# RE: distribution assumption of Eta in NONMEM

From: Mats Karlsson <mats.karlsson>
Date: Tue, 1 Jun 2010 07:19:36 +0200

Leonid and Mike,

Leonid - you understood the idea.

Mike below are 3 tested transformations from Petersson et al. Pharm Res. =
2009 Sep;26(9):2174-85

Box-Cox transformation
TVCL=THETA(1)
BXPAR=THETA(2)
PHI = EXP(ETA(1))
ETATR = (PHI**BXPAR-1)/BXPAR
CL=TVCL*EXP(ETATR)

Heavy tailed transformation
TVCL=THETA(1)
HTPAR=THETA(2)
ETATR=ETA(1)*SQRT(ETA(1)*ETA(1))**HTPAR
CL=TVCL*EXP(ETATR)

Logit transformation
TVCL=THETA(1)
LGPAR1 = THETA(2)
LGPAR2 = THETA(3)
PHI = LOG(LGPAR1/(1-LGPAR1))
PAR1 = EXP(PHI+ETA(1))
ETATR = (PAR1/(1+PAR1)-LGPAR1)*LGPAR2
CL=TVCL*EXP(ETATR)

Mats

Mats Karlsson, PhD
Professor of Pharmacometrics
Dept of Pharmaceutical Biosciences
Uppsala University
Box 591
751 24 Uppsala Sweden
phone: +46 18 4714105
fax: +46 18 471 4003

-----Original Message-----
From: owner-nmusers
On Behalf Of Michael Fossler
Sent: Tuesday, June 01, 2010 3:54 AM
To: Leonid Gibiansky; Nick Holford; nmusers
Subject: RE: [NMusers] distribution assumption of Eta in NONMEM

Interesting topic. Can anyone provide specific transformations of ETAs =
that they have found useful?

Mike Fossler
GSK

-----Original Message-----
From: owner-nmusers
On Behalf Of Leonid Gibiansky
Sent: Monday, May 31, 2010 5:31 PM
To: Nick Holford; nmusers
Subject: Re: [NMusers] distribution assumption of Eta in NONMEM

Nick,
I think, transformation idea is the following:
Assume that your (true) model is

CL=POPCL*exp(ETAunif)

where ETAunif is the random variable with uniform distribution.
Assume that you have transformation TRANS that converts normal to
uniform. Then ETAunif can be presented (exactly) as

ETAunif=TRANS(ETAnorm).

Therefore, the true model can be presented (again, exactly) as

CL=POPCL*exp(TRANS(ETAnorm))

This model should be used for estimation and according to Mats, should
provide you the lowest OF

Leonid

--------------------------------------
Leonid Gibiansky, Ph.D.
President, QuantPharm LLC
web: www.quantpharm.com
e-mail: LGibiansky at quantpharm.com
tel: (301) 767 5566

Nick Holford wrote:
> Leonid,
>
> The result is what I expected. NONMEM just estimates the variance of =
the
> random effects. It doesn't promise to tell you anything about the
> distribution.
>
> It is indeed bad news for simulation if your simulation relies heavily =

> on the assumption of a normal distribution and the true distribution =
is
> quite different.
>
> I think you have to be very careful looking at posthoc ETAs. They are
> not informative about the true ETA distribution unless you can be sure =

> that you have low shrinkage. If shrinkage is not low then a true =
uniform
> will become more normal looking because the tails will collapse.
>
> The approach that Mats seems to suggest is to try different
> transformations of NONMEM's ETA variables to try to lower the OFV. =
What
> is not clear to me is why these transformations which lower the OFV =
will
> make the simulation better when the ETA variables that are used for =
the
> simulation are required to be normally distributed.
>
> Imagine I use this for estimation:
> CL=POPCL*EXP(ETA(1)) where the true ETA is uniform
> If I now use the estimated OMEGA(1,1) which will be a good estimate of =

> the uniform distribution variance, uvar, for simulation then I am =
using
> CL=POPCL*EXP(N(0,uvar))
> which will be wrong because I am now assuming a normal distribution =
but
> using the variance of a uniform.
>
> Now suppose I try:
> CL=POPCL*TRANS(ETA(1)) where TRANS is some transformation that =
lowers
> the OFV to the lowest I can find but the true ETA is still uniform
> If I now use the same transformation for simulation with an OMEGA(1,1) =

> estimate of the variance transvar
> CL=POPCL*TRANS(N(0,transvar)) which uses a normal distribution then =
why
> should I expect the simulated distribution of CL to resemble the true
> distribution with a uniform ETA?
>
> Nick
>
> Leonid Gibiansky wrote:
>> Hi Nick,
>> I think, I understood it from your original e-mail, but it was so
>> unexpected that I asked to confirm it.
>>
>> Actually, not a good news from your example.
>>
>> Nonmem cannot distinguish two models:
>> with normal distribution, and
>> with uniform distributions
>> as long as they have the same variance.
>>
>> So if you simulate from the model, you will end up with very =
different
>> results: either simular to the original data (if by chance, your
>> original problem happens to be with normal distribution) or very
>> different (if original distribution was uniform).
>>
>> This shows the need to investigate normality of posthoc ETAs very
>> carefully.
>>
>> Very interesting example
>> Thanks
>> Leonid
>>
>> --------------------------------------
>> Leonid Gibiansky, Ph.D.
>> President, QuantPharm LLC
>> web: www.quantpharm.com
>> e-mail: LGibiansky at quantpharm.com
>> tel: (301) 767 5566
>>
>>
>>
>>
>> Nick Holford wrote:
>>> Leonid,
>>>
>>> I meant by OMEGA(1) the OMEGA value estimated by NONMEM. I suppose I =

