16-3

Two-sample problems (see Section 7.2) are among the most common in statistics. The most useful nonparametric significance test compares two distributions. Here is an example of this setting.

Figure 16.2 is a back-to-back stemplot that compares the markup percents of the 12 Used Auto consumers and the eight Midwest Auto consumers in our sample. The Used Auto distribution has a high outlier, and the Midwest Auto distribution is strongly skewed to the right. If data show neither outliers nor skewness, then two-sample procedures are fairly robust, even for a total combined sample size of 20 as we have here. However, it is not so clear we can rely on the robustness of the two-sample test for the data of Case 16.1. We prefer to consider an alternative test that is robust to departures in Normality.

16-4

The rank transformation

We first rank all 20 observations together. To do this, arrange them in order from smallest to largest:

The boldface entries in the list are the sales at the Used Auto lot. The idea of rank tests is to look just at position in this ordered list. To do this, replace each observation by its order, from 1 (smallest) to 20 (largest). These numbers are the ranks:

Moving from the original observations to their ranks is a transformation of the data, like moving from the observations to their logarithms. The rank transformation retains only the ordering of the observations and makes no other use of their numerical values. Working with ranks allows us to dispense with specific assumptions about the shape of the distribution, such as Normality.

16-5

The Wilcoxon rank sum test

If Used Car consumers tend to have a higher markup percent relative to Midwest Car consumers, we expect the ranks of the Used Car consumers to be larger than the ranks of the Midwest consumers. Let’s compare the sums of the ranks from the two groups:

These sums measure how much the ranks of the Used Auto consumers as a group exceed those of the Midwest Auto consumers. In fact, the sum of the ranks from 1 to 20 is always equal to 210, so it is enough to report the sum for one of the two groups. If the sum of the ranks for the Used Auto consumers is 155, the ranks for the Midwest Auto consumers must add to 55 because .

Because there are more Used Auto consumers than Midwest Auto consumers, we would expect the sum of the Used Auto ranks to be greater than the sum of the Midwest Auto ranks if there are no systematic price discrimination differences. But how much greater? Here are the facts we need stated in general form.

In the used car consumer study of Case 16.1, we want to test

against the one-sided alternative

Our test statistic is the rank sum for the Used Auto consumers.

To calculate the -value, , we need to know the sampling distribution of the rank sum when the null hypothesis is true. This distribution depends on the two sample sizes and . Tables are, therefore, a bit unwieldy, though you can find them in handbooks of statistical tables. Most statistical software will give you -values, as well as carry out the ranking and calculate . However, some software gives only approximate -values. You must learn what your software offers.

16-7

The two-sample test gives a somewhat more significant result than the Wilcoxon test in Example 16.2 (, ). We hesitate to trust the test because of the skewness of one sample and the outlier in the other sample along with the fact that the sample sizes are not large.

The Normal approximation

The rank sum statistic becomes approximately Normal as the two sample sizes increase. We can then form yet another statistic by standardizing :

Use standard Normal probability calculations to find -values for this statistic. Because takes only whole-number values, the continuity correction can improve the accuracy of the approximation. The idea of continuity correction was first introduced with binomial probability calculations.

16-8

We recommend using either the exact distribution (from software or tables) or the continuity correction for the rank sum statistic W. The exact distribution is safer for small samples. As Example 16.3 illustrates, however, the Normal approximation with the continuity correction is often adequate.

What hypotheses do the Wilcoxon test?

Our null hypothesis is that the markup percents of Used Auto consumers and Midwest Auto consumers do not differ systematically. Our alternative hypothesis is that Used Auto consumers’ markup percents are higher. If we are willing to assume that markup percents are Normally distributed, or if we have reasonably large samples, we use the two-sample test for means. Our hypotheses then become

When the distributions may not be Normal, we might restate the hypotheses in terms of population medians rather than means:

The Wilcoxon rank sum test does test hypotheses about population medians, but only if an additional assumption is met: both populations must have distributions of the same shape. That is, the density curve for markup percents for Used Auto consumers must look exactly like that for markup percents for Midwest Auto consumers, except that it may slide to a different location on the scale of markup percents. The Minitab output in Figure 16.4 states the hypotheses in terms of population medians (which it denotes as ) and also gives a confidence interval for the difference between the two population medians.

