Introductory Biostatistics

HA:p 1 <p 2 or p 10 p 2

How large a total sample should be used to conduct this vaccine study?
Suppose that it is important to detect a reduction of infection rate

d¼p 2 p 1

If we decide to preset the size of the study ata¼ 0 :05 and want the power
(1b) to detect the di¤erenced, the required sample size is given by the com-
plicated formula

N¼ 4 ðz 1 aþz 1 bÞ^2

pð 1 pÞ

ðp 2 p 1 Þ^2

or

z 1 b¼

ffiffiffiffiffi

N

p

jp 2 p 1 j

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pð 1 pÞ

p z 1 a

In this formula the quantitiesz 1 aandz 1 bare defined as in Section 12.9.1 and
pis the average proportion:

p¼

p 1 þp 2

2

It is obvious that the problem of planning sample size is more di‰cult and that
a good solution requires a deeper knowledge of the scientific problem: a good
idea of the magnitude of the proportionsp 1 andp 2 themselves.

Example 12.10 Suppose that we wish to conduct a clinical trial of a new
therapy where the rate of successes in the control group was known to be about
5%. Further, we would consider the new therapy to be superior—cost, risks,
and other factors considered—if its rate of successes is about 15%. Suppose
also that we decide to preseta¼ 0 :05 and want the power to be about 90%
(i.e.,b¼ 0 :10). In other words, we use

z 1 a¼ 1 : 96

z 1 b¼ 1 : 28

From this information, the total sample size required is

N¼ð 4 Þð 1 : 96 þ 1 : 28 Þ^2

ð 0 : 10 Þð 0 : 90 Þ

ð 0 : 15  0 : 05 Þ^2

F378 or 189 patients in each group

SAMPLE SIZE DETERMINATION FOR PHASE III TRIALS 465