# RE: distribution assumption of Eta in NONMEM

From: Mats Karlsson <mats.karlsson>
Date: Sun, 30 May 2010 08:40:19 +0200

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 =

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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Received on Sun May 30 2010 - 02:40:19 EDT

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