Introductory Biostatistics

Example 12.9 Suppose that in Example 12.8, the researcher wanted to design
a study such that its power to detect a di¤erence between means of 3 is 90% (or
b¼ 0 :10). In addition, the variance of cholesterol reduction (with placebo) is
not known precisely, but it is reasonable to assume that it does not exceed 50.
As in Example 12.8, let’s seta¼ 0 :05, leading to

a¼ 0 : 05 !z 1 a¼ 1 : 96 ðtwo-sided testÞ

b¼ 0 : 10 !z 1 b¼ 1 : 28

Then using the upper bound for variance (i.e., 50), the required total sample
size is

N¼ð 4 Þð 1 : 96 þ 1 : 28 Þ^2

50

ð 3 Þ^2

F 234

Each group will have 117 subjects.
Suppose, however, that the study was actually conducted with only 180
subjects, 90 randomized to each group (it is a common situation that studies
are underenrolled). From the formula for sample size, we can solve and ob-
tained

z 1 b¼

ffiffiffiffiffiffiffiffiffiffiffi

N

4

d^2

s^2

r

z 1 a

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

180

4

32

50

r

 1 : 96

¼ 0 : 886

corresponding to a power of approximately 81%.

12.9.2 Comparison of Two Proportions

In many other studies, the endpoint may be on a binary scale; so let us consider
a similar problem where we want to design a study to compare two propor-
tions. For example, a new vaccine will be tested in which subjects are to be
randomized into two groups of equal size: a control (nonimmunized) group
(group 1), and an experimental (immunized) group (group 2). Subjects in both
control and experimental groups will be challenged by a certain type of bacteria
and we wish to compare the infection rates. The null hypothesis to be tested is

H 0 :p 1 ¼p 2

versus

464 STUDY DESIGNS