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-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On Behalf Of Eleveld, DJ Sent: Sunday, May 30, 2010 1:20 AM To: Nick Holford; nmusers@globomaxnm.com 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 _____ 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.
