Chapter 7: Correlation

Chapter 7: CorrelationBivariate distribution: a distribution that shows the relation between two variables

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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.

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8 13

8 7

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14 11

7 1

21 5

22 15

19 26

30 28 5 10 15 20 25 30 35

5

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Stress

Eat

ing

Diff

icul

ties

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

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21 5

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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: nzz

r yx

68.0nzz

r yx

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8 13

8 7

20 18

14 11

7 1

21 5

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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.

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Stress

Eat

ing

Diff

icul

ties

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

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10 20 30

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25

Stress

Eat

ing

Diff

icul

ties

r = 0.68

0 20 40

110

120

130

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StressE

atin

g D

iffic

ultie

s

r = 0.68

nzz

r 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

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StressE

atin

g D

iffic

ultie

s

r = 0.68

0 20 400

5

10

15

20

25

30

Stress

Eat

ing

Diff

icul

ties

r = 0.68

5 10 15 20 25 300

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20

30

Stress

Eat

ing

Diff

icul

ties

r = 0.68

Guess that correlation!

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Average of parent's height (in)

Stu

dent

's h

eigh

t (in

)

n = 90, r = 0.34


58 60 62 64 66 68 70 72

66

68

70

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Father‘s height (in)

Mal

e st

uden

t's h

eigh

t (in

)

n = 21, r = 0.34

50 55 60 65 70 75 80 85

50

55

60

65

70

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Mother's height (in)

Fem

ale

stud

ent's

hei

ght (

in)

n = 70, r = 0.68


2.5 3 3.5 42.5

3

3.5

4

High School GPA

UW

GPA

n = 90, r = 0.19


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Caffeine (cups/day)

Sle

ep (h

ours

/nig

ht)

n = 91, r = -0.12


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0

5

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15

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25

30

Caffeine (cups/day)

Drin

ks (p

er w

eek)

n = 91, r = 0.01


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0

200

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Facebook friends

Drin

ks (p

er w

eek)

n = 91, r = 0.10


30 40 50 60 70 80 90

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1

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2.5

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Favorite outdoor temperature (F)

Vid

eo g

ame

play

ing

(hou

rs/w

eek)

n = 91, r = -0.19

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70

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90

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120

130

140

x

y

r = -0.56


10 20 30 40 50 60

105

110

115

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140

145

150

x

y

r = 0.94


10 20 30 40 50 60 70 80 90

100

110

120

130

140

150

160

x

y

r = 0.08


-20 -15 -10 -5 0 5

135

140

145

150

155

x

y

r = -1.00


-40 -30 -20 -10 0 10 20 30 40

80

90

100

110

120

130

140

x

y

r = -0.08


-50 0 50 10080

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x

y

r = 0.49


-20 -10 0 10 20 30 40 50 60 700

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x

y

r = -0.92


-40 -20 0 20 40 60130

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x

y

r = -0.77


r is a measure of the linear relation between two variables

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x

y

r = 0.01

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-1

-0.5

0

0.5

1

x

y

r = 0.00


-1 -0.5 0 0.5 1-1

-0.8

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0

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1

x

y

r = 0.91


nzz

r yx

Z-Score formula for calculating r (intuitive, but not very practical)

Deviation-Score formula for calculating r: (somewhat intuitive, somewhat more practical)

YX SnSYYXX

r

))((

Substituting the formula for z:

XSXXz

Computational formula for calculating r: (less intuitive, more practical)

YX SSSSYYXX

r

))((

Computational formula for calculating r: (less intuitive, more practical)

YX SSSSYYXX

r

))((

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

YXXYr

nXXSSx

22

nYYSSY

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(6

1 2

2

nnD

rs

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?

http://en.wikipedia.org/wiki/Orange_County,_California

http://en.wikipedia.org/wiki/Stanford

http://en.wikipedia.org/wiki/Otter_Pops

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(6

1 2

2

nnD

rs

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.

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Caffeine (cups/day)

Sle

ep (h

ours

/nig

ht)

n = 91Pearson's r = -0.12

Spearman's rs = 0.02

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5

6

7

8

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11

Caffeine (cups/day)

Sle

ep (h

ours

/nig

ht)

n = 89Pearson's r = 0.06

Spearman's rs = 0.07

Only the rank order matters for the Spearman coefficient

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1

X

Y

Pearson r: 0.92Spearman r s: 1.00

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Chapter 7: Correlation