15.2 The Wilcoxon Signed Rank Test

Data set icon for Vtm.

We use the one-sample t procedures for inference about the mean of one population or for inference about the mean difference in a matched pairs setting. The matched pairs setting is more important because good studies are generally comparative. We previously discussed the sign test for this setting. We now meet a nonparametric procedure that uses ranks.

Example 15.13 Storytelling and reading.

Data set icon for story.

Refer to the study of early childhood education that asked kindergarten students to retell two fairy tales that had been read to them earlier in the week (see Exercise 15.2, page 15-14. The first (Story 1) had been read to them, and the second (Story 2) had been read and also illustrated with pictures. An expert listened to recordings of the children retelling each story and assigned a score for certain uses of language. Higher scores are better. Here are the data for the five kindergarten students in the low-progress group:8

Child 1 2 3 4 5
Story 2 0.77 0.49 0.66 0.28 0.38
Story 1 0.40 0.72 0.00 0.36 0.55
Difference 0.37 -0.23 0.66 -0.08 -0.17

Do illustrations improve how the children retell a story?

Because this is a matched pairs design, we base our inference on the differences. The matched pairs t test gives t=0.635 with one-sided P-value equal to 0.280. Displays of the data (Figure 15.6) suggest some lack of Normality. Therefore, we prefer to use a rank test.

A normal quantile plot and histogram.

Figure 15.6 Normal quantile plot and histogram for the five differences in story scores, Example 15.13.

Example 15.14 Ranks for storytelling and reading.

Data set icon for story.

Positive differences in Example 15.3 indicate that the child performed better telling Story 2. For the sign test, our test statistic was the number of positive differences. This ignored the size of the difference. We now use a test that also takes into account the size or magnitude. If scores are generally higher with illustrations, the positive differences should be farther from zero in the positive direction than the negative differences are in the negative direction. We, therefore, compare the absolute values of the differences—that is, their magnitudes without a sign. Here they are, with boldface indicating the positive values:

0.37 0.23 0.66 0.08 0.17

Arrange these in increasing order and assign ranks, keeping track of which values were originally positive. Tied values receive the average of their ranks. If there are cases with zero differences, discard them before ranking.

Absolute value 0.08 0.17 0.23 0.37 0.66
Rank 1 2 3 4 5

The ranks of the absolute values of the differences that are positive are also highlighted in bold face.

Now that we have ranks for both the positive and negative differences, we proceed in a manner similar to the Wilcoxon rank sum test. Here are the details of this test procedure.

Example 15.15 Wilcoxon signed rank statistic for storytelling and reading.

Data set icon for story.

Here are the ranks for the absolute values of the differences in Example 15.14:

Absolute value 0.08 0.17 0.23 0.37 0.66
Rank 1 2 3 4 5

The Wilcoxon signed rank statistic is the sum of the ranks of the positive differences highlighted in bold. Its value here is W+=4+5=9.

In this example, we have chosen to use the positive differences when summing the ranks. We could equally well use the sum of the ranks of the negative differences. In that case, W+=1+2+3=6. This is similar to the Wilcoxon rank sum test, and these two test statistics should sum to total sum of the ranks.

To determine the P-value, we need to know the null and alternative hypotheses.

Example 15.16 Storytelling and reading hypotheses.

Our question of interest is whether illustrations improve how children retell a story. Thus, we would like to test the null hypothesis

H0: Scores have the same distribution for both stories.

versus the alternative

Ha: Scores are systematically higher for Story 2.

The alternative hypothesis is a one-sided alternative, so now we can proceed to computing the P-value. In general, we will use software to either compute the exact P-value or approximate it using the Normal approximation.

Example 15.17 Software output.

Data set icon for story.

In the storytelling study of Example 15.13, our observed value W+=9 is only slightly larger than this mean. The one-sided P-value is P(W+≥9).

