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
*

*>
*

This message and any attachments are solely for the intended recipient. =

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Received on Tue Jun 01 2010 - 02:13:08 EDT

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:

collapse.

lowers

assumption.

contributes.

SIG=3

FILE=uni.fit

This message and any attachments are solely for the intended recipient. =

If you are not the intended recipient, disclosure, copying, use or =

distribution of the information included in this message is prohibited =

--- Please immediately and permanently delete.

Received on Tue Jun 01 2010 - 02:13:08 EDT