Date: Jan 31, 2013 10:29 AM Author: David Jones Subject: Re: R^2 for linearized regression

"Darek" wrote in message

news:9362f26d-9e21-46ca-9dd4-accb562606f5@l13g2000yqe.googlegroups.com...

Hi all!

I would like to ask about R^2 in linearized regression where Y value

is transformed e.g.:

http://en.wikipedia.org/wiki/Nonlinear_regression#Linearization

If we apply power function (Y=a*b^X) for regression in Excel or SPSS

the R^2 (sum of squares etc.) is calculated using linearized function

i.e.: ln(Y)=a+ln(X)

I think that comparison of R^2 for the same dataset for various

regression functions (e.g.between linear and power function) where Y

is transformed is not proper method of selection of best regression

model.

I think that in the case described above if we would like to compare

various functions of regression, R^2 should be calculated using

function Y=a*b^X not function after linearization ln(Y)=a+ln(X).

Could you give your opinion on this matter?

Thanks in advance.

Darek

======================================

It is important to be clear about how the value of R^2 that you use is

calculated when you use it. Just using values from individual fitting

modules may well not be enough.

See http://en.wikipedia.org/wiki/Coefficient_of_determination

You should try calculating R^2 directly from the sets of observed and

corresponding values predicted values, where

(i) "observed" is the original observations and "predicted" is either the

predictions from linear regression or the exponential of the predictions

from the regression model for the log-ed data (it is also possible to

include a "bias adjusted" version of the latter)

and

(ii) "observed" is the log-ed original observations and "predicted" is

either the predictions from linear regression on the log-ed data or the

logarithm of the predictions from the regression model for the original

data.

This gives at least 4 values to compare. You can also try introducing an

additional linear regression step, for example where in (i) you could fit a

linear model for the observed data based on the exponentiated predictions

from the linear model for the log-ed observations.

If you have time you could construct a pair of scatter plots of observed

versus predicted values in both original and transformed spaces.

But there is no definite generally applicable answer to your question,

except hat you should definitely have a comparison of R^2 values calculated

for the same transformation of the observed data. From a theoretical point

of view , if the usual model-checks for regression models suggest that the

transformed model is better then you should be using the R^2 calculated for

the log-ed data. But, if practical/real-world considerations suggest that

the "importance" of errors of prediction is equal on the non-transformed

scale, then R^2 calculated for the untransformed observations may be more

closely aligned to what you are trying to use the predictions for.

David Jones