Bioequivalence and Bioavailability Forum

Hi D. Labes, Helmut, and KR.

I'm far more of a PK scientist than a statistician. When I first read the FDA paper on scaled reference limits, I did simple algebra so I wouldn't have to alter my SAS program in ways that would make my head spin. The FDA asked for:

a 95% upper bound for (µT - µR)² / (sig_wr)² < theta

or

a 95% upper bound for (µT - µR)² / (sig_wr)² - theta < 0

The FDA wants you to use this formula to calculate theta:

theta = [ln(1.25)]² / sig_w₀², where sig_w₀ = 0.25.
theta = 0.796689

Theta is a constant, and sig_wr is also a constant once you calculate what it is.

For convenience, I re-arranged the 95% upper bound formula to:

a 95% upper bound for (µT - µR) < sqrt[theta × (sig_wr)²]
Now you can use your PROC MIXED program on the ln-transformed data as is without the headache. All you need to do is change "alpha = 0.1" to "alpha = 0.05" in the ESTIMATE statement to get SAS to spit out a 95% confidence interval instead of the typical 90% confidence interval. Here is my PROC MIXED code, nothing proprietary about this:

proc mixed data=ABEexample1;
  class Subject Sequence Period Treatment;
  model logcmax = Sequence Period Treatment / DDFM=SATTERTH;
  random Treatment/TYPE=FA0(2) SUB=Subject G;
  repeated/Group=Treatment SUB=Subject;
  lsmeans Treatment / pdiff;
  estimate 'Average Bioequivalence for log(cmax)' Treatment 1 -1/CL alpha=0.05;
  ods output estimates=se_lncmax;
  make "CovParms" out=cov_lncmax;
  ods output "Least Squares Means"=LSM_lncmax;
run;

If you examine the "CovParms" output you'll see the mean square residual of the reference in there (sig_wr²).

However, since the FDA isn't asking for the 95%CI on (µT-µR), I'm assuming you can calculate it back to what they want to see, or you can just get a biostatistician to write you a proper program rather than use a work-around.

My program (FARTSSIE) does calculate theta, but that's not what I report in big green numbers - I report the limits that would have been imposed on (uT-uR). In retrospect, looking at what the FDA wants to see, this isn't terrifically helpful for people. I can't exactly get it to calculate your adjusted 95% confidence interval though, since Excel can't perform PROC MIXED.

Does this make any sense? Has anyone else performed this analysis more elegantly? I'd be interested to see some SAS code from someone who's done the job properly and knows what they're doing.

Best, -Dave