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Re: distribution assumption of Eta in NONMEM

From: Nick Holford <n.holford>
Date: Sun, 30 May 2010 09:45:55 +0200


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:

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
> (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
> [mailto:owner-nmusers
> *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

Received on Sun May 30 2010 - 03:45:55 EDT

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