Posted on behalf of Joe Hilbe.
Alex:
For some reason my response was not posted. This has been happening more and
more lately. Strange.
Anyhow, here is my response. It may get to the list eventually, but if not,
I'll not worry about it.
Best, Joe
=====================================
If I understand what you are asking correctly, it appears that you are
confusing mu, x*Beta, and the linear predictor. x*Beta, in matrix form, is
the linear predictor; i.e.
x*Beta = B_0 + x_1*B_1 + x_2*B_2 + ... x_n*B_n
The identity link refers to the case when x*Beta = mu. That is, the linear
perdictor is identical to the fitted value, mu.
The linear predictor is also called eta. So, for the identiry link, eta=mu.
Remember that the canonical or natural link of the Gaussian family is the
identity link. If, however, we want to use the non-canonical log link
(non-canonical for the Gaussian family), then we must exponentiate the
linear predictor, eta or x*Beta, to obtain the fitted value, mu.
Thus, for the log linked Gaussian, mu=exp(x*Beta), or mu=exp(eta). Note
that the initialization of the log link in the GLM algorithm is eta=ln(mu).
Look on page 61 of the text for the log-linked algorithm.
The difficulty in seeing these relationships with the Gaussian is that the
identity linked model is identical to OLS and requires only one iteration to
solve. It is unlike any of the other GLM members. My suggestion is to read
through the next chapter on gamma regression and I believe you will
understand the relationships.
I hope that I answered what you were looking for. Joe Hilbe
--Alex Cavallo
Lexecon, Inc.
332 South Michigan Avenue
Suite 1300
Chicago, IL 60604
(312) 322-0208 voice
(312) 322-0218 fax
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