This is more slowly, and more clearly as well. I hate to be the one to break
the news, but some time ago Man invented Mathematics to make these descriptions
even clearer.
It seems to me you are saying
prob(Y=1|Z) = .2(Z+2.5)
where Z is continuously distributed between -2.5 and + 2.5 (either as
a truncated normal or as a uniform or as whatever). You are asking about
R^2 in this situation (and perhaps about the maximum and minimum of
R^2 for different distributions of Z).
Thus, for the joint,
prob(Y=1,Z)=(1/2+Z/5)p(z)
prob(Y=0,Z)=(1/2-Z/5)p(z)
which means
prob(Y=1)=(1/2+M/5)
prob(Y=0)=(1/2-M/5)
where M=E(Z). Thus
E(Y) = (1/2+M/5)
V(Y) = (1/2+M/5)(1/2-M/5)
E(ZY)= M/2 + (S^2+M^2)/5
where S^2 = V(Z). Thus
C(ZY) = M/2 + (S^2+M^2)/5 - M(1/2+M/5)=S^2/5
which means
R(XY) = S / 5*SQRT((1/2+M/5)(1/2-M/5)).
I usually make mistakes in these calculations, so this may very well
be wrong, but at least it is an answer. If M=0, then R(ZY)= 2S/5. This
is minimal for a uniform, where R(ZY)=2/sqrt(12)=.577 and maximal for
a two-point distribution where R(ZY)=1.
It is easy to calculate for a truncated normal, but somebody else can do that.
At 5:07 AM +0000 2/7/00, Milo Schield wrote:
>Rich Ulrich wrote in message ...
> > ...could you explain the question more slowly?
>
>
>Sorry for not giving enough motivation or context. Yes, I'm not comparing
>the fit for continuous data with the fit for it's binary equivalent.
>
>I have been thinking about the small values of R-squared values for logistic
>regressions involving z (the standard normal) on the horizontal axis. I
>realize that the binary nature of the data precludes good fits, but that per
>se does not mean that a given regression model can't bring about a
>substantial improvement in fit -- when compared with the fit using just the
>mean as the model. When data is linearly distributed on the horizontal
>axis, even though the data is binary on the vertical axis, a simple Ordinary
>Least Squares model can bring about a great improvement in the standard
>deviation when comparing the explained [original - unexplained] with the
>original.
>
>I don't have any idea whether a normal distribution of data on the
>horizontal axis would make it very much more difficult to get any
>improvement in modeling binary data on the vertical axis -- due to the
>non-linear weighting on the horizontal axis. That is what I want to learn.
>
>To simplify matters, I wanted to consider a linear model of the expected
>value of Y with values ranging from 0 to 1. I'll entertain more complex
>models later.
>
>As a basis for comparison, I began with data linearly distributed on the
>horizontal axis. In the horizontal linear case , I let X range from 0 to 1
>and let Y_continuous = X. Y_discrete was assigned so that the fraction of
>ones in an adjacent group of points had a value similar to that of the
>continuous model (Y_c) in that region of X.
>
>In the horizontal normal case, I let Z range from -2.5 to +2.5 and I let
>Y_continuous = .2*(Z + 2.5) so that Y_continuous ranged from 0 to 1 over
>this interval. I then assigned values to Y_discrete so that the fraction
>of ones in an adjacent group of points had a value similar to that for the
>continuous model (Y_c) in that region of Z. I presume that the R-squared
>will vary with the range of Z values involved since Z range from -oo to +oo
>might involve some real challenges in fitting a linear distribution over
>that interval.
>
>I presume that there is a theoretical/analytical solution for the value of
>R-squared given an infinite number of points in the intervals on the X axis.
>I expect that my crude discrete simulations give some indication of these
>values. My question is this:
>
>"What is the limiting value of R-squared as the number of discrete points in
>the X-axis interval approaches infinity WHEN
>1. the data is normally distributed on the X axis [say-2.5 < Z < +2.5] and
>2. the binary data values on the Y axis models a linear distribution going
>from 0 to 1 over the interval on the X axis?"
>
>
>
>
>
>
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===
Jan de Leeuw; Professor and Chair, UCLA Department of Statistics;
US mail: 8142 Math Sciences Bldg, Box 951554, Los Angeles, CA 90095-1554
phone (310)-825-9550; fax (310)-206-5658; email: [EMAIL PROTECTED]
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