# RE: distribution assumption of Eta in NONMEM

From: Elassaiss - Schaap, J. - <jeroen.elassaiss>
Date: Tue, 1 Jun 2010 08:13:08 +0200

Leonid, Nick,

Plotting the uniform distribution w/o exponentation was useful to me (R
code):
hist(runif(100))
hist(runif(1000))
hist(exp(runif(100)))
hist(exp(runif(1000)))
hist(exp(runif(10000)))
- Also after exponentation, the uniform distribution has very sharp
edges. I have never encountered such data distributions myself. And such
sharp edges seem pretty difficult to capture in a continuous model.
- You need an excessive amount of data to pinpoint the shape of a
distribution exactly

On a more general note: the more informative a dataset is on a
distribution, the less assumptions you have to make about it. From
limited to very rich informativeness one could go from untransformed via
exponential (*), semi-parametric and splines to non-parametric
approaches in order to describe the distribution, if needed.

My guess is that in most real-life cases we will have to live with
making assumptions about the shape of the distribution.

Best regards,
Jeroen

Modeling & Simulation Expert
Pharmacokinetics, Pharmacodynamics & Pharmacometrics (P3) - DMPK
MSD
PO Box 20 - AP1112
5340 BH Oss
The Netherlands
jeroen.elassaiss
T: +31 (0)412 66 9320
M: +31 (0)6 46 101 283
F: +31 (0)412 66 2506
www.msd.com

(*) or vice versa, from exponential via untransformed, as exponential
transformation often makes more sense and describes data better in PK-PD
analyses

-----Original Message-----
From: owner-nmusers
On Behalf Of Leonid Gibiansky
Sent: Monday, 31 May, 2010 23:31
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 don'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 ONEHEADER NOPRINT =
FILE=uni.fit
>>>>>
>>>>> 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
>

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