Pearson Correlation, Spearman Correlation &Linear Regression

Correlation (Pearson & Spearman)
& Linear Regression

Azmi Mohd Tamil

Key Concepts

Correlation as a statistic
Positive and Negative Bivariate Correlation
Range Effects
Outliers
Regression & Prediction
Directionality Problem ( & cross-lagged panel)
Third Variable Problem (& partial correlation)

Assumptions

Related pairs
Scale of measurement. For Pearson, data
should be interval or ratio in nature.
Normality
Linearity
Homocedasticity

Example of Non-Linear Relationship
Yerkes-Dodson Law – not for correlation

Better

Performance

Worse

Low
Stress High

Correlation

X Y
Stress Illness

Correlation – parametric & non-para

2 Continuous Variables - Pearson
linear relationship
e.g., association between height and weight

1 Continuous, 1 Categorical Variable
(Ordinal) Spearman/Kendall
–e.g., association between Likert Scale on work
satisfaction and work output
–pain intensity (no, mild, moderate, severe) and
severe
dosage of pethidine

Pearson Correlation

2 Continuous Variables
– linear relationship
– e.g., association between height and weight, +

measures the degree of linear association
between two interval scaled variables
analysis of the relationship between two
quantitative outcomes, e.g., height and weight,

Pearson Correlation, Spearman Correlation &Linear Regression

How to calculate r?

df = np - 2

How to calculate r?
a

b c

Example

•∑ x = 4631 ∑ x2 = 688837
•∑ y = 2863 ∑ y2 = 264527
•∑ xy = 424780 n = 32

•a=424780-(4631*2863/32)=10,450.22
•b=688837-46312/32=18,644.47
•c=264527-28632/32=8,377.969
•r=a/(b*c)0.5
=10,450.22/(18,644.47*83,77.969)0.5
=0.836144

•t= 0.836144*((32-2)/(1-0.8361442))0.5
t = 8.349436 & d.f. = n - 2 = 30,
p < 0.001

We refer to Table A3.
so we use df=30 .
t = 8.349436 > 3.65 (p=0.001)

Therefore if t=8.349436, p<0.001.

Correlation

Two pieces of information:
The strength of the relationship
The direction of the relationship

Strength of relationship

r lies between -1 and 1. Values near 0
means no (linear) correlation and values
near ± 1 means very strong correlation.

-1.0 0.0 +1.0
Strong Negative No Rel. Strong Positive

How to interpret the value of r?

Correlation ( + direction)

Positive correlation:
high values of one
variable associated with
high values of the other
Example: Higher
course entrance exam
scores are associated
with better course
grades during the final Positive and Linear
exam.

Correlation ( - direction)

Negative correlation: The
negative sign means that
the two variables are
inversely related, that is,
as one variable increases
the other variable
decreases.
Example: Increase in
body mass index is
associated with reduced Negative and Linear
effort tolerance.

Pearson’s r

A 0.9 is a very strong positive association (as one
variable rises, so does the other)
A -0.9 is a very strong negative association
(as one variable rises, the other falls)

r=0.9 has nothing to do with 90%
r=correlation coefficient

Coefficient of Determination
Defined

Pearson’s r can be squared , r 2, to derive a
coefficient of determination.

Coefficient of determination – the portion of
variability in one of the variables that can be
accounted for by variability in the second
variable


Pearson’s r can be squared , r 2, to derive a
coefficient of determination.

