Wilcoxon Matched-Pairs Signed Ranks Test

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


************
wilcoxon.doc
************
Background:  The Wilcoxon Matched-Pairs Ranks test is a non-
             parametric test that is often regarded as being 
             similar to a matched pairs t-test.  The Wilcoxon
             Matched-Pairs Ranks test is used to determine
             differences between groups of paired data when
             the data do not meet the rigor associated with
             a parametric test. 
              
             Unlike less robust nonparametric tests such as the 
             sign test:

             -- The Wilcoxon test is used to determine the 
                magnitude of difference between matched groups.

             -- The Wilcoxon test is used to determine more than
                only the direction of difference.  


Scenario:    This file examines if separating students who
             previously worked on programming assignments  
             as a pair has any impact on project quality  
             when:  

             -- One member of the pair is required to
                work on their own,

             -- And the other member of the pair is assigned 
                a new programming partner.

             Dr. Kieta teaches Cobol I and Cobol II at a 
             local community college.  There are 60 students in
             Cobol I, with instruction offered in a laboratory-
             type classroom setting with only 30 computers.
             Logically, Dr. Kieta used random assignment to
             segregate the 60 students into 30 sets of paired
             students.  

             During the 10-day break between Cobol I and Cobol
             II, more computers are brought into the classroom.
             Further, many of the students who enrolled in
             Cobol I immediately enrolled for Dr. Kieta's
             class in Cobol II.

             Seeing an opportunity for a research project on
             group vs. individual work with the introduction of
             these new computers, Dr. Kieta makes new assignments:
           
             -- 20 students who were enrolled in Cobol I, working
                with an assigned partner, now have the chance to
                work on their own in Cobol II

             -- their counterparts from Cobol I are now assigned
                a new partner for Cobol II

             This test will determine if there are differences
             in the Cobol II final examination among these 40
             Cobol I and Cobol II students.

             A summary of the study is presented in Table 1.


             Table 1

             Final Exam Scores in Cobol II of Students Who Were
             Previously Paired Together in Cobol I
             ===================================================    

                                Exam Score for Previously
                                     Paired Students 
                        ----------------------------------------      

                            Pair Member:     Pair Member:  
                            Worked on Own    Worked With a New
                 Pair       in Cobol II      Partner in Cobol II
             ---------------------------------------------------    

                   01            090            083 
                   02            084            089 
                   03            078            058 
                   04            079            093 
                   05            065            049 
                   06            049            082 
                   07            092            093 
                   08            071            100 
                   09            085            083 
                   10            076            055 
                   11            068            065 
                   12            071            083 
                   13            088            081 
                   14            089            088 
                   15            076            083 
                   16            061            078 
                   17            088            088 
                   18            058            080 
                   19            073            070 
                   20            082            083 
             _______________________________________________________


Ho:          Null Hypothesis:  There is no difference in final   
             examination test scores in Cobol II between students  
             who were previously paired together in Cobol I, with
             the pairs now redesigned so that one student in Cobol 
             II now works on their own and their matched counterpart 
             now works with another student in Cobol II (p <= .05).

Files:       1.  wilcoxon.doc

             2.  wilcoxon.dat

             3.  wilcoxon.r01

             4.  wilcoxon.o01

             5.  wilcoxon.con

             6.  wilcoxon.lis


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

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


************
wilcoxon.dat
************
                   01            090            083 
                   02            084            089 
                   03            078            058 
                   04            079            093 
                   05            065            049 
                   06            049            082 
                   07            092            093 
                   08            071            100 
                   09            085            083 
                   10            076            055 
                   11            068            065 
                   12            071            083 
                   13            088            081 
                   14            089            088 
                   15            076            083 
                   16            061            078 
                   17            088            088 
                   18            058            080 
                   19            073            070 
                   20            082            083 

************
wilcoxon.r01
************
SET WIDTH      = 80
SET LENGTH     = NONE
SET CASE       = UPLOW
SET HEADER     = NO
TITLE          = Wilcoxon Matched Pairs Signed Ranks Test
COMMENT        = This file examines if separating students who
                 previously worked on programming assignments  
                 as a pair has any impact on project quality  
                 when:  one member of the pair is required to
                 work on their own and the other member of the
                 pair is assigned a new programming partner.

