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Re: distribution assumption of Eta in NONMEM

From: Nick Holford <n.holford>
Date: Sun, 30 May 2010 09:57:11 +0200

Hi,

Sorry - I meant to write this in reference to Stuart Beal's suggestion:
"He proposed the term "apparent coefficient of variation" as a way of
**NOT* *implying a normal distribution of ETA."

This is analogous to the term "apparent clearance" which is often used
to refer to a description of clearance without assuming that the extent
of bioavailability is 1.

Nick

Nick Holford wrote:
> 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
> <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
>> [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
> 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.holford
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford


Received on Sun May 30 2010 - 03:57:11 EDT

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