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

CHAPTER 4 ST 520, A. TSIATIS and D. Zhang
which implies that the estimator is unbiased. The variance of the estimator is given by
Simple randomization
var( ¯ YA − ¯ YB) = var(¯ǫA) + var(¯ǫB) = σ 2
�
2
n
�
2
+
n
4σ2 . (4.3)
n
With simple randomization the counts nA1 conditional on nA and nB1 conditional on nB follow
a binomial distribution. Specifically,
and
nA1|nA, nB ∼ b(nA, θ) (4.4)
nB1|nA, nB ∼ b(nB, θ), (4.5)
where θ denotes the proportion of the population in stratum 1. In addition, conditional on
nA, nB, the binomial variables nA1 and nB1 are independent of each other.
The estimator given by (4.2) has expectation equal to
Because of (4.4)
Similarly
Hence,
E( ¯ YA − ¯ �
YB) = β + α E
E
� �
nA1
nA
= E
� �
nA1
nA
− E
� ��
nB1
nB
� � ��
nA1
E |nA = E
nA
E
� �
nB1
nB
= θ.
E( ¯ YA − ¯ YB) = β;
+ E(¯ǫA − ¯ǫB). (4.6)
� �
nAθ
= θ.
that is, with simple randomization, the estimator ¯ YA − ¯ YB is an unbiased estimator of the
treatment difference β.
In computing the variance, we use the formula for iterated conditional variance; namely
var( ¯ YA − ¯ YB) = E{var( ¯ YA − ¯ YB|nA, nB)} + var{E( ¯ YA − ¯ YB|nA, nB)}.
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nA