# [S] model of proportions

Lorenz Gygax (lgygax@amath.unizh.ch)
Tue, 1 Dec 1998 10:47:54 +0100 (MET)

Dear all,

can someone enlighten me on the following or point me to the right

Imagine, that I have a data set where the response variable is a
proportion (say the proportion of a volume) and with some explanatory
variable(s). The proportion of the response is not made up of several
entities (thus a classical logistic regression with 0s and 1s doesn't
work). [I also know that there are recent special methods to work with
compositional data e.g. developed by Peter Guttorp.] But let's assume for
the moment that I am either only interested in one component or that there
are only two components.

Now, I made up a simple example and tried two things:

Firstly, I fed the data into glm (Y ~ X, family= binomial). This seems to
give a sensible curve (though the tests might be wrong?) but the handbook
is very clear about that the binomial family expects only two different
values in the response.

Secondly, I transformed Y into Z = log (Y/1-Y). Then I fitted a line by
lm (Z ~ X). If I look at the estimated values they are very similar
(though the tests of the coefficients are not).

What I don't see/understand is:
- How exactly are the two methods related (is the former just a least
square and the latter a loglikelihood estimate)?
- Which of the two is (more?) correct?
- Is there a better way to do this with some "standard methods" (I am
trying to give some advice to someone else for this)?
- What happens if I cover the whole range between 0 and 1 (including quite
a few 0s and 1s)? E.g. I expect problems with the second version as
I assume that the resiudals won't be normally distributed anymore.

Regards, Lorenz

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Lorenz Gygax

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