# PharmPK Discussion - Allometry: linear regression of log-transformed data or nonlinear fitting

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• On 25 Oct 2010 at 18:45:56, Bernard Murray (Bernard.Murray.-at-.gilead.com) sent the message
`Hello there, I was hoping to solicit the list's opinions to help with a debate  between myself and some of my colleagues.  The question concerns the  "best" way to generate an allometric fit for data.  The two main options  being discussed are; 1) log (usually log10) transformation of the data  followed my nonlinear least squares regression (typically without  weighting the data), or 2) non-linear fitting to the allometric model  (with or without various weightings). As an example, our (somewhat arbitrary) test data set is the collection  of hepatic blood flow data in the Davies & Morris (1993) physiological  parameters paper.  We are using the five preclinical species and  attempting to predict the human value for a 70 kg individual (listed in  that reference as 1450 mL/min). Simple linear regression of the log-transformed data gives:r-squared 0.988slope (allometric exponent) 0.869 +/- 0.056intercept (log allometric coefficient) 1.73 +/- 0.06so, allometric coefficient 53.7 mL/minextrapolated human flow 2148 mL/min Ordinary least squares fit of untransformed data gives:r-squared 0.975allometric exponent 0.537 +/- 0.093 (0.241 - 0.834)allometric coefficient 92.0 mL/min +/- 17.0 (37.8 - 146.1)extrapolated human flow 901 mL/min Ordinary least squares fit of transformed data with weighting of 1/BW  gives:r-squared 0.921allometric exponent 0.765 +/- 0.118 (0.391 - 1.140)allometric coefficient 62.2 mL/min +/- 13.8 (18.3 - 106.1)extrapolated human flow 1605 mL/min Mean +/- SEmean and (low - high 95% confidence intervals) where easily  calculable A quick survey of the literature I have to hand reveals that, when the  method for allometric analysis is stated explicitly, the authors appear  to use option (1) above (linear regression of log transformed data).  On  many occasions the actual method is not described.  In some cases the  descriptions are ambiguous.  I have not seen any discussion of weighting  schemes.  When comments on the absolute value of the allometric exponent  are being made option (1) also seems to be implied. I realise that the "best" method is the one that gives the correct  result.  However, I am not a statistician and so my naiive question was  if there was any theoretical justification for the various potential  methods (including any others I may not have covered).  This is not  intended to be a question about the pros and cons of allometric scaling  per se. I didn't see this topic covered in the PharmPK archive. All the very best, Bernard Bernard Murray, Ph.D.Senior Research Scientist, Drug MetabolismGilead Sciences, Foster City CA`
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• On 26 Oct 2010 at 17:22:39, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message
`The following message was posted to: PharmPKBernard,> The question concerns the "best" way to generate an allometric fit for  data.  The two main options being discussed are; 1) log (usually log10)  transformation of the data followed my nonlinear least squares  regression (typically without weighting the data), or 2) non-linear  fitting to the allometric model (with or without various weightings).Sheiner (1984) proposed a distinction between a naive pooled data (NPD) approach and a population approach for estimating PK parameters. The NPD method assumes (or ignores) between subject variability (BSV) and treats all variation around the fitted line as residual error while the population approach tries to distinguish BSV from residual error. The population approach is also known as a mixed effects modeling approach because it distinguishes predictable sources of variability (e.g. body mass) from unpredictable variability (BSV and residual error).Imagine you could measure blood flow with reasonable precision  (e.g. replicate CV of 10%) and you measured blood blow in 100 animals which all happened to have the same weight. Typical BSV for human clearance of high extraction ratio drugs is about 50%. So for 100 humans the BSV would be much bigger than the measurement error (25 times bigger in variance terms). On the other hand if you had 100 cloned rats you might find the BSV was much smaller e.g. 10% so now the measurement error and rat to rat variability are very similar.The population approach would be able to distinguish between BSV in different species and perhaps different residual error (due to technical measurement problems in different sized animals).The kind of regression methods you describe above are NPD and may be misleading when describing biology. They contribute to the ongoing controversy about what is the 'correct' allometric exponent when experimental observations are used to test theoretical predictions.It is important to remember that allometry is only about using mass to describe differences in variables such as hepatic blood flow. This means that other variables which may be correlated with mass but are functionally different from mass (e.g. age, species, disease) must be properly controlled for before it is possible to claim the estimated exponent is describing only the relationship with mass.In your example of 5 non-human species there may be species differences in blood flow caused by factors such as diet (vegetarian, carnivore, etc) which will be confounded with mass e.g. cows are vegetarian and are much bigger than carnivorous cats.So the statistical solution to your question is pretty simple -- use a mixed effects model (population approach) rather than NPD. The hard part is interpreting the number you get for an allometric exponent and trying to convince sceptics (like me) that the number is a 'true' allometric exponent reflecting only the influence of mass.Good luck!NickSheiner LB. The population approach to pharmacokinetic data analysis: rationale and standard data analysis methods. Drug Metab Rev. 1984;15(1-2):153-71.`
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• On 26 Oct 2010 at 14:19:40, (junfang.xu.-a-.novartis.com) sent the message
`Dear Bernard,In my group, allometry is based on log transformed data (log plot CL/V  vs BW). There are other adjustments, eg. log plot CL/V*fu vs BW, or log  plot CL/V*MLP/BrWT vs BW), or FCIM method. It is quite simple to  calculate log transformed data by simple linear regression (assuming CL  or V is a linear function of size).Nonlinear regression in theory should come up to the same result.  However as you mentioned, higher body weigh will give higher  contribution to the estimation, and thus a weight of 1/BW or 1/BW^2 will  be added.Best,Junfang Xu, M.D.GPKPD / CNIBR`
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