In a 3-way table, of say religion by party by
cohort, the first order terms mean that there are
different numbers in the categories of the marginal distributions
of each variable. For instance, cohort size is not a constant,
different parties have different levels of support, etc. .
The two-way terms, e.g., religion by cohort
mean that there is an association between religion and
cohort: the distribution of religion is different
in the different cohorts.
Having more than one 2-way term means that there is more than
one 2-way association, even when you take the other(s) into
account.
You may find an apparent association between say,
religion and party (e.g., if you created the
religion by party table and tested for
independence), but this is really due to associations between
religion and cohort and between party
and cohort. That is, you may have marginal
association but not conditional association.
This can arise by many means: for instance, assume there is
no mechanism associating religion and party,
but that the distribution of these states changes over time
(across cohort). Let's say there is an increase in religious
non-participation and an entirely unrelated rise in the vote for
the Social Democratic Party. Early cohorts will have lower
non-participation and lower SDP, later will have higher. In the
religion by party marginal table this will look
like association, but in each cohort-specific religion
by party subtable there will be no association.
