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Spearman’s Ranked Correlation
Spearman Ranked Correlation
If the data are not normally distributed one can use ranked data to determine the correlation coefficient.
where d2 is the squared difference in ranks for each i observation and n is the sample size. Remember to rank within the columns.
nn
d
r
n
i
i
s
3
1
26
1
Remember that when the ranks are tied we assign the average rank to those values.
Technically, if there are many tied ranks we could use the Pearson equation using the ranks as observations, since it is a more powerful test.
However, either one could be used… just recognize thatthe Spearman test is less powerful with many tied ranks.
If there are few or no tied ranks, the Spearman test is moreappropriate.
Emergency Room Hospital Admissions for AsthmaPM10 Non-Attainment Event: December 8, 1992
Tests of Normality
.171 69 .000 .865 69 .000
.211 69 .000 .904 69 .000
Asthma
PM10
Stat istic df Sig. Stat istic df Sig.
Kolmogorov-Smirnova
Shapiro-Wilk
Lillief ors Signif icance Correctiona.
Asthma Stem-and-Leaf Plot
Frequency Stem & Leaf
20.00 0 . 11111111111111111111
13.00 0 . 2222222233333
10.00 0 . 4445555555
8.00 0 . 66677777
9.00 0 . 888999999
3.00 1 . 111
2.00 1 . 33
2.00 1 . 44
1.00 1 . 6
1.00 1 . 8
Stem width: 10
Each leaf: 1 case(s)
PM10 Stem-and-Leaf Plot
Frequency Stem & Leaf
3.00 Extremes (=<18)
5.00 0 . 33333
5.00 0 . 55555
4.00 0 . 7777
8.00 0 . 99999999
9.00 1 . 111111111
19.00 1 . 3333333333333333333
13.00 1 . 5555555555555
3.00 1 . 777
Stem width: 100
Each leaf: 1 case(s)
Asthma PM10
Asthma
Rank
Pm10
Rank di di2 Asthma PM10
Asthma
Rank
Pm10
Rank di di2
1 17.5 10.5 2 8.5 72.25 1 134 10.5 44 -33.5 1122.25
2 17.5 24.5 2 22.5 506.25 1 134 10.5 44 -33.5 1122.25
2 17.5 24.5 2 22.5 506.25 1 134 10.5 44 -33.5 1122.25
1 37.5 10.5 6 4.5 20.25 1 134 10.5 44 -33.5 1122.25
1 37.5 10.5 6 4.5 20.25 1 134 10.5 44 -33.5 1122.25
3 37.5 31 6 25 625 2 134 24.5 44 -19.5 380.25
3 37.5 31 6 25 625 3 134 31 44 -13 169
4 37.5 35 6 29 841 4 134 35 44 -9 81
1 57 10.5 11 -0.5 0.25 5 134 39.5 44 -4.5 20.25
1 57 10.5 11 -0.5 0.25 6 134 45 44 1 1
2 57 24.5 11 13.5 182.25 7 134 49 44 5 25
5 57 39.5 11 28.5 812.25 7 134 49 44 5 25
9 57 57.5 11 46.5 2162.25 8 134 53 44 9 81
1 76 10.5 15.5 -5 25 9 134 57.5 44 13.5 182.25
1 76 10.5 15.5 -5 25 9 134 57.5 44 13.5 182.25
6 76 45 15.5 29.5 870.25 11 134 62 44 18 324
7 76 49 15.5 33.5 1122.25 13 134 64.5 44 20.5 420.25
1 95 10.5 21.5 -11 121 16 134 68 44 24 576
1 95 10.5 21.5 -11 121 1 154 10.5 60 -49.5 2450.25
1 95 10.5 21.5 -11 121 2 154 24.5 60 -35.5 1260.25
5 95 39.5 21.5 18 324 2 154 24.5 60 -35.5 1260.25
6 95 45 21.5 23.5 552.25 3 154 31 60 -29 841
8 95 53 21.5 31.5 992.25 4 154 35 60 -25 625
9 95 57.5 21.5 36 1296 5 154 39.5 60 -20.5 420.25
13 95 64.5 21.5 43 1849 5 154 39.5 60 -20.5 420.25
1 115 10.5 30 -19.5 380.25 8 154 53 60 -7 49
1 115 10.5 30 -19.5 380.25 9 154 57.5 60 -2.5 6.25
2 115 24.5 30 -5.5 30.25 9 154 57.5 60 -2.5 6.25
2 115 24.5 30 -5.5 30.25 11 154 62 60 2 4
3 115 31 30 1 1 14 154 66.5 60 6.5 42.25
5 115 39.5 30 9.5 90.25 18 154 69 60 9 81
5 115 39.5 30 9.5 90.25 1 173 10.5 68 -57.5 3306.25
11 115 62 30 32 1024 7 173 49 68 -19 361
14 115 66.5 30 36.5 1332.25 7 173 49 68 -19 361
1 134 10.5 44 -33.5 1122.25
31.0
6914.01
328440
5.2270771
6969
)6)(25.37846(1
3
s
s
s
s
r
r
r
r
nn
d
r
n
i
i
s
3
1
26
1
1 nrt s
To test the significance of the Spearman’s correlation coefficient, use the t statistic calculated as:
where rs is the correlation coefficient and n is the sample size.
