Bioequivalence and Bioavailability Forum

❝ ❝ Has anyone thought about using type=FA0(1) instead? Which in this case amounts to the same model but correctly specified.

❝

❝ Interesting! Can you please a bit more specific:

• Why do you think that it is the same model?
• Why do you think it is correctly specified? Seems to me a logical trap: same models, one correct, the other overspecified.

Dear Detlew,
thanks for your comments.
If we look at the effects of the two random statements,
random tmt / sub=subject type=FA0(2); /* FDA */
random tmt / sub=subject type=FA0(1); /* new model */
then the corresponding G matrices are: (trying to display matrices in a linear form)
G1: 1st line: a², a*b; 2nd line: a*b, b²+c²
G2: 1st line: a², a*b; 2nd line: a*b, b²
(a, b, c vary freely, -1≤r≤1)

The repeated statement leads to the following R matrix:
For the RTRT/TRTR design, R=diag(X,Y,X,Y) for any subject of sequence RTRT
For the RTR/TRR/RRT design, R=diag(X,Y,X) for any subject of sequence RTR
where X and Y must be nonnegative numbers.

Within the same subject, we have, for the first random statement:

• the variance of an R observation is a²+X
  
• the variance of a T observation is b²+c²+Y
  
• the covariance of two different R observations is a²
  
• the covariance of two different T observations is b²+c²
  
• the covariance of an R and a T observation is a*b

For the RTRT/TRTR design, this works well.

For the RTR/TRR/RRT design, however, the covariance of two different T observations (in the same subject; above in red) does not appear, and therefore the model is over-specified: the parameters c and Y appear only together, in the form c²+Y. As a consequence, Proc Mixed will often have difficulties in estimating the model, in particular it may fail to converge.

Remedy: we should replace c²+Y with a single value, i.e. delete either c or Y (or fix it at zero). Getting rid of Y does not seem to be easily possible, but c² can be removed by using FA0(1) instead of FA0(2).

(Note: both models are mathematically correct, one is just overspecified which makes it much more difficult to estimate and less stable numerically. A better version of Proc Mixed might detect this itself and handle it correctly.)

Hope I have explained it in an understandable way?
Best regards,
Wolfgang Seewald