From: Nick Holford <*n.holford*>

Date: Mon, 31 May 2010 21:27:51 +0200

Mats,

I agree that trying to learn anything from the EBE distribution is a

largely uninformative activity. If the shrinkage is too small then the

EBEs are driven primarily by the data distribution (which is probably

happening in the example I reported) while if they are too big they

shrink to the population mean (with no information about the

distribution at the limit). NONMEM7 claims that the ETA shrinkage for

this experiment is about -40% (whatever that might mean).

What do I think this exercise contributes? Well it seems (with this

rather limited example) to show that NONMEM FOCE can estimate the

variance of the true ETA distribution (and also a non-linear fixed

effect parameter) quite accurately even if the true ETA distribution is

clearly not normal and no normalizing transformation is used.

It has also made me think more carefully what I mean by ETA. There is

the true ETA used in a simulation that represents the random deviation

of a parameter from the population value and that random deviation might

arise from many different distributions. There is the value of NONMEM's

ETA variable which arises from a distribution defined with mean zero and

variance OMEGA but without any assumption about its distribution being

normal when used for estimation (according to Stuart). And finally there

is some transformed value of NONMEM's ETA variable which influences the

objective function.

Which of these kinds of ETA were you referring to when you wrote this?

" If you use a method like FOCE, and try different transformation of you

parameters, you will find that OFV will be lowest and other goodness of

fit best, when the transformation is such that ETA is normally

distributed. "

Are you saying that in my simulation experiment in which the true ETAs

are known to be uniform that if I apply a transformation involving the

NONMEM ETA(*) variable which makes the transformed random effect

normally distributed then the OFV cannot be made lower? So if I try

different transformations and find the transformation with the lowest

OFV and if I know what distribution this transformation of the true ETA

turns into a normal distribution that I can then learn the nature of the

true ETA distribution?

Do you know what transformation can I apply with NM-TRAN that will

transform a function of uniform ETA(*) into a normal distribution?

Implementations of things like the Box-Mueller transform require the use

of additional random uniform numbers so that won't work for estimation

e.g.

http://stackoverflow.com/questions/75677/converting-a-uniform-distribution-to-a-normal-distribution

Nick

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

Received on Mon May 31 2010 - 15:27:51 EDT

Date: Mon, 31 May 2010 21:27:51 +0200

Mats,

I agree that trying to learn anything from the EBE distribution is a

largely uninformative activity. If the shrinkage is too small then the

EBEs are driven primarily by the data distribution (which is probably

happening in the example I reported) while if they are too big they

shrink to the population mean (with no information about the

distribution at the limit). NONMEM7 claims that the ETA shrinkage for

this experiment is about -40% (whatever that might mean).

What do I think this exercise contributes? Well it seems (with this

rather limited example) to show that NONMEM FOCE can estimate the

variance of the true ETA distribution (and also a non-linear fixed

effect parameter) quite accurately even if the true ETA distribution is

clearly not normal and no normalizing transformation is used.

It has also made me think more carefully what I mean by ETA. There is

the true ETA used in a simulation that represents the random deviation

of a parameter from the population value and that random deviation might

arise from many different distributions. There is the value of NONMEM's

ETA variable which arises from a distribution defined with mean zero and

variance OMEGA but without any assumption about its distribution being

normal when used for estimation (according to Stuart). And finally there

is some transformed value of NONMEM's ETA variable which influences the

objective function.

Which of these kinds of ETA were you referring to when you wrote this?

" If you use a method like FOCE, and try different transformation of you

parameters, you will find that OFV will be lowest and other goodness of

fit best, when the transformation is such that ETA is normally

distributed. "

Are you saying that in my simulation experiment in which the true ETAs

are known to be uniform that if I apply a transformation involving the

NONMEM ETA(*) variable which makes the transformed random effect

normally distributed then the OFV cannot be made lower? So if I try

different transformations and find the transformation with the lowest

OFV and if I know what distribution this transformation of the true ETA

turns into a normal distribution that I can then learn the nature of the

true ETA distribution?

Do you know what transformation can I apply with NM-TRAN that will

transform a function of uniform ETA(*) into a normal distribution?

Implementations of things like the Box-Mueller transform require the use

of additional random uniform numbers so that won't work for estimation

e.g.

http://stackoverflow.com/questions/75677/converting-a-uniform-distribution-to-a-normal-distribution

Nick

Mats Karlsson wrote:

--

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 Mon May 31 2010 - 15:27:51 EDT