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

CHAPTER 2 ST 520, A. TSIATIS and D. Zhang
sample space ( X = {0, 1, . . ., n} )
n
k
0
Figure 2.1: Exact confidence intervals
≤ α/2
A(π)
≤ α/2
πL(k) π πU(k)
parameter space (π)
0 1
Another way of viewing a (1 − α)-th confidence interval is to find, for each realization X = k,
all the values π ∗ for which the value k would not reject the hypothesis H0 : π = π ∗ . Therefore, a
(1 −α)-th confidence interval is sometimes more appropriately called a (1 −α)-th credible region
(interval).
If X ∼ b(n, π), then when X = k, the (1 − α)-th confidence interval is given by
C(k) = [πL(k), πU(k)],
where πL(k) denotes the lower confidence limit and πU(k) the upper confidence limit, which are
defined as
and
⎛
n� ⎜
PπL(k)(X ≥ k) = ⎝
j=k
n
⎞
⎟
⎠ πL(k)
j
j {1 − πL(k)} n−j = α/2,
⎛
k� ⎜
PπU(k)(X ≤ k) = ⎝
j=0
n
⎞
⎟
⎠ πU(k)
j
j {1 − πU(k)} n−j = α/2.
The values πL(k) and πU(k) need to be evaluated numerically as we will demonstrate shortly.
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