Introductory Biostatistics

di¤erent way, besides the level of significanceaand the desired powerð 1 bÞ,
the required total sample size depends on the ratiod=s. You will see this simi-
larity in the design of a case–control study with a continuous risk factor. When
the risk factor under investigation in a case–control study is measured on a
continuous scale, the data are analyzed using the method of logistic regression
(Chapter 9). However, as pointed out in Section 9.3, when the cases and con-
trols are assumed to have the same variances^2 , the logistic regression model
can be written as

logit¼ln

px

1 px

¼constantþ

m 1 m 0

s^2

x

Under this model, the log of the odds ratio associated with a 1-unit increase in
the value of the risk factor is

b 1 ¼

m 1 m 0

s^2

Therefore, the log of the odds ratio associated with a 1-standard deviation
increase in the value of the risk factor is

b 1 ¼

m 1 m 0

s

which is the same as the ratiod=sabove.
In the design of case–control studies with a continuous risk factor, the key
parameter is the log of the odds ratioyassociated with a 1-standard deviation
change in the value of the covariate. Consider a level of significanceaand sta-
tistical power 1b.

1. If we plan a balanced study with each group consisting ofn¼N=2 sub-
   jects, the total sample sizeNis given by

N¼

4

ðlnyÞ^2

ðz 1 aþz 1 bÞ^2

2. If we allocate di¤erent sizes to the cases and the controls,

n 1 ¼w 1 N

n 0 ¼w 0 N

w 1 þw 0 ¼ 1 : 0

474 STUDY DESIGNS