# 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.