Spearman's Rank-Difference Coefficient of Correlation
© 1998 by Dr. Thomas W. MacFarland -- All Rights Reserved
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spearman.doc
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Background: Consider the degree of association between the
Grade Point Average of an individual student and
the family income for this student:
-- A student who comes from a wealthy family
may not have to work after school, but can
instead use this free time to study and
do extra homework problems.
-- A student who comes from a poor family
probably does not have this free time, but
instead needs to work after school and on
weekends to help the family with daily
expenses.
-- A student who comes from a wealthy family
quite possibly has their own computer and
is able to prepare programming homework problems
at any convenient time.
-- A student who comes from a poor family
probably does not have a computer at home,
but instead has to schedule "at computer"
time during open laboratory hours, which
may quite possibly limit time-on-task for
attempts at creative programming projects.
As such, it is not at all surprising that there
is historically a strong positive correlation
between Grade Point Average and family income:
-- Overally, for groups of students, Grade
Point Average increases as family income
increases.
-- Correspondingly, for groups of students,
Grade Pont Average decreases as family
income decreases.
The key points here are that:
-- Family income does not "cause" the Grade
Point Average. Instead, there is merely
an association between the two phenonema.
-- This degree of association is for a
collective "group" of subjects. Any one
individual could have results that are
different than group observations.
With the importance of relationships between
data in mind, it is often helpful to determine
if there is a positive or negative relationship
(i.e., association) between various phenomena.
Consider the following scenario:
1. There is a high "negative" correlation between
miles of paved roads and infant mortality:
-- Let X equal miles of paved roads
-- Let Y equal incidence of infant mortality
2. As X increases Y decreases. That is to
say, as the miles of paved roads in an area
increases, the rate of infant mortality
decreases.
Why? Well, that is a totally different concern. Do
not assume that cause and effect is in place here.
Sure, a few mothers may get to the hospital faster
(and therefore have children who survive birth), but
that is hardly the reason for the association.
Other factors such as economic wealth, societal
issues, and the general infrastructure are also prime
concerns.
Spearman's Rank-Difference Coefficient of Correlation
is viewed as the nonparametric test for determining
if there is an association between phenomena.
Please recall that the negative (- or decrease) and
positive (+ or increase) signs in correlation are
only used to suggest direction. The negative sign
does not mean "bad" and the positive sign does not
mean "good".
Y |*
| *
| *
| * As X increases Y decreases
| *
----------- Negative Correlation
X
Y | *
| *
| *
| * As X increases Y increases
|*
----------- Positive Correlation
X
Scenario: This study examines any possible correlation between
student performance on a mathematics mastery test and
later performance in a final examination in a C++
programming course.
The mathematics mastery test was prepared by faculty
at Warren County Community College. Because the
teacher conducting this analysis knows that this test
has not been subjected to any estimates of reliability
and validity, she thinks that it is best to view these
test scores as ordinal data (data are ordered, but not
with the precision of interval data).
Accordingly, she will use the nonparametric Spearman's
test, instead of Pearson's product moment correlation
coefficient test which requires the use of interval
data, to test the association between the two
phenonema:
-- Mathematics mastery test score
-- C++ final examination score
Data associated with this investigation are presented
in Table 1.
Table 1
Mathematics Mastery Test Scores and C++ Final
Examination Test Scores at Warren County
Community College
====================================================
Test Score
================================
Student Number Mathematics Test C++ Exam
----------------------------------------------------
01 089 082
02 084 078
03 083 092
04 093 085
05 076 091
06 086 075
07 092 091
08 090 086
09 085 082
10 082 087
11 085 093
12 069 072
13 088 069
14 073 080
15 064 076
16 076 084
17 081 063
18 068 082
19 078 073
20 095 092
----------------------------------------------------
Ho: Null Hypothesis: There is no association between
test scores on a mathematics mastery test and later
final examination test scores in a C++ programming
course final among students at Warren County
Community College (p = ,05).
