MEASURES OF ASSOCIATION IN CASE-CONTROL STUDIES
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The odds ratio is the principal measure of association in a case-control
study. One of the most useful features of the odds ratio, and the reason for
its use in case-control study designs, is that it can be estimated from a
response-based sampling design, even if the incidence of the exposure and
outcome in the underlying population remain unknown.
Likelihood of Suicide and Gun Ownership
Suppose, for example, that one wishes to learn how the likelihood of
suicide varies with gun ownership in a population of 1,000,000 persons for
whom there were the following number of suicides among gun owners and
nongun owners in the course of one year:
Suicide = yes Suicide = no Total
Gun owner A = 60 B = 399,940 A + B = 400,000
Not gun owner C = 40 D = 599,960 C + D = 600.000
Total A + C = 100 B + D = 999,900 1,000,000
In this population, the incidence of suicide among gun owners is A/
(A+B), or 60 per 400,000 per year, and the incidence of suicide among
nongun owners is C/(C+D), or 40 per 600,000 per year. To compare these
two probabilities, we could calculate the relative risk, which can be defined
as the incidence of the outcome in the exposed group divided by the incidence
of the outcome in the unexposed group, namely:
(1) RR =
incidence of outcome in exposed group
=
A/(A+B)
incidence of outcome in unexposed group C/(C+D)
In our example, the relative risk of suicide among gun owners compared
with nongun owners would be (60/400,000)/(40/600,000), which equals
2.25.
However, another relative measure of association is the odds ratio. The
odds in favor of a particular event are defined as the frequency with which
the event occurs, divided by the frequency with which it does not occur. In
our sample population, the odds of suicide among gun owners were 60/
399,940, and the odds of suicide among nongun owners were 40/599,960.
The odds ratio can then be defined as the odds in favor of the outcome in
the exposed group, divided by the odds in favor of the outcome in the
unexposed group.
odds ratio =
odds of outcome in exposed group
=
A/B
odds of outcome in unexposed group C/D
In our example, the odds ratio of suicide for gun owners relative to nongun
owners would be (60/399,940) / (40/599,960), which is about 2.2502. As
the outcome becomes more rare, (B) approaches (A + B) and (D) approaches
(C + D), and the odds ratio approaches the risk ratio. As a rule of thumb,
the odds ratio can be used as a direct approximation for the risk ratio
whenever the incidence of the outcome falls below about 10 percent. This
“rare outcome assumption” holds true in most studies of completed suicide.
Although the rare outcome assumption is not required for the odds
ratio to be a valid measure of association in its own right (Miettinen, 1976;
Hennekens and Buring, 1987), the odds ratio does diverge from the risk
ratio as the outcome becomes more common.
Of what use is this estimate? Why not just calculate the risk ratio
directly? It turns out that the odds ratio has several attractive mathematical
properties, but the most important property is that the ratio that we have
just calculated as (a/b)/(c/d), is equivalent to (a/c)/(b/d). In our example, the
odds ratio we calculated is therefore exactly equal to the ratio of gun
owners to nonowners among the suicide victims (60/40) divided by the
ratio of gun owners to nonowners among population members who have
not committed suicide: (399,940/599,960). This sleight of hand means that
the odds ratio of exposure, given the outcome, which is the measure of
(2)
association obtained from a case-control study, can be used to estimate the
odds ratio of the outcome, given exposure, which is usually the question of
interest.
To see how this works, suppose that we now conduct a case-control
study in the population in order to estimate the association between gun
ownership and suicide. We might do this by selecting all 100 suicides that
occurred during the study year, and by drawing a random sample of 100
control subjects who did not commit suicide during the study year. The
results of the case-control study might be as follows:
Outcome Outcome
Present Absent
Exposure a = 60 b = 40
Present
Exposure c = 40 d = 60
Absent
a + c = 100 b + d = 100
= total cases = total controls
Even though the control group in the case-control study now contains only
100 subjects, we have selected these subjects so that they are representative
of the frequency of exposure to firearms in the population of nonsuicides
from which the control sample was drawn. So the odds ratio for our casecontrol
study is:
(3) odds ratio = (a/c)/(b/d) = (60/40)/(40/60) ≈2.25
Prospective studies can measure the frequency of the outcome among persons
with different levels of exposure; retrospective case-control studies
measure the frequency of exposure among persons with different levels of
the outcome. But the symmetry of the odds ratio allows us to estimate the
risk of the outcome, given exposure, from information about the odds of
exposure, given the outcome.
Attributable Risk
In fact, by themselves, neither the odds ratio nor the risk ratio can assist
policy makers who need to compare the number of occurrences that could
be altered through intervention with the costs of the intervention. Policy
makers would prefer to know the attributable risk, which can be defined as
the difference between the incidence of the outcome among the exposed
and the incidence of the outcome among the unexposed:
(4) AR = A/(A+B) – C/(C+D).
To see the problem with the odds ratio and the relative risk, consider two
populations, one in which the suicide probability conditional on owning a
firearm is 0.02 per person per year and that conditional on not owning a
firearm is 0.01 per person per year, and another in which these two probabilities
are 0.0002 and 0.0001, respectively. The odds ratio and the relative risk
are the same in both scenarios, but if guns are causal, then removal of guns
from the population might avert 0.01 deaths per person per year in the first
scenario, but only 0.0001 deaths per person per year in the second.
In a case-control study, this limitation can be overcome by using
information from other sources. When a case-control study is population
based—that is, when all or a known fraction of cases in a particularly
community are identified and a random sample of unaffected individuals
are selected as controls—or when information about the incidence of
outcome and exposure are available from other sources, it is possible to
calculate the incidence rates and attributable risk from the information
derived from the study (see, for example, Manski and Lerman, 1977;
Hsieh et al., 1985).
In our example, suppose that we already know that the cases represent
all of the suicides occurring in the population in a given year, and suppose
that we know the size of the population. We know, from the case-control
study itself, that 40 percent of control households in random sample own
firearms, and the study has revealed an odds ratio of (about) 2.25 to 1. The
“rare outcome” assumption is satisfied, which simplifies the calculations;
we can treat the odds ratio as a risk ratio and calculate incidence rates and
attributable risks as follows:
The total incidence of suicide in the population is equal to the incidence
of suicide among gun owners, times the probability of being a gun owner,
plus the incidence of suicide among nongun owners, times the probability
of not being a gun owner, i.e.:
(5.1) 10/100,000 = (A/(A+B))(.40) + (C/(C+D))(.60)
A, B, C, and D are the unobserved “true” frequencies of events in the
population. But from the risk ratio of 2.25 we also know that:
(5.2) A/(A+B) = 2.25(C/(C+D))
So: (5.3) 10/100,000 = (2.25)(C/C+D)(.40) + (C/C+D)(.60)
= (.90+.60)(C/C+D)
= (1.50)(C/C+D)
Therefore, the probability of suicide among nongun owners = C/(C+D) =
(10/100,000)/(1.50) ≈6.67 per 100,000 persons per year; and the probability
of suicide among gun owners = (2.25)(C/C+D) = 15 per 100,000 persons
per year.
The attributable risk is the difference between the probability of suicide
among gun owners, and the probability of suicide among nongun owners:
15 – 6.67 ≈8.33 suicides per 100,000 attributable to gun ownership. The
interpretation of this attributable risk depends on the actual causal mechanism
linking exposure and outcome. In our example, there would be about
8.33 suicides per 100,000 that might be preventable by restricting access to
guns, if guns were to play a causal role in the risk of suicide.