1 nrt s
681
56.2
)246.8(31.0
16931.0
ndf
t
t
t
tcritical = 1.995 Since 2.56 > 1.995 reject Ho.
Asthma hospital admissions and zipcode average PM10 levelsduring the 1992 Denver non-attainment event were significantlycorrelated (t2.56,0.02 > p > 0.01, rs = 0.31)
Correlations
1.000 .287*
. .017
69 69
.287* 1.000
.017 .
69 69
Correlation Coef f icient
Sig. (2-tailed)
N
Correlation Coef f icient
Sig. (2-tailed)
N
Asthma
PM10
Spearman's rho
Asthma PM10
Correlation is signif icant at the 0.05 level (2-tailed).*.
The difference between our results (rs = 0.31) and the SPSS results (rs = 0.287) is that SPSS provides a correction factor for the tied ranks.
SPSS Output
.12
)(
.12
)(
26/)(26/)(
)6/)((
3
3
33
23
tiesYgroupofnumbertheistwherett
T
and
tiesXgroupofnumbertheistwherett
T
where
TnnTnn
TTdnnr
i
ii
y
i
ii
x
yx
yxi
s
Spearman’s r Correction for Tied Ranks
Asthma PM10
Asthma
Rank
Pm10
Rank di di2 Asthma PM10
Asthma
Rank
Pm10
Rank di di2
1 17.5 10.5 2 8.5 72.25 4 154 35 60 -25 625
1 37.5 10.5 6 4.5 20.25 5 57 39.5 11 28.5 812.25
1 37.5 10.5 6 4.5 20.25 5 95 39.5 21.5 18 324
1 57 10.5 11 -0.5 0.25 5 115 39.5 30 9.5 90.25
1 57 10.5 11 -0.5 0.25 5 115 39.5 30 9.5 90.25
1 76 10.5 15.5 -5 25 5 134 39.5 44 -4.5 20.25
1 76 10.5 15.5 -5 25 5 154 39.5 60 -20.5 420.25
1 95 10.5 21.5 -11 121 5 154 39.5 60 -20.5 420.25
1 95 10.5 21.5 -11 121 6 76 45 15.5 29.5 870.25
1 95 10.5 21.5 -11 121 6 95 45 21.5 23.5 552.25
1 115 10.5 30 -19.5 380.25 6 134 45 44 1 1
1 115 10.5 30 -19.5 380.25 7 76 49 15.5 33.5 1122.25
1 134 10.5 44 -33.5 1122.25 7 134 49 44 5 25
1 134 10.5 44 -33.5 1122.25 7 134 49 44 5 25
1 134 10.5 44 -33.5 1122.25 7 173 49 68 -19 361
1 134 10.5 44 -33.5 1122.25 7 173 49 68 -19 361
1 134 10.5 44 -33.5 1122.25 8 95 53 21.5 31.5 992.25
1 134 10.5 44 -33.5 1122.25 8 134 53 44 9 81
1 154 10.5 60 -49.5 2450.25 8 154 53 60 -7 49
1 173 10.5 68 -57.5 3306.25 9 57 57.5 11 46.5 2162.25
2 17.5 24.5 2 22.5 506.25 9 95 57.5 21.5 36 1296
2 17.5 24.5 2 22.5 506.25 9 134 57.5 44 13.5 182.25
2 57 24.5 11 13.5 182.25 9 134 57.5 44 13.5 182.25
2 115 24.5 30 -5.5 30.25 9 154 57.5 60 -2.5 6.25
2 115 24.5 30 -5.5 30.25 9 154 57.5 60 -2.5 6.25
2 134 24.5 44 -19.5 380.25 11 115 62 30 32 1024
2 154 24.5 60 -35.5 1260.25 11 134 62 44 18 324
2 154 24.5 60 -35.5 1260.25 11 154 62 60 2 4
3 37.5 31 6 25 625 13 95 64.5 21.5 43 1849
3 37.5 31 6 25 625 13 134 64.5 44 20.5 420.25
