Introductory Biostatistics

end of the study, thebettertreatment will be the one with thelargersample
mean. This goal is achieved by imposing a condition,

Prðx 2 bx 1 jm 2 m 1 ¼dÞ¼ 1 a

For example, if we want to be 99% sure that the better treatment will be the
one with the larger sample mean, we can preseta¼ 0 :01. To do that, the total
sample size must be at least

N¼ 4 ðz 1 aÞ^2

s^2

d^2

assuming that we conduct a balanced study with each group consisting of
n¼N=2 subjects. To calculate this minimum required total sample size, we
need the variances. The exact value ofs^2 is unknown; we may depend on
prior knowledge about one of the two arms from a previous study or use some
upper bound.

12.7.2 Binary Endpoints

When the primary outcome of a trial is measured on a binary scale, the focus is
on a proportion, the response rate. At the end of the study, we select the treat-
ment or schedule with thelargersample proportion. But first we have to define
what we mean bybetter treatment. Suppose that treatment 2 is said to bebetter
than treatment 1 if

p 2 p 1 bd

wheredis the magnitude of the di¤erence betweenp 2 andp 1 that is deemed to
be important; the quantitydis often called the minimum clinical significant
di¤erence. Then we want to make therightselection by making sure that at the
end of the study, thebettertreatment will be the one with thelargersample
proportion. This goal is achieved by imposing a condition,

Prðp 2 bp 1 jp 2 p 1 ¼dÞ¼ 1 a

where thep’s are sample proportions. For example, if we want to be 99% sure
that the better treatment will be the one with the larger sample proportion, we
can preseta¼ 0 :01. To do that, the total sample size must be at least

N¼ 4 ðz 1 aÞ^2

pð 1 pÞ

ðp 2 p 1 Þ^2

assuming that we conduct a balanced study with each group consisting of

458 STUDY DESIGNS