Re: Maximizing coefficient of determination

From: LSheiner <lewis@c255.ucsf.edu> Subject: Re: Maximizing coefficient of determination Date: Thu, 08 Feb 2001 08:42:30 -0800 Dear Erik, Before anything else, we all need to understand what you are doing and why. I do not understand, for example, your statement: "Both yi and yhati are given by models that contain parameters to be estimated." From the usual definition of r2, yi would be the observations, and they would be fixed. In which case your objective function is a linear transformation of the sum of squares sum((yi-yhati)^2, and maximizing it is equivalent to minimizing the latter. Perhaps, though, you are transforming yi (say g(yi)) and trying to estimate the parameters of the transformation g at the same time as the parameters of the model yhat? That would make your statement "minimizing the sum of squares would lead to parameter values that give the optimal, but meaningless yi = yhati = constant" understandable. In this case maximizing the (squared) correlation between yhati and g(yi) subject to the constraint that var(g(yi))=1 is indeed reasonable, and is implemented in the ACE algorithm (See Breiman, L. and J. H. Friedman (1985). Estimating optimal transformations for multiple regression and correlation. J Am Stat Assoc 80: 580-598.), available in S+. For parametric g and yhati, a likelihood-based approach is more flexible ("Transform Both Sides" - TBS - see Carroll & Ruppert, Transformation and Weighting in Regression, NY, Chapman & Hall, 1988) and can be adapted to hierarchical models in NONMEM. LBS. -- _/ _/ _/_/ _/_/_/ _/_/_/ Lewis B Sheiner, MD (lewis@c255.ucsf.edu) _/ _/ _/ _/_ _/_/ Professor: Lab. Med., Bioph. Sci., Med. _/ _/ _/ _/ _/ Box 0626, UCSF, SF, CA, 94143-0626 _/_/ _/_/ _/_/_/ _/ 415-476-1965 (v), 415-476-2796 (fax)