🚀 go-pugleaf

kadie@cs.uiuc.edu (Carl M. Kadie) writes:

>I'm (trying to) design an experiment. The input, x, is a real number.
>The output, y, is random, but either 0 or 1. I believe that either x
>or x^2 has linear relationship with the expected value of y. How can I
>test this and find the best model?

Since E(y) is a probability for the event y=1, it must lie in the
interval [0,1].  But all linear and quadratic functions of x
eventually move outside this interval, so your hypothesis cannot be
correct unless the range of x is limited.  Even then, it is not
usually appropriate to fit a linear or quadratic model.  A bounded
(between 0 and 1) monotone transformation of the linear or quadratic
function is usually fit instead.

However, unless you know more about how the data are generated it is
not possible to make a definitive statement about which approaches
"work" and which do not.  You need a more general model of how the
probability for y=1 depends on x in order to construct a formal test
of your specific hypothesis.

Standard approaches to this problem include the binary "logit" and
"probit" models, which differ only in their choice of the monotone
transformation listed above.  There are many, many references.  One
starting point is the book "Analysis of Binary Data" by D.R. Cox.
--
T. Scott Thompson              email:  thompson@atlas.socsci.umn.edu
Department of Economics        phone:  (612) 625-0119
University of Minnesota        fax:    (612) 624-0209