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Suppose that instead of a CI, we had wished to test a hypothesis about yu. For
Ho: [k= lg versus Hy: A Lo, the ¢ test at level .10 specifies that Hy should be
rejected if t is either > 1.711 or < —1.711, where
x- 100 — py 100 — py
pee Ho SS Bo (14.3)
s/V¥25 20/V25 4
Consider now the null value fig = 95. Then t = 1.25, so Ho is not rejected.
Similarly, if fig = 104, then t= —1, so again Ho is not rejected. However,
if fo = 90, then t = 2.5, so Ho is rejected, and if fig = 108, then t = —2, so Ho
is again rejected. By considering other values of jg and the decision resulting
from each one, the following general fact emerges: Every number inside the
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772 = cuarrer 14 Alternative Approaches to Inference
interval (14.2) specifies a value of [io for which t of (14.3) leads to nonrejection of
Hp, whereas every number outside interval (14.2) corresponds to a t for which Hy is
rejected. That is, for the fixed values of n, X, and s, the interval (14.2) is precisely
the set of all jy values for which testing Ho: 4 = fg versus Hy: pe # Mp results in
not rejecting Ho.
PROPOSITION Suppose we have a level x test procedure for testing Ho: 0 = 09 versus
H,: 0 4 0. For fixed sample values, let A denote the set of all values
Qo for which Hp is not rejected. Then A is a 100(1 — ~)% CI for 0.
There are actually pathological examples in which the set A defined in the
proposition is not an interval of @ values, but instead the complement of an interval
or something even stranger. To be more precise, we should really replace the notion
of a CI with that of a confidence set. In the cases of interest here, the set A does
turn out to be an interval.
The Wilcoxon Signed-Rank Interval
To test Hg: 1 = Up versus H,: ft # [lg using the Wilcoxon signed-rank test, where
u is the mean of a continuous symmetric distribution, the absolute values
[x1 —Hols-++;|%n — Hol are ordered from smallest to largest, with the smallest
receiving rank 1 and the largest, rank n. Each rank is then given the sign of its
associated x; — lo, and the test statistic is the sum of the positively signed ranks.
The two-tailed test rejects Ho if s, is either > c or < n(n + 1)/2 — c, where c is
obtained from Appendix Table A.12 once the desired level of significance « is
specified. For fixed x), ...,x,, the 100(1 — «)% signed-rank interval will consist of
all fo for which Ho: 4 = [lo is not rejected at level %. To identify this interval, it is
convenient to express the test statistic S, in another form.
S., = the number of pairwise averages (X; + Xj) /2 withi <j (144)
that are > pW ,
That is, if we average each x; in the list with each 4; to its left, including (x; + ;)/2
(which is just x;), and count the number of these averages that are > Lo, 5, results.
In moving from left to right in the list of sample values, we are simply averaging
every pair of observations in the sample [again including (x; + x))/2] exactly once,
so the order in which the observations are listed before averaging is not important.
The equivalence of the two methods for computing s, is not difficult to verify. The
number of pairwise averages is (3) +n (the first term due to averaging of different
observations and the second due to averaging each x; with itself), which equals
n(n + 1)/2. If either too many or too few of these pairwise averages are > [o,
Ap is rejected.
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14.3 Distribution-Free Confidence Intervals 773
The following observations are values of cerebral metabolic rate for rhesus monkeys:
X= 4.51, x2 = 4.59, x3 = 4.90, x4 = 4.93, x5 = 6.80, x6 = 5.08, x7 = 5.67. The
28 pairwise averages are, in increasing order,
451 455 459 4.705 4.72 4.745 4.76 4.795 4.835 4.90
4.915 4.93 4.99 5.005 5.08 5.09 5.13 5.285 5.30 5.375
5.655 5.67 5.695 5.85 5.865 5.94 6.235 6.80
The first few and the last few of these are pictured on a measurement axis in Figure 14.2.
5, = 2 s,=2
s,=27 54
5 = 8B YY 355,525 Xs. =0
So rn
jee¢}—_teeees|_—-- fees} copte—o
45 146 47 48 55 5.75 16
OO
' At level .0469, Hy is '
i not rected for pip in here '
Figure 14.2 Plot of the data for Example 14.4
Because of the discreteness of the distribution of S,, «= .05 cannot be
obtained exactly. The rejection region {0, 1, 2, 26, 27, 28} has « = .046, which
is as close as possible to .05, so the level is approximately .05. Thus if the number of
pairwise averages > plo is between 3 and 25, inclusive, Ho is not rejected. From
Figure 14.2 the (approximate) 95% CI for yu is (4.59, 5.94). i |
In general, once the pairwise averages are ordered from smallest to largest,
the endpoints of the Wilcoxon interval are two of the “extreme” averages. To
express this precisely, let the smallest pairwise average be denoted by X(,), the next
smallest by Xj), ... , and the largest by X(a(n41)/2)-
PROPOSITION If the level « Wilcoxon signed-rank test for Ho: 1 = [lo versus Hy: 1 F [lp is to
reject Ho ifeithers, > cors, < n(n + 1)/2 — c,thena100(1 — %)% Cl for wis
(Rnn41)/2-c41))%(e)) (14.5)
In words, the interval extends from the dth smallest pairwise average to the dth
largest average, where d = n(n + 1)/2—c + 1. Appendix Table A.14 gives the
values of c that correspond to the usual confidence levels for n = 5,6, ... , 25.
For n = 7, an 89.1% interval (approximately 90%) is obtained by using c = 24
(Example 14.4 (since the rejection region {0, 1, 2, 3, 4, 24, 25, 26, 27, 28} has « = .109). The
continued) interval is (X2g—2441);X(24)) = (is); X(24)) = (4.72, 5.85), which extends from the
fifth smallest to the fifth largest pairwise average. a
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