Bioequivalence and Bioavailability Forum

Hi G. C.,

❝ You have compared two methods and based on AIC value you have concluded the result. Which software you have used to calculate these two cases ?

Phoenix NLME 1.3. The AIC is available in all software I know (except Excel, of course).

❝ My meaning for tool is that.. if my tool's result value is different then FDA's tool's result values then will it be acceptable for them ?

Heck, do you really believe the FDA insists in reproducing a result from obsolete software which is 20+ years old? It’s somehow difficult to speak about “correct” or “false”. But what do we have: One result obtained by the (in)famous P-Pharm and others which are almost identical across five other software packages and six fitting algos – but different to what’s given in the guidance. There is no democracy in science but the odds are >5:1 against the FDA. I’m pretty sure you would get similar results in other software as well (hey guys: SAS, S+, Kinetica, NCSS, SPSS, MatLab, STaTa, STATISTICA, OO Calc, …)
Maybe back in ’94/95 the FDA was hit by the Pentium bug? The time window would fit just perfectly!

We could set up an experimentum crucis. Generate data from known parameters and compare results. In my example the data are exact, i.e., without error. Do you agree that we should expect almost identical results? E_max = –50, ED₅₀ = 1.

  x        y
 1/4    -10.0
 1/3    -12.5
 1      -25.0
 1+1/2  -30.0
 2+1/3  -35.0
 4      -40.0
 5+1/3  -42.5
 9      -45.0

In all packages I provided initial estimates of –10 and 2. R/nls by design does not allow fitting error-free data. I added ε [0, 10^–6].

                                   Emax        ED50
Excel solver (Newton)          -50.00001791  0.99999968
             (gradient)        -50.00009120  1.00000539
WNL PD 101 Gauß-Newton
           (Levenberg-Hartley) -50.00003052  0.99999860
WNL PD 101 Nelder-Mead
           (simplex)           -49.99991035  0.99998636
PHX/WNL 101                    -50.00000000  1.00000000
PHX/NLME                       -50.00000000  1.00000000
Gnumeric                       -50.00000000  1.00000000
R/nls                          -50.00000000  1.00000000

Of course this is not a proof. Unfortunately the E_max-model cannot be solved in closed form. Let’s simplify: \(\small{a=E_\text{max}}\) and \(\small{ED_{50}=b}\). Then we can write \(\small{y=ax}/(x+b)\), getting $$a=y(x+b)/x\tag{1}$$ and $$b=(xy-ax)/y\tag{2}$$ Now \(\small{a}\) depends on \(\small{b}\) and \(\small{b}\) depends on \(\small{a}\). ↺ End of story. You could cheat. For any of the \(\small{x|y}\) above setting \(\small{b=1}\) you will get \(\small{a=-50}\) from \(\small{(1)}\). Or the other way ’round: For any \(\small{x|y}\) setting \(\small{a=-50}\) you will get \(\small{b=1}\) from \(\small{(2)}\). But, wait a minute. The data are exact, right? We can plug in two data pairs, two equations, two unknowns, doable:
\(\small{-25=a/(1+b)}\)
\(\small{-40=4a/(4+b)}\)
Simsalabim: \(\small{a=-50,\,b=1}\)
At least the comparison should give you strong confidence that these software packages perform well and what the FDA reported is poo.

See also this post.