3 115 31 30 1 1 14 115 66.5 30 36.5 1332.25
3 134 31 44 -13 169 14 154 66.5 60 6.5 42.25
3 154 31 60 -29 841 16 134 68 44 24 576
4 37.5 35 6 29 841 18 154 69 60 9 81
4 134 35 44 -9 81
20 tied ranks
5.781
12
)22()22()33()66()33()55()33()77()33()55()88()2020( 333333333333
x
x
T
T
Do this for each set of tied ranks.
886
12
)33()1313()1919()99()88()44()55()55()33( 333333333
y
y
T
T
287.0
4.53072
25.15226
)52968)(53177(
25.15226
)886(26/)6969()5.781(26/)6969(
8865.78125.37846)6/)6969((
33
3
s
s
s
s
r
r
r
r
Correlations
1.000 .287*
. .017
69 69
.287* 1.000
.017 .
69 69
Correlation Coef f icient
Sig. (2-tailed)
N
Correlation Coef f icient
Sig. (2-tailed)
N
Asthma
PM10
Spearman's rho
Asthma PM10
Correlation is signif icant at the 0.05 level (2-tailed).*.
If there are a significant number of tied ranks (as in our example),the Pearson’s correlation methods can be used on the ranks andthe results will be quite close:
Correlations
1 .286*
.017
69 69
.286* 1
.017
69 69
Pearson Correlation
Sig. (2-tailed)
N
Pearson Correlation
Sig. (2-tailed)
N
AsthmaRank
PM10Rank
AsthmaRank PM10Rank
Correlation is signif icant at the 0.05 level (2-tailed).*.
Correlations
1.000 .287*
. .017
69 69
.287* 1.000
.017 .
69 69
Correlation Coef f icient
Sig. (2-tailed)
N
Correlation Coef f icient
Sig. (2-tailed)
N
Asthma
PM10
Spearman's rho
Asthma PM10
Correlation is signif icant at the 0.05 level (2-tailed).*.
Correlation Among Dichotomous Variables:Cramér Coefficient (φ)
The Cramér φ (pronounced fy as in simplify) statistic is used to determine whether correlation exists between two dichotomous variables:
• Yes, no• Presence, absence• Etc…
Such analysis is often called contingency correlation since the data are represented by a 2x2 contingency table.
Just as with Pearson’s correlation, Cramér’s correlation coefficient ranges between -1 and 1, with 0 meaning the complete absence of association.
The variables are positively associated if most of the data falls along the left to right diagonal cells and negatively associated if most of the data falls along the right to left diagonal cells.
The variables are considered to have no association if most of the data fall along the rows or columns.
2121
21122211
RRCC
ffff
where f are the cell values, C are the column totals, and R are the row totals.
Always set f11 and f22 to be the frequencies of agreement between the two variables.
