Introductory Biostatistics

and from

p¼

0 : 3 þ 0 : 5625

2

¼ 0 : 4313

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

b¼ 0 : 10 !z 1 b¼ 1 : 28

we obtain a required total sample size of

N¼ 4 ðz 1 aþz 1 bÞ^2

pð 1 pÞ

ðp 1 p 0 Þ^2

¼ð 4 Þð 1 : 96 þ 1 : 28 Þ^2

ð 0 : 4313 Þð 0 : 5687 Þ

ð 0 : 2625 Þ^2

F 146

or 73 cases and 73 controls are needed.

Example 12.15 Suppose that all specifications are the same as in Example
12.14, but we design a study in which the size of the control group is four times
the number of the cases. Here we have

p¼ð 0 : 3 Þð 0 : 8 Þþð 0 : 5625 Þð 0 : 2 Þ

¼ 0 : 3525

leading to a required total sample sizeN, satisfying

ffiffiffiffiffi

N

p

ð 0 : 5625  0 : 3 Þ¼ð 1 : 96 Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð 0 : 3525 Þð 0 : 6475 Þð 5 þ 1 : 25 Þ

p

þð 1 : 28 Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð 0 : 3 Þð 0 : 7 Þð 1 : 25 Þþð 0 : 5625 Þð 0 : 4375 Þð 5 Þ

p

NF 222

or 45 cases and 177 controls. It can be seen that the study requires a larger
number of subjects, 222 as compared to 146 subjects in Example 12.14; how-
ever, it may be easier to implement because it requires fewer cases, 45 to 73.

12.10.2 Matched Designs for a Binary Exposure

The design of a matched case–control study of a binary exposure is also speci-
fied by the very same two parameters: the exposure rate of the control groupp 0
and the relitive risk associated with the exposurey. The problem, however,
becomes more complicated because the analysis of a matched case–control
study of a binary exposure uses onlydiscordant pairs, pairs of subjects where

SAMPLE SIZE DETERMINATION FOR CASE–CONTROL STUDIES 471