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Newsgroups: bit.listserv.stat-l
Date:         Wed, 29 Jul 1992 11:22:32 EST
From:         "D. Glen McPherson" <mcphers@HILBERT.MATHS.UTAS.EDU.AU>
Subject:      Association and interpretation in log-linear modeling
X-To:         "stat-l list" <stat-l@vm1.mcgill.ca>

Given models based on a three-way table of frequencies, where the variables
are designated A, B and C, it is intuitively reasonable that the table formed
by collapsing over levels of C to give an A x B table is not generally
meaningful if there is a three-way association between variables A, B and C.
The reason is clear - a three-way association implies that the relation
between A and B varies across the levels of C. Hence the association
portrayed in the collapsed AxB table is a composite measure of association.
Not only is information about the AxB association lost by the combination,
but the composite measure of the AxB association is dependent on such,
commonly irrelevant, information as the relative numbers of sample members at
each level of C.

My problems arise when there is no three-way association between A, B and C,
but the simplest model which fits the data must include all two-way
associations, i.e. AxB, AxC and BxC. A number of authors establish that, in
such circumstances, the parameter(s) defining the AxB association under the
model which fits all three variables, are not the same parameters as those
which apply when only the two-variable model for A and B is fitted to the
collapsed table. (Technically, the three-way table is not 'strictly'
collapsible' in this case). I am comfortable with the distinction - under the
model based only on the AxB table, the parameters measure the association
between A and B, whereas under the model based on the AxBxC table, the
parameters measure the 'partial' association between A and B adjusting for
relationships of A and B with C.

My problem is how do I estimate these 'partial' association parameters? From
my reading, none of the books which counsel against collapsing the table over
C to estimate the parameters defining the AxB association, actually tell me
how to estimate these parameters. It is as though most authors in this area
never venture beyond the world of academia, and they are quite content to say
'Stop the analysis if you reach this situation!'.

Can someone point me to a reference which might throw some light on the way
to estimate these parameters, please?

My ultimate test of the usefulness of methodology is its practical
application. This is one example of many difficulties I have in attempting to
develop skills in the practical application of log-linear methodology for my
non-statistics majors, and in interpreting results of analyses for consulting
clients. Do others have the same problem or am I missing out on what should
be intuitively obvious? Following is my non-fashionable way of viewing model
formation and interpretation. I would appreciate comments on the approach.


The usual parameterizations employed with linear models are both intuitively
appealing for non-statistics majors and can be quickly introduced. By
contrast, standard log-linear parameterizations are not easily grasped by
non-mathematically oriented students, and require both a lengthy introduction
and a long absorption period. The fact that log-linear models come within the
framework of Generalized liner models, has led to the presentation of the
linear predictor form of presentation of the equations. It seems to me that
the multiplicative form is more appealing since it is a simple extension of
the notion of probabilistic independence which is easily motivated. Thus, in
a model based only on the two variables, A and B,

the independence model includes the equation

        Pr(A=i and B=j) = pi(A)i x pi(B)j

where pi(A)i is the marginal probability for variable A, and pi(B)j is the
marginal probability for the  variable B. If the variables are not
independent, there is need to adjust for this fact by introducing  'AxB
association' parameters delta(AB)ij, i.e.

        Pr(A=i and B=j) = pi(A)i x pi(B)j x delta(AB)ij

This is the analogous development to the introduction of interactions in
linear models, and, as such, sits comfortably with students who have
previously been introduced to the concept of interactions in additive
equations. The test of no association becomes a test that all delta(AB)ij =
1.

The approach extends simply and naturally to three variables. Thus, pairwise
independence among all pairs is represented by the equation

        Pr(A=i and B=j and C=k) = pi(A)i x pi(B)j x pi(C)k.

The inclusion of a component which allows for an association between A and B,
but independence between the other pairs, is represented by the equation

        Pr(A=i and B=j and C=k) = pi(A)i x pi(B)j x pi(C)k x delta(AB)ij.

Further associations are readily and obviously introduced with the inclusion
of additional multiplicative components.

The role of model comparisons in hypothesis testing is readily tied to
components in the multiplicative equations.

Given a model which fits the data, the matter of practical interpretation can
follow. As part of this process, reparameterization can be discussed . The
interpretive value of considering odds as ratios of 'pi' components, and odds
ratios based on 'delta' components can be introduced - for their practical
value rather than as mathematical artifacts.

Do others share my concerns about teaching the basis and application of this
methodology to non-statistics majors? Am I missing something - can the
log-linear approach be made intuitive and easy?
--
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|  Glen McPherson            |                                   |  _--_|\  |
|  Department of Mathematics |-----------------------------------| /      \ |
|  University of Tasmania    |               E-Mail:             | \_.--._/ |
|  Australia.                | mcphers@hilbert.maths.utas.edu.au |       *  |
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