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252 CHAPTER 7 HYPOTHESIS TESTING
where d is the sample mean difference, m d0
is the hypothesized population mean difference,
s d = s d >1n, n is the number of sample differences, and s d is the standard
deviation of the sample differences. When H 0 is true, the test statistic is distributed
as Student’s t with n - 1 degrees of freedom.
Although to begin with we have two samples—say, before levels and after levels—
we do not have to worry about equality of variances, as with independent samples, since
our variable is the difference between readings in the same individual, or matched individuals,
and, hence, only one variable is involved. The arithmetic involved in performing
a paired comparisons test, therefore, is the same as for performing a test involving
a single sample as described in Section 7.2.
The following example illustrates the procedures involved in a paired comparisons
test.
EXAMPLE 7.4.1
John M. Morton et al. (A-14) examined gallbladder function before and after fundoplication—a
surgery used to stop stomach contents from flowing back into the esophagus
(reflux)—in patients with gastroesophageal reflux disease. The authors measured gallbladder
functionality by calculating the gallbladder ejection fraction (GBEF) before and
after fundoplication. The goal of fundoplication is to increase GBEF, which is measured
as a percent. The data are shown in Table 7.4.1. We wish to know if these data
provide sufficient evidence to allow us to conclude that fundoplication increases GBEF
functioning.
Solution:
We will say that sufficient evidence is provided for us to conclude that the
fundoplication is effective if we can reject the null hypothesis that the population
mean change m d is different from zero in the appropriate direction.
We may reach a conclusion by means of the ten-step hypothesis testing
procedure.
1. Data. The data consist of the GBEF for 12 individuals, before
and after fundoplication. We shall perform the statistical analysis on
the differences in preop and postop GBEF. We may obtain the differences
in one of two ways: by subtracting the preop percents from
the postop percents or by subtracting the postop percents from the
preop percents. Let us obtain the differences by subtracting the preop
TABLE 7.4.1 Gallbladder Function in Patients with Presentations of
Gastroesophageal Reflux Disease Before and After Treatment
Preop (%) 22 63.3 96 9.2 3.1 50 33 69 64 18.8 0 34
Postop (%) 63.5 91.5 59 37.8 10.1 19.6 41 87.8 86 55 88 40
Source: John M. Morton, Steven P. Bowers, Tananchai A. Lucktong, Samer Mattar, W. Alan Bradshaw,
Kevin E. Behrns, Mark J. Koruda, Charles A. Herbst, William McCartney, Raghuveer K. Halkar, C. Daniel
Smith, and Timothy M. Farrell, “Gallbladder Function Before and After Fundoplication,” Journal of
Gastrointestinal Surgery, 6 (2002), 806–811.