                 Dr. Kieta teaches Cobol I and Cobol II at a 
                 local community college.  There are 60 students in
                 Cobol I, with instruction offered in a laboratory-
                 type classroom setting with only 30 computers.
                 Logically, Dr. Kieta used random assignment to
                 segregate the 60 students into 30 sets of paired
                 students.  

                 During the 10-day break between Cobol I and Cobol
                 II more computers are brought into the classroom.
                 Further, many of the students who enrolled in
                 Cobol I immediately enrolled for Dr. Kieta's
                 class in Cobol II.

                 Seeing an opportunity for a research project on
                 group vs. individual work with the introduction of
                 these new computers, Dr. Kieta makes new assignments:
           
                 -- 20 students who were enrolled in Cobol I, working
                    with an assigned partner, now have the chance to
                    work on their own in Cobol II

                 -- their counterparts from Cobol I are now assigned
                    a new partner for Cobol II

                 This test will determine if there are differences
                 in the Cobol II final examination among these 40
                 Cobol I and Cobol II students.

DATA LIST FILE = 'wilcoxon.dat' FIXED
     / Pair       20-21
       Wk_Own     34-36
       Wk_New     49-51

Variable Labels
       Pair       "Pair Number"  
     / Wk_Own     "Exam Score:  Student Who Worked on Their Own"
     / Wk_New     "Exam Score:  Student Assigned a New Partner"

NPAR TESTS WILCOXON = Wk_Own WITH Wk_New      


************
wilcoxon.o01
************
   1  SET WIDTH      = 80
   2  SET LENGTH     = NONE
   3  SET CASE       = UPLOW
   4  SET HEADER     = NO
   5  TITLE          = Wilcoxon Matched Pairs Signed Ranks Test
   6  COMMENT        = This file examines if separating students who
   7                   previously worked on programming assignments
   8                   as a pair has any impact on project quality
   9                   when:  one member of the pair is required to
  10                   work on their own and the other member of the
  11                   pair is assigned a new programming partner.
  12
  13                   Dr. Kieta teaches Cobol I and Cobol II at a
  14                   local community college.  There are 60 students in
  15                   Cobol I, with instruction offered in a laboratory-
  16                   type classroom setting with only 30 computers.
  17                   Logically, Dr. Kieta used random assignment to
  18                   segregate the 60 students into 30 sets of paired
  19                   students.
  20
  21                   During the 10-day break between Cobol I and Cobol
  22                   II more computers are brought into the classroom.
  23                   Further, many of the students who enrolled in
  24                   Cobol I immediately enrolled for Dr. Kieta's
  25                   class in Cobol II.
  26
  27                   Seeing an opportunity for a research project on
  28                   group vs. individual work with the introduction of
  29                   these new computers, Dr. Kieta makes new assignments:
  30
  31                   -- 20 students who were enrolled in Cobol I, working
  32                      with an assigned partner, now have the chance to
  33                      work on their own in Cobol II
  34
  35                   -- their counterparts from Cobol I are now assigned
  36                      a new partner for Cobol II
  37
  38                   This test will determine if there are differences
  39                   in the Cobol II final examination among these 40
  40                   Cobol I and Cobol II students.
  41
  42  DATA LIST FILE = 'wilcoxon.dat' FIXED
  43       / Pair       20-21
  44         Wk_Own     34-36
  45         Wk_New     49-51
  46

This command will read 1 records from wilcoxon.dat

Variable   Rec   Start     End         Format

PAIR         1      20      21         F2.0
WK_OWN       1      34      36         F3.0
WK_NEW       1      49      51         F3.0

  47  Variable Labels
  48         Pair       "Pair Number"
  49       / Wk_Own     "Exam Score:  Student Who Worked on Their Own"
  50       / Wk_New     "Exam Score:  Student Assigned a New Partner"
  51
  52  NPAR TESTS WILCOXON = Wk_Own WITH Wk_New

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



- - - - - Wilcoxon Matched-Pairs Signed-Ranks Test

     WK_OWN    Exam Score:  Student Who Worked on Their
with WK_NEW    Exam Score:  Student Assigned a New Part


