# RE: distribution assumption of Eta in NONMEM

From: Mats Karlsson <mats.karlsson>
Date: Tue, 1 Jun 2010 07:48:05 +0200

Nick,

In your example you simulate and estimate only one random effect. As we =
are talking about situations where we have several levels of random =
effects, I redid you example with a full model which simulates and =
estimates with 1 THETA, 1 OMEGA and 1 SIGMA. The data were still made =
highly informative about ETAs (low shrinkage). Two observations each in =
100 subjects. When simulating with a uniform ETA distribution, =
estimation resulted in biased parameters (see below). When simulating =
with a normal ETA distribution of the same variance, estimation =
resulted in unbiased parameters (see below).

What you would want to do for the case where data is simulated with a =
uniform distribution, is a transform that makes the uniform to a normal. =
I don’t know of a transform that can be implemented into =
estimation that does this, but I implemented a logit transformation (see =
below) which decreased OFV by about 50 units. Also, see below.

\$PROB EBE

\$INPUT ID DV UNIETA

\$DATA uni2.csv; 100 subjects with 2 obs each

\$THETA 5 ; HILL

\$OMEGA 0.083333333 ; PPV_HILL = 1/12

\$SIGMA 0.0001 ; EPS1

\$SIM (1234) (5678 UNIFORM) NSUB=10

\$EST METHOD=COND MAX=9990 SIG=3 PRINT=1 ;MSFO=msf

\$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)+EPS(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

simulated normal =
simulated uniform

TRUE 5 0.001 =
0.083333 5 0.001 =
0.083333

Average 5.040598 9.63339E-05 0.078359 =
4.965324 1E-08 0.094581

SD 0.068156 7.00218E-06 =
0.005206 0.08444 1.74E-24 0.001808

LGPAR1 = THETA(2)

LGPAR2 = THETA(3)

PHI = LOG(LGPAR1/(1-LGPAR1))

PAR1 = EXP(PHI+ETA(1))

ETATR = (PAR1/(1+PAR1)-LGPAR1)*LGPAR2

HILL=THETA(1)*EXP(ETATR)

Y=1.1**HILL/(1.1**HILL+1) + EPS(1)

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
On Behalf Of Nick Holford
Sent: Monday, May 31, 2010 9:28 PM
To: nmusers
Cc: 'Marc Lavielle'
Subject: Re: [NMusers] distribution assumption of Eta in NONMEM

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).
I didn’t suggest that learning from the EBE distribution in =
general is a bad idea, I just didn’t understand how your example =

It can be informative to inspect EBEs. Just like you did in your =
example, you learnt that the underlying parameters were more uniformly =
than normally distributed. The distribution of the parameters is often =
we want to know something about whenever possible. (Maybe you agree =
since you think it is only “largely” uninformative).

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.

I think your example is too simple with simulating and estimating only =
one level of random effects.

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. "

I don’t think that ETA of the 2nd type exist. I don’t =
understand why you say “transformed” for the third type, =
so I won’t go for that one either. I talk about the deviations =
between the typical parameter value and the individual parameters under =
the model.

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-distributio=
n-to-a-normal-distribution

No, so if you want to simulate – reestimate to understand this =
better, I suggest you choose an underlying distribution that has a =
simpler transform to the normal.

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 =
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
On Behalf Of Nick Holford
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

Received on Tue Jun 01 2010 - 01:48:05 EDT

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