Most statistical software uses the differences between the two variables, with the signs, as input. Alternatively, the differences can sometimes be calculated within the software. Figure 15.7 displays the output from three statistical programs. Each does things a little differently. The JMP output gives the one-sided (Prob>t in the “Signed-Rank” column) P=0.4063. The Minitab output gives P=0.394 for the one-sided Wilcoxon signed rank test with n=5 observations and W+=9.0. The SPSS output gives P=0.686 for testing the two-sided alternative. We divide this by 2(0.686/2=0.343) to obtain the P-value for the one-sided alternative.

Results reported in the three outputs lead us to the same qualitative conclusion: the data do not provide evidence to support the idea that the Story 2 scores are higher than (or not equal to) the Story 1 scores. Different methods and approximations are used to compute the P-values. With larger sample sizes, we would not expect so much variation in the P-values. Note that the t test results reported by JMP also gives the same conclusion, P=0.2899.

JMP, Minitab, and SPSS outputs of test data.

Figure 15.7 JMP, Minitab, and SPSS outputs for the storytelling data, Example 15.17.

When the sampling distribution of a test statistic is symmetric, we can use output that gives a P-value for a two-sided alternative to compute a P-value for a one-sided alternative. Check that the effect is in the direction specified by the one-sided alternative and then divide the P-value by 2.

Check-in

  1. 15.8 The effect of altering a software parameter. Example 7.7 (page 394) describes a study in which researchers studied sensor software used in the measurement of complex machine parts. They were interested in the possibility of improving productivity by unchecking one particular software option. They measured 76 parts both with and without the option. Use the data to investigate the effect of the option. Formulate this question in terms of null and alternative hypotheses. Then compute the differences and find the value of the Wilcoxon signed rank statistic W+. Data set icon for geparts.

  2. 15.9 Cold plasma technology. Check-in question 7.10 (page 397) discusses a study related to the use of cold plasma (CP) technology for extending the shelf-life of food. For it to become widely used, however, CP’s effects on food quality must be understood. Consider the following study, which compared the taste of fruit juice with and without CP treatment. Each juice was rated by a set of taste experts, using a 0 to 100 scale, with 100 being the highest rating. For each expert, a coin was tossed to see which juice was tasted first.

    Expert
    1 2 3 4 5 6 7 8 9 10
    With CP: 93 65 37 78 89 49 88 55 63 62
    Without CP: 92 67 47 79 84 52 80 52 67 69

    Use the the Wilcoxon signed rank test to examine whether the CP technology changes the taste of juice. Data set icon for juice.

The Normal approximation

The distribution of the signed rank statistic when the null hypothesis (no difference) is true becomes approximately Normal as the sample size becomes large. We can then use Normal probability calculations (with the continuity correction) to obtain approximate P-values for W+. Let’s see how this works in the storytelling example, even though n=5 is certainly not a large sample.

Example 15.18 The Normal approximation for storytelling and reading.

We first find the mean and the standard deviation for the distribution of W+ under the null hypothesis. The mean is

μW+=n(n+1)4=5(6)4=7.5

and the standard deviation is

σW+=n(n+1)(2n+1)24=(5)(6)(11)24=13.75=3.708

Using the continuity correction, we calculate the P-value P(W+≥9) as P(W+≥8.5), treating the value W+=9 as occupying the interval from 8.5 to 9.5. We find the Normal approximation for the P-value by standardizing and using the standard Normal table:

P(W+≥8.5)=P(W+-7.53.708≥8.5-7.53.708)=P(Z≥0.27)=0.394

Despite the small sample size, the Normal approximation gives a result quite close to the exact value P=0.406.

Check-in

  1. 15.10 Significance test for altering a software parameter. Refer to Check-in question 15.8 (page 15-19). Find μW+, σW+, and the Normal approximation for the P-value for the Wilcoxon signed rank test. Data set icon for geparts.

