Mann-Whitney U-Test

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


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mann_whi.doc
************
Background:  The Mann-Whitney U test is often viewed as the 
             nonparametric equivalent of Student's t-test. 
             Like the parametric Student's t-test, the non-
             parametric Mann-Whitney U test:

             -- is used to determine if a difference exists 
                between two "groups," however you define 
                "group"

             -- is ideally dependent on random selection of
                subjects into their respective group

             The major difference between the Mann-Whitney U
             Test and Student's t-Test involves the concept of 
             normal distribution:
  
             -- Mann-Whitney is a nonparametric test. 

             -- Normal distribution of data is not necessary 
                for use of this test.

             There is a table of U values in many statistics
             texts.  If you use this table:

             -- The column number should be the number of the 
                larger sample.

             -- The row number should be the number of the smaller 
                sample.

             -- Samples of equal size will use n by n for 
                determining the criterion U statistic.

             If you use SPSS (or many of the other statistical 
             packages) for data analysis you may find it far more 
             convenient to use either z values or p values in the
             output file to determine significance.  

             When using z values:

             -- If the observed z value does not equal or exceed 
                the critical z value of 1.96 (p <= .05 critical z 
                value for a two-tailed test), then you can assume 
                that the null hypothesis is correct and that there 
                is no difference between groups.  

             -- If the z value, however, exceeds 1.96 then you 
                have evidence to reject the null hypothesis.  

             Or, you may find it more convenient to observe
             the printed p value.  


Scenario:    This study examines if there any differences in 
             outcomes in a Pascal programming course between
             students who received  Computer Based Training,
             as opposed to students who received traditional 
             lecture:

             -- Differences (if indeed they exist) between the 
                two teaching formats will be measured by student 
                performance on a common final examination.

             -- Based on prior experience with the test 
                instrument, it is suspected that outcomes are 
                not normally distributed (e.g., bell-shaped curve) 
                but are instead skewed to the right.  Accordingly, 
                Student's t-Test is not the appropriate test for 
                difference between the two groups.  Instead, this
                study will be based on the use of the Mann-Whitney
                U Test.

             In this study random selection was used to assign the
             30 students in Mr. Seeger's Pascal programming course
             into one of two groups:

             1.  Students in group 1 received instruction
                 through the use of Computer Based Training
                 (CBT).

             2.  Subjects in group 2 received instruction
                 through the use of traditional lecture.

             A summary of the study is presented in Table 1.


             Table 1

             Pascal Programming Course Final Examination 
             Scores:  Breakouts by Computer Based Training and 
             Traditional Lecture
             ====================================================   
                              Assigned Group
                              ==============
                                1 = CBT
             Student Number     2 = Lecture    Exam Score
             ----------------------------------------------------

                   01              1            080 
                   02              1            082 
                   03              1            091 
                   04              1            100 
                   05              1            076 
                   06              1            065 
                   07              1            085 
                   08              1            088 
                   09              1            097 
                   10              1            055 
                   11              1            069 
                   12              1            088 
                   13              1            075 
                   14              1            097 
                   15              1            081 
                   16              2            072 
                   17              2            089 
                   18              2            086 
                   19              2            085 
                   20              2            099 
                   21              2            047 
                   22              2            079 
                   23              2            088 
                   24              2            100 
                   25              2            076 
                   26              2            083 
                   27              2            094 
                   28              2            084 
                   29              2            082 
                   30              2            093 
             ----------------------------------------------------


Ho:          Null Hypothesis:  There is no difference in final
             examination test scores in a Pascal programming 
             course between students who received Computer
             Based Training and students who received traditional
             lecture (p <= .05).


Files:       1.  mann_whi.doc

             2.  mann_whi.dat

             3.  mann_whi.r01

             4.  mann_whi.o01

             5.  mann_whi.con

             6.  mann_whi.lis


Command:     At the Unix prompt (%), key:

             %spss -m < mann_whi.r01 > mann_whi.o01


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mann_whi.dat
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                   01              1            080 
                   02              1            082 
                   03              1            091 
                   04              1            100 
                   05              1            076 
                   06              1            065 
                   07              1            085 
                   08              1            088 
                   09              1            097 
                   10              1            055 
                   11              1            069 
                   12              1            088 
                   13              1            075 
                   14              1            097 
                   15              1            081 
                   16              2            072 
                   17              2            089 
                   18              2            086 
                   19              2            085 
                   20              2            099 
                   21              2            047 
                   22              2            079 
                   23              2            088 
                   24              2            100 
                   25              2            076 
                   26              2            083 
                   27              2            094 
                   28              2            084 
                   29              2            082 
                   30              2            093 


************
mann_whi.r01
************
SET WIDTH      = 80
SET LENGTH     = NONE
SET CASE       = UPLOW
SET HEADER     = NO
TITLE          = Sign Test
COMMENT        = This file examines if Computer Based Training
                 is as equally effective as traditional lecture
                 in a Pascal programming course.  Differences 
                 between the two teaching formats will be 
                 measured by student performance on a common
                 final examination.
DATA LIST FILE = 'mann_whi.dat' FIXED
     / Stu_Code   20-21
       Group         36
       Score      49-51

Variable Labels
       Stu_Code   "Subject Code"
     / Group      "Assigned Group:  CBT or Traditional"
     / Score      "Common Final Examination Score"

Value Labels
       Group    1 'Computer Based Training'
                2 'Traditional Lecture'

NPAR TESTS M-W = Score BY Group (1,2) 