Present Absent Column
Total
Present f11 f12 R1
Absent f21 f22 R2
Row Total C1 C2
Cramér’s φ equation:
Mining Town Gold Silver
Peru Y N
Argentine N Y
Chihuahua Y Y
Tiger Y N
Swanville Y N
Parkville Y Y
Rexford Y N
Swandyke Y N
Spencerville Y N
Lincoln Y N
Dyersville Y N
Montgomery Y Y
Quartzville N Y
Dudley N Y
Silverheels Y N
Tarryall Y N
Hamilton Y N
Park City N N
Holland Y N
Sacramento N Y
Georgetown Y Y
Montezuma Y N
Breckenridge Y Y
Alma N Y
Fairplay Y Y
Delaware Flats Y Y
Buckskin Joe Y N
Swan City Y N
Wild Irishman N Y
Mitchell Cabins Y Y
Saints John Y Y
Silver Plume N Y
Conger Y Y
Waldorf Y Y
Santiago Y Y
Horseshoe N Y
Mudsill N Y
Leavick N Y
Mosquito Y N
Colorado Mining Towns (late 1800s)
Research Question:Is there correlation between the occurrence of gold and silveramong the mining towns.
Ho: There is no association between the occurrence of gold and silver among the mining towns.
Ha: There is association between the occurrence of gold and silver among the mining towns.
Mining Town Gold Silver Cell Code
Peru Y N 21
Argentine N Y 12
Chihuahua Y Y 11
Tiger Y N 21
Swanville Y N 21
Parkville Y Y 11
Rexford Y N 21
Swandyke Y N 21
Spencerville Y N 21
Lincoln Y N 21
Dyersville Y N 21
Montgomery Y Y 11
Quartzville N Y 12
Dudley N Y 12
Silverheels Y N 21
Tarryall Y N 21
Hamilton Y N 21
Park City N N 22
Holland Y N 21
Sacramento N Y 12
Georgetown Y Y 11
Montezuma Y N 21
Breckenridge Y Y 11
Alma N Y 12
Fairplay Y Y 11
Delaware Flats Y Y 11
Buckskin Joe Y N 21
Swan City Y N 21
Wild Irishman N Y 12
Mitchell Cabins Y Y 11
Saints John Y Y 11
Silver Plume N Y 12
Conger Y Y 11
Waldorf Y Y 11
Santiago Y Y 11
Horseshoe N Y 12
Mudsill N Y 12
Leavick N Y 12
Mosquito Y N 21
Gold (YES) Gold (NO) Row TotalSilver (YES) 12 10 22Silver (NO) 16 1 17Column Total 28 11 39
Contingency Table
436.0
4.339
148
)17)(22)(11)(28(
)16)(10()1)(12(
The probability of Cramer’s Φ can be determined by employing theχ2 statistic for the contingency table using the equation:
where fij are the observed values for row i and column j, and are the expected values for row i and column j which are calculated for each cell as:
Degrees of Freedom (DF) = (R-1)(C-1)
ij
ijij
f
ff
ˆ
)ˆ( 2
2
i jf̂
n
CR ji
ij
))((ˆ f
OBSERVED Gold (YES) Gold (NO) Row TotalSilver (YES) 12 10 22Silver (NO) 16 1 17Column Total 28 11 39
EXPECTED Gold (YES) Gold (NO)
Silver (YES) 15.8 6.2
Silver (NO) 12.2 4.8
841.3
1)12)(12(
42.7
00.333.218.191.0
8.4
)8.41(
2.6
)2.610(
2.12
)2.1216(
8.15
)8.1512(
8.439
171121.12
39
17282.6
39
22118.15
39
2228
2
2
2
22222
22211211
Critical
df
ffff
Since 7.42 > 3.841 reject Ho.
There is an inverse association between the occurrence of gold and silver among the mining towns (χ2
7.42, 0.01 > p > 0.005, φ = -0.436).
Symmetric Measures
-.436 .006
.436 .006
39
Phi
Cramer's V
Nominal by
Nominal
N of Valid Cases
Value Approx. Sig.
Not assuming the null hy pothesis.a.
Using the asymptotic standard error assuming the null
hypothesis.
b.
SPSS Results
In SPSS Cramer’s φ is found under:
Analyze > Descriptive Statistics > Crosstabs > Statistics
