chapter 7: correlation bivariate distribution: a distribution that shows the relation between two...
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Chapter 7: CorrelationBivariate distribution: a distribution that shows the relation between two variables
-2 -1.9 -1.8 -1.7 -1.6 -1.5 -1.4 -1.30.4
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Area of primary visual cortex
Vis
ual A
cuity
Left hemisphereRight hemisphere
This graph is called a scatter plot or scatter diagram
How do we quantify the strength of the relationship between the two variables in a bivariate distribution?
How do we quantify the strength of the relationship between the two variables in a bivariate distribution?
Example from the book:Two measures made for each subject – stress level and eating difficulties
Stress E.D.
17 9
8 13
8 7
20 18
14 11
7 1
21 5
22 15
19 26
30 28 5 10 15 20 25 30 35
5
10
15
20
25
Stress
Eat
ing
Diff
icu
ltie
s
The most common way to quantify the relation between the two variables in a bivariate distribution is the Pearson correlation coefficient, labeled r. r is always between -1 and 1.
The z-score formula is the most intuitive formula:
17 9
8 13
8 7
20 18
14 11
7 1
21 5
22 15
19 26
30 28
X Y
16.60
7.02
13.30
8.28
mx =
sx =
my =
sy =
zx zy zxzy
0.06 -0.52 -0.03
-1.23 -0.04 0.04
-1.23 -0.76 0.93
0.48 0.57 0.27
-0.37 -0.28 0.10
-1.37 -1.48 2.03
0.63 -1.00 -0.63
0.77 0.21 0.16
0.34 1.53 0.52
1.91 1.77 3.39
yxzz 6.68
raw scores z scores
Example: use the z-score formula to calculate r: n
zzr yx
68.0n
zzr yx
17 9
8 13
8 7
20 18
14 11
7 1
21 5
22 15
19 26
30 28
x y
0.06 -0.52 -0.03
-1.23 -0.04 0.04
-1.23 -0.76 0.93
0.48 0.57 0.27
-0.37 -0.28 0.10
-1.37 -1.48 2.03
0.63 -1.00 -0.63
0.77 0.21 0.16
0.34 1.53 0.52
1.91 1.77 3.39
zx zy zxzy
How does each data point contribute to the correlation value?
30
mx
my
Points in the upper right or lower left quadrants add to the correlation valuePoints in the upper left or lower right subtract to the correlation value.
5 10 15 20 25 30 35
5
10
15
20
25
Stress
Eat
ing
Diff
icu
ltie
s
r = 0.68
Fun fact about the Pearson correlation statistic
Since the z-scores do not change when you add or multiply the raw scores, the Pearson correlation doesn’t change either.
multiplying y by 2 and adding
100
10 20 30
5
10
15
20
25
Stress
Ea
ting
Diff
icu
ltie
s
r = 0.68
0 20 40
110
120
130
140
150
StressE
atin
g D
iffic
ulti
es
r = 0.68
n
zzr yx
Similarly, the correlation stays the same no matter how you stretch your axes:
As a rule, you should plot your axes with an equal scale.
10 20 30
5
10
15
20
25
StressE
atin
g D
iffic
ulti
es
r = 0.68
0 20 400
5
10
15
20
25
30
Stress
Eat
ing
Diff
icu
ltie
s
r = 0.68
5 10 15 20 25 300
10
20
30
Stress
Eat
ing
Diff
icu
ltie
s
r = 0.68
Guess that correlation!
50 55 60 65 70 75 80
55
60
65
70
75
Average of parent's height (in)
Stu
dent
's h
eig
ht (
in)
n = 90, r = 0.34
Guess that correlation!
58 60 62 64 66 68 70 72
66
68
70
72
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76
78
Father‘s height (in)
Mal
e s
tud
ent's
hei
ght (
in)
n = 21, r = 0.34
50 55 60 65 70 75 80 85
50
55
60
65
70
75
Mother's height (in)
Fe
ma
le s
tude
nt's
hei
ght (
in)
n = 70, r = 0.68
Guess that correlation!
2.5 3 3.5 4
2.5
3
3.5
4
High School GPA
UW
GP
A
n = 90, r = 0.19
Guess that correlation!
0 5 10 15 20 25
5
6
7
8
9
10
11
Caffeine (cups/day)
Sle
ep
(ho
urs/
nigh
t)
n = 91, r = -0.12
Guess that correlation!
0 5 10 15 20 25
0
5
10
15
20
25
30
Caffeine (cups/day)
Drin
ks (
per
we
ek)
n = 91, r = 0.01
Guess that correlation!
0 2 4 6 8
0
200
400
600
800
1000
1200
1400
1600
1800
2000
Facebook friends
Drin
ks (
per
we
ek)
n = 91, r = 0.10
Guess that correlation!
