Kruskal-Wallis H-Test for Oneway ANOVA by Ranks
© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved
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kruskalw.doc
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Background: The Kruskal-Wallis H-test is often viewed as
the nonparametric equivalent of the parametric
One Way Analysis of Variance (Oneway ANOVA),
with both tests used to serve the same purpose
of comparing possible differences between various
"groups." The Kruskal-Wallis test is used when
the data do not meet the rigor of interval data
associated with the parametric Oneway ANOVA test.
It may help to think of the Kruskal-Wallis H test
as an ANOVA test by ranks.
Scenario: This file examines possible differences in graded
performance to three separate activities (e.g.,
final examination score, composite score for all
homework problems, final project score) in a high
school Logo programming language class.
Because the teacher conducting this analysis
has a concern that homework scores and final
project scores are ordinal data (data are ordered,
but not with the precision of interval data), it
is best to use the non-parametric K-W H test
instead of the Oneway Analysis of Variance (ANOVA),
which is based on the use of interval data.
A summary of the study is presented in Table 1.
Table 1
Summary Data of Final Examination Scores, Homework
Problems, and Final Project Scores in a High School
Logo Programming Class
====================================================
Graded Activity
===============
1 = Final Exam
2 = Homework
Student Number 3 = Final Project Score
----------------------------------------------------
01 1 085
01 2 090
01 3 075
02 1 088
02 2 092
02 3 082
03 1 091
03 2 074
03 3 055
04 1 088
04 2 093
04 3 067
05 1 086
05 2 083
05 3 077
06 1 072
06 2 091
06 3 062
07 1 057
07 2 059
07 3 066
08 1 093
08 2 079
08 3 072
09 1 082
09 2 088
09 3 081
10 1 077
10 2 071
10 3 063
11 1 085
11 2 082
11 3 093
12 1 072
12 2 073
12 3 062
13 1 074
13 2 094
13 3 090
14 1 057
14 2 075
14 3 048
15 1 079
15 2 082
15 3 063
16 1 083
16 2 093
16 3 098
17 1 068
17 2 078
17 3 082
18 1 078
18 2 082
18 3 088
19 1 074
19 2 078
19 3 000
20 1 085
20 2 089
20 3 094
21 1 094
21 2 093
21 3 083
22 1 088
22 2 081
22 3 073
23 1 083
23 2 081
23 3 071
24 1 089
24 2 092
24 3 080
25 1 088
25 2 079
25 3 068
26 1 081
26 2 088
26 3 092
27 1 091
27 2 094
27 3 083
28 1 095
28 2 092
28 3 084
29 1 095
29 2 093
29 3 100
30 1 095
30 2 096
30 3 098
31 1 073
31 2 079
31 3 064
32 1 081
32 2 085
32 3 091
33 1 087
33 2 081
33 3 081
34 1 069
34 2 075
34 3 076
35 1 077
35 2 088
35 3 083
----------------------------------------------------
Ho: Null Hypothesis: There is no difference in graded
performance to three separate activities (e.g.,
final examination score, composite score for all
homework problems, final project score) between
students in a high school Logo programming language
class (p = .05).
Files: 1. kruskalw.doc
2. kruskalw.dat
3. kruskalw.r01
4. kruskalw.o01
5. kruskalw.con
6. kruskalw.lis
Command: At the Unix prompt (%), key:
%spss -m < kruskalw.r01 > kruskalw.o01
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kruskalw.dat
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01 1 085
01 2 090
01 3 075
02 1 088
02 2 092
02 3 082
03 1 091
03 2 074
03 3 055
04 1 088
04 2 093
04 3 067
05 1 086
05 2 083
05 3 077
06 1 072
06 2 091
06 3 062
07 1 057
07 2 059
07 3 066
08 1 093
08 2 079
08 3 072
09 1 082
09 2 088
09 3 081
10 1 077
10 2 071
10 3 063
11 1 085
11 2 082
11 3 093
12 1 072
12 2 073
12 3 062
13 1 074
13 2 094
13 3 090
14 1 057
14 2 075
14 3 048
15 1 079
15 2 082
15 3 063
16 1 083
16 2 093
16 3 098
17 1 068
17 2 078
17 3 082
18 1 078
18 2 082
18 3 088
19 1 074
19 2 078
19 3 000
20 1 085
20 2 089
20 3 094
21 1 094
21 2 093
21 3 083
22 1 088
22 2 081
22 3 073
23 1 083
23 2 081
23 3 071
24 1 089
24 2 092
24 3 080
25 1 088
25 2 079
25 3 068
26 1 081
26 2 088
26 3 092
27 1 091
27 2 094
27 3 083
28 1 095
28 2 092
28 3 084
29 1 095
29 2 093
29 3 100
30 1 095
30 2 096
30 3 098
31 1 073
31 2 079
31 3 064
32 1 081
32 2 085
32 3 091
33 1 087
33 2 081
33 3 081
34 1 069
34 2 075
34 3 076
35 1 077
35 2 088
35 3 083
************
kruskalw.r01
************
SET WIDTH = 80
SET LENGTH = NONE
SET CASE = UPLOW
SET HEADER = NO
TITLE = Kruskal-Wallis Oneway Anova by Ranks
COMMENT = This file examines possible differences
in graded performance to three separate
activities (e.g., final examination score,
composite score for all homework problems,
final project score) in a high school Logo
programming language class.
Because the teacher conducting this analysis
has a concern that homework scores and final
project scores are ordinal data (data are ordered,
but not with the precision of interval data), it
is best to use the non-parametric K-W H test
instead of the Oneway Analysis of Variance (ANOVA)
based on the use of interval data.
DATA LIST FILE = 'kruskalw.dat' FIXED
/ Stu_Code 20-21
Activity 36
Score 51-53
Variable Lables
Stu_Code "Student Code"
/ Activity "Graded Activity"
/ Score "Score on Graded Activity"
Value Labels
Activity 1 'Final Examination'
2 'Homework Problems'
3 'Final Project'
NPAR TESTS K-W = Score by Activity(1,3)
************
kruskalw.o01
************
1 SET WIDTH = 80
2 SET LENGTH = NONE
3 SET CASE = UPLOW
4 SET HEADER = NO
5 TITLE = Kruskal-Wallis Oneway Anova by Ranks
6 COMMENT = This file examines possible differences
7 in graded performance to three separate
8 activities (e.g., final examination score,
9 composite score for all homework problems,
10 final project score) in a high school Logo
11 programming language class.
12
13 Because the teacher conducting this analysis
14 has a concern that homework scores and final
15 project scores are ordinal data (data are ordered,
16 but not with the precision of interval data), it
17 is best to use the non-parametric K-W H test
18 instead of the Oneway Analysis of Variance (ANOVA)
19 based on the use of interval data.
20 DATA LIST FILE = 'kruskalw.dat' FIXED
21 / Stu_Code 20-21
22 Activity 36
23 Score 51-53
24
This command will read 1 records from kruskalw.dat
Variable Rec Start End Format
STU_CODE 1 20 21 F2.0
ACTIVITY 1 36 36 F1.0
SCORE 1 51 53 F3.0
25 Variable Lables
26 Stu_Code "Student Code"
27 / Activity "Graded Activity"
28 / Score "Score on Graded Activity"
29
30 Value Labels
31 Activity 1 'Final Examination'
32 2 'Homework Problems'
33 3 'Final Project'
34
35 NPAR TESTS K-W = Score by Activity(1,3)
***** Workspace allows for 18724 cases for NPAR tests *****
- - - - - Kruskal-Wallis 1-Way Anova
SCORE Score on Graded Activity
by ACTIVITY Graded Activity
Mean Rank Cases
54.47 35 ACTIVITY = 1 Final Examination
60.49 35 ACTIVITY = 2 Homework Problems
44.04 35 ACTIVITY = 3 Final Project
---
105 Total
Corrected for ties
Chi-Square D.F. Significance Chi-Square D.F. Significance
5.2238 2 .0734 5.2328 2 .0731
************
kruskalw.con
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Outcome: Computed H = 5.2234 (K-W H approximates Chi-Square)
df = k-1 = 3-1 = 2
Criterion H (alpha = .05, df = 2) = 5.9915
Computed H (5.2238) < Criterion H (5.9915)
Therefore, the null hypothesis is accepted and
it can be claimed that there is no difference in
the graded performance to three separate activities
(e.g., final examination score, composite score for
all homework problems, final project score) in a
high school Logo programming language class (p =
.05).
The p value is another way to view differences in
the three graded activities:
-- The calculated p value is .0731.
-- The delcared p value is .05.
The calculated p value exceeds the declared p value
and there is, accordingly, no difference in scores
of the three graded activities at this level of
significance (p = .05). Differences in mean rankings
of scores for all three graded activities are due
only to chance.
Note: Although the test statistic for Kruskal-Wallis
analysis is "H," you will notice that chi-square
values are used for data analysis. H approximates
the chi-square distribution.
Note: There is often disagreement in the profession about
using parametric analysis for data that simply do not
follow normal distribution:
-- Grades on a standardized test, such as a well-
constructed final examination, likely follow
a normal distribution along the bell-shaped curve.
-- Grades awarded to homework assignments are a
typical example of data that are not distributed
along a normal curve, since students rarely turn
in a continuum of "bad" to "good" homework
assignments.
Should parametric analyses be used for nonparametric
data? There is no easy answer to this question. Yet,
if the purpose of statistical analysis is to offer
possible answers to the otherwise unknown, then expect
to see parametric analyses for data that may not meet
the assumptions associated with tests that rely on
normal distribution.
Even so, this problem provides a "typical" use of
nonparametric data in educational research. In
turn, the Kruskal-Wallis H test was indeed the
appropriate test for this data set.
************
kruskalw.lis
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% minitab
MTB > outfile 'kruskalw.lis'
Collecting Minitab session in file: kruskalw.lis
MTB > # MINITAB addendum to kruskalw.dat
MTB > read 'kruskalw.dat' c1 c2 c3
Entering data from file: kruskalw.dat
105 rows read.
MTB > print c1 c2 c3
ROW C1 C2 C3
1 1 1 85
2 1 2 90
3 1 3 75
4 2 1 88
5 2 2 92
6 2 3 82
7 3 1 91
8 3 2 74
9 3 3 55
10 4 1 88
11 4 2 93
12 4 3 67
13 5 1 86
14 5 2 83
15 5 3 77
16 6 1 72
17 6 2 91
18 6 3 62
Continue? y
19 7 1 57
20 7 2 59
21 7 3 66
22 8 1 93
23 8 2 79
24 8 3 72
25 9 1 82
26 9 2 88
27 9 3 81
28 10 1 77
29 10 2 71
30 10 3 63
31 11 1 85
32 11 2 82
33 11 3 93
34 12 1 72
35 12 2 73
36 12 3 62
37 13 1 74
38 13 2 94
39 13 3 90
40 14 1 57
41 14 2 75
Continue? y
42 14 3 48
43 15 1 79
44 15 2 82
45 15 3 63
46 16 1 83
47 16 2 93
48 16 3 98
49 17 1 68
50 17 2 78
51 17 3 82
52 18 1 78
53 18 2 82
54 18 3 88
55 19 1 74
56 19 2 78
57 19 3 0
58 20 1 85
59 20 2 89
60 20 3 94
61 21 1 94
62 21 2 93
63 21 3 83
64 22 1 88
Continue? y
65 22 2 81
66 22 3 73
67 23 1 83
68 23 2 81
69 23 3 71
70 24 1 89
71 24 2 92
72 24 3 80
73 25 1 88
74 25 2 79
75 25 3 68
76 26 1 81
77 26 2 88
78 26 3 92
79 27 1 91
80 27 2 94
81 27 3 83
82 28 1 95
83 28 2 92
84 28 3 84
85 29 1 95
86 29 2 93
87 29 3 100
Continue? y
88 30 1 95
89 30 2 96
90 30 3 98
91 31 1 73
92 31 2 79
93 31 3 64
94 32 1 81
95 32 2 85
96 32 3 91
97 33 1 87
98 33 2 81
99 33 3 81
100 34 1 69
101 34 2 75
102 34 3 76
103 35 1 77
104 35 2 88
105 35 3 83
MTB > name c1 'Stu_Code' c2 'Activity' c3 'Score'
MTB > kruskal-wallis test for data in c3, levels in c2
LEVEL NOBS MEDIAN AVE. RANK Z VALUE
1 35 83.00 54.5 0.35
2 35 83.00 60.5 1.78
3 35 80.00 44.0 -2.13
OVERALL 105 53.0
H = 5.22 d.f. = 2 p = 0.074
H = 5.23 d.f. = 2 p = 0.074 (adj. for ties)
MTB > stop
--------------------------
Disclaimer: All care was used to prepare the information in this
tutorial. Even so, the author does not and cannot guarantee the
accuracy of this information. The author disclaims any and all
injury that may come about from the use of this tutorial. As
always, students and all others should check with their advisor(s)
and/or other appropriate professionals for any and all assistance
on research design, analysis, selected levels of significance, and
interpretation of output file(s).
The author is entitled to exclusive distribution of this tutorial.
Readers have permission to print this tutorial for individual use,
provided that the copyright statement appears and that there is no
redistribution of this tutorial without permission.
Prepared 980316
Revised 980914
end-of-file 'kruskalw.ssi'