Introductory Biostatistics

the exposure histories of the case and his or her matched control are di¤erent.
We have to go through two steps, first to calculate the number of discordant
pairsm, then the number of pairsM; the total number of subjects isN¼ 2 M,
Mcases andMcontrols.
The exposure rate of the cases is first calculated using the same previous
formula:

p 1 ¼

yp 0

1 þðy 1 Þp 0

Then, given specified levels of type I and type II errorsaandb, the number of
discordant pairsmrequired to detect a relative risky, treated as an approxi-
mate odds ratio, is obtained from

m¼

½ðz 1 a= 2 Þþz 1 b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pð 1 PÞ

p

Š^2

ðP 0 : 5 Þ^2

where

P¼

y

1 þy

Finally, the total number of pairsMis given by

M¼

m

p 0 ð 1 p 1 Þþp 1 ð 1 p 0 Þ

Example 12.16 Suppose that an investigator is considering designing a case–
control study of a potential association between endometrial cancer and expo-
sure to estrogen (whether ever taken). Suppose also that the exposure rate of
controls is estimated to be about 40% and that a relative risk ofy¼4is
hypothesized. We also decide to preseta¼ 0 :05 and to design a study large
enough so that its power regarding the hypothesized relative risk above is 90%
(orb¼ 0 :10). We also plan a 1:1 matched design; matching criteria are age,
race, and county of residence.
First, we obtain the exposure rate of the cases and thezvalues using speci-
fied levels of type I and type II errors:

p 1 ¼

yp 0

1 þðy 1 Þp 0

¼

ð 4 Þð 0 : 4 Þ

1 þð 3 Þð 0 : 4 Þ

¼ 0 : 7373

472 STUDY DESIGNS