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774 ~~ cuarrer 14 Alternative Approaches to Inference
The derivation of the interval depended on having a single sample from a continuous
symmetric distribution with mean (median) jz. When the data is paired, the interval
constructed from the differences d,, ds, ... , d,, is a CI for the mean (median)
difference ftp. In this case, the symmetry of X and Y distributions need not be
assumed; as long as the X and Y distributions have the same shape, the X — Y
distribution will be symmetric, so only continuity is required.
For n > 20, the large-sample approximation (Exercise 6) to the Wilcoxon
test based on standardizing S, gives an approximation to c in (14.5). The result
[for a 100(1 — «)% interval] is
_ n(n+ 1) n(n + 1)(2n + 1)
ce a. + 24/2 ny a
The efficiency of the Wilcoxon interval relative to the r interval is roughly the
same as that for the Wilcoxon test relative to the ¢ test. In particular, for large
samples when the underlying population is normal, the Wilcoxon interval will tend
to be slightly longer than the ft interval, but if the population is quite nonnormal
(symmetric but with heavy tails), then the Wilcoxon interval will tend to be
much shorter than the ¢ interval. And as we emphasized earlier in our discussion
of bootstrapping, in the presence of nonnormality the actual confidence level of
the ¢ interval may differ considerably from the nominal (e.g., 95%) level.
The Wilcoxon Rank-Sum Interval
The Wilcoxon rank-sum test for testing Ho : {; — fl) = Ao is carried out by first
combining the (X; — Ao)’s and Y;’s into one sample of size m + n and ranking them
from smallest (rank 1) to largest (rank m + n). The test statistic W is then the sum of
the ranks of the (X; — Ao)’s. For the two-sided alternative, Ho is rejected if w is
either too small or too large.
To obtain the associated CI for fixed x;’s and y,’s, we must determine the set
of all Ap values for which Hp is not rejected. This is easiest to do if we first express
the test statistic in a slightly different form. The smallest possible value of W is
m(m + 1)/2, corresponding to every (X; — Ao) less than every Y;, and there are mn
differences of the form (X; — Ao) — Y;. A bit of manipulation gives
W = |number of (X; — Y; — Ao)’s > 0] ant)
2 (14.6)
m(m + 1)
= (number of (X; — Y;)’s > Ao] + a
Thus rejecting Ho if the number of (x; — y,)’s > Ao is either too small or too large
is equivalent to rejecting Hy for small or large w.
Expression (14.6) suggests that we compute x; — yj for each i and j and order
these mn differences from smallest to largest. Then if the null value Ao is neither
smaller than most of the differences nor larger than most, Ho: (4; — fy = Ao is not
rejected. Varying Aj now shows that a CI for jt; — fy will have as its lower
endpoint one of the ordered (x; — y;)’s, and similarly for the upper endpoint.
--- Trang 788 ---
14.3 Distribution-Free Confidence Intervals 775
PROPOSITION Letxy,...,X, and yj, ..., y, be the observed values in two independent samples
from continuous distributions that differ only in location (and not in shape).
With dj; = x; — y;and the ordered differences denoted by dijc1), dia), « -- » diftmnys
the general form of a 100(1 — «)% CI for jt; — po is
(Aij(nne1) die)) (14.7)
where c is the critical constant for the two-tailed level x Wilcoxon rank-sum test.
Notice that the form of the Wilcoxon rank-sum interval (14.7) is very similar to the
Wilcoxon signed-rank interval (14.5); (14.5) uses pairwise averages from a single
sample, whereas (14.7) uses pairwise differences from two samples. Appendix
Table A.15 gives values of ¢ for selected values of m and n.
The article “Some Mechanical Properties of Impregnated Bark Board” (Forest
Products J., 1977: 31-38) reports the following data on maximum crushing strength
(psi) for a sample of epoxy-impregnated bark board and for a sample of bark board
impregnated with another polymer:
Epoxy (x’s) 10,860 11,120 11,340 12,130 14,380 13,070
Other (y’s) 4,590 4,850 6,510 5,640 6,390
Obtain a 95% CI for the true average difference in crushing strength between the
epoxy-impregnated board and the other type of board.
From Appendix Table A.15, since the smaller sample size is 5 and the larger
sample size is 6, c = 26 for a confidence level of approximately 95%. The d,;’s
appear in Table 14.4. The five smallest dj;’s [dijay, - -- , dics] are 4350, 4470, 4610,
4730, and 4830; and the five largest dj;’s are (in descending order) 9790, 9530,
8740, 8480, and 8220. Thus the CT is (dics), dijc26)) = (4830, 8220).
Table 14.4 Differences (dj) for the rank-sum interval in Example 14.6
yi
4590 4850 5640 6390 6510
10,860 6270 6010 5220 4470 4350
11,120 6530 6270 5480 4730 4610
x 11,340 6750 6490 5700 4950 4830
12,130 7540 7280 6490 5740 5620
13,070 8480 8220 7430 6680 6560
14,380 9790 9530 8740 7990 7870
a
When m and v are both large, the Wilcoxon test statistic has approximately a