I received an interesting email recently. Someone was studying the
relationship between firearm suicides and veterans to answer the
question: "Are veterans more likely to commit suicide with a firearm than
non-veterans?" The data was arranged in a table that looked something
like this (I changed the numbers to simplify the calculations) :
Then this person proceeded to find seven different formulas for the
odds ratio and tried all of them. How I wish more people were this
adventurous! Here are the results.
Actually, there may be a couple more formulas, but this is still an
amazing list. So how come some of the formulas produce an odds ratio of 2
and some of them produce an odds ratio of 0.5? Is there are rule about
how to layout a 2 by 2 table? What are the rows? What are the columns?
What order do the rows and columns go in?
The odds ratio is literally a ratio of odds. But there is more than
one way to compute the odds and more than one way to compute the ratio.
The first formula is simply a cross-product ratio. It is simple and
fast, but does not have a good intuitive explanation.
The second formula computes the odds of a firearm suicide versus
non-firearm suicide in vets (2.5 odds or 5 to 2 in favor of a firearm
suicide) and compares it to the odds of a firearm suicide versus
non-firearm suicide in non-vets (1.25 odds or 5 to 4 in favor of a
firearm suicide). This produce a ratio of odds of 2.0.
The third formula computes the odds that a firearm suicide is a
veteran versus non-veteran (0.5 odds or 2 to 1 against veterans) and
compares it to the odds that a non-firearm suicide is a veteran versus
non-veteran (0.25 odds or 4 to 1 against veterans).
The fourth formula computes the odds that odds that a non-veteran is a
non-firearm suicide (0.8 odds or 5 to 4 against a non-firearm suicide)
and compares it to the odds that a veteran is a non-firearm suicide (0.4
odds or 5 to 2 against a non-firearm suicide).
Now wait, isn't that backwards? Why would anyone compute an odds ratio
this way? It turns out that this is a perfectly fine way to compute an
odds ratio, but it sounds awkward because of the lengthy terms
non-firearms suicide and non-veteran. If we substitute different words (I
realize these words are not quite accurate), the statement would appear
perfectly reasonable. So this formula compute the odds of a passive
suicide among civilians compared to the odds of a passive suicide among
non-civilians.
So why do the formulas (6) and (7) produce different results? Notice
that formula (6) is identical to formula (2) except that A and C are
swapped and B and D are swapped. You could get this effect if you
switched the first and second rows of your table.
The order of your table is arbitrary. You listed the rows in
alphabetical order, but there is nothing stopping you from listing the
rows in reverse alphabetical order. And since there is more than one way
to label the rows, you could still get a different order even if you
rigorously enforced alphabetical ordering. The labels "passive suicide"
and "violent suicide" have a different alphabetical order than "firearm
suicide" and "non-firearm suicide".
The reason that switching the two rows changes the odds ratio is that
you are effectively swapping the numerator and denominator of the odds
ratio. And an estimate that was 2.0 becomes 0.5. It is not a coincidence
that these two numbers are reciprocals.
So what one estimate is telling you is that being a veteran increases
your odds of being a firearm suicide. The odds are twice as high in the
veteran group than the non-veteran group. The other odds ratio is telling
you that being a non-veteran decreases your odds of being a firearm
suicide. The odds in the non-veteran group is about half of the odds in
the veteran group.
So your numbers are perfectly consistent, even though they look wildly
inconsistent.
So which odds ratio do you use? It's "dealer's choice" and no one will
complain about any of these choices. Certain groups, such as the Cochran
collaboration have set up rules. In the Cochran collaboration, they
always orient their tables so that an odds ratio less than one represents
benefit for the new therapy versus the standard therapy. But you do not
have to follow their convention.
I have some informal rules about how to orient a 2 by 2 table, but
these rules are based on making the table easier to comprehend rather
than on any rigorous criteria or mathematical formula.
So the good news is that you can do whatever you like. If you do a lot
of work with odds ratios (and other ratios like the relative risk), you
just have to get comfortable with the fact that an odds ratio of 2.0 and
an odds ratio of 0.5 are just the opposite sides of the same coin.
07/08/2008.