  2. 15.11 Significance test for the plasma technology. Refer to Check-in question 15.9 (page 15-19). Find μW+, σW+, and the Normal approximation for the P-value for the Wilcoxon signed rank test. Data set icon for juice.

Ties

Ties among the absolute differences are handled by assigning average ranks. A tie within a pair gives a difference of zero. The usual procedure is to simply drop such pairs from the sample. A consequence of this convention is to drop observations that favor the null hypothesis (no difference). caution If there are many ties, the test may be biased in favor of the alternative hypothesis.

As in the case of the Wilcoxon rank sum procedure, ties between nonzero absolute differences complicate the process of finding a P-value. The standard deviation σW+ must be adjusted for the ties before we can use the Normal approximation. Software will do this. Here is an example.

Example 15.19 Golf scores of a women’s golf team.

Data set icon for golf.

Here are the golf scores of 12 members of a college women’s golf team in two rounds of tournament play. A golf score is the number of strokes required to complete the course, so low scores are better.

Player 1 2 3 4 5 6 7 8 9 10 11 12
Round 2 94 85 89 89 81 76 107 89 87 91 88 80
Round 1 89 90 87 95 86 81 102 105 83 88 91 79
Difference 5 -5 2 -6 -5 -5 5 -16 4 3 -3 1

Because we subtract Round 1 scores from Round 2 scores, negative differences indicate better (lower) scores on the second round. We see that 6 of the 12 golfers improved their scores. We would like to test the hypotheses that in a large population of collegiate women golfers

H0: Scores have the same distribution in Rounds 1 and 2.
Ha: Scores are systematically lower or higher in Round 2.

A Normal quantile plot of the differences (Figure 15.8) shows some irregularity and a low outlier. We will use the Wilcoxon signed rank test. The absolute values of the differences, with boldface indicating those that are negative, are

5 5 2 6 5 5 5 16 4 3 3 1

Arrange these in increasing order and assign ranks, keeping track of which values were originally negative. Tied values receive the average of their ranks.

Absolute value 1 2 3 3 4 5 5 5 5 5 6 16
Rank 1 2 3.5 3.5 5 8 8 8 8 8 11 12

The Wilcoxon signed rank statistic is the sum of the ranks of the negative differences. (We could equally well use the sum of the ranks of the positive differences.) Its value is W+=50.5.

A normal quantile plot of difference in golf score versus normal score.

Figure 15.8 Normal quantile plot of the difference in scores for two rounds of a golf tournament, Example 15.19.

Example 15.20 Software output for P-values.

Data set icon for golf.

Here are the two-sided P-values for the Wilcoxon signed rank test for the golf score data from three statistical programs:

Program P-value
JMP P=0.388
Minitab P=0.388
SPSS P=0.363

All lead to the same practical conclusion: these data give no evidence for a systematic change in scores between rounds. However, the P-value reported by SPSS differs a bit from the other two. The reason for the variation is that the programs use slightly different versions of the approximate calculations needed when ties are present. The reported P-value depends on which version is used.

For the golf data, the matched pairs t test gives t=−0.9314 with P=0.3716. Once again, t and W+ lead to the same conclusion.

Testing a hypothesis about the median of a distribution

Let’s take another look at how the Wilcoxon signed rank test works. The analysis starts by computing the differences between the two variables. At this stage, we can think of our data as consisting of a single variable. Under the null hypothesis, the population median of the differences is zero. The alternative is that the median is not zero. Now think about starting the analysis at the stage with just a single variable and keep in mind that we are interested in testing a hypothesis about the median of the variable’s distribution.

The value specified by the null hypothesis does not need to be zero. If you can’t specify a value other than zero with your software, you can simply subtract that value from each observation before you start the analysis. Exercise 15.14 is an example.

Section 15.2 SUMMARY

  • The Wilcoxon signed rank test applies to matched pairs studies. It tests the null hypothesis that there is no systematic difference within pairs against alternatives that assert a systematic difference (either one-sided or two-sided).

