Kruskal-Wallis Test and Runs Test
In previous Concepts, we learned how to conduct nonparametric tests, including the sign test, the sign rank test, the rank sum test, and the rank correlation test. These tests allowed us to test hypotheses using data that did not meet the assumptions of being normally distributed or having homogeneity with respect to variance. In addition, each of these non-parametric tests had parametric counterparts.
In this Concept, we will examine another nonparametric test-the Kruskal-Wallis one-way analysis of variance (also known simply as the Kruskal-Wallis test). This test is similar to the ANOVA test, and the calculation of the test statistic is similar to that of the rank sum test. In addition, we will also explore something known as the runs test, which can be used to help decide if sequences observed within a data set are random.
Evaluating Hypotheses Using the Kruskal-Wallis Test
The Kruskal-Wallis test is the analog of the one-way ANOVA and is used when our data set does not meet the assumptions of normality or homogeneity of variance. However, this test has its own requirements: it is essential that the data set has identically shaped and scaled distributions for each group.
As we learned in Chapter 11, when performing the one-way ANOVA test, we establish the null hypothesis that there is no difference between the means of the populations from which our samples were selected. However, we express the null hypothesis in more general terms when using the Kruskal-Wallis test. In this test, we state that there is no difference in the distributions of scores of the populations. Another way of stating this null hypothesis is that the average of the ranks of the random samples is expected to be the same.
Like most nonparametric tests, the Kruskal-Wallis test relies on the use of ranked data to calculate a test statistic. In this test, the measurement observations from all the samples are converted to their ranks in the overall data set. The smallest observation is assigned a rank of 1, the next smallest is assigned a rank of 2, and so on. Similar to this procedure in the rank sum test, if two observations have the same value, we assign both of them the same rank.
It is easy to use Microsoft Excel or a statistical programming package, such as SAS or SPSS, to calculate this test statistic and evaluate our hypothesis. However, for the purposes of this example, we will perform this test by hand.
Performing the Kruskal-Wallis Test
Suppose that a principal is interested in the differences among final exam scores from Mr. Red, Ms. White, and Mrs. Blue’s algebra classes. The principal takes random samples of students from each of these classes and records their final exam scores as shown:
|Mr. Red||Ms. White||Mrs. Blue|
Determine if there is a difference between the final exam scores of the three teachers.
To test this hypothesis, we need to calculate our test statistic. To calculate this statistic, it is necessary to assign and sum the ranks for each of the scores in the table above as follows:
|Mr. Red||Overall Rank||Ms. White||Overall Rank||Mrs. Blue||Overall Rank|
Using this information, we can calculate our test statistic as shown:
Determining the Randomness of a Sample Using the Runs Test
We often use the runs test in studies where measurements are made according to a ranking in either time or space. In these types of scenarios, one of the questions we are trying to answer is whether or not the average value of the measurement is different at different points in the sequence. For example, suppose that we are conducting a longitudinal study on the number of referrals that different teachers give throughout the year. After several months, we notice that the number of referrals appears to increase around the time that standardized tests are given. We could formally test this observation using the runs test.
Using the laws of probability, it is possible to estimate the number of runs that one would expect by chance, given the proportion of the population in each of the categories and the sample size. Since we are dealing with proportions and probabilities between discrete variables, we consider the binomial distribution as the foundation of this test. When conducting a runs test, we establish the null hypothesis that the data samples are independent of one another and are random. On the contrary, our alternative hypothesis states that the data samples are not random and/or not independent of one another.
Performing the Runs Test
1. In the following sequence, we would have 5 runs. We could also say that the sequence of the data switched five times.
With this information, we are able to calculate our test statistic using the following formulas:
2. A teacher is interested in assessing if the seating arrangement of males and females in his classroom is random. He observes the seating pattern of his students and records the following sequence:
To calculate the test statistic, we first record the number of runs and the number of each type of observation as shown:
With these data, we can easily compute the test statistic as follows:
Determine whether the following sequence of binary numbers is random:
1 0 0 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 0 0
There are eight runs, with a total of twelve 1's and and 8 0's.
With these data, we can easily compute the test statistic as follows:
- Suppose scores for 17 students from 3 schools in intermural competitions are as given below. Use the Kruskal-Wallis Test to test at the 5% level whether average scores for students from the three schools are the same.
School A: 29, 23, 33, 25, 20, 19
School B: 24, 31, 19, 26, 16, 18
School C: 26, 14, 13, 16, 30
- An investigator randomly sorts 21 wine aficionados into three groups, A, B and C. Each subject is interviewed and are asked to rank the overall quality of each of three wines on a 10-point scale, with 1 at the bottom of the scale and ten at the top. The three wines are the same for all subjects. What changes is the way in which the interview is conducted. The interview is designed to encourage a high expectation from group A, a low expectation in members of group C and a neutral expectation for members of group B. At the end of the study, each subject’s ratings are averaged across all three wines. The table below gives these averages for each subject in each group.
|Group A||Group B||Group C|
The means are, A: 8.2, B: 5.5, C: 4.9.
a) State the null and alternative hypotheses.
b) Conduct a Kruskal-Wallis test.
c) What is your conclusion?
- Students are randomly assigned to groups that are taught French using three different methods. The scores of the final exam for the three groups are:
Method 1: 94 88 93 76 88 99
Method 2: 87 84 81 86 63 74 82
Method 3: 91 69 74 78 71
Use the Kruskal Wallis test statistic to determine if there is a significant difference in the mean score between these groups.
- A drug company is interested in testing three forms of a pain relief medicine. 27 volunteers were selected and 9 were randomly assigned to one of the three drug formulations. The subjects were instructed to take the drug during their next episode and to report pain on a scale of 1 to 10 (10 is the most pain). Following is the data:
Drug A: 4 5 4 3 2 4 3 4 4
Drug B: 8 10 6 7 6 8 7 9 8
Drug C: 8 9 8 8 9 7 9 7 7
Use the Kruskal Wallis test to determine if there is a significant difference among the three formulations of the drug.
- Below is a ranking of course averages for males (m) and females (f), ranked from high to low. Test whether the arrangement is random, at the 10% level.
f f m m m f m f f f m m m m m f f m m m f m m m f
For 6-10, determine whether the given series is random:
- 2 2 1 1 1 1 2 1 2 1 1 1 2 1 2 2 2 1 1 2 1 1 1 1 2 2 1 1 1 1 2 1 1 1 2 1
- 1 0 0 1 0 1 1 0 1 0 0 1 0 0 0 0 0 1 0 0
- 0 1 0 0 0 0 0 0 1 1 1 1 0 1 1 0 0 0 0 0
- 1 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 1 0
- 1 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 0 1
To view the Review answers, open this PDF file and look for section 12.3.