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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:
slope (allometric exponent) 0.869 +/- 0.056
intercept (log allometric coefficient) 1.73 +/- 0.06
so, allometric coefficient 53.7 mL/min
extrapolated human flow 2148 mL/min
Ordinary least squares fit of untransformed data gives:
allometric 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:
allometric 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 Murray, Ph.D.
Senior Research Scientist, Drug Metabolism
Gilead Sciences, Foster City CA
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The following message was posted to: PharmPK
> 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.
Sheiner 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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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.
Junfang Xu, M.D.
GPKPD / CNIBR
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