bug in Phoenix WinNonlin’s Bioequivalence module.

BTW: Running model 1 of my 85 data sets (5,004 subjects) in Bioequivalence takes more than ten hours and sucks up almost my entire 16 GB RAM (memory leak?). Direct execution in Linear Mixed Effects takes five minutes (max. RAM consumption 175 MB).
Much slower than R, which takes five seconds for model 1, model 2, model 3 (for each group), and model 3 (pooled).

I was wrong. Has nothing to do with unbalanced sequences and/or unequal group sizes.