146 CHAPTER 9 Nonparametric MethodsThere is a price paid for this, however. That price is that some
information in the data is ignored. In the case of rank tests, we ignore
the numerical values of the observations and just consider how they
are ordered. So when a parametric model is consistent with the data,
the maximum likelihood estimates make effi cient use of the data and
provide a more powerful test than its nonparametric counterpart, which
cannot exploit the information in the distribution ’ s family.
9.1 RANKING DATA
We use rank tests when we want to make inferences about two or more
populations and we don ’ t have a good parametric model (that theoretical
or empirical work would suggest). Suppose, for example, that we have
samples from two populations. Our null hypothesis is that the two dis-
tributions are identical. In this case, we pool the observations and order
the pooled data from the smallest value to the largest value. This is like
temporarily forgetting the population the data points were taken from.
Under the null hypothesis, this shouldn ’ t matter, since the distributions
are the same. So the data should be well mixed (i.e., there will not be
a tendency for the sample from population 1 to have mostly high ranks
or mostly low ranks). In fact, we would expect that the average rank
for each population would be nearly the same. On the other hand, if the
populations were different, then the one with the larger median would
tend to have more of the higher ranks than the one with the lower median.
This is the motivation for the Wilcoxon rank - sum test. Comparing
the average of the ranks is very similar to comparing the sum of the
ranks (if the sample sizes are equal or nearly so). The Wilcoxon rank -
sum test (equivalent to the Mann – Whitney test) compares the sum of
the ranks from, say, population one, and compares it with the expected
value for that rank sum under the null hypothesis. This test is the topic
of the next section. It can be generalized to three or more populations.
The generalization is called the Kruskal – Wallis test.
9.2 WILCOXON RANK - SUM TEST
The Wilcoxon rank - sum test is a nonparametric analog to the unpaired
t - test. See Conover (1999) for additional information about this test.