Files: 1. spearman.doc
2. spearman.dat
3. spearman.r01
4. spearman.o01
5. spearman.con
6. spearman.lis
Command: At the Unix prompt (%), key:
%spss -m < spearman.r01 > spearman.o01
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spearman.dat
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01 089 082
02 084 078
03 083 092
04 093 085
05 076 091
06 086 075
07 092 091
08 090 086
09 085 082
10 082 087
11 085 093
12 069 072
13 088 069
14 073 080
15 064 076
16 076 084
17 081 063
18 068 082
19 078 073
20 095 092
************
spearman.r01
************
SET WIDTH = 80
SET LENGTH = NONE
SET CASE = UPLOW
SET HEADER = NO
TITLE = Spearman Rank Correlation Coefficient
COMMENT = This file examines any possible correlation
between student performance on a mathematics
mastery test and later performance in a
final examination in a C++ programming
course.
The mathematics mastery test was prepared by
faculty at Warren County Community College.
Because the teacher conducting this analysis
knows that this test has not been subjected to
any estimates of reliability and validity, she
thinks that it is best to view these test scores
as ordinal data (data are ordered, but not with
the precision of interval data).
Accordingly, she will use the nonparametric
Spearman's test, instead of Pearson's product moment
correlation coefficient test, which requires the use
of interval data.
DATA LIST FILE = 'spearman.dat' FIXED
/ Stu_Code 20-21
Math_Tst 40-42
Cpp_Test 55-57
Variable Lables
Stu_Code "Student Code"
/ Math_Tst "Score - Mathematics Mastery Test"
/ Cpp_Test "Score - C++ Test"
NONPAR CORR VARIABLES = Math_Tst WITH Cpp_Test
/ PRINT = SPEARMAN
************
spearman.o01
************
2 SET WIDTH = 80
3 SET LENGTH = NONE
4 SET CASE = UPLOW
5 SET HEADER = NO
6 TITLE = Spearman Rank Correlation Coefficient
7 COMMENT = This file examines any possible correlation
8 between student performance on a mathematics
9 mastery test and later performance in a
10 final examination in a C++ programming
11 course.
12
13 The mathematics mastery test was prepared by
14 faculty at Warren County Community College.
15 Because the teacher conducting this analysis
16 knows that this test has not been subjected to
17 any estimates of reliability and validity, she
18 thinks that it is best to view these test scores
19 as ordinal data (data are ordered, but not with
20 the precision of interval data).
21
22 Accordingly, she will use the nonparametric
23 Spearman's test, instead of Pearson's product moment
24 correlation coefficient test, which requires the use
25 of interval data.
26 DATA LIST FILE = 'spearman.dat' FIXED
27 / Stu_Code 20-21
28 Math_Tst 40-42
29 Cpp_Test 55-57
30
This command will read 1 records from spearman.dat
Variable Rec Start End Format
STU_CODE 1 20 21 F2.0
MATH_TST 1 40 42 F3.0
CPP_TEST 1 55 57 F3.0
31 Variable Lables
32 Stu_Code "Student Code"
33 / Math_Tst "Score - Mathematics Mastery Test"
34 / Cpp_Test "Score - C++ Test"
35
36 NONPAR CORR VARIABLES = Math_Tst WITH Cpp_Test
37 / PRINT = SPEARMAN
*** Workspace allows for 26214 cases for nonparametric correlation problem ***
- - - S P E A R M A N C O R R E L A T I O N C O E F F I C I E N T S - - -
CPP_TEST
MATH_TST .3601
N( 20)
Sig .119
************
spearman.con
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Outcome: Computed r = + .3601 or + .360
Criterion r (alpha = .05, n = 25) = + or - .377
Computed r (+ .360) < Criterion r (+ .377)
Therefore, there is no association between the test
score on a mathematics mastery test and the later
final examination test score in a C++ programming
class at Warren County Community College (p = .05).
The p value is another way to view differences in
the three graded activities:
-- The calculated p value is .119.
-- The delcared p value is .05.
The calculated p value exceeds the declared p value
and there is, accordingly, no association between
the two measures: mastery test scores and final
examination scores in a C++ programming course.
