Kruskal-Wallis Test

16.4 Kruskal-Wallis Test

In Chapters 13, 14, and 15, the technique of analysis of variance was prominent as an analytical technique for testing equality of k ≥ 2 population means. Again, however, the reader should recall that normality must be assumed in order for the F-test to be theoretically correct. In this section, we investigate a nonparametric alternative to analysis of variance.

The Kruskal-Wallis test, also called the Kruskal-Wallis H test, is a gen- eralization of the rank-sum test to the case of k > 2 samples. It is used to test the null hypothesis H 0 that k independent samples are from identical populations. Introduced in 1952 by W. H. Kruskal and W. A. Wallis, the test is a nonpara- metric procedure for testing the equality of means in the one-factor analysis of variance when the experimenter wishes to avoid the assumption that the samples were selected from normal populations.

Let n i (i = 1, 2, . . . , k) be the number of observations in the ith sample. First, we combine all k samples and arrange the n = n 1 +n 2 +···+n k observations in ascending order, substituting the appropriate rank from 1, 2, . . . , n for each obser- vation. In the case of ties (identical observations), we follow the usual procedure of replacing the observations by the mean of the ranks that the observations would have if they were distinguishable. The sum of the ranks corresponding to the n i observations in the ith sample is denoted by the random variable R i . Now let us consider the statistic

which is approximated very well by a chi-squared distribution with k − 1 degrees of freedom when H 0 is true, provided each sample consists of at least 5 observations. The fact that h, the assumed value of H, is large when the independent samples come from populations that are not identical allows us to establish the following

decision criterion for testing H 0 :

Kruskal-Wallis To test the null hypothesis H 0 that k independent samples are from identical Test populations, compute

where r i is the assumed value of R i , for i = 1, 2, . . . , k. If h falls in the critical region H > χ 2 α with v = k − 1 degrees of freedom, reject H 0 at the α-level of

significance; otherwise, fail to reject H 0 .

Example 16.6: In an experiment to determine which of three different missile systems is preferable, the propellant burning rate is measured. The data, after coding, are given in Table

16.5. Use the Kruskal-Wallis test and a significance level of α = 0.05 to test the hypothesis that the propellant burning rates are the same for the three missile systems.

16.4 Kruskal-Wallis Test 669

Table 16.5: Propellant Burning Rates

Missile System

2. H 1 : The three means are not all equal.

4. Critical region: h > χ 2 0.05 = 5.991, for v = 2 degrees of freedom.

5. Computations: In Table 16.6, we convert the 19 observations to ranks and sum the ranks for each missile system.

Table 16.6: Ranks for Propellant Burning Rates

Missile System

Now, substituting n 1 = 5, n 2 = 6, n 3 = 8 and r 1 = 61.0, r 2 = 63.5, r 3 = 65.5, our test statistic H assumes the value

6. Decision: Since h = 1.66 does not fall in the critical region h > 5.991, we have insufficient evidence to reject the hypothesis that the propellant burning rates are the same for the three missile systems.

670 Chapter 16 Nonparametric Statistics

Exercises

16.15 A cigarette manufacturer claims that the tar selected at random and given additional instruction by content of brand B cigarettes is lower than that of the teacher. The results on the final examination are brand A cigarettes. To test this claim, the follow- as follows: ing determinations of tar content, in milligrams, were

Grade recorded:

Additional

Brand A

87 69 78 91 80 Brand B

1 12 9 13 11 14 Instruction

8 10 7 No Additional

Use the rank-sum test with α = 0.05 to test whether

75 88 64 82 93 79 67 the claim is valid.

Instruction

Use the rank-sum test with α = 0.05 to determine if the additional instruction affects the average grade.

16.16 To find out whether a new serum will arrest leukemia, nine patients, who have all reached an ad-

16.20 The following data represent the weights, in vanced stage of the disease, are selected. Five patients kilograms, of personal luggage carried on various flights

receive the treatment and four do not. The survival by a member of a baseball team and a member of a times, in years, from the time the experiment com- basketball team. menced are

Luggage Weight (kilograms) Treatment

Basketball Player No treatment

2.1 5.3 1.4 4.6 0.9 Baseball Player

1.9 0.5 2.8 3.1 16.3 20.0 18.6 15.4 16.3 Use the rank-sum test, at the 0.05 level of significance,

18.1 15.0 15.4 17.7 18.1 to determine if the serum is effective.

16.17 The following data represent the number of 17.7 19.1 17.4 15.0 13.6 hours that two different types of scientific pocket cal-

16.3 13.6 14.8 15.9 16.3 culators operate before a recharge is required.

Calculator A 5.5 5.6 6.3 4.6 5.3 5.0 6.2 5.8 5.1 Use the rank-sum test with α = 0.05 to test the null hy- Calculator B

3.8 4.8 4.3 4.2 4.0 4.9 4.5 5.2 4.5 pothesis that the two athletes carry the same amount of luggage on the average against the alternative hy-

Use the rank-sum test with α = 0.01 to determine if pothesis that the average weights of luggage for the two calculator A operates longer than calculator B on a full athletes are different. battery charge.

16.21 The following data represent the operating 16.18 A fishing line is being manufactured by two times in hours for three types of scientific pocket cal- processes. To determine if there is a difference in the culators before a recharge is required: mean breaking strength of the lines, 10 pieces manu- factured by each process are selected and then tested for breaking strength. The results are as follows:

Calculator A B C Process 1

9.5 11.0 9.8 10.5 9.9 Use the Kruskal-Wallis test, at the 0.01 level of signif- Use the rank-sum test with α = 0.1 to determine icance, to test the hypothesis that the operating times

if there is a difference between the mean breaking for all three calculators are equal. strengths of the lines manufactured by the two pro- cesses.

16.22 In Exercise 13.6 on page 519, use the Kruskal- Wallis test at the 0.05 level of significance to determine

16.19 From a mathematics class of 12 equally capable if the organic chemical solvents differ significantly in students using programmed materials, 5 students are sorption rate.

16.5 Runs Test 671