Introductory Biostatistics

where

p 1 ¼

yp 0

1 þðy 1 Þp 0

The problem is more complicated if we plan for groups with unequal sample
sizes. First, we have to specify the allocation of sizes:

n 1 ¼w 1 N

n 0 ¼w 0 N

w 1 þw 0 ¼ 1 : 0

whereNis the total sample size needed. For example, if we want the size of the
control group to be three times the number of the cases, w 1 ¼ 0 :25 and
w 0 ¼ 0 :75. Then the the total sample size neededNcan be obtained from the
formula

ffiffiffiffiffi

N

p

jp 1 p 0 j¼z 1 a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pð 1 pÞðw 11 þw 01 Þ

q

þz 1 b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p 1 ð 1 p 1 Þw 11 þp 0 ð 1 p 0 Þw 01

q

wherepis the weighted average:

p¼w 1 p 1 þw 0 p 0

Example 12.14 Suppose that an investigator is considering designing a case–
control study of a potential association between congenital heart defect and the
use of oral contraceptives. Suppose also that approximately 30% of women of
childbearing age use oral contraceptives within three months of a conception,
and suppose that a relative risk ofy¼3 is hypothesized. We also decide to
preseta¼ 0 :05 and want to design a study with equal sample sizes so that its
power to the hypothesized relative risk ofy¼3 is 90% (orb¼ 0 :10).
First, the exposure rate for the cases, the percent of women of childbear-
ing age who use oral contraceptives within three months of a conception, is
obtained from

p 1 ¼

yp 0

1 þðy 1 Þp 0

¼

ð 3 Þð 0 : 3 Þ

ð 3  1 Þð 0 : 3 Þ

¼ 0 : 5625

470 STUDY DESIGNS