Odds Ratios
First of all, what are odds? Odds are simply a ratio of the probability that an
event will occur versus the probability that the event will not occur, or probability / (1-probability).
For
example, if you go fishing and you catch 3 largemouth bass and 1 trout, then
the odds of catching a trout = [(1/4)/(3/4)] = 1/3 = 0.33.
Note
that this differs from risk (or probability): the risk of catching a trout is equal to
(#
of trout caught) / (total # of fish caught) = 1/4 = 0.25.
For
example, if you compare your luck with fishing with no bait versus fishing with
Billy Bob's Bassassinator Bait and cast your line 100 times using each method,
a 2x2 table would show the following:
|
|
# of times caught |
# of times not caught |
Total # of casts |
|
Bassassinator |
50 |
50 |
100 |
|
No bait |
2 |
98 |
100 |
The
odds of catching a fish with the
Bassassinator is 50/50 or 1.0.
The
odds of catching a fish with no bait
is 2/98 or 0.02.
Therefore,
the odds ratio for catching a fish
with the Bassassinator vs. no bait is
1.0/0.02
= 50.
The
probability of catching a fish with the Bassassinator is
50/100 or 0.50.
The
probability of catching a fish with no bait is 2/100 or 0.02.
Therefore,
the relative risk for catching a fish
with the Bassassinator vs. no bait is 0.50/0.02 = 25.
So what exactly does this odds ratio tell us? Odds ratio
can be used to give us an idea of how strongly
a given variable may be associated with the outcome of interest compared to
other variables. Odds ratios are simply a different way of expressing this
association than relative risk since they compare odds rather than risk of
an event; however, they are sometimes very close to each other, such as when
the outcome of interest is rare.
For
example, in the example above, we can say that using the Bassassinator Bait is
strongly associated with catching fish compared with using no bait. If we also
compared another group, such as using worms for bait, and found that 25 of 100
casts using worms yielded fish, the odds for catching a fish using worms would
be 0.33, with an OR = 16.5. We could
conclude that Bassassinator Bait is better than worms at catching fish, and
worms are better than nothing.
Assuming that this trial is
in fact a RCT, we can conclude that Bassassinator Bait improves the odds of catching a fish by 50 times
compared to using no bait; or we could say that the risk or chance of catching a fish is increased 25 times in
fishermen using Bassassinator Bait compared to those using no bait.
Why do we use OR instead of RR in case-control
studies? To be able to calculate relative risk, we compare the risks of
outcome in different groups. In case-control studies, we already know what the
outcome is and we separate groups into those with the outcome vs. controls. Our
objective in such studies is to try to identify risk factors that are more
strongly associated with one group than the other; thus, risk and therefore relative
risk cannot be calculated from these studies. We use odds ratios instead, which can give us a measure of how strongly
the risk factor is associated with the outcome.
For
example, if we suspect that Bassassinator Bait is associated with catching more
fish, then we could take 100 successful fishermen and compare them with 100
fishermen who were unable to catch any fish and find out how many in each group
used Bassassinator Bait. Since we
select the outcome in both groups, we cannot calculate the relative chance
(risk) of catching fish in the Bassassinator Bait users because we do not know
the chance of catching fish in the general population (who are
non-Bassassinator users), and therefore we have no comparison group. However,
we can compare the odds of the use of
Bassassinator Bait in those who caught fish vs. those who were unable to catch
fish by calculating the odds ratio.
So for example:
|
|
Bassassinator use |
No Bassassinator |
|
Caught fish |
40 |
60 |
|
Caught nothing |
20 |
80 |
Odds of
Bassassinator use in those who caught fish = 40/60 = 0.67.
Odds of
Bassassinator use in those who caught nothing = 20/80 = 0.25.
Odds ratio of
Bassassinator use in successful vs. unsuccessful fishermen = 0.67/0.25 = 2.7.
You can say that the odds of use of Bassassinator were 2.7
times greater in successful fishermen vs. unsuccessful fishermen in this study.
This implies an association between use of Bassassinator bait and catching
fish. However, remember that many other things could have contributed to this apparent
association: chance alone could have accounted for this difference (helpful to
know the 95% CI for the OR); the sample selected for both groups could have
been skewed to favor Bassassinator users in the group that caught fish; or
perhaps the successful fishermen were more likely to recall using Bass. Bait
(recall bias).
Please note that we cannot conclude that Bassassinator increases the risk of catching
fish by 2.7 times;
all we can conclude is that successful fishermen were 2.7 times more likely to
have used Bassassinator Bait than unsuccessful fisherman in this study only, which would lead us to believe that there could be an association with
Bassassinator use and successfully catching fish. We would need to do a RCT or
prospective cohort study to be able to estimate the magnitude of the
effectiveness of the Bassassinator Bait.
QUICK HITTERS:
1.
Odds = Probability /
(1-probability).
2.
Odds ratio (OR) = ratio of odds of
event occurring in exposed vs. unexposed group.
3.
Odds ratio are used to estimate how
strongly a variable is associated with the outcome of interest; in prospective
trials, it is simply a different way of expressing this association than
relative risk.
4.
In case-control studies, we separate
groups by their outcomes and retrospectively try to identify variables
that appear to be more associated with one outcome than another. Therefore, we
cannot deduce a calculable risk
because the outcome has already been predetermined. We therefore use odds
ratios instead to estimate the strength of association of the variable with the
outcome of interest.