  • The test is based on the Wilcoxon signed rank statistic W+, which is the sum of the ranks of the positive (or negative) differences when we rank the absolute values of the differences. The matched pairs t test and the sign test are alternative tests in this setting.

  • P-values for the signed rank test are based on the sampling distribution of W+ when the null hypothesis is true. You can find P-values from special tables, software, or a Normal approximation (with continuity correction).

Section 15.2 EXERCISES

  1. 15.13 Fuel efficiency. Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the driver recorded the vehicle’s mpg each time the gas tank was filled and then reset the computer. In addition to the computer calculating mpg, the driver also recorded the mpg by dividing the miles driven by the number of gallons at fill-up.9 The driver wants to determine if these calculations are different. Data set icon for mpg7.

    Fill-up 1 2 3 4 5 6 7
    Computer 42.2 43.2 44.6 48.4 46.4 46.8 39.2
    Driver 39.2 38.8 44.5 45.4 45.3 45.7 34.2

    1. For each of the eight fill-ups, find the difference between the computer mpg and the driver mpg.

    2. Find the absolute values of the differences you found in part (a).

    3. Order the absolute values of the differences that you found in part (b) from smallest to largest and underline those absolute differences that came from positive differences in part (a).

  2. 15.14 Comparison of two energy drinks. Consider the following study comparing two popular energy drinks. For each subject, a coin was flipped to determine which drink to rate first. Each drink was rated on a 0 to 100 scale, with 100 being the highest rating. Data set icon for enerdr6.

    Drink Subject
    1 2 3 4 5 6
    A 43 83 66 87 78 67
    B 45 78 64 79 71 62

    1. Inspect the data. Is there a tendency for these subjects to prefer one of the two energy drinks?

    2. Use the matched pairs t test from Chapter 7 (page 393) to compare the two drinks.

    3. Use the Wilcoxon signed rank test to compare the two drinks.

    4. Write a summary of your results and explain why the two tests give different conclusions.

  3. 15.15 Find the Wilcoxon signed rank statistic. Using the work that you performed in Exercise 15.13, find the value of the Wilcoxon signed rank statistic W+. Data set icon for mpg7.

  4. 15.16 Number of friends on Facebook. A study examined all active Facebook users (more than 10% of the global population) and determined that the average user has 190 friends. This distribution takes only integer values, so it is certainly not Normal. It is also highly skewed to the right, with a median of 100 friends.10 Consider the following SRS of n=30 Facebook users from your large university: Data set icon for facefr.

    594  60 417 120 132 176 516 319 734   8
     31 325  52  63 537  27 368  11  12 190
     85 165 288  65  57  81 257  24 297 148

    1. Use the Wilcoxon signed rank test to test the null hypothesis that the median number of Facebook friends for Facebook users at your university is 190. Describe the steps in the procedure and summarize the results.

    2. Analyze these data using the matched pairs t test of Chapter 7 (page 393) and compare the results with those that you found in part (a).

  5. 15.17 State the hypotheses. Refer to Exercise 15.13. State the null hypothesis and the alternative hypothesis for this setting.

  6. 15.18 Comparison of two energy drinks with an additional subject. Refer to Exercise 15.14. Let’s suppose that there is an additional subject who expresses a strong preference for energy drink B. Here is the new data set: Data set icon for enerdr7.

    Drink Subject
    1 2 3 4 5 6 7
    A 43 83 66 87 78 67 90
    B 45 78 64 79 71 62 60

    Answer the questions given in Exercise 15.14. Write a summary comparing this exercise with Exercise 15.14. Include a discussion of what you have learned regarding the choice of the t test versus the Wilcoxon signed rank test for different sets of data.

  7. 15.19 Find the mean and the standard deviation. Refer to the odd-numbered exercises from Exercise 15.13 through Exercise 15.17. Use the sample size to find the mean and the standard deviation of the sampling distribution of the Wilcoxon signed rank statistic W+ under the null hypothesis.

