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
*

*>>>>> 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
*

*>
*

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

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:

the

is

uniform

What

will

the

using

but

lowers

why

different

example)

N(0,

contributes.

transform

Zealand

Zealand

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