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st: re: binomial regression


From   Kit Baum <baum@bc.edu>
To   statalist@hsphsun2.harvard.edu
Subject   st: re: binomial regression
Date   Fri, 3 Aug 2007 14:25:50 -0400

I know little about the issues raised in this discussion in terms of relative risks and odds ratios, but I must speak out about the seemingly loose use of the notion of link functions.

Marcello said
...
Finally, yes the cdf of the uniform distribution, as opposed to the
logistic distribution, say, is what gives you a straight line
transformation between the limits of the uniform. So, yes, a uniform is
involved.


But the link function is not a CDF. The uniform distribution over some finite range maps a value on the real line into a constant number. That is not what link(identity) means. From [R] glm,

The link function is \eta = g(\mu).
The identity link is \eta = \mu, so that g(x) = x.
The log link is \eta = \log(\mu). One would not argue that \log(x) is somehow related to a CDF.

The logit link is \eta = \log ( \mu / (1 - \mu)), the natural log of the odds.
The probit link is the INVERSE Gaussian cdf.

So if the link function has a relation to a CDF, it is to an inverse CDF. But I don't think it makes much sense to describe the identity link or the log link as having something to do with a CDF. After all, using the identity link, we have y = X b + u, OLS regression, and as we know there are no distributional assumptions whatsoever (other than \sigma^2_u > 0) for the method of least squares. If we do make a distributional assumption in estimating a linear regression (e.g., estimating with maximum likelihood) we do not consider the uniform distribution.



Kit Baum, Boston College Economics and DIW Berlin
http://ideas.repec.org/e/pba1.html
An Introduction to Modern Econometrics Using Stata:
http://www.stata-press.com/books/imeus.html


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