Re: 2x2x2 contingency cube
- From: Richard Ulrich <Rich.Ulrich@xxxxxxxxxxx>
- Date: Mon, 15 May 2006 16:24:39 -0400
On 15 May 2006 10:34:38 -0700, hhdave@xxxxxxx wrote:
My colleagues and I are interested in examining relationships among 3
dichotomous variables A, B and C. There are 2,801 total categories
under consideration (a fixed total). The observed frequencies are:
In my usage, I would report that the "categories" are
0/1 (or whatever) pairs of levels of A, B, and C; and that
there are 2801 cases or events or instances.
A B C Count
------------------------
N N N 2677
N N Y 22
N Y N 73
N Y Y 1
Y N N 9
Y N Y 3
Y Y N 5
Y Y Y 11
We are mainly interested in answering the following:
Given that there are 14 categories that are both A:Yes and C:Yes, how
significant is it that 11 of them will also be B:Yes? (What is the
appropriate p-value to report?)
Well, what is the hypothesis? Certainly, the 3-way table
is "needed" to explain the relations, because there are
two 2-way tables that are *very* dissimilar:
2677+22 versus 73 +1 shows no trend apart from the margins;
9+3 versus 5+11 shows bigger numbers in the main diagonal,
a strong positive trend.
First, you cannot readily see similarities between the two tables,
since the margins are so very different. Do you want to look
at the second table by itself? ... for some purposes, that would
be reasonable.
Next: you cannot ordinarily blame the interaction of a 2x2 table
on a single cell, like the "11". Any one cell of a 2x2 table can be
changed to make its proportions similar, if you are not constrained
to make the margins the same. If you keep the margins the same,
every cell has the same difference in units, (Expected minus
Observed). On the other hand, considering the full 2x2x2 table,
the "YYY" category would the rare one, so it would have the smallest
expectation.
What is the best way to tackle this? Should we be looking to compare
odds ratios (A x B when C=Yes, vs. A x B when C=No)? Are we really
looking for the significance of ABC interaction?
Individual 2x2 Fisher's Exact tests were plain enough, showing very
strong associations between A and B, A and C, & B and C. But we want
to find an appropriate measure of three-way association.
We plugged the data into VassarStats' Log-Linear Analysis for a 2x2x2
Table of Cross-Categorized Frequency Data
(http://faculty.vassar.edu/lowry/log.html) but I'm not sure how to
interpret the results or if they even answer the question. We also ran
a loglinear analysis with SPSS, using the multinomial option (since the
2,801 is a fixed total), but again, I'm not sure which part of the
results is relevant to our question.
Any insight into this analysis would be greatly appreciated!
I don't think you can finalize the testing without becoming
explicit about what A, B, and C represent. For instance, *I*
would lean toward looking at the second 2x2 table by itself,
saying that conditioning on the first Y seems too overwhelming,
so you have to do it. That does not give a very extreme p-value
for the "11" because the counts are fairly small.
But I might favor some using the three-way interaction test,
or some other test based on another 2x2 partition, depending
on what those groupings are.
Or, I might figure out that the log-linear model is not the
appropriate one, if the question were more explicit.
--
Rich Ulrich, wpilib@xxxxxxxx
http://www.pitt.edu/~wpilib/index.html
.
- References:
- 2x2x2 contingency cube
- From: hhdave
- 2x2x2 contingency cube
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