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
ONEHEADER NOPRINT FILE=uni2.fit
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-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On
Behalf Of Nick Holford
Sent: Monday, May 31, 2010 9:28 PM
To: nmusers@globomaxnm.com
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 showed your point.
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-distribution-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-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On
Behalf Of Nick Holford
Sent: Monday, May 31, 2010 6:05 PM
To: nmusers@globomaxnm.com
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.holf...@auckland.ac.nz
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.holf...@auckland.ac.nz
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