>>> should have written OMEGA(1,1) to be more precise -- sorry!
>>>
>>> Nick
>>>
>>> Leonid Gibiansky wrote:
>>>> Nick, Mats
>>>>
>>>> I would guess that nonmem should inflate variance (for this =
example)
>>>> trying to fit the observed uniform (-0.5, 0.5) into some normal =
N(0,
>>>> ?). This example (if I read it correctly) shows that Nonmem somehow =

>>>> estimates variance without making distribution assumption.
>>>> Nick, you mentioned:
>>>>
>>>> "the mean estimate of OMEGA(1) was 0.0827"
>>>>
>>>> does it mean that Nonmem-estimated OMEGA was close to 0.0827 or you =

>>>> refer to the variances of estimated ETAs?
>>>>
>>>> Thanks
>>>> Leonid
>>>>
>>>>
>>>> --------------------------------------
>>>> Leonid Gibiansky, Ph.D.
>>>> President, QuantPharm LLC
>>>> web: www.quantpharm.com
>>>> e-mail: LGibiansky at quantpharm.com
>>>> tel: (301) 767 5566
>>>>
>>>>
>>>>
>>>>
>>>> Mats Karlsson wrote:
>>>>> Nick,
>>>>>
>>>>>
>>>>>
>>>>> It has been showed over and over again that empirical Bayes
>>>>> estimates, when individual data is rich, will resemble the true
>>>>> individual parameter regardless of the underlying distribution.
>>>>> Therefore I donâ€™t understand what you think this exercise =
contributes.
>>>>>
>>>>>
>>>>>
>>>>> Best regards,
>>>>>
>>>>> Mats
>>>>>
>>>>>
>>>>>
>>>>> Mats Karlsson, PhD
>>>>>
>>>>> Professor of Pharmacometrics
>>>>>
>>>>> Dept of Pharmaceutical Biosciences
>>>>>
>>>>> Uppsala University
>>>>>
>>>>> Box 591
>>>>>
>>>>> 751 24 Uppsala Sweden
>>>>>
>>>>> phone: +46 18 4714105
>>>>>
>>>>> fax: +46 18 471 4003
>>>>>
>>>>>
>>>>>
>>>>> *From:* owner-nmusers
>>>>> [mailto:owner-nmusers
>>>>> *Sent:* Monday, May 31, 2010 6:05 PM
>>>>> *To:* nmusers
>>>>> *Cc:* 'Marc Lavielle'
>>>>> *Subject:* Re: [NMusers] distribution assumption of Eta in NONMEM
>>>>>
>>>>>
>>>>>
>>>>> Hi,
>>>>>
>>>>> I tried to see with brute force how well NONMEM can produce an
>>>>> empirical Bayes estimate when the ETA used for simulation is
>>>>> uniform. I attempted to stress NONMEM with a non-linear problem
>>>>> (the average DV is 0.62). The mean estimate of OMEGA(1) was 0.0827 =

>>>>> compared with the theoretical value of 0.0833.
>>>>>
>>>>> The distribution of 1000 EBEs of ETA(1) looked much more uniform
>>>>> than normal.
>>>>> Thus FOCE show no evidence of normality being imposed on the EBEs.
>>>>>
>>>>> \$PROB EBE
>>>>> \$INPUT ID DV UNIETA
>>>>> \$DATA uni1.csv ; 100 subjects with 1 obs each
>>>>> \$THETA 5 ; HILL
>>>>> \$OMEGA 0.083333333 ; PPV_HILL = 1/12
>>>>> \$SIGMA 0.000001 FIX ; EPS1
>>>>>
>>>>> \$SIM (1234) (5678 UNIFORM) NSUB=10
>>>>> \$EST METHOD=COND MAX=9990 SIG=3
>>>>> \$PRED
>>>>> IF (ICALL.EQ.4) THEN
>>>>> IF (NEWIND.LE.1) THEN
>>>>> CALL RANDOM(2,R)
>>>>> UNIETA=R-0.5 ; U(-0.5,0.5) mean=0, variance=1/12
>>>>> HILL=THETA(1)*EXP(UNIETA)
>>>>> Y=1.1**HILL/(1.1**HILL+1)
>>>>> ENDIF
>>>>> ELSE
>>>>>
>>>>> HILL=THETA(1)*EXP(ETA(1))
>>>>> Y=1.1**HILL/(1.1**HILL+1) + EPS(1)
>>>>> ENDIF
>>>>>
>>>>> REP=IREP
>>>>>
>>>>> \$TABLE ID REP HILL UNIETA ETA(1) Y
>>>>>
>>>>> I realized after a bit more thought that my suggestion to =
transform
>>>>> the eta value for estimation wasn't rational so please ignore that =

>>>>> senior moment in my earlier email on this topic.
>>>>>
>>>>> Nick
>>>>>
>>>>>
>>>>> --
>>>>>
>>>>> Nick Holford, Professor Clinical Pharmacology
>>>>>
>>>>> Dept Pharmacology & Clinical Pharmacology
>>>>>
>>>>> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New
>>>>> Zealand
>>>>>
>>>>> tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
>>>>>
>>>>> email: n.holford
>>>>>
>>>>> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
>>>>>
>>>
>>> --
>>> Nick Holford, Professor Clinical Pharmacology
>>> Dept Pharmacology & Clinical Pharmacology
>>> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New =
Zealand
>>> tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
>>> email: n.holford
>>> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
>>>
>
> --
> Nick Holford, Professor Clinical Pharmacology
> Dept Pharmacology & Clinical Pharmacology
> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New =
Zealand
> tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53
> email: n.holford
> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
>

Received on Tue Jun 01 2010 - 01:19:36 EDT

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