     Mean Rank    Cases

          8.78        9  - Ranks (WK_NEW LT WK_OWN)
         11.10       10  + Ranks (WK_NEW GT WK_OWN)
                      1    Ties  (WK_NEW EQ WK_OWN)
                     --
                     20    Total

         Z =    -.6439            2-Tailed P =  .5197

************
wilcoxon.con
************

Outcome:     It is certainly possible to determine the status
             of the Null Hypothesis by comparing the calculated
             z statistic against the criterion z statistic:

             z = |-0.644| (p = .520)

             z |-0.644| < |-1.960| (criterion z with alpha = .05)

             Note.  The | and | characters are used to indicate
                    absolute value.
             
             The null hypothesis is accepted at p <= .05.  Any
             differences in final examination scores in Cobol II
             between students who were previously paired in 
             Cobol I are due to chance.

             Of course, you can also observe that the calculated
             p value of .5197 exceeded the declared p value of .05
             to determine the probability level of difference.
        
Conclusion:  In further analysis of the output file and the 
             "practical" significance of p values:

             1.  There were nine cases where students who were      
                 assigned to a new partner had lower final exam     
                 scores in Cobol II than their original partner   
                 in Cobol I.


          8.78        9  - Ranks (WK_NEW LT WK_OWN)
         11.10       10  + Ranks (WK_NEW GT WK_OWN)
                      1    Ties  (WK_NEW EQ WK_OWN)

             2.  There were ten cases where students who were      
                 assigned to a new partner had higher final exam     
                 scores in Cobol II than their original partner   
                 in Cobol I.

             3.  There was one case where scores between the two
                 original partners from Cobol I were the same in    
                 Cobol II.

             With final examination scores all over the range,
             such as what you see here, it would be surprising 
             if differences between the two groups were due to
             anything but chance.


************
wilcoxon.lis
************
% minitab

 MTB > outfile 'wilcoxon.lis'
 Collecting Minitab session in file: wilcoxon.lis
 MTB > # MINITAB addendum for wilcoxon.dat
 MTB > read 'wilcoxon.dat' c1 c2 c3
 Entering data from file: wilcoxon.dat
      20 rows read.
 MTB > print c1 c2 c3
 
 
  ROW    C1    C2     C3
 
    1     1    90     83
    2     2    84     89
    3     3    78     58
    4     4    79     93
    5     5    65     49
    6     6    49     82
    7     7    92     93
    8     8    71    100
    9     9    85     83
   10    10    76     55
   11    11    68     65
   12    12    71     83
   13    13    88     81
   14    14    89     88
   15    15    76     83
   16    16    61     78
   17    17    88     88
   18    18    58     80
 Continue? y
   19    19    73     70
   20    20    82     83
 
 MTB > describe c2 c3
 
                 N     MEAN   MEDIAN   TRMEAN    STDEV   SEMEAN
 C2             20    76.15    77.00    76.78    11.75     2.63
 C3             20    79.20    83.00    79.72    13.34     2.98
 
               MIN      MAX       Q1       Q3
 C2          49.00    92.00    68.75    87.25
 C3          49.00   100.00    72.00    88.00
 
 MTB > winterval c2 c3
 
                 ESTIMATED    ACHIEVED
              N     MEDIAN  CONFIDENCE  CONFIDENCE INTERVAL
 C2          20       76.7        95.0  (    70.5,    82.0)
 C3          20       81.5        95.0  (    73.0,    85.5)
 MTB > mann-whitney c2 c3
 
 Mann-Whitney Confidence Interval and Test
 
 C2         N =  20     Median =       77.00
 C3         N =  20     Median =       83.00
 Point estimate for ETA1-ETA2 is       -4.00
 95.0 pct c.i. for ETA1-ETA2 is (-12.00,5.00)
 W = 377.0
 Test of ETA1 = ETA2  vs.  ETA1 n.e. ETA2 is significant at 0.3793
 The test is significant at 0.3785 (adjusted for ties)
 
 Cannot reject at alpha = 0.05
 

--------------------------
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 'wilcoxon.ssi'

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

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