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RE: distribution assumption of Eta in NONMEM

From: Mats Karlsson <mats.karlsson>
Date: Sun, 30 May 2010 10:54:06 +0200

Nick,


Clearly the choice of distributional assumption you make regarding the =
parameters have an impact on the estimation (parameters, =
goodness-of-fit, predictions and simulations). Simulations showing that =
is presented in the Petersson paper and many others. Therefore I =
don’t know what results Stuart were basing his thoughts on, do =
you? Maybe the keyword in Stuart’s sentence is =
“largely”.

 

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: Sunday, May 30, 2010 9:46 AM
To: nmusers
Cc: 'Marc Lavielle'
Subject: Re: [NMusers] distribution assumption of Eta in NONMEM

 

Mats,

This is a helpful and interesting response but I think it is an answer =
to a different kind of question. My understanding of the original =
question was does NONMEM assume somewhere in its estimation procedure =
that some quantity is normally distributed regardless of the (mis) =
specification of the model by the user.

You describe ways of describing the shape of a parameter distribution =
with different models. Associated with these transformations there may =
be an interpretation of the resulting parameter distribution which would =
obtain if the ETA distribution was indeed normal.

Stuart Beal wrote about this issue in 1997 and cautioned that the =
interpretation is in the eye of the user because NONMEM does not require =
ETAs to be normally distributed:
"Many discussions state that ETA is assumed to be normal, but these are =
often misleading. While there are sometimes good reasons for making this =
assumption, the NONMEM methodology largely avoids the assumption."
He proposed the term "apparent coefficient of variation" as a way of =
implying a normal distribution of ETA.
"Since we do not need to make the normality assumption, it does not =
follow that the "extra accuracy" given by the lognormal formula really =
represents extra accuracy; it can just as well be garbage. Suppose we =
want to really do the right thing, and CV is large (perhaps as a =
pragmatic matter, we will judge the CV to be large when the results from =
the two formulas differ substantially). Then we should probably avoid =
reporting the CV as a "CV", but report it as an "apparent CV"."

Note: I had thought that Stuart's posting was originally to nmusers but =
the Cognigen search engine failed to find it for me. Here it is on an =
AACP site: http://gaps.cpb.ouhsc.edu/nm/91sep2697.html


Mats Karlsson wrote:

Dear Douglas and all,

 

We always have some knowledge about our parameter distribution. It comes =
from two sources: prior information and the data, under the model. Prior =
information almost always tell us that parameters must be non-normally =
distributed. That’s why we enforce different types of fixed =
transformations. Usually exponential transformation for parameters that =
has to be non-negative and logit transformation for fractions and =
probabilities. We then often have introduced what prior knowledge we =
have regarding the shape of the distribution. However, also our data =
contain information about the parameter distribution under the model we =
choose and one distribution may describe data better than another. We =
can explore this by choosing different fixed transformation. We may also =
allow the data to speak to the shape of the distribution as part of the =
estimation process. The latter approach was introduced into our field by =
Davidian&Gallant (J Pharmacokinet Biopharm. 1992 Oct;20(5):529-56) using =
polynomials and a specialized software. We recently explored other =
transformation that could be easily introduced into NONMEM and other =
standard programs (Petersson et al., Pharm Res. 2009 Sep;26(9):2174-85). =
If you want to explore deviations from normality under your fixed =
transformation, these semi-parametric* methods may be a good =
alternative. Below is code for a simple box-cox transformation on top =
of a fixed exponential transformation. Positive values of SHP indicates =
right-skewed distribution (compared to a exponential transformation), =
negative a left-skewed. If the transformation offers no improvement in =
fit over an exponential distribution, the goodness-of-fit will be =
similar to that of a simpler model (CL=THETA(1)*EXP(ETA(1))).

 

 

SHP = THETA(2)

TETA = ((EXP(ETA(1))**SHP-1)/SHP

CL = THETA(1)*EXP(TETA)

 

(Semi-parametric is the traditionally used word for these methods, it =
probably comes from the fact that it lies between the standard =
parametric methods where the shape is prescribed by the model, and =
non-parametric methods where very little distributional assumption is =
being made. Semi-parametric methods are essentially parametric but =
parameters are estimated that relates not just the magnitude, but also =
the shape of the distribution.)

 

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 Eleveld, DJ
Sent: Sunday, May 30, 2010 1:20 AM
To: Nick Holford; nmusers
Cc: Marc Lavielle
Subject: RE: [NMusers] distribution assumption of Eta in NONMEM

 

I'd like to interject a slightly different point of view to the =
distributional assumption question here.

When I hear people speak in terms of the “distribution =
assumptions of some estimation method” I think its easy for =
people to jump to the conclusion that the normal distribution assumption =
is just one of many possible, equally justifiable distributional =
assumptions that could potentially be made. And that if the normal =
distribution is the “wrong” one then the results from =
such an estimation method would be “wrong”. This is =
what I used to think, but now I believe this is wrong and I'd like to =
help others from wasting as much time thinking along this path, as I =
have.

From information theory, information is gained when entropy decreases. =
So if you have data from some unknown distribution and if you must make =
some distribution assumption in order to analyze the data, you should =
choose the highest entropy distribution you can. This insures that your =
initial assumptions, the ones you do before you actually consider your =
data, are the most uninformative you can make. This is the principle of =
Maximum Entropy which is related to Principle of Indifference and the =
Principle of Insufficient Reason.

A normal distribution has the highest entropy of all real-valued =
distributions that share the same mean and standard deviation. So if =
you assume your data has some true SD, then the best distribution to =
assume would be normal distribution. So we should not think of the =
normal distribution assumption as one of many equally justifiable =
choices, it is really the “least-bad” assumption we can =
make when we do not know the true distribution. Even if normal is the =
“wrong” distribution, it still remains the =
“best”, by virtue of being the =
“least-bad”, because it is the most uninformative =
assumption that can be made (assuming a some finite true variance).

In the real-word we never know the true distribution and so it makes =
sense to always assume a normal distribution unless we have some =
scientifically justifiable reason to believe that some other =
distribution assumption would be advantageous.

The Cauchy distribution is a different animal though since its has an =
infinite variance, and is therefore an even weaker assumption than the =
finite true SD of a normal distribution. It would possibly be even =
better than a normal distribution because its entropy is even higher =
(comparing the standard Cauchy and standard normal). It would be very =
interesting if Cauchy distributions could be used in NONMEM. Actually, =
the ratio of two N(0,1) random variables is Cauchy distributed. Maybe =
this property could be used trick NONMEM into making a Cauchy (or =
nearly-Cauchy) distributed random variable?

Douglas Eleveld

  _____

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--
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 Sun May 30 2010 - 04:54:06 EDT

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