The first step in implementing this test is to rank all the data together. For a small data set, this can be done by hand. Here’s how.

It would not be unusual in the baseball example to have sampled from a day where more than one game had the same number of hits. We will discuss how to handle ties later in this section.

Check-in

1. 15.1 Numbers of rooms in top meeting hotels. The software company Cvent ranks meeting hotels in the United States and lists the top 100, with characteristics of each hotel.¹ We let Group A be the 25 top-ranked hotels and let Group B be the hotels ranked 26 to 50. A simple random sample of 5 hotels was taken from each group, and the number of rooms in each selected hotel was recorded. Here are the data:

   Group A	1303	1522	2019	1260	1996
   Group B	1564	1036	3933	1214	1096

   Rank all the observations together and make a list of the ranks for Group A and Group B.

2. 15.2 The effect of Caesars Palace on the result. Refer to the previous Check-in question. Caesars Palace in Las Vegas, with 3933 rooms, was the third hotel selected in Group B. Suppose, instead, a different hotel, with 1600 rooms, fewer than half as many, had been selected. Replace the observation 3933 in Group B with 1600. Use the modified data to make a list of the ranks for Groups A and B combined. What changes?

If the American League games have more hits than the National League games, we’d expect the ranks of the American League games to generally be higher than those of the National League games. Let’s compare the sums of these ranks for this example.

If there were no difference between the leagues, we would expect the sum of the ranks for each league to be 18 (half of the total sum of 36). The sum of the ranks is higher for the American League, but is this difference in ranks statistically significant? Here are the facts we need to complete the test for the general setting where the two samples need not be the same size.

To calculate the P-value P(W≥21), we need to know the sampling distribution of the rank sum W when the null hypothesis is true. This distribution depends on the two sample sizes n1 and n2. Most statistical software will give you the P-values and also carry out the ranking and calculate W. However, some software gives only approximate P-values.

The rank sum statistic W becomes approximately Normal as the two sample sizes increase. Thus, by standardizing W, we have a new z statistic:

Use standard Normal probability calculations to find P-values for this statistic. Because W takes only whole-number values, the continuity correction improves the accuracy of the approximation.

We have chosen n1=n2=4 so that it would be easy to illustrate the mechanics of the Wilcoxon rank sum test. With such small sample sizes, we do not have much power to reject the null hypothesis. Therefore, we should not be very surprised with this result. Larger sample sizes may very well give different conclusions.

Example 15.7 Software output.

Figure 15.2 shows the Wilcoxon rank sum test output from JMP. The sum of the ranks for the American League is given as W=21. The value for the National League is W=15. Dividing these sums by the sample sizes, both 4 in this example, gives the means displayed in the “Score Mean” column in the output. JMP uses the continuity correction for its calculations. A z statistic is given for each league. These have the same values but with opposite signs. The P-value for the two-sided alternative is 0.4705. For the one-sided alternative that American League games have more hits than the National League games, we divide the P-value by 2 giving P=0.24, which agrees with the value that we calculated in Example 15.6.

Example 15.8 More software output.

Figure 15.3 shows the test output for the baseball data from two additional statistical programs. Minitab gives the Normal approximation P-value, and it refers to the method as the Mann-Whitney test. This is an alternative form of the Wilcoxon rank sum test. It also reports the 95% confidence interval for the difference in medians. SPSS uses the exact calculation for the P-value. Both report the significance test for the two-sided alternative only.

Our null hypothesis is that the distribution of hits is the same in the two leagues. Our alternative hypothesis is that there are more hits in the American League than in the National League. If we are willing to assume that hits are Normally distributed, or if we have reasonably large samples, we use the two-sample t 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: the two populations must have distributions of the same shape and spread. That is, the density curve for hits in the American League must look exactly like that for the National League except that it may be shifted to the left or to the right.

The same-shape assumption is too strict to be reasonable in practice. Recall that our preferred version of the two-sample t 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. Let X1 be the random variable for the first distribution and let X2 be the random variable for the second distribution. Then X1 is systematically larger than X2 if

for any value of x.

This exact statement of the hypotheses we are testing is a bit awkward.² 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 computer 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. “The number of American League hits per game is systematically higher than the number of National League hits per game” is easy to understand and is a proper statement of the effect that the Wilcoxon test examines.

The exact distribution for the Wilcoxon rank sum is obtained assuming that all observations in both samples take different values. This allows us to uniquely rank them. In practice, however, we often find 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:

The exact distribution for the Wilcoxon rank sum W changes if the data contain ties. Moreover, the standard deviation σW 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.

