[R] nls: different results if applied to normal or linearized data

[Wolfgang Waser]

Thanks for your comments!

> Yes.  You are fitting by least-squares on two different scales:
> differences in y and differences in log(y) are not comparable.
>
> Both are correct solutions to different problems.  Since we have no idea
> what 'x' and 'y' are, we cannot even guess which is more appropriate in
> your context.

I'm fitting metabolic rate data from small fish (oxygen consumption in 
nmol/min vs. body weight in g).
The b coefficient is the interesting part and is generally somewhere around 
0.75.
The one calculated for my data using option (a) is therefore 'better' than 
(b,c), but which one is the correct to use? Log-transformation of metabolic 
rate data is (was) normally performed to be able to determine a and b by 
simple linear regression (or even on paper).


> The two approaches assume two different models.
> 
>         Model (1) is y = a*x^b + E (where different errors are independent  
> and identically
>         --- usually normally --- distributed).
> 
>         Model (2) is y = a*(x^b)*E (and you are usually tacitly assuming  
> that ln E is normally distributed).
> 
>         The point estimates of a and b will consequently be different ---  
> although usually not hugely
>         different.  Their distributional properties will be substantially  
> different.

So in view of my context (metabolic rate data) would Model (1) be the 
appropriate model to use? 


> > Dear all,
> >
> > I did a non-linear least square model fit
> >
> > y ~ a * x^b
> >
> > (a) > nls(y ~ a * x^b, start=list(a=1,b=1))
> >
> > to obtain the coefficients a & b.
> >
> > I did the same with the linearized formula, including a linear model
> >
> > log(y) ~ log(a) + b * log(x)
> >
> > (b) > nls(log10(y) ~ log10(a) + b*log10(x), start=list(a=1,b=1))
> > (c) > lm(log10(y) ~ log10(x))
> >
> > I expected coefficient b to be identical for all three cases. Hoever,
> > using my dataset, coefficient b was:
> > (a) 0.912
> > (b) 0.9794
> > (c) 0.9794
> >
> > Coefficient a also varied between option (a) and (b), 107.2 and 94.7,
> > respectively.
> >
> > Is this supposed to happen? Which is the correct coefficient b?
> >
> > Regards,
> >
> > Wolfgang