Thanks to all of you who have responded to my question.

Here is a list of the responses, since it seems I'm not the only person who
has to deal with this problem:


Robert Newcombe wrote:

This sounds a most unusual problem! The first step must be to redefine the
problem
into something more intelligible.

One possible scenario that would lead to data like yours is as follows.
You have a large group of people. Some continuous outcome variable such as
income is
measured for each person. Each person is also classified according to
whether they
have educational qualifications such as GCSE or O level in each of k
subjects. This
gives you data on income for each of k sets of subjects, i=1 ... k, set
number i is
defined as those who have a qualification in subject i.

Clearly these sets will overlap greatly. One possible solution to this
problem
is to look at each subject separately and to estimate the difference between
those
with and without a qualification in that subject, then to seek to test the
heterogeneity
of those differences in some way - though it isn't obvious how this should
be done.

Another, probably better approach is to model the outcome variable by k
binary variables
simultaneously, variable i representing whether the respondent has or
doesn't have a
qualification in subject i. This regression model has k degrees of freedom,
and allows
qualifications in different subjects to have different effects on income.
Then fit an
alternative model with 1 degree of freedom, in which the k coefficients are
constrained
to be equal. This model has 1 df, and represents the hypothesis that income
is a linear
function of the number of qualifications, all of which carry the same
weight. Test the
difference in fit between these models by a k-1 df test.

Other possible 1 df models you could consider are to split subjects into any

qualification vs. none and into all qualifications vs. not all, and proceed
similarly, though I guess that these will be less useful. Even what I
suggest
above might not work, if the degree of collinearity (i.e. the degree of
overlap
between groups) is too great.

Hope this helps.

Robert Newcombe.

..........................................
Robert G. Newcombe, PhD, CStat, Hon MFPHM
Reader in Medical Statistics
University of Wales College of Medicine
Heath Park
Cardiff CF14 4XN, UK.
Phone 029 2074 2329 or 2311
Fax 029 2074 3664
Email [log in to unmask]
Web:
http://www.uwcm.ac.uk/epidemiology_statistics/research/statistics/newcombe.h
tm



Bryan Bayfield wrote:

I don't know if there's a better method, but here's what I'd do:

        - Define new groups based on membership of existing groups. If
there's
           10 groups g1, ... , g10; then consider for example g1. If it is
possible in
           the sample to be a member of g1 and none, one or both of g2 and
g3 then that
           gives you g1, g1g2, g1g3, g1g2g3 as your new groups.

        - Plot the response classified by new groupings to look for
structure in the data.
           What would be of particular interest would be if being a member
of, say, group 1
           and 2 had any effect of practical significance upon the response
over the subject
           being just a member of say group 1.

        - Choose an appropriate method of comparison given the above.


Hope this is of some use.

Bryan.



Paul Seed wrote:

What you have is not groups in the usual sense. You need to think about
exactly what question you
are trying to answer & why.

One approach is to define a conventional groups structure & do comparisons
based on that.
You could drop all subjects that are members of more than on group. You
could create a
new group of "multiples", or several such groups with different criteria.

You have to decide what will make sense and answer the research questions of
those who
produced the data.

The other approach is to come up with an analysis method that uses all the
information.
You can create dummy variables for each "group", rename them risk factors,
and fit a
multiple regression model. Differences in the parameters for the different
groups will
be equal to the differences in the means of each group, with an adjustment
for
membership of other groups. Formal tests and CI can be worked out for such
comparisons.




Stephen Duffy Wrote:

As a starting point, you could have a separate dummy variable for each
group,
ie treating membership of each group as a binary covariate. You might want
to
progress to some fancier, hierarchical modelling later.

Stephen