Re: [NMusers] distribution assumption of Eta in NONMEM

[Nick Holford]

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 0.08333 ; PPV_HILL = 1/12
$SIGMA 0.01 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

RE: [NMusers] distribution assumption of Eta in NONMEM

[Mats Karlsson]

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 0.08333 ; PPV_HILL = 1/12
$SIGMA 0.01 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

Re: [NMusers] distribution assumption of Eta in NONMEM

[Leonid Gibiansky]

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-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 0.08333 ; PPV_HILL = 1/12
$SIGMA 0.01 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

Re: [NMusers] distribution assumption of Eta in NONMEM

[Nick Holford]

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 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 0.08333 ; PPV_HILL = 1/12
$SIGMA 0.01 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

Re: [NMusers] distribution assumption of Eta in NONMEM

[Nick Holford]

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-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 0.08333 ; PPV_HILL = 1/12
$SIGMA 0.01 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

Re: [NMusers] distribution assumption of Eta in NONMEM

[Nick Holford]

Sorry - I misread my results. The ETA shrinkage was -0.4% not -40%. I 
had forgotten that NONMEM reports the shrinkage as a % not as a fraction.


Nick Holford wrote:

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).


--
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

Re: [NMusers] distribution assumption of Eta in NONMEM

[Nick Holford]

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-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 0.08333 ; PPV_HILL = 1/12
$SIGMA 0.01 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

Re: [NMusers] distribution assumption of Eta in NONMEM

[Leonid Gibiansky]

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-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

RE: [NMusers] distribution assumption of Eta in NONMEM

[Michael Fossler]

Interesting topic. Can anyone provide specific transformations of ETAs that 
they have found useful?

Mike Fossler
GSK



-Original Message-
From: owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] 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 ov

RE: [NMusers] distribution assumption of Eta in NONMEM

[Mats Karlsson]

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-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] 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-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] 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) th

RE: [NMusers] distribution assumption of Eta in NONMEM

[Elassaiss - Schaap, J. (Jeroen)]

Mike,

For some proprietary analyses I have applied the logit transformation
with succes to normalize the posthocs. It also made the model more
stable, and made it possible to get a covariance step.

This was an example with clearly censored randomization, therefore the
logit shape made a lot of sense.

Jeroen

Modeling & Simulation Expert
Pharmacokinetics, Pharmacodynamics & Pharmacometrics (P3) - DMPK
MSD
PO Box 20 - AP1112
5340 BH Oss
The Netherlands
jeroen.elassa...@merck.com
T: +31 (0)412 66 9320
M: +31 (0)6 46 101 283
F: +31 (0)412 66 2506
www.msd.com


-Original Message-
From: owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com]
On Behalf Of Michael Fossler
Sent: Tuesday, 01 June, 2010 3:54
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-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com]
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 

RE: [NMusers] distribution assumption of Eta in NONMEM

[Mats Karlsson]

 

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 0.08333 ; 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

TRUE5 0.001 
0.085 0.001  0.08

Average   5.040598  9.63339E-050.078359 
 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

RE: [NMusers] distribution assumption of Eta in NONMEM

[Elassaiss - Schaap, J. (Jeroen)]

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(1)))
- 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.elassa...@merck.com
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-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com]
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
>>
>> --
>> L