"Understanding Correlation and Regression: Key Concepts for Data Analysis"

Correlation &
Correlation &
Regression
Regression

Correlation
Correlation
Correlation is a statistical technique
used to determine the degree to which
two variables are related

• Rectangular coordinate
• Two quantitative variables
• One variable is called independent (X) and
the second is called dependent (Y)
• Points are not joined
• No frequency table
Scatter diagram

Scatter diagram of weight and systolic blood
Scatter diagram of weight and systolic blood
pressure
pressure
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
wt (kg)
SBP(mmHg)

80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
Wt (kg)
SBP(mmHg)
Scatter diagram of weight and systolic blood pressure

Scatter plots
The pattern of data is indicative of the type of
relationship between your two variables:
 positive relationship
 negative relationship
 no relationship

Positive relationship
Positive relationship

0
2
4
6
8
10
12
14
16
18
0 10 20 30 40 50 60 70 80 90
Age in Weeks
Height
in
CM

Negative relationship
Negative relationship
Reliability
Age of Car

Correlation Coefficient
Correlation Coefficient
Statistic showing the degree of relation
between two variables

Simple Correlation coefficient
Simple Correlation coefficient (r)
(r)
 It is also called Pearson's correlation
It is also called Pearson's correlation
or product moment correlation
or product moment correlation
coefficient.
coefficient.
 It measures the
It measures the nature
nature and
and strength
strength
between two variables of
between two variables of
the
the quantitative
quantitative type.
type.

The
The sign
sign of
of r
r denotes the nature of
denotes the nature of
association
association
while the
while the value
value of
of r
r denotes the
denotes the
strength of association.
strength of association.

 If the sign is
If the sign is +ve
+ve this means the relation
this means the relation
is
is direct
direct (an increase in one variable is
(an increase in one variable is
associated with an increase in the
associated with an increase in the
other variable and a decrease in one
other variable and a decrease in one
variable is associated with a
variable is associated with a
decrease in the other variable).
decrease in the other variable).
 While if the sign is
While if the sign is -ve
-ve this means an
this means an
inverse or indirect
inverse or indirect relationship (which
relationship (which
means an increase in one variable is
means an increase in one variable is
associated with a decrease in the other).
associated with a decrease in the other).

 The value of r ranges between ( -1) and ( +1)
The value of r ranges between ( -1) and ( +1)
 The value of r denotes the strength of the
The value of r denotes the strength of the
association as illustrated
association as illustrated
by the following diagram.
by the following diagram.
-
1 1
0
-0.25
-0.75 0.75
0.25
strong strong
intermediate intermediate
weak weak
no relation
perfect
correlation
perfect
correlation
Direct
indirect

If
If r
r = Zero
= Zero this means no association or
this means no association or
correlation between the two variables.
correlation between the two variables.
If
If 0 <
0 < r
r < 0.25
< 0.25 = weak correlation.
= weak correlation.
If
If 0.25 ≤
0.25 ≤ r
r < 0.75
< 0.75 = intermediate correlation.
= intermediate correlation.
If
If 0.75 ≤
0.75 ≤ r
r < 1
< 1 = strong correlation.
= strong correlation.
If
If r
r = l
= l = perfect correlation.
= perfect correlation.





















 
 

 
n
y)
(
y
.
n
x)
(
x
n
y
x
xy
r
2
2
2
2
How to compute the simple correlation
coefficient (r)

Example
Example
:
:
A sample of 6 children was selected, data about their
A sample of 6 children was selected, data about their
age in years and weight in kilograms was recorded as
age in years and weight in kilograms was recorded as
shown in the following table . It is required to find the
shown in the following table . It is required to find the
correlation between age and weight.
correlation between age and weight.
serial
No
Age
(years)
Weight
(Kg)
1 7 12
2 6 8
3 8 12
4 5 10
5 6 11
6 9 13

These 2 variables are of the quantitative type, one
These 2 variables are of the quantitative type, one
variable (Age) is called the independent and
variable (Age) is called the independent and
denoted as (X) variable and the other (weight)
denoted as (X) variable and the other (weight)
is called the dependent and denoted as (Y)
is called the dependent and denoted as (Y)
variables to find the relation between age and
variables to find the relation between age and
weight compute the simple correlation coefficient
weight compute the simple correlation coefficient
using the following formula:
using the following formula:




















 
 

 
n
y)
(
y
.
n
x)
(
x
n
y
x
xy
r
2
2
2
2

Regression Analyses
Regression Analyses
Regression: technique concerned with predicting
some variables by knowing others
The process of predicting variable Y using
variable X

Regression
Regression
 Uses a variable (x) to predict some outcome
Uses a variable (x) to predict some outcome
variable (y)
variable (y)
 Tells you how values in y change as a function
Tells you how values in y change as a function
of changes in values of x
of changes in values of x

Correlation and Regression
Correlation and Regression
 Correlation describes the strength of a
Correlation describes the strength of a linear
relationship between two variables
 Linear means “straight line”
 Regression tells us how to draw the straight line
described by the correlation

Regression
 Calculates the “best-fit” line for a certain set of data
Calculates the “best-fit” line for a certain set of data
The regression line makes the sum of the squares of
The regression line makes the sum of the squares of
the residuals smaller than for any other line
the residuals smaller than for any other line
Regression minimizes residuals
80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
Wt (kg)

Regression Equation
 Regression equation
describes the regression
line mathematically
 Intercept
 Slope 80
100
120
140
160
180
200
220
60 70 80 90 100 110 120
Wt (kg)
SBP(mmHg)

Linear Equations
Linear Equations
Y
Y = bX + a
a = Y-intercept
X
Change
in Y
Change in X
b = Slope
bX
a
