From: "Eleveld, DJ" d.j.eleveld@anest.umcg.nl
Subject: [NMusers] Difference between typical values and geometric mean of posthoc values 
Date:  Wed, March 30, 2005 5:54 am 

Hello everyone, 

Many thanks for all those who reacted to my question about the difference between
the typical (theta) values and the geometric mean of the posthoc values.  The
comments seemed to have three main points:

1) The underlying data might not really be log-normal distributed 
2) The sample being fitted will have sampling-error (I think thats what its called) 
3) The thetas are the 'most-probable values', not the geometric mean 

The data I have been fitting comes from monte-carlo simulation so the parameters
are really log-normally distributed.  So 1) does not apply in this case.  I see
the point that this would definetly be applicable when fitting 'real' measured
data (i.e. not simulated data).

I think sampling error has been handeled as well as i can.  The monte-carlo simulations
are fittings of 10 simulated data sets from a 3 compartment PK model.  The fitting are
repeated 100 times, each time with different underlying PK parameters.  If the difference
between the typical values and the geometric mean of the posthoc values is a result of
sampling-error then one would expect an 'on average' difference between these values to
be zero.  For one of the model parameters (CL) there is a 10% difference.  This seems to
me to be unusually large to be just 'random'.

I'm not sure what exactly the difference is between the 'most-probable' values and the
geometric mean.  I thought that 'most-probable' value would be the maximum-likelihood value
and that the maximum likelihood value and the geometric mean are the same for a log-normal
distribution, although I dont have a definition handy.  I'll try to find out this for sure.

I remember reading that nonmem uses a linearization technique for estimation.  Could this
result in the bias i am seeing?  Might LAPLACIAN make a difference here?

Thank you very much for your replies, 

Doug Eleveld 

Just to check that I havent made any obvious errors the control stream is: 

$PROB  Test fitting 
$DATA  mcra.dat 
    IGNORE=C 
$INPUT ID TIME WGT AMT RATE DV 
$SUBROUTINES ADVAN11 TRANS4 
$PK 
    V1=THETA(1)*EXP(ETA(1)) 
    V2=THETA(2)*EXP(ETA(2)) 
    V3=THETA(3)*EXP(ETA(3)) 
    CL=THETA(4)*EXP(ETA(4)) 
    Q3=THETA(6)*EXP(ETA(6)) 
    Q2=(THETA(2)*(THETA(6)/THETA(3) + THETA(5)))*EXP(ETA(5)) 
    S1=V1 
$ERROR 
    Y=F*EXP(ERR(1)) 

; Starting at the exact values 
$THETA (0, 3.64)(0, 3.01)(0, 6.44)(0, 0.51)(0, 0.048)(0, 0.051) 
$OMEGA 
0.0533 
0.1217 
0.1063 
0.1245 
0.2215 
0.0650 
$SIGMA 0.01 
;SIGMA 0.04 Some (12, 37, 44, 49, 54, 71, 84, 92) used this to get convergence 

$ESTIMATION MAX=9999 SIG=6 METHOD=COND NOABORT POSTHOC 
$TABLE TIME V1 V2 V3 CL Q2 Q3 DV 
$SCATTER PRED VS DV UNIT 

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From:  "Eleveld, DJ" d.j.eleveld@anest.umcg.nl
Subject: RE: [NMusers] Difference between typical values and geometric mean of posthoc values 
Date:  Fri, April 1, 2005 9:14 am 

Thanks to everyone who replied to my question about the difference between typical 
values and geometric mean of posthoc values.  I am afraid I am still a bit lost. 

In monte-carlo tests I am seeing a bias of 10% between the typical values and 
geometric mean of posthoc values for CL.  Other parameters have lower biases.  
This is for 100 estimations of 10 individuals each.  I am using ADVAN11, TRANS4 
and the FOCE method.  Interestingly the geometric mean of posthoc values is closer 
to the 'real' values than the typical (THETA) values.  I have no idea why this 
is. 

I believe that the maximum likelihood estimate of a log-normal distribution is the 
geometric mean.  So if the typical values (THETA) are the maximum likelihood values 
then there should be no bias with the 'real' values. 