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.

Example 15.11 Exergaming in Canada.

Exergames are active video games such as rhythmic dancing, virtual bicycles, balance board simulators, and virtual sports simulators that require a screen and a console. A study of exergaming in students from grades 10 and 11 in Montreal, Canada, examined many factors related to participation in exergaming.³ In Exercise 14.25 (page 14-21), we used logistic regression to examine the relationship between exergaming and time spent viewing television. Here are the data displayed in a two-way table of counts:

Exergamer	TV time (hours per day)
	None	Some but less than two hours	Two hours or more
Yes	6	160	115
No	48	616	255

How do we approach the analysis of these data using the Wilcoxon test? Before answering this, let’s perform a two-way table analysis.

As with any other significance test, we start with the hypotheses. We have two distributions of TV viewing: one for the exergamers and one for those who are not exergamers. The null hypothesis states that these two distributions are the same. The alternative hypothesis uses the fact that the responses are ordered from no TV to two hours or more per day. It states that one of the groups watches more TV than the other.

The alternative hypothesis is two-sided. Because the responses can take only three values, there are very many ties. All 6+48=54 students who watch no TV are tied. Similarly, all students in each of the other two columns of the table are tied. The graphical display that you prepared in Check-in question 15.7 suggests that the exergamers watch more TV than those who are not exergamers. Is this difference statistically significant?

Example 15.12 Software output.

Look at Figure 15.4, which gives JMP output for the Wilcoxon test. The rank sum for the exergamers (using average ranks for ties) is W=187,748 (Score Sum for Yes). The expected rank sum under the null hypothesis is 168,741 (Expected Score for Yes in the output). So the exergamers have a higher rank sum than we would expect. The Normal approximation test statistic is z=4.46794, and the two-sided P-value is reported as P<0.0001. There is very strong evidence of a difference. Students who are exergamers watch more TV than the students who are not exergamers.

We can use our framework of “systematically larger” (page 15-9) to summarize these data. For the exergamers, 98% watch some TV and 41% watch two or more hours per day. The corresponding percents for the students who are not exergamers are 95% and 28%. The difference is statistically significant (z=4.468, P<0.0001.)

In our discussion of TV viewing and exergaming, we have expressed results in terms of the amount of TV watched. In fact, we do not have the actual hours of TV watched by each student in the study. Only data with the hours classified into three groups are available. Many government surveys summarize quantitative data categorized into ranges of values. When summarizing the analysis of data, it is very important to explain clearly how the data are recorded. In this setting, we have chosen to use phrases such as “watch more TV” because they express the findings based on the data available.

Note that the two-sample t test would not be appropriate in this setting. If we coded the TV-watching categories as 1, 2, and 3, the average of these coded values would not be meaningful.

On the other hand, we frequently encounter variables measured in scales such as “strongly agree,” “agree,” “neither agree nor disagree,” “disagree,” and “strongly disagree.” In these circumstances, many would code the responses with the integers 1 to 5 and then use standard methods such as a t test or ANOVA. Whether to do this or not is a matter of judgment. Rank tests avoid the dilemma because they use only the order of the responses, not their actual values. Some statisticians use t procedures when there is not a fully meaningful scale of measurement, but others avoid them.

The two-sample t procedures are the most commonly used procedures 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 sample means have distributions that are approximately Normal. Tests based on Normality and nonparametric rank procedures apply in many other settings as well. How do these approaches compare in general?

• Moving from the actual data values to their ranks allows us to find an exact sampling distribution for rank statistics such as the Wilcoxon rank sum W when the null hypothesis is true. When our samples are small, are truly random samples from the populations, and show non-Normal distributions of the same shape, the Wilcoxon test is more reliable than the two-sample t test. In other situations, the robustness of t procedures implies that we can obtain reasonably accurate P-values.
• Normal tests compare means and are accompanied by simple confidence intervals for means or differences between means. When we use rank tests to compare medians, we can also give confidence intervals for medians. However, the usefulness of rank tests is clearest in settings when they do not simply compare medians; see the discussion “What hypotheses does Wilcoxon test?” (page 15-9). Rank methods focus on significance tests, not confidence intervals.
• Inference based on ranks is largely restricted to simple settings. Normal inference extends to methods for use with complex experimental designs and multiple regression, but nonparametric rank procedures do not. We stress Normal inference in part because it leads to more advanced statistics.

In general, Normal distribution methods are more useful than rank tests. We recommend rank tests only for very small samples that are clearly non-Normal. Other alternatives include the permutation and bootstrap methods discussed in Chapter 16.