30 40 50 60 70 80 90
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Favorite outdoor temperature (F)
Vid
eo
gam
e p
layi
ng (
hou
rs/w
eek)
n = 91, r = -0.19
0 20 40 60 80 100
70
80
90
100
110
120
130
140
x
y
r = -0.56
Guess that correlation!
10 20 30 40 50 60
105
110
115
120
125
130
135
140
145
150
x
y
r = 0.94
Guess that correlation!
10 20 30 40 50 60 70 80 90
100
110
120
130
140
150
160
x
y
r = 0.08
Guess that correlation!
-20 -15 -10 -5 0 5
135
140
145
150
155
x
y
r = -1.00
Guess that correlation!
-40 -30 -20 -10 0 10 20 30 40
80
90
100
110
120
130
140
x
y
r = -0.08
Guess that correlation!
-50 0 50 100
80
100
120
140
160
180
200
220
240
x
y
r = 0.49
Guess that correlation!
-20 -10 0 10 20 30 40 50 60 700
10
20
30
40
50
60
70
x
y
r = -0.92
Guess that correlation!
-40 -20 0 20 40 60
130
140
150
160
170
180
190
200
210
220
x
y
r = -0.77
Guess that correlation!
r is a measure of the linear relation between two variables
-2 -1 0 1 2
0
0.5
1
1.5
2
2.5
3
3.5
4
x
y
r = 0.01
-1.5 -1 -0.5 0 0.5 1 1.5
-1
-0.5
0
0.5
1
x
y
r = 0.00
Guess that correlation!
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
x
y
r = 0.91
Guess that correlation!
n
zzr yx
Z-Score formula for calculating r (intuitive, but not very practical)
Deviation-Score formula for calculating r: (somewhat intuitive, somewhat more practical)
YX SnS
YYXXr
))((
Substituting the formula for z:
XS
XXz
Computational formula for calculating r: (less intuitive, more practical)
YX SSSS
YYXXr
))((
Computational formula for calculating r: (less intuitive, more practical)
YX SSSS
YYXXr
))((
A little algebra shows that:
n
YXXYYYXX ))((
Computational raw score formula for calculating r: (least intuitive, most practical)
YX SSSSn
YXXY
r
Using the Computational raw-score formula:
n X Y X2 Y2 XY10 17 9 289 81 153
8 13 64 169 1048 7 64 49 56
20 18 400 324 36014 11 196 121 154
7 2 49 4 1421 5 441 25 10522 15 484 225 33019 26 361 676 49430 28 900 784 840
Totals 166 134 3248 2458 2610
SSX 492.4SSy 662.4
r 0.675
yxSSSSn
YXXY
r
n
XXSSx
22
n
YYSSY
22
A second measure of correlation, called the Spearman Rank-Order Coefficient is appropriate for ordinal scores. It is calculated by:
Where D is the difference between each pair of ranks.
Most often used when:
a) At least one variable is an ordinal scaleb) One of the distributions is very skewed or has outliers
)1(
61
2
2
nn
Drs
Fact: (According to Wikipedia anyway)
In 1995, National Pax had planned to replace the "Sir Isaac Lime" flavor with "Scarlett O'Cherry," until a group of Orange County, California fourth-graders created a petition in opposition and picketed the company's headquarters in early 1996. The crusade also included an e-mail campaign, in which a Stanford professor reportedly accused the company of "Otter-cide." After meeting with the children, company executives relented and retained the Sir Isaac Lime flavor.[1]
Example: Is there a correlation between your preference for Otter Pops® flavors and mine?
Example: Suppose two wine experts were asked to rank-order their preference for eight wines. How can we measure the similarity of their rankings?
X Y Rank X Rank Y D D2
1 2 1 2 -1 12 1 2 1 1 13 5 3 5 -2 44 3 4 3 1 15 4 5 4 1 16 7 6 7 -1 17 8 7 8 -1 18 6 8 6 2 4
n=8 14 2D
833.)18(8
)14)(6(1
2
sr
)1(
61
2
2
nn
Drs
Pearson correlation is much more sensitive to outlying values than the Spearman coefficient.
From: http://en.wikipedia.org/wiki/Spearman%27s_rank_correlation_coefficient
Pearson correlation is much more sensitive to outlying values than the Spearman coefficient.
0 5 10 15 20
5
6
7
8
9
10
11
Caffeine (cups/day)
Sle
ep
(ho
urs/
nigh
t)
n = 91Pearson's r = -0.12
Spearman's rs = 0.02
0 5 10 15 20
5
6
7
8
9
10
11
Caffeine (cups/day)
Sle
ep
(ho
urs/
nigh
t)
n = 89Pearson's r = 0.06
Spearman's rs = 0.07
Only the rank order matters for the Spearman coefficient
-0.5 0 0.5
-0.5
0
0.5
1
X
Y
Pearson r: 0.92Spearman r s: 1.00
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