I dont think using INTERACTION should be used beacuse all invididuals have the same 
error variance. 

I dont think there are, at least there shouldn't be, model complexity problems.  The 
data is simulated and there is no model misspecification.  

So I am still a bit lost.  Why should we rely on THETA values to describe central 
tendency of our parameter distributions when the POSTHOC values are available and 
seem to allow better accuracy? 

Thank you, 

Doug Eleveld 
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From: "Wang, Yaning" WangYA@cder.fda.gov
Subject: RE: [NMusers] Difference between typical values and geometricmea n of posthoc values
Date:  Fri, April 1, 2005 1:08 pm

Eleveld:
Try a simple linear mixed effect model in NONMEM, like
Yij=Intercepti+slopei*tij, to see whether you still have
this kind of observation. 
 
Here are some of my opinions about nonlinear mixed
effect modeling regarding your questions. 
 
1. Nonlinear mixed effect modeling (parametric) is not a
pure maximum likelihood estimation method. The likelihood
is approximated in NONMEM (also in other softwares). Furthermore,
there are different methods for approximation. The impact of
these approximation on the consistency of the parameter estimates
is not clear yet (as far as I know).
 
2. Log-normal distribution is also approximated as a result of
likelihood approximation or in order to approximate the likelihood
depending on whether you start with linearization of the random
effect or Laplacian approximation of the likelihood. Therefore, even
though you simulated log-normal samples, they are not fitted as
log-normal, but constant CV normal.
 
3. The quality of posthoc estimates depends on both the typical
values (THETA) and the individual data (how many data points per
subject and where they are collected). Therefore, the observation
that "the geometric mean of posthoc values is closer to the 'real'
values than the typical (THETA) values" may not always hold. 
 
4. With "INTERACTION" or without "INTERACTION" in NONMEM is another
source of approximation. If your residual error model is Y=F*(1+ERR(1))
or Y=F*(1+ERR(1))+ERR(2) or Y=F*EXP(ERR(1)). You should have INTERACTION
for FOCE. Otherwise, you introduced more approximation during the estimation process.
 
I hope this helps
 
Yaning
 
-----------------------------------------------------------------------------
Yaning Wang, Ph.D. 
Pharmacometrician 
Office of Clinical Pharmacology and Biopharmaceutics 
Center of Drug Research and Evaluation 
Food and Drug Administration 
Office: 301-827-9763 
----------------------------------------------------------------------------- 
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From:  "Eleveld, DJ" d.j.eleveld@anest.umcg.nl
Subject: RE: [NMusers] Difference between typical values and geometricmea n of posthoc values
Date:  Mon, April 4, 2005 8:47 am 

Hello everyone, 

Thank you all again for your helpful reponses to my questions.  I seem to have solved my bias problem by 
using INTERACTION and now the the geometric mean of the posthoc individual parameters is (on average) much 
closer to the typical values as estimated by nonmem. 

The root of the problem was my interpretation of when INTERATION should be used.  From searching the 
NONMEM usergroup archives I found the text: 

        If your data appear to have a constant error distribution and this is the same across 
        subjects then you may have a homoscedastic (constrant variance) structure in which case 
        you don't need INTERACTION. 

I this context I interpreted 'error' as the nonmem ERR() variable which, for my problem, is 
homoscedastic.  So I assumed that INTERACTION shouldnt be used.  Unfortunatly, my interpretation was 
wrong.  A better description of when INTERACTION is appropriate might be when heteroscedastic *residuals* 
are expected.  If I had seen it described like this I would have used INTERCTION at the start. 

Matt Hutmacher and Yaning Wang made it clear in emails to me that you should use INTERACTION whenever the 
residual term is not additive.  This is much clearer to me than what I could find the the NONMEM manuals 
about INTERACTION.  Although I would prefer this to be so plainly stated in the manuals, I guess that it 
will (with this email) now be archived and searchable in the usersgroup archive so others dont fall into 
the same trap as I have. 

Thank you for your help in clearing this up, 

Doug Eleveld 
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