Example of depression and CGPA
– Pearson’s r shows negative correlation, r=-0.5
– r2=0.25

– In this example we can say that 1/4 or 0.25 of the variability
in CGPA scores can be accounted for by depression
(remaining 75% of variability is other factors, habits, ability,
motivation, courses studied, etc)

and Pearson’s r

Pearson’s r can be squared , r 2

If r=0.5, then r2=0.25
If r=0.7 then r2=0.49

Thus while r=0.5 versus 0.7 might not look so
different in terms of strength, r2 tells us that r=0.7
accounts for about twice the variability relative to
r=0.5

A study was done to find the association between
the mothers’ weight and their babies’ birth weight.
The following is the scatter diagram showing the
relationship between the two variables.
5.00
The coefficient of
correlation (r) is
4.00
0.452
The coefficient of
Baby's Birthweight

determination (r2)
3.00 is 0.204
Twenty percent of
2.00
the variability of
the babies’ birth
weight is
1.00 determined by the
R Sq Linear = 0.204 variability of the
mothers’ weight.
0.00
0.0 20.0 40.0 60.0 80.0 100.0

Mother's Weight

Causal Silence:
Correlation Does Not Imply Causality

Causality – must demonstrate that variance in one
variable can only be due to influence of the other
variable

Directionality of Effect Problem

Third Variable Problem

CORRELATION DOES NOT MEAN
CAUSATION

A high correlation does not give us the evidence to make a cause-
and-effect statement.
A common example given is the high correlation between the cost of
damage in a fire and the number of firemen helping to put out the
fire.
Does it mean that to cut down the cost of damage, the fire
department should dispatch less firemen for a fire rescue!
The intensity of the fire that is highly correlated with the cost of
damage and the number of firemen dispatched.
The high correlation between smoking and lung cancer. However,
one may argue that both could be caused by stress; and smoking
does not cause lung cancer.
In this case, a correlation between lung cancer and smoking may be
a result of a cause-and-effect relationship (by clinical experience +
common sense?). To establish this cause-and-effect relationship,
controlled experiments should be performed.

More
Big Fire Firemen
Sent

More
Damage


X Y

X Y

X Y


X Y
Class Higher
Higher
Attendance Grades

X Y
Class Higher
Higher
Attendance Grades


X Y
Aggressive Behavior Viewing Violent TV
Viewing Violent TV

X Y
Aggressive Behavior Viewing Violent TV
Viewing Violent TV

Aggressive children may prefer violent programs or
Violent programs may promote aggressive behavior

Methods for Dealing with
Directionality

Cross-Lagged Panel design
– A type of longitudinal design
– Investigate correlations at several points in time
– STILL NOT CAUSAL

Example next page

Cross-Lagged Panel

Pref for violent TV .05 Pref for violent TV
3rd grade 13th grade

.21 .31 .01 -.05
TV to later Aggression to later TV
aggression
Aggression Aggression
3rd grade .38 13th grade


X Y

Z

Class Exercise

Identify the

third variable

that influences both X and Y


+
Class GPA
Attendance

Motivation


+
Number of Crime
Mosques Rate

Size of
Population


+
Ice Cream Number of
Consumed Drownings

Temperature


+
Reading Score Reading
Comprehension

IQ

Data Preparation - Correlation

Screen data for outliers and ensure that
there is evidence of linear relationship, since
correlation is a measure of linear
relationship.
Assumption is that each pair is bivariate
normal.
If not normal, then use Spearman.

Correlation In SPSS

For this exercise, we will be
using the data from the CD,
under Chapter 8, korelasi.sav
This data is a subset of a case-
control study on factors
affecting SGA in Kelantan.
Open the data & select -
>Analyse
>Correlate
>Bivariate…

Correlation in SPSS

We want to see whether
there is any association
between the mothers’ weight
and the babies’weight. So
select the variables (weight2
& birthwgt) into ‘Variables’.
Select ‘Pearson’ Correlation
Coefficients.
Click the ‘OK’ button.

Correlation Results

Correlations

WEIGHT2 BIRTHWGT
WEIGHT2 Pearson Correlation 1 .431*
Sig. (2-tailed) . .017
N 30 30
BIRTHWGT Pearson Correlation .431* 1
Sig. (2-tailed) .017 .
N 30 30
*. Correlation is significant at the 0.05 level (2-tailed).

The r = 0.431 and the p value is significant at
0.017.
The r value indicates a fair and positive linear
relationship.