************
mann_whi.o01
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   1  SET WIDTH      = 80
   2  SET LENGTH     = NONE
   3  SET CASE       = UPLOW
   4  SET HEADER     = NO
   5  TITLE          = Sign Test
   6  COMMENT        = This file examines if Computer Based Training
   7                   is as equally effective as traditional lecture
   8                   in a Pascal programming course.  Differences
   9                   between the two teaching formats will be
  10                   measured by student performance on a common
  11                   final examination.
  12  DATA LIST FILE = 'mann_whi.dat' FIXED
  13       / Stu_Code   20-21
  14         Group         36
  15         Score      49-51
  16

This command will read 1 records from mann_whi.dat

Variable   Rec   Start     End         Format

STU_CODE     1      20      21         F2.0
GROUP        1      36      36         F1.0
SCORE        1      49      51         F3.0

  17  Variable Labels
  18         Stu_Code   "Subject Code"
  19       / Group      "Assigned Group:  CBT or Traditional"
  20       / Score      "Common Final Examination Score"
  21
  22  Value Labels
  23         Group    1 'Computer Based Training'
  24                  2 'Traditional Lecture'
  25
  26  NPAR TESTS M-W = Score BY Group (1,2)

***** Workspace allows for 18724 cases for NPAR tests *****



- - - - - Mann-Whitney U - Wilcoxon Rank Sum W Test

     SCORE     Common Final Examination Score
  by GROUP     Assigned Group:  CBT or Traditional


     Mean Rank    Cases

         14.53       15  GROUP = 1  Computer Based Train
         16.47       15  GROUP = 2  Traditional Lecture
                     --
                     30  Total

                                  Exact             Corrected for ties
           U             W      2-Tailed P           Z      2-Tailed P
          98.0         218.0       .5668          -.6020       .5472


************
mann_whi.con
************

Outcome:     Significance can of course be verified by using the 
             computed test statistic (e.g., U) and comparing 
             this statistic to the criterion (i.e., table) value.  
             It is often much easier, however, to use the output
             file to verify interpretation of significance:

             p = .5472

             By interpretation of the p (probability) value, it
             is observed that p = .55, which exceeds the Null
             Hypothesis declaration that p <= .05.

             There is certainly sufficient information to accept
             the Null Hypothesis and to declare that there is
             no difference between the two training groups in 
             terms of final examination scores.


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mann_whi.lis
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% minitab
 
 MTB > outfile 'mann_whi.lis'
 Collecting Minitab session in file: mann_whi.lis
 MTB > # MINITAB addendum to mann_whi.dat
 MTB > read 'mann_whi.dat' c1 c2 c3
 Entering data from file: mann_whi.dat
      30 rows read.
 MTB > print c1 c2 c3
 
 
  ROW    C1   C2     C3
 
    1     1    1     80
    2     2    1     82
    3     3    1     91
    4     4    1    100
    5     5    1     76
    6     6    1     65
    7     7    1     85
    8     8    1     88
    9     9    1     97
   10    10    1     55
   11    11    1     69
   12    12    1     88
   13    13    1     75
   14    14    1     97
   15    15    1     81
   16    16    2     72
   17    17    2     89
   18    18    2     86
 Continue? y
   19    19    2     85
   20    20    2     99
   21    21    2     47
   22    22    2     79
   23    23    2     88
   24    24    2    100
   25    25    2     76
   26    26    2     83
   27    27    2     94
   28    28    2     84
   29    29    2     82
   30    30    2     93
 
 MTB > # I will now UNSTACK the data to get distinct
 MTB > # groups.
 MTB > unstack (c2-c3) into (c5-c6) (c8-c9);
 SUBC> subscripts c2.
 MTB > print c1 c2 c3 c4 c5 c6 c7 c8 c9
 
 
  ROW    C1   C2     C3   C5     C6   C8     C9
 
    1     1    1     80    1     80    2     72
    2     2    1     82    1     82    2     89
    3     3    1     91    1     91    2     86
    4     4    1    100    1    100    2     85
    5     5    1     76    1     76    2     99
    6     6    1     65    1     65    2     47
    7     7    1     85    1     85    2     79
    8     8    1     88    1     88    2     88
    9     9    1     97    1     97    2    100
   10    10    1     55    1     55    2     76
   11    11    1     69    1     69    2     83
   12    12    1     88    1     88    2     94
   13    13    1     75    1     75    2     84
   14    14    1     97    1     97    2     82
   15    15    1     81    1     81    2     93
   16    16    2     72                        
   17    17    2     89                        
   18    18    2     86                        
 Continue? y
   19    19    2     85                        
   20    20    2     99                        
   21    21    2     47                        
   22    22    2     79                        
   23    23    2     88                        
   24    24    2    100                        
   25    25    2     76                        
   26    26    2     83                        
   27    27    2     94                        
   28    28    2     84                        
   29    29    2     82                        
   30    30    2     93                        
 
 * NOTE  * One or more variables are undefined.
 
 MTB > mannwhitney c6 c9
 
 Mann-Whitney Confidence Interval and Test
 
 C6         N =  15     Median =       82.00
 C9         N =  15     Median =       85.00
 Point estimate for ETA1-ETA2 is       -2.00
 95.4 pct c.i. for ETA1-ETA2 is (-11.00,6.00)
 W = 218.0
 Test of ETA1 = ETA2  vs.  ETA1 n.e. ETA2 is significant at 0.5614
 The test is significant at 0.5611 (adjusted for ties)
 
 Cannot reject at alpha = 0.05
 
 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 'mann_whi.ssi'

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

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