12.10 SAMPLE SIZE DETERMINATION FOR CASE–CONTROL
STUDIES
In a typical case–control study, cases of a specific disease are ascertained as
they arise from population-based registers or lists of hospital admissions, and
controls are sampled either as disease-free people from the population at risk,
or as hospitalized patients having a diagnosis other than the one under study.
Then in the analysis, we compare the exposure histories of the two groups. In
other words, a typical case–control study fits the framework of a two-arm
randomized phase III trials. However, the sample determination is somewhat
more complicated, for three reasons:
- Instead of searching for a di¤erence of two means or proportions as in
the case of a phase III trial, the alternative hypothesis of a case–control
study is postulated in the form of a relative risk. - It must be decided whether to design a study with equal or unequal sam-
ple sizes because in epidemiologic studies, there are typically a small
number of cases and a large number of potential controls to select from. - It must be decided whether to design a matched or an unmatched study.
For example, we may want to design a case–control study to detect a rela-
tive risk, due to a binary exposure, of 2.0, and the size of the control group is
twice the number of the cases. Of course, the solution also depends on the
endpoint and its measurement scale; so let’s consider the two usual categories
one at a time and some simple configurations.
12.10.1 Unmatched Designs for a Binary Exposure
As mentioned previously, the data analysis is similar to that of a phase III trial
where we want to compare two proportions. However, in the design stage, the
alternative hypothesis is formulated in the form of a relative risky. Since we
cannot estimate or investigate relative risk using a case–control design, we
would treat the given numberyas anodds ratio, the ratio of the odds of being
exposed by a casep 1 =ð 1 p 1 Þdivided by the odds of being exposed by a con-
trolp 0 =ð 1 p 0 Þ. In other words, from the information given, consisting of the
exposure rate of the control groupp 0 and the approximated odds ratio due to
exposurey, we can obtain the two proportionsp 0 andp 1. Then the process of
sample size determination can proceed similar to that of a phase III trial. For
example, if we want to plan for a study with equal sample size,N=2 cases and
N=2 controls, the total sample size needed should be at least
N¼ 4 ðz 1 aþz 1 bÞ^2pð 1 pÞ
ðp 1 p 0 Þ^2SAMPLE SIZE DETERMINATION FOR CASE–CONTROL STUDIES 469