I have a non-linear model of the following form:
$y = a*x^b$
I can fit it using logarithms and a linear model or directly with a non-linear model.
First approach, logarithms and linear model:
lmfit <- lm(log(y)~log(x))
Second approach, non-linear model:
nlsfit <- nls(y~a*x^b, start=list(a=200, b=1.6))
In the first case I can simply get the $R^2$ value from the linear model or calculate it myself by:
rsquared <- var(fitted(lmfit)) / var(log(y))
In the second case there is no $R^2$ value generated, but I can obtain one $pseudoR^2$ value myself by:
pseudorsquared <- var(fitted(nlsfit)) / var(y)
In a linear model I can calculate the fraction of variance unexplained by simply doing $1-R^2$. I have read that this is not applicable to non-linear regressions. I would like to know if there is an equivalent version of this measure, so that I can compare both regressions and use the best one.
As an extra information, I would like to add that this is a regression of physical variables, and that the non-linear approach is providing more close-to-literature results for the coeficients, whereas the linear approach gives better statistical performance ($R^2$, bias, etc.).