1. Test to See if Samples Come From Same Population
Lesson
Test to See if Samples Come From Same Population

2. Objectives
Test a claim using the Kruskal–Wallis test

3. Vocabulary
Kruskal–Wallis Test -- nonparametric procedure used to test the claim that k (3 or more) independent samples come from populations with the same distribution.

4. Test of Means of 3 or more groups
Parametric test of the means of three or more groups:
Compared the corresponding observations by subtracting one mean from the other
Performed a test of whether the mean is 0
Nonparametric case for three or more groups:
Combine all of the samples and rank this combined set of data
Compare the rankings for the different groups

5. Kruskal-Wallis Test Assumptions:
Samples are simple random samples from three or more populations
Data can be ranked
We would expect that the values of the samples, when combined into one large dataset, would be interspersed with each other
Thus we expect that the average relative ratings of each sample to be about the same

6. Test Statistic for Kruskal–Wallis Test
A computational formula for the test statistic is
where
Ri is the sum of the ranks of the ith sample
R²1 is the sum of the ranks squared for the first sample
R²2 is the sum of the ranks squared for the second sample, and so on
n1 is the number of observations in the first sample
n2 is the number of observations in the second sample, and so on
N is the total number of observations (N = n1 + n2 + … + nk)
k is the number of populations being compared.
ni(N + 1) ²
H = Ri
N(N + 1) ni Σ
R² R² R²k
H = … (N + 1)
N(N + 1) n n nk

7. Test Statistic (cont)
Large values of the test statistic H indicate that the Ri’s are different than expected
If H is too large, then we reject the null hypothesis that the distributions are the same
This always is a right-tailed test

8. Critical Value for Kruskal–Wallis Test
Small-Sample Case
When three populations are being compared and when the sample size from each population is 5 or less, the critical value is obtained from Table XIV in Appendix A.
Large-Sample Case
When four or more populations are being compared or the sample size from one population is more than 5, the critical value is χ²α with k – 1 degrees of freedom, where k is the number of populations and α is the level of significance.

9. Hypothesis Tests Using Kruskal–Wallis Test
Step 0 Requirements:
1. The samples are independent random samples.
2. The data can be ranked.
Step 1 Box Plots: Draw side-by-side boxplots to compare the sample data from the populations. Doing so helps to visualize the differences, if any, between the medians.
Step 2 Hypotheses: (claim is made regarding distribution of three or more populations)
H0: the distributions of the populations are the same
H1: the distributions of the populations are not the same
Step 3 Ranks: Rank all sample observations from smallest to largest. Handle ties by finding the mean of the ranks for tied values. Find the sum of the ranks for each sample.
Step 4 Level of Significance: (level of significance determines the critical value)
The critical value is found from Table XIV for small samples. The critical value is χ²α with k – 1 degrees of freedom (found in Table VI) for large samples.
Step 5 Compute Test Statistic:
Step 6 Critical Value Comparison: We reject the null hypothesis if the test statistic is greater than the critical value.
R² R² R²k
H = … (N + 1)
N(N + 1) n n nk

10. Kruskal–Wallis Test Hypothesis
In this test, the hypotheses are
H0: The distributions of all of the populations are the same
H1: The distributions of all of the populations are not the same
This is a stronger hypothesis than in ANOVA, where only the means (and not the entire distributions) are compared

11. Example 1 from 15.7 S 20-29 40-49 60-69 1 54 (29) 61 (31.5) 44 (18) 2
43 (16)
41 (14)
65 (34.5) 3
38 (11.5)
62 (33) 4
30 (2)
47 (21)
53 (27.5) 5
33 (3)
51 (26) 6
29 (1)
49 (22.5) 7
35 (7.5)
59 (30) 8
34 (4.5)
42 (15) 9
39 (13) 10
46 (20)
74 (36) 11
50 (24.5)
37 (10) 12
Medians (Sums) 41
(194.5)
45.5 (225.5)
46.5 (246)

12. Example 1 (cont) Critical Value: (Large-Sample Case)
R² R² R²k
H = … (N + 1)
N(N + 1) n n nk
² ² ²
H = (36 + 1) = 1.009
36(36 + 1)
Critical Value: (Large-Sample Case)
χ²α with 2 (3 – 1) degrees of freedom, where 3 is the number of populations and 0.05 is the level of significance
CV= 5.991
Conclusion: Since H < CV, therefore we FTR H0 (distributions are the same)

13. HyCCI for Example
Hyp: H0: Distributions of all of the populations are the same Ha: Distributions of all of the populations are not the same
Conditions: Assume an independent random sample Data can be ranked
Calculations:
Critical Value: (Large-Sample Case) χ²α with 2 (3 – 1) degrees of freedom, where 3 is the number of populations and 0.05 is the level of significance CV= 5.991
Interpretation: Since H < CV, therefore we Fail to Reject H0 (distributions are the same)
R² R² R²k
H = … (N + 1)
N(N + 1) n n nk
² ² ²
H = (36 + 1) = 1.009
36(36 + 1)

14. Summary and Homework Summary Homework
The Kruskal-Wallis test is a nonparametric test for comparing the distributions of three or more populations
This test is a comparison of the rank sums of the populations
Critical values for small samples are given in tables
The critical values for large samples can be approximated by a calculation with the chi-square distribution
Homework
problems 3, 5, 7, 10 from the CD

15. Homework Problem 3 Sorts and Ranks Problem 3 9 1.5 1 2 Values Ranks 11
Subject Nr X Y Z 12
4.5 4 13 16
6.5 10 5 18 14 8 6 17 7 15
Ri =
Sum of the Ranks
23.5
31.5 23
R²i =
552.25
992.25
529
ni =
N =
i = 1
i = 2
i = 3
H =
0.875
Hcr =
5.6923
FTR

16. Homework Problem5 Problem 5 Ranks Subject Nr Mon Tues Wed Thurs Fri48
226
144
194.5
207.5
R²i =
2304
51076
20736
ni = 8
N = 40
i = 1
i = 2
i = 3
i = 4
i = 5
H =
Hcr =
9.488
Reject

17. Homework Problem 7 Sorts and Ranks Problem 3 9 1.5 1 2 Values Ranks 11
Subject Nr X Y Z 12
4.5 4 13 16
6.5 10 5 18 14 8 6 17 7 15
Ri =
Sum of the Ranks
23.5
31.5 23
R²i =
552.25
992.25
529
ni =
N =
i = 1
i = 2
i = 3
H =
0.875
Hcr =
5.6923
FTR

18. Homework Problem 10 Sort & Rank Problem 10 456 1 458 2 Values Ranks
480 3
Subject Nr CA DN US CN
485 4
578
568
506 24 21 8
491 5
548
530
518 17
13.5 11
492 6
521
571 12 23
502 7
555
569 18 22
563
16.5 20
513
9.5 9
535 15 10
561 19
Ri =
Sum of the Ranks
114
143
43.5 13
R²i =
12996
20449 14
ni =
N =
i = 1
i = 2
i = 3 16
H =
Hcr =
9.21
Reject