Introductory Biostatistics

in the formula above is the percentile of the standard normal distribution

associated with a choice ofa; for example,z 1 a¼ 1 :96 whena¼ 0 :05 is

chosen. In the process of sample size determination, statistical tests, such

as two-samplettest, are usually planned as two-sided. However, if a one-

sided test is planned, this step is changed slightly; for example, we use

z¼ 1 :65 whena¼ 0 :05 is chosen.

2. The desired powerð 1 bÞ(or probability of committing a type II error
   b). This value is also selected by the investigator; a power of 80 or 90% is
   often used.


3. The quantity

d¼jm 2 m 1 j

which is the magnitude of the di¤erence betweenm 1 andm 2 that is deemed

to be important. The quantitydis often called the minimum clinical sig-

nificant di¤erence and its determination is a clinical decision, not a sta-

tistical decision.

4. The variances^2 of the population. This variance is the only quantity that
   is di‰cult to determine. The exact value ofs^2 is unknown; we may use
   information from similar studies or past studies or use an upper bound.
   Some investigators may even run a preliminary or pilot study to estimate
   s^2 ; but estimate from a small pilot study may only be as good as any
   guess.

Example 12.8 Suppose that a researcher is studying a drug which is used to
reduce the cholesterol level in adult males aged 30 or over and wants to test it
against a placebo in a balanced randomized study. Suppose also that it is
important that a reduction di¤erence of 5 be detected (d¼5). We decide to
preseta¼ 0 :05 and want to design a study such that its power to detect a dif-
ference between means of 5 is 95% (orb¼ 0 :05). Also, the variance of choles-
terol reduction (with placebo) is known to be abouts^2 F36.

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

b¼ 0 : 05 !z 1 b¼ 1 : 65

leading to the required total sample size:

N¼ð 4 Þð 1 : 96 þ 1 : 65 Þ^2

36

ð 5 Þ^2

F 76

Each group will have 38 subjects.

SAMPLE SIZE DETERMINATION FOR PHASE III TRIALS 463