  8. 15.20 A summer language institute for teachers. A matched pairs study of the effect of a summer language institute on the ability of teachers to comprehend spoken French had these improvements in scores between the pretest and the posttest for 20 teachers: Data set icon for sumlang.

    2 0 6 6 3 3 2 3 −6 6
    6 6 3 0 1 1 0 2 3 3

    1. Show how you rank these data.

    2. Calculate the signed rank statistic W+. Be sure to show your work. Remember that zeros are dropped from the data before ranking, so that n is the number of nonzero differences within pairs.

    3. Perform the significance test and write a short summary of your conclusions.

  9. 15.21 Find and interpret the P-value. Refer to the odd-numbered exercises from Exercise 15.13 through Exercise 15.19. Find the P-value for the Wilcoxon signed rank statistic using the Normal approximation with the continuity correction. Data set icon for mpg7.

  10. 15.22 Read the output. The data in Exercise 15.13 are a subset of a larger set of data. Figure 15.9 gives Minitab output for the analysis of this larger set of data. Data set icon for mpgcomp.

    1. How many pairs of observations are in the larger data set?

    2. What is the value of the Wilcoxon signed rank statistic W+?

    3. Report the P-value for the significance test and give a brief statement of your conclusion.

    4. The output reports an estimated median. Explain how this statistic is calculated from the data.

    A Minitab output of test data.

    Figure 15.9 Minitab output for the fuel efficiency data, Exercise 15.22.

  11. 15.23 The full moon and behavior. Can the full moon influence behavior? A study observed 15 nursing-home patients with dementia. The number of incidents of aggressive behavior was recorded each day for 12 weeks. Here we call a day a “moon day” if it is the day of a full moon or the day before or after a full moon. Here are the average numbers of aggressive incidents for moon days and other days for each subject:11 Data set icon for moon.

    Patient Moon days Other days
    1 3.33 0.27
    2 3.67 0.59
    3 2.67 0.32
    4 3.33 0.19
    5 3.33 1.26
    6 3.67 0.11
    7 4.67 0.30
    8 2.67 0.40
    9 6.00 1.59
    10 4.33 0.60
    11 3.33 0.65
    12 0.67 0.69
    13 1.33 1.26
    14 0.33 0.23
    15 2.00 0.38

    The matched pairs t test gives P<0.000015, and a permutation test (Chapter 16) gives P=0.0001. Does the Wilcoxon signed rank test, based on ranks rather than means, agree that there is strong evidence that there are more aggressive incidents on moon days?

  12. NAEP 15.24 Radon detectors. How accurate are radon detectors of a type sold to homeowners? To answer this question, university researchers placed 12 detectors in a chamber that exposed them to 105 picocuries per liter (pCi/l) of radon.12 The detector readings are as follows: Data set icon for radon.

     91.9  97.8 111.4 122.3 105.4  95.0
    103.8  99.6  96.6 119.3 104.8 101.7

    We wonder if the median reading differs significantly from the true value 105.

    1. Graph the data and comment on skewness and outliers. A rank test is appropriate.

    2. We would like to test hypotheses about the median reading from home radon detectors:

      H0: median=105Ha: median≠105

      To do this, apply the Wilcoxon signed rank statistic to the differences between the observations and 105. (This is the one-sample version of the test.) What do you conclude?

  13. 15.25 Vitamin C in wheat-soy blend. The U.S. Agency for International Development provides large quantities of wheat-soy blend (WSB) for development programs and emergency relief in countries throughout the world. One study collected data on the vitamin C content of five bags of WSB at the factory and five months later in Haiti.13 Here are the data: Data set icon for wsbvitc.

    Sample 1 2 3 4 5
    Before 73 79 86 88 78
    After 20 27 29 36 17

We want to know if vitamin C has been lost during transportation and storage. Describe what the data show about this question. Then use the Wilcoxon signed rank test to see whether there has been a significant loss.