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)
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