16-9

The same-shape assumption is too strict to be reasonable in practice. Recall that our preferred version of the two-sample test does not require that the two populations have the same standard deviation—that is, it does not make a same-shape assumption. Fortunately, the Wilcoxon test also applies in a much more general and more useful setting. It tests hypotheses that we can state in words as

Here is a more exact statement of the systematically larger alternative hypothesis. Take to be the markup percent paid by a randomly chosen Used Auto consumer and to be the markup percent paid by a randomly chosen Midwest Auto consumer. These markup percents are random variables. That is, every time we choose a Used Auto consumer at random, the consumer’s markup percent is a value of the variable . The probability that a Used Auto consumer’s markup percent is more than 10% is . Similarly, is the corresponding probability for a randomly chosen Midwest Auto consumer. If the markup percents for Used Auto consumers are “systematically larger” than those of Midwest consumers, getting getting a markup percent greater than 10% should be more likely for Used Auto consumers. That is, we should have

The alternative hypothesis says that this inequality holds not just for 10% markup but for any markup percent.³

This exact statement of the hypotheses we are testing is a bit awkward. The hypotheses really are “nonparametric” because they do not involve any specific parameter such as the mean or median. If the two distributions do have the same shape, the general hypotheses reduce to comparing medians. Many texts and software outputs state the hypotheses in terms of medians, sometimes ignoring the same-shape requirement. We recommend that you express the hypotheses in words rather than symbols. “Used Auto consumers pay a systematically higher markup percent than Midwest Auto consumers” is easy to understand and is a good statement of the effect that the Wilcoxon test looks for.

Ties

The exact distribution for the Wilcoxon rank sum is obtained assuming that all observations in both samples take different values. In theory, with continuous distributions, the probability is 0 that we encounter observations of exactly the same value. Having different values allows us to rank them all. In practice, however, we can encounter observations tied at the same value. What shall we do? The usual practice is to assign all tied values the average of the ranks they occupy. Here is an example with six observations:

The tied observations occupy the third and fourth places in the ordered list, so they share rank 3.5.

The exact distribution for the Wilcoxon rank sum applies only to data without ties. Moreover, the standard deviation must be adjusted if ties are present. The Normal approximation can be used after the standard deviation is adjusted. Statistical software will detect ties, make the necessary adjustment, and switch to the Normal approximation. In practice, software is required if you want to use rank tests when the data contain tied values.

16-10

It is sometimes useful to use rank tests on data that have very many ties because the scale of measurement has only a few values. Here is an example.

We should first ask if the subjects in Case 16.2 are a random sample of people who attend fairs, at least in the Midwest. The researcher visited 11 different fairs. She stood near the entrance and stopped every 25th adult who passed. Because no personal choice was involved in choosing the subjects, we can reasonably treat the data as coming from a random sample. (As usual, there was some nonresponse, which could create bias.)

Here are the data (variable “sfair” in the associated data file), presented as a two-way table of counts:

Comparing row percents shows that the women in the sample are more concerned about food safety than the men:

Is the difference between the genders statistically significant?

We might apply the chi-square test (Chapter 9). It is highly significant (, , ). Although the chi-square test answers our general question, it ignores the ordering of the responses and so does not use all of the available information. We would really like to know whether men or women are more concerned about the safety of the food served. This question depends on the ordering of responses from least concerned to most concerned. We can use the Wilcoxon test for the hypotheses:

16-11

The alternative hypothesis is two-sided. Because the responses can take only five values, there are very many ties. All 35 people who chose “very rarely” are tied at 1, and all 165 who chose “once in a while” are tied at 2.

With more than 100 observations in each group and no outliers, we might use the two-sample even though responses take only five values. In fact, the results for Example 16.5 are with . The -value for two-sample is the same as that for the Wilcoxon test. There is, however, another reason to prefer the rank test in this example. The statistic treats the response values 1 through 5 as meaningful numbers. In particular, the possible responses are treated as though they are equally spaced. The difference between “very rarely” and “once in a while” is the same as the difference between “once in a while” and “often.” This may not make sense. The rank test, on the other hand, uses only the order of the responses, not their actual values. The responses are arranged in order from least to most concerned about safety, so the rank test makes sense. Some statisticians avoid using procedures when there is not a fully meaningful scale of measurement.

Rank versus tests

The two-sample procedures are the most common methods for comparing the centers of two populations based on random samples from each. The Wilcoxon rank sum test is a competing procedure that does not start from the condition that the populations have Normal distributions. Both are available in almost all statistical software. How do these two approaches compare in general?