In the study reported in Example 7.3, blood pressure was measured from a
group of women before and after each took an oral contraceptive. In Exercise
7.2, the insulin level in the blood was measured from dogs before and after
some kind of nerve stimulation. In another exercise, we compared self-reported
versus measured height. A popular application is an epidemiological design
called apair-matched case–control study. In case–control studies, 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
individuals 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, or
race.
Data from matched or before-and-after experiments should never be con-
sidered as coming from two independent samples. The procedure is to reduce
the data to a one-sample problem by computing before-and-after (or case-and-
control) di¤erence for each subject or pairs of matched subjects. By doing this
with paired observations, we get a set of di¤erences, each of which is indepen-
dent of the characteristics of the person on whom measurements were made.
The analysis of pair-matched data with a continuous measurement can be seen
as follows. What we really want to do is to compare the means, before versus
after or cases versus controls, and use of the sample of di¤erencesfdig, one for
each subject, helps to achieve that. With large sample size and assuming that
the null hypothesisH 0 ofno di¤erenceis true, the meandof theses di¤erences
is distributed asnormalwith mean and variance given by
md¼ 0sd^2 ¼sd^2
nrespectively. The extra needed parameter, the variancesd^2 , has to be estimated
from our data by the sample variances^2 d. In other words, the analysis of pair-
matched data with a continuous measurement can be seen as a special case of
the one-sample problem of Section 7.1 withm 0 ¼0. Recalling the form of the
test statistic of Section 7.1, we have
t¼d 0
sd=ffiffiffi
npand the rejection region is determined using thetdistribution atn1 degrees
of freedom. This test is referred to as theone-sample t test, the same one-sample
ttest as in Section 7.1.
Example 7.2 Trace metals in drinking water a¤ect the flavor of the water, and
unusually high concentrations can pose a health hazard. Table 7.1 shows trace-
ANALYSIS OF PAIR-MATCHED DATA 249