Types of correlation coefficients

Saudi Board of Preventive Medicine, Riyadh Ministry of Health, KSA
Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course
Correlation
Dr. S. A. Rizwan, M.D.,
Public Health Specialist (MOH),
Lecturer, Saudi Board of Preventive
Medicine - Joint Program Riyadh
Nov 2019 1

Learning objectives
• Describe the conditions in which correlation is
used
• Describe the steps in calculation of correlation
• Describe the various interpretations of
correlation coefficient
Nov 2019 2

Outline
• Questions answered by correlation
• Scatterplots
• An example
• The correlation coefficient
• Other kinds of correlations
• Factors affecting correlations
• Testing for significance
Nov 2019 3

The Question
• Are two variables related?
– Does one increase as the other increases?
• e. g. skills and income
– Does one decrease as the other increases?
• e. g. health problems and nutrition
• How can we get a numerical measure of the
degree of relationship?
Nov 2019 4

Scatterplots
• AKA scatter diagram or scattergram
• Graphically depicts the relationship between
two variables in two dimensional space
Nov 2019 5

Scatterplot:Video Games and Alcohol Consumption
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25
Average Hours of Video Games Per Week
AverageNumberofAlcoholicDrinks
PerWeek
Direct Relationship
Nov 2019 6

Scatterplot: Video Games and Test Score
0
10
20
30
40
50
60
70
80
90
100
0 5 10 15 20
Average Hours of Video Games Per Week
ExamScore
Inverse Relationship
Nov 2019 7

An Example
• Does smoking cigarettes increase systolic
blood pressure?
• Plotting number of cigarettes smoked per day
against systolic blood pressure
– Fairly moderate relationship
– Relationship is positive
Nov 2019 8

Trend?
SMOKING
3020100
SYSTOLIC
170
160
150
140
130
120
110
100
Nov 2019 9

Smoking and BP
• Note relationship is moderate, but real.
• Why do we care about relationship?
– What would conclude if there were no
relationship?
– What if the relationship were near perfect?
– What if the relationship were negative?
Nov 2019 10

Heart Disease and Cigarettes
• Data on heart disease
and cigarette smoking in
21 developed countries
(Landwehr and Watkins,
1987)
Country Cigarettes CHD
1 11 26
2 9 21
3 9 24
4 9 21
5 8 19
6 8 13
7 8 19
8 6 11
9 6 23
10 5 15
11 5 13
12 5 4
13 5 18
14 5 12
15 5 3
16 4 11
17 4 15
18 4 6
19 3 13
20 3 4
21 3 14
Nov 2019 11

Scatterplot of Heart Disease
• CHD Mortality goes on ordinate (Y axis)
– Why?
• Cigarette consumption on abscissa (X axis)
– Why?
• What does each dot represent?
• Best fitting line included for clarity
Nov 2019 12

Cigarette Consumption per Adult per Day
12108642
30
20
10
0
{X = 6, Y =
11}
Nov 2019 13

What Does the Scatterplot Show?
• As smoking increases, so does coronary heart
disease mortality
• Relationship looks strong
• Not all data points on line
– This gives us “residuals” or “errors of prediction”
Nov 2019 14

Correlation
• Co-relation
• The relationship between two variables
• Measured with a correlation coefficient
• Most popularly seen correlation coefficient:
Pearson Product-Moment Correlation
Nov 2019 15

Types of Correlation
• Positive correlation
– High values of X tend to be associated with high
values of Y.
– As X increases, Y increases
• Negative correlation
– High values of X tend to be associated with low values
of Y.
– As X increases, Y decreases
• No correlation
• No consistent tendency for values on Y to
increase or decrease as X increases
Nov 2019 16

Correlation Coefficient
• A measure of degree of relationship.
• Between 1 and -1
• Sign refers to direction.
• Based on covariance
– Measure of degree to which large scores on X go
with large scores on Y, and small scores on X go
with small scores on Y
– Think of it as variance, but with 2 variables instead
of 1
Nov 2019 17

Dr. S. A. Rizwan, M.D.Demystifying statistics series: Meta-analysis course Nov 2019 18

Covariance
• Remember that variance is:
• The formula for co-variance is:
2
( ) ( )( )
1 1
X
X X X X X X
Var
N N
    
 
 
