RE: distribution assumption of Eta in NONMEM
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
Quoted reply history
From: [email protected] [mailto:[email protected]] On
Behalf Of Eleveld, DJ
Sent: Sunday, May 30, 2010 1:20 AM
To: Nick Holford; [email protected]
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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