148 CHAPTER 9 Nonparametric Methodsthe Wilcoxon test cannot reject the null hypothesis that the distributions
are the same, the t - test (one sided) rejected the null hypothesis that the
means are equal. Why do we get confl icting results? First of all, we
made two very dubious assumptions when applying the t - test. They
were (1) both populations have normal distributions, and (2) the distri-
butions have the same variances. Standard tests for normality such as
Wilk – Shapiro or Anderson – Darling would reject normality, and the
control group standard deviation is about 2.5 times larger than the treat-
ment group. The F - test for equality of variances would likely reject
equality of variances.
The t - test is therefore not reliable. So it should not be a surprise
that the test could give erroneous results. Since neither assumption is
needed for a nonparametric rank test, it is more trustworthy. The fact
that the result is nonsignifi cant may just be an indication that the sample
size is too small. Recall also that the Wilcoxon test is not the most
powerful.
We shall now look at another simpler analog to the two - sample
independent t - test. It is called the sign test, and just looks at the sign
of the difference between the two means.
Table 9.2
Pig Blood Loss Data (mL)
Control group pigs (pooled ranks) Treatment group pigs (pooled ranks)
786 (9) 543 (5)
375 (1) 666 (7)
4446 (19) 455 (3)
2886 (16) 823 (11)
478 (4) 1716 (14)
587 (6) 797 (10)
434 (2) 2828 (15)
4764 (20) 1251 (13)
3281 (17) 702 (8)
3837 (18) 1078 (12)
Sample mean = 2187.40 (rank - sum = 112) Sample mean = 1085.90
(rank - sum = 98)
Sample SD = 1824.27 Sample SD = 717.12