Introductory Biostatistics

ingful maximum tolerated error for estimation of the coe‰cient of correlation.
Instead of controlling the width of a desired confidence interval, the sample size
determination ia approached in terms of controlling the risk of making a type
II error. The decision is concerned with testing a null hypothesis,

H 0 :r¼ 0

against an alternative hypothesis,

HA:r¼rA

in whichrAis the investigator’s hypothesized value for the coe‰cient of corre-
lationrof interest. With a given level of significancea(usually, 0.05) and a
desired statistical power (1b;bis the size of type II error associated with the
alternativeHA), the required sample size is given by

ffiffiffiffiffiffiffiffiffiffiffi

n 3

p

FðrAÞ¼z 1 aþz 1 b

where

FðrÞ¼

1

2

ln

1 þr

1 r

The transformation fromrtoFðrÞis often referred to asFisher’s transforma-
tion, the same transformation used in forming confidence intervals in Chapter

4. Obviously, to detect a true correlationrAgreater than 0.5, a small sample
   size would su‰ce, which is suitable in the context of phase II trials.

Example 12.3 Suppose that we decide to preseta¼ 0 :05. To design a study
such that its power to detect a true correlationrA¼ 0 :6 is 90% (orb¼ 0 :10),
we would need only

FðrAÞ¼

1

2

ln

1 þrA

1 rA

¼

1

2

ln

1 : 6

0 : 4

¼ 0 : 693

ffiffiffiffiffiffiffiffiffiffiffi

n 3

p

FðrAÞ¼z 1 aþz 1 b

ffiffiffiffiffiffiffiffiffiffiffi

n 3

p

ð 0 : 693 Þ¼ 1 : 96 þ 1 : 28

orn¼25 subjects.

SAMPLE SIZES FOR OTHER PHASE II TRIALS 455