Introductory Biostatistics

the total sample sizeNis given by

N¼

1

ðlnyÞ^2

1

w 1

þ

1

w 0



ðz 1 aþz 1 bÞ^2

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 andw 0 ¼ 0 :75.

Example 12.17 Suppose that an investigator is considering designing a case–
control study of a potential association between coronary heart disease and
serum cholesterol level. Suppose further that it is desirable to detect an odds
ratio ofy¼ 2 :0 for a person with a cholesterol level 1 standard deviation above
the mean for his or her age group using a two-sided test with a significance level
of 5% and a power of 90%. From

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

b¼ 0 : 10 !z 1 b¼ 1 : 28

the required total sample size is:

N¼

4

y^2

ðz 1 aþz 1 bÞ^2

¼

4

ðln 2Þ^2

ð 1 : 96 þ 1 : 28 Þ^2

F 62

if we plan a balanced study with each group consisting of 31 subjects.

Example 12.18 Suppose that all specifications are the same as in Example
12.17 but we design a study in which the size of the control group is three times
the number of cases. Here we have

N¼

1

½lnðyފ^2

1

w 1

þ

1

w 0



ðz 1 aþz 1 bÞ^2

¼

1

½lnð 2 ފ^2

1

0 : 25

þ

1

0 : 75



ð 1 : 96 þ 1 : 28 Þ^2

F 84

or 21 cases and 63 controls.

SAMPLE SIZE DETERMINATION FOR CASE–CONTROL STUDIES 475