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

[Martin Elff]

On Wednesday 05 March 2008 (14:53:27), Wolfgang Waser wrote:
> 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.

Models (a) and (b) entail different distributions of the dependent variable y
and different ranges of values that y may take.
(a) implies that y has, conditionally on x, a normal distribution and
has a range of feasible values from -Inf to +Inf.
(b) and (c) imply that log(y) has a normal distribution, that is, 
y has a log-normal distribution and can take values from zero to +Inf.

> Is this supposed to happen? 
Given the above considerations, different results with respect to the 
intercept are definitely to be expected.

> Which is the correct coefficient b? 
That depends - is y strictly non-negative or not ...

Just my 20 cents...