Douglas,
Thanks for your thoughtful and insightful comments on why anyone might
be interested in the answer to the question "Does NONMEM assume a normal
distribution for estimation?".
In fact one has no choice but to use whatever assumptions are built into
the estimation algorithm. So a more practical question might be "Are
there situations when models built with this assumption might be
misleading?". It is known that NONMEM parameter estimates obtained with
FOCE may be a bit biased compared true values used for simulation. But
is this due to the approximation to the likelihood used by FOCE or is
because of an assumption of normality? It has been my understanding that
it is due to the likelihood approximation.
On a somewhat unrelated issue - there is one part of the estimation
process that can be misleading if a normal assumption is made and that
is the use of estimated standard errors to compute confidence intervals
(CIs). If likelihood profiling (Holford & Peace 1992) or bootstraps
(Matthews et al. 2004) are used to obtain CIs then it not uncommon to
find the CI is asymmetrical and this cannot be predicted from the
asymptotic standard error estimate. Computation of CIs with standard
errors typically assumes a normal distribution of the uncertainty and
this leads to a misleading impression of the uncertainty that can be
only be discovered by methods which do not make this normal assumption.
This is not just a problem with NONMEM - it is a problem with any
procedure that only provides a standard error as an estimate of uncertainty.
Nick
Holford, N. H. G. and K. E. Peace (1992). "Results and validation of a
population pharmacodynamic model for cognitive effects in Alzheimer
patients treated with tacrine." Proceedings of the National Academy of
Sciences of the United States of America 89(23): 11471-11475.
Matthews, I., C. Kirkpatrick, et al. (2004). "Quantitative justification
for target concentration intervention - Parameter variability and
predictive performance using population pharmacokinetic models for
aminoglycosides." British Journal of Clinical Pharmacology 58(1): 8-19.
Eleveld, DJ wrote:
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
------------------------------------------------------------------------
De inhoud van dit bericht is vertrouwelijk en alleen bestemd voor de
geadresseerde(n). Anderen dan de geadresseerde(n) mogen geen gebruik
maken van dit bericht, het niet openbaar maken of op enige wijze
verspreiden of vermenigvuldigen. Het UMCG kan niet aansprakelijk
gesteld worden voor een incomplete aankomst of vertraging van dit
verzonden bericht.
The contents of this message are confidential and only intended for
the eyes of the addressee(s). Others than the addressee(s) are not
allowed to use this message, to make it public or to distribute or
multiply this message in any way. The UMCG cannot be held responsible
for incomplete reception or delay of this transferred message.
--
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.holf...@auckland.ac.nz
http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford
