Kruskal-Wallis H-Test for Oneway ANOVA by Ranks

© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved


************
kruskalw.doc
************
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


************
kruskalw.dat
************
                   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
************

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
************
% 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'

Please send comments or suggestions to Dr. Thomas W. MacFarland

There have been [an error occurred while processing this directive] visitors to this page since February 1, 1999.