The ranking procedure described in Exercise 35 is somewhat asymmetric, because the smallest observation receives rank 1, whereas the largest receives rank 2, and so on. Suppose both the smallest and the largest receive rank 1, the second smallest and second largest receive rank 2, and so on, and let Wʺ be the sum of the X ranks. The null distribution of Wʺ is not identical to the null distribution of W, so different tables are needed. Consider the case m = 3, n = 4. List all 35 possible orderings of the three X values among the seven observations (e.g., 1, 3, 7 or 4, 5, 6), assign ranks in the manner described, compute the value of W0 for each possibility, and then tabulate the null distribution of Wʺ. What is the P-value if wʺ = 9? This is the Ansari-Bradley test; for additional information, see the book by Hollander and Wolfe in the chapter bibliography.
In Exercise 35
Suppose we wish to test.
H0: the X and Y distributions are identical versus
Ha: the X distribution is less spread out than the Y distribution
The accompanying figure pictures X and Y distributions for which Ha is true. The Wilcoxon rank-sum test is not appropriate in this situation because when Ha is true as pictured, the Y's will tend to be at the extreme ends of the combined sample (resulting in small and large Y ranks), so the sum of X ranks will result in a W value that is neither large nor small.

Consider modifying the procedure for assigning ranks as follows: After the combined sample of m + n observations is ordered, the smallest observation is given rank 1, the largest observation is given rank 2, the second smallest is given rank 3, the second largest is given rank 4, and so on. Then if Ha is true as pictured, the X values will tend to be in the middle of the sample and thus receive large ranks. Let Wʹ denote the sum of the X ranks and consider an upper-tailed test based on this test statistic. When H0 is true, every possible set of X ranks has the same probability, so Wʹ has the same distribution as does W when H0 is true. The accompanying data refers to medial muscle thickness for arterioles from the lungs of children who died from sudden infant death syndrome (x's) and a control group of children (y's). Carry out the test of H0 versus Ha at level .05.

Transcribed Image Text:

X distribution Y distribution Ranks3 5... 6 4 i SIDS 4.04.44.84.9 Control3.74.1 4.3 5. 5.6