Scatter Diagram
3.6
3.4
3.2
3.0
2.8 If the correlation is
2.6
significant, it is best to
2.4
2.2
include the scatter
2.0 diagram.
1.8
1.6
The r square indicated
1.4 mothers’ weight
1.2 contribute 19% of the
BIRTHWEIGHT

1.0
.8
variability of the babies’
.6 weight.
.4
.2
0.0 Rsq = 0.1861
0 10 20 30 40 50 60 70 80 90 100

MOTHERS' WEIGHT

Spearman/Kendall Correlation

To find correlation between a related pair of
continuous data (not normally distributed); or
Between 1 Continuous, 1 Categorical
Variable (Ordinal)
e.g., association between Likert Scale on work
satisfaction and work output.

Spearman's rank correlation coefficient

In statistics, Spearman's rank correlation coefficient, named
for Charles Spearman and often denoted by the Greek letter ρ
(rho), is a non-parametric measure of correlation – that is, it
assesses how well an arbitrary monotonic function could
describe the relationship between two variables, without
making any assumptions about the frequency distribution of the
variables. Unlike the Pearson product-moment correlation
coefficient, it does not require the assumption that the
relationship between the variables is linear, nor does it require
the variables to be measured on interval scales; it can be used
for variables measured at the ordinal level.

Example
•Correlation between
sphericity and visual
acuity.
•Sphericity of the eyeball
is continuous data while
visual acuity is ordinal
data (6/6, 6/9, 6/12,
6/18, 6/24), therefore
Spearman correlation is
the most suitable.
•The Spearman rho
correlation coefficient is
-0.108 and p is 0.117. P
is larger than 0.05,
therefore there is no
significant association
between sphericity and
visual acuity.

Example 2
•- Correlation between glucose level and systolic
blood pressure.
•Based on the data given, prepare the following
table;
•For every variable, sort the data by rank. For ties,
take the average.
•Calculate the difference of rank, d for every pair
and square it. Take the total.
•Include the value into the following formula;

•∑ d2 = 4921.5 n = 32
•Therefore rs = 1-((6*4921.5)/(32*(322-1)))
= 0.097966.
This is the value of Spearman correlation
coefficient (or ϒ).
•Compare the value against the Spearman table;
•p is larger than 0.05.
•Therefore there is no association between systolic
BP and blood glucose level.

Spearman’s
table

•0.097966 is the value
of Spearman correlation
coefficient (or ρ).
•Compare the value
against the Spearman
table;
•0.098 < 0.364 (p=0.05)
•p is larger than 0.05.
•Therefore there is no
association between
systolic BP and blood
glucose level.

SPSS Output

Correlations

GLU BPS1
Spearman's rho GLU Correlation Coefficient 1.000 .097
Sig. (2-tailed) . .599
N 32 32
BPS1 Correlation Coefficient .097 1.000
Sig. (2-tailed) .599 .
N 32 32

Regression

X Y
Predictor Criterion
Variable Variable

Predicting Y based on a given value of X

Regression and Prediction

X Y
Height Weight

REGRESSION

Regression
– one variable is a direct cause of the other
– or if the value of one variable is changed, then as
a direct consequence the other variable also
change
– or if the main purpose of the analysis is prediction
of one variable from the other

REGRESSION

– Regression: the dependence of dependent
variable (outcome) on the independant variable
(RF)
– relationship is summarised by a regression
equation.
– y = a + bx

REGRESSION

Regression
– The regression line
x - independent variable - horizontal axis
y - dependent variable - vertical axis
– Regression equation
y = a + bx
– a = intercept at y axis
– b = regression coefficient

– Test of significance - b is not equal to zero

The Least Squares (Regression)
Line

A good line is one that minimizes
the sum of squared differences between the
points and the line.

