Introductory Biostatistics

and controls are sampled either as disease-free persons from the population at
risk or as hospitalized patients having a diagnosis other than the one under
investigation. As a technique to control confounding factors, individual cases
are matched, often one to one, to controls chosen to have similar values for
confounding variables such as age, gender, and race.
For pair-matched data with a single binary exposure (e.g., smoking versus
nonsmoking), data can be represented by a 2
2 table (Table 6.1) whereðþ;Þ
denotes the (exposed, nonexposed) outcome. In this 2
2 table,adenotes the
number of pairs with two exposed members,bdenotes the number of pairs
where the case is exposed but the matched control is unexposed,cdenotes the
number of pairs where the case is unexposed but the matched control is
exposed, andddenotes the number of pairs with two unexposed members. The
analysis of pair-matched data with a single binary exposure can be seen, heu-
ristically, as follows. What we really want to do is to compare the incidence of
exposure among the cases versus the controls; the parts of the data showing no
di¤erence, the numberaof pairs with two exposed members, and the numberd
of pairs with two unexposed members would contribute nothing as evidence
in such a comparison. The comparison therefore relies solely on two other fre-
quencies,bandc; under the null hypothesis that the exposure has nothing to do
with the disease, weexpectthatb¼c orb=ðbþcÞ¼ 0 :5. In other words,
the analysis of pair-matched data with a single binary exposure can be seen as
a special case of the one-sample problem with binary of Section 6.1 with
n¼bþc,x¼b, andp 0 ¼ 0 :5. Recall the form of the test statistic of Section
6.1; we have

z¼

pp 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 0 ð 1 p 0 Þ=ðbþcÞ

p

¼

½b=ðbþcފ^12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

2



1 ^12



=ðbþcÞ

q

¼

bc

ffiffiffiffiffiffiffiffiffiffiffi

bþc

p

The decision is based on the standardizedzscore and referring to the percen-
tiles of the standard normal distribution or, in the two-sided form, the square of
the statistic above, denoted by

TABLE 6.1

Control

Case þ

þ ab

 cd

ANALYSIS OF PAIR-MATCHED DATA 211