ST 520 Statistical Principles of Clinical Trials - NCSU Statistics ...

CHAPTER 6 ST 520, A. TSIATIS and D. Zhang
A clinical trial is conducted where n1 patients are assigned treatment 1 and n2 patients are
assigned treatment 2 and the number of patients who respond to treatments 1 and 2 are denoted
by X1 and X2 respectively. As usual, we assume
and
X1 ∼ b(n1, π1)
X2 ∼ b(n2, π2),
and that X1 and X2 are statistically independent. If we let π1 = π2 +∆, then the distribution of
X1 and X2 is characterized by the test parameter ∆ and the nuisance parameter π2. If we denote
the sample proportions by p1 = X1/n1 and p2 = X2/n2, then we know from the properties of a
binomial distribution that
This motivates the test statistic
E(p1) = π1, var(p1) = π1(1 − π1)
,
n1
E(p2) = π2, var(p2) = π2(1 − π2)
.
Tn =
p1 − p2
�
¯p(1 − ¯p) � 1 1 + n1 n2
n2
�� 1/2 ,
where ¯p is the combined sample proportion for both treatments; i.e. ¯p = (X1 + X2)/(n1 + n2).
Note: The statistic T 2 n
We can also write
is the usual chi-square test used to test equality of proportions.
¯p = p1n1 + p2n2
n1 + n2
� � � �
n1 n2
= p1 + p2 .
n1 + n2 n1 + n2
As such, ¯p is an approximation (consistent estimator) for
� � � �
n1 n2
π1 + π2 = ¯π,
n1 + n2 n1 + n2
where ¯π is a weighted average of π1 and π2. Thus
Tn ≈
p1 − p2
�
¯π(1 − ¯π) � 1 1 + n1 n2
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�� 1/2 .