The Least Squares (Regression)
Line
Sum of squared differences = (2 - 1)2 + (4 - 2)2 +(1.5 - 3)2 + (3.2 - 4)2 = 6.89
Sum of squared differences = (2 -2.5)2 + (4 - 2.5)2 + (1.5 - 2.5)2 + (3.2 - 2.5)2 = 3.99
Let us compare two lines
4 (2,4)
The second line is horizontal
(4,3.2)
3
2.5
2
(1,2) (3,1.5)
1 The smaller the sum of
squared differences
the better the fit of the
1 2 3 4
line to the data.

Linear Regression

Come up with a Linear Regression Model to predict
a continous outcome with a continous risk factor, i.e.
predict BP with age. Usually LR is the next step after
correlation is found to be strongly significant.
y = a + bx; a = y - bx
– e.g. BP = constant (a) + regression coefficient (b) * age

b=

Testing for significance

test whether the slope is significantly different from zero by:

t = b/SE(b)

Regression Line

In a scatterplot showing the association
between 2 variables, the regression line is
the “best-fit” line and has the formula
y=a + bx
a=place where line crosses Y axis
b=slope of line (rise/run)
Thus, given a value of X, we can predict a
value of Y

Regression Graphic – Regression Line
120

100

80

60
Total ball toss points

y’=47
40

y’=20 20

0 Rsq = 0.6031
8 10 12 14 16 18 20 22 24 26

Distance from target if x=18 if x=24
then… then…

Regression Equation

y’= a + bx
– y’ = predicted value of y
– b = slope of the line
– x = value of x that you enter
– a = y-intercept (where line crosses y axis)
In this case….
– y’ = 125.401 - 4.263(x)

So if the distance is 20 feet
– y’ = -4.263(20) + 125.401
20
– y’ = -85.26 + 125.401
– y’ = 40.141

Regression Line
(Defined)

Regression line is the line where absolute values of
vertical distances between points on scatterplot and
a line form a minimum sum (relative to other possible
lines)

Positive and Linear Negative and Linear

Example

b=

∑x = 6426 ∑ x2 = 1338088
∑y = 4631 ∑ xy = 929701
n = 32
b = (929701-(6426*4631/32))/
(1338088-(64262/32)) = -0.00549
Mean x = 6426/32=200.8125
mean y = 4631/32=144.71875
y = a + bx
a = y – bx (replace the x, y & b value)
a = 144.71875+(0.00549*200.8125)
= 145.8212106
Systolic BP = 144.71875 - 0.00549.chol

Example
∑x = 6426 ∑ x2 = 1338088
∑ y = 4631 ∑ xy = 929701
n = 32
b = (929701-(6426*4631/32))/
(1338088-(64262/32)) = -0.00549
Mean x = 6426/32=200.8125
mean y = 4631/32=144.71875
a = 144.71875+(0.00549*200.8125)
= 145.8212106
Systolic BP = 145.82 - 0.005.chol

•“Criterion,”
•y-axis variable,
SPSS Regression Set-up •what you’re trying
to predict

•“Predictor,”
•x-axis variable,
•what you’re
basing the
prediction on

Getting Regression Info from
SPSS
Model Summary

Adjusted Std. Error of
Model R R Square R Square the Estimate
1 .777 a .603 .581 18.476
a. Predictors: (Constant), Distance from target
y’ = a + b (x)
y’ = 125.401 - 4.263(20)
20
a a
Coefficients

Unstandardized Standardized
Coefficients Coefficients
Model B Std. Error Beta t Sig.
1 (Constant) 125.401 14.265 8.791 .000
Distance from target -4.263 .815 -.777 -5.230 .000
a. Dependent Variable: Total ball toss points

b

Birthweight=1.615+0.049mBMI

Coefficientsa

Unstandardized Standardized
Coefficients Coefficients
Model B Std. Error Beta t Sig.
1 (Constant) 1.615 .181 8.909 .000
BMI .049 .007 .410 6.605 .000
a. Dependent Variable: Birth weight

Pearson Correlation, Spearman Correlation &Linear Regression

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