3.20 FAQ-245 Why are fitted parameters different when fitting data are transformed?

EffectOfRSS

Last Update: 2/3/2015

When we transform fitting data and use the transformed fitting function to fit, sometimes we may find fitted parameters are different, although it seems like both fit data well.

For example, for fitting function y=a/(x-b), we transform y to 1/y, and the fitting function becomes linear, v=c*x-d

where v=1/y, a=1/c and b=d/c .

a and b calculated from two fitted results may be different. This is due to different contributions to residual sum of squares (RSS) for points in two models. For the function y=a/(x-b), y is large at small x and small at large x, so points at small x contribute more to RSS, and points at large x contribute less to RSS. And for the function v=c*x-d, v is large at large x and small at small x, so the contribution to RSS is reverse.

This may also occur for other transforms, e.g. y=a*exp(-b*x) is transformed to log(y)=log(a)-log(e)*b*x, $y=ax^b\!$ is transformed to log(y)=log(a)+b*log(x) .