A useful result is in the current Statistics in Medicine:
@Article{buy00r2,
author = {Buyse, Marc},
title = {$R^2$: {A} useful measure of model performance when
predicting a dichotomous outcome},
journal = SM,
year = 2000,
volume = 19,
pages = {271-274},
note = {Letter to the Editor regarding {\em Statistics in
Medicine} 18:375-384; 1999}
}
Milo Schield wrote:
> QUESTION: What is the theoretical maximum value of R-sq ** when binary data
> (Y) is obtained from a simple linear model?
>
> The data is binary with Y values taken from a linear model going from 0 to 1
> over the range of X.
>
> The binary sequences of Y values are organized to minimize* the standard
> deviation around the model.
>
> TYPE REGRESSION DISTRIBUTION OF X VALUES
> a. OLS linear
> b. OLS normal [width truncated at 6
> sigma?]
> c. Logistic linear
> d. Logistic normal [width truncated at 6
> sigma?]
>
> Based on some discrete trials, I get the following estimates for R-sq:
> a. 99%
> b. 16%
> c. 96%
> d. 16%
>
> * On the selection of binary Y values. Suppose the X values are linearly
> distributed from 0 to 1 and the Model is Y=X. In the discrete case with 100
> points, the first 5 would be all zeros and the last 5 would be all ones. At
> the center, half the points would be zeroes and the other half would be
> ones.
>
> ** R^2 = (S^2 around mean - S^2 around model) / (S^2 around mean)
--
Frank E Harrell Jr
Professor of Biostatistics and Statistics
Division of Biostatistics and Epidemiology
Department of Health Evaluation Sciences
University of Virginia School of Medicine
http://hesweb1.med.virginia.edu/biostat
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