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

CHAPTER 6 ST 520, A. TSIATIS and D. Zhang
power of a test and to compute sample sizes, we need to not only specify the clinically important
difference ∆A, but also plausible values of the nuisance parameters.
2. It is often the case that under the alternative hypothesis the standard deviation σ∗(∆A, θ)
will be equal to (or approximately equal) to one. If this is the case, then the mean under the
alternative φ(n, ∆A, θ) is referred to as the non-centrality parameter.
For example, when testing the equality in mean response between two treatments with normally
distributed continuous data, we often use the t-test
Tn =
sY
¯Y1 − ¯ Y2
� ≈ 1/2
1 + n1 n2
� 1
σY
¯Y1 − ¯ Y2
� , 1/2
1 + n1 n2
which is approximately distributed as a standard normal under the null hypothesis. Under the
alternative hypothesis HA : µ1 −µ2 = ∆ = ∆A, the distribution of Tn will also be approximately
normally distributed with mean
and variance
Hence
Thus
and
⎧
⎪⎨
Hence
⎪⎩
EHA (Tn)
⎧
⎪⎨
≈ E
⎪⎩
∆A �
1
σY + n1 1
n2 σY
¯Y1 − ¯ Y2
� 1
n1
+ 1
n2
� 1/2
⎫
⎪⎬
⎪⎭ =
σY
� 1
µ1 − µ2
�
1
varHA (Tn) = {var(¯ Y1) + var( ¯ Y2)}
σ2 � � =
1 1
Y + n1 n2
σ2 Y
σ2 Y
⎛
n1
� = 1/2
1
+ n2
� 1
n1 �
1
⎞
n1
HA ⎜ ∆A ⎟
Tn ∼ N ⎝ � � , 1 1/2 ⎠.
1 1
σY + n1 n2
φ(n, ∆A, θ) =
σY
� 1
n1
∆A
σY
� 1
n1
∆A
�
1 + n2�
= 1.
1 + n2
� , 1/2
1
+ n2
� , (6.1)
1/2
1 + n2
⎫
⎪⎬
σ∗(∆A, θ) = 1. (6.2)
�1/2 is the non-centrality parameter.
⎪⎭
Note: In actuality, the distribution of Tn is ⎧a
non-central t distribution with n1 +n2 −2 degrees
⎪⎨
∆A
of freedom and non-centrality parameter �
⎪⎩ 1
σY +
n1 1
⎫
⎪⎬
�1/2 . However, with large n this is well
⎪⎭
n2 approximated by the normal distribution given above.
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