1
))((



N
YYXX
CovXY
Nov 2019 19

Example
Country X (Cig.) Y (CHD) ( )X X ( )Y Y ( )X X * ( )Y Y
1 11 26 5.05 11.48 57.97
2 9 21 3.05 6.48 19.76
3 9 24 3.05 9.48 28.91
4 9 21 3.05 6.48 19.76
5 8 19 2.05 4.48 9.18
6 8 13 2.05 -1.52 -3.12
7 8 19 2.05 4.48 9.18
8 6 11 0.05 -3.52 -0.18
9 6 23 0.05 8.48 0.42
10 5 15 -0.95 0.48 -0.46
11 5 13 -0.95 -1.52 1.44
12 5 4 -0.95 -10.52 9.99
13 5 18 -0.95 3.48 -3.31
14 5 12 -0.95 -2.52 2.39
15 5 3 -0.95 -11.52 10.94
16 4 11 -1.95 -3.52 6.86
17 4 15 -1.95 0.48 -0.94
18 4 6 -1.95 -8.52 16.61
19 3 13 -2.95 -1.52 4.48
20 3 4 -2.95 -10.52 31.03
21 3 14 -2.95 -0.52 1.53
Mean 5.95 14.52
SD 2.33 6.69
Sum 222.44Nov 2019 20

Example
.&
( )( ) 222.44
11.12
1 21 1
cig CHD
X X Y Y
Cov
N
  
  
 
Nov 2019 21

• Pearson’s Product Moment Correlation
• Symbolized by r
• Covariance ÷ (product of the 2 SDs)
• Correlation is a standardized covariance
YX
XY
ss
Cov
r 
Nov 2019 22

Calculation for Example
• CovXY = 11.12
• sX = 2.33
• sY = 6.69
cov 11.12 11.12
.713
(2.33)(6.69) 15.59
XY
X Y
r
s s
   
Nov 2019 23

Example
• Correlation = 0.713
• Sign is positive
– Why?
• If sign were negative
– What would it mean?
– Would not alter the degree of relationship.
Nov 2019 24

Other Kinds of Correlation
Attractiveness Symmetry
3 2
4 6
1 1
2 3
5 4
6 5
rsp = 0.77
• Spearman Rank-Order
(rsp)
– used with 2
ranked/ordinal variables
– uses the same Pearson
formula
Nov 2019 25

Attractiveness Date?
3 0
4 0
1 1
2 1
5 1
6 0
rpb = -0.49
• Point biserial
(rpb)
– used with one
continuous scale and
one nominal or ordinal
or dichotomous scale.
formula
Nov 2019 26

Attractiveness Date?
0 0
1 0
1 1
1 1
0 0
1 1
F = 0.71
• Phi coefficient (F)
– used with two
dichotomous scales.
formula
Nov 2019 27

Factors Affecting r
• Range restrictions
– Looking at only a small portion of the total scatter plot
(looking at a smaller portion of the scores’ variability)
decreases r.
– Reducing variability reduces r
• Nonlinearity
– The Pearson r (and its relatives) measure the degree
of linear relationship between two variables
– If a strong non-linear relationship exists, r will provide
a low, or at least inaccurate measure of the true
relationship
Nov 2019 28

Factors Affecting r
• Heterogeneous subsamples
– Everyday examples (e.g. height and weight using
both men and women)
• Outliers
– Overestimate Correlation
– Underestimate Correlation
Nov 2019 29

Countries With Low Consumptions
Data With Restricted Range
Truncated at 5 Cigarettes Per Day
Cigarette Consumption per Adult per Day
5.55.04.54.03.53.02.5
CHDMortalityper10,000
20
18
16
14
12
10
8
6
4
2
Nov 2019 30

Truncation
Nov 2019 31

Non-linearity
Nov 2019 32

Heterogenous samples
Nov 2019 33

Outliers
Nov 2019 34

Testing Correlations
• So you have a correlation. Now what?
• In terms of magnitude, how big is big?
– Small correlations in large samples are “big”
– Large correlations in small samples aren’t always
“big”
• Depends upon the magnitude of the
• AND
• The size of your sample
Nov 2019 35

Testing r
• Population parameter = 
• Null hypothesis H0:  = 0
– Test of linear independence
• Alternative hypothesis (H1)   0
– Two-tailed
Nov 2019 36

Tables of Significance
• We can convert r to t and test for significance:
• Where DF = N - 2
t = r
N -2
1- r2
Nov 2019 37

Tables of Significance
• In our example r was 0.71
• N - 2 = 21 – 2 = 19
• T-crit (19) = 2.09
• Since 6.90 is larger than 2.09, reject r = 0.
2 2
2 19 19
.71* .71* 6.90
1 1 .71 .4959
N
t r
r

   
 
Nov 2019 38

Take home messages
• Correlation coefficient describes the
relationship between two variables
• There are different types of correlation
coefficient
• Pearson’s correlation coefficient is used for
two continuous variables
• It ranges from -1 to +1
Nov 2019 39

THANK YOU
Kindly send your queries to sarizwan1986@gmail.com
Nov 2019 40

Types of correlation coefficients

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