Note. As a general "rule of thumb," correlation is
often viewed along the following continuum:
+ .00 to + .30 = no positive correlation
between X and Y
- .00 to - .30 = no negative correlation
between X and Y
+ .40 to + .70 = mild positive correlation
between X and Y
- .40 to - .70 = mild negative correlation
between X and Y
+ .80 to + .99 = strong positive correlation
between X and Y
- .80 to - .99 = strong negative correlation
between X and Y
At the most, a correlation coefficient can only
reach -1.0 or + 1.0.
It may also be helpful to consider the sample
size when considering the efficacy or the "practical"
significance of a correlation design:
1. Correlation studies are very sensitive to n. That
is to say, a correlation of - .425 is not significant
(p = .05) when n = 15. But, a correlation of
- .425 is significant when n = 17.
Refer to a table on critical values of Spearman's
Rank Correlation Coefficient for criterion values.
2. Do not automatically think, however, that
increasing n will give you greater validity in
developing conclusions. A trivial study,
regardless of the magnitude of n, is still a
trivial study.
3. Finally, do not allow "cause and effect" to
creep into your decisions. Use caution when
making inferences about correlation.
Correlation does not imply cause and effect.
************
spearman.lis
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% minitab
MTB > outfile 'spearman.lis'
Collecting Minitab session in file: spearman.lis
MTB > # MINITAB addendum to spearman.dat
MTB > read 'spearman.dat' c1 c2 c3
Entering data from file: spearman.dat
20 rows read.
MTB > name c1 'Stu_Code' c2 'Math_Tst' c3 'Cpp_Test'
MTB > print c1 c2 c3
ROW Stu_Code Math_Tst Cpp_Test
1 1 89 82
2 2 84 78
3 3 83 92
4 4 93 85
5 5 76 91
6 6 86 75
7 7 92 91
8 8 90 86
9 9 85 82
10 10 82 87
11 11 85 93
12 12 69 72
13 13 88 69
14 14 73 80
15 15 64 76
16 16 76 84
17 17 81 63
18 18 68 82
Continue? y
19 19 78 73
20 20 95 92
MTB > # It is always helpful to plot data, when appropriate,
MTB > # to gain a sense of the association between variables.
MTB >
MTB > plot c3 c2
- *
- * * * *
90+
- *
Cpp_Test- * *
- *
- * * *
80+ *
- *
- * *
- *
- *
70+ *
-
-
- *
-
------+---------+---------+---------+---------+---------+Math_Tst
66.0 72.0 78.0 84.0 90.0 96.0
MTB > # And you can see the general way data appear when plotted.
MTB >
MTB > # With the Spearman test, you must first rank data before
MTB > # you can use the correlation command.
MTB >
MTB > rank the data in c2 and put into c6
MTB > rank the data in c3 and put into c7
MTB >
MTB > print c1-c7
ROW Stu_Code Math_Tst Cpp_Test C6 C7
1 1 89 82 16.0 10.0
2 2 84 78 11.0 7.0
3 3 83 92 10.0 18.5
4 4 93 85 19.0 13.0
5 5 76 91 5.5 16.5
6 6 86 75 14.0 5.0
7 7 92 91 18.0 16.5
8 8 90 86 17.0 14.0
9 9 85 82 12.5 10.0
10 10 82 87 9.0 15.0
11 11 85 93 12.5 20.0
12 12 69 72 3.0 3.0
13 13 88 69 15.0 2.0
14 14 73 80 4.0 8.0
15 15 64 76 1.0 6.0
16 16 76 84 5.5 12.0
17 17 81 63 8.0 1.0
18 18 68 82 2.0 10.0
Continue? y
19 19 78 73 7.0 4.0
20 20 95 92 20.0 18.5
* NOTE * One or more variables are undefined.
MTB > # Be sure to notice that rank was used, not the sort
MTB > # command.
MTB >
MTB > # Now that the data are in rank order, it is a simple
MTB > # task to use the correlation command. Be sure to
MTB > # remember that the data are ranked, so the correlation
MTB > # output will be the nonparametric Spearman and not
MTB > # the parametric Pearson coefficient of correlation.
MTB >
MTB > correlation of the data in c6 and c7
Correlation of C6 and C7 = 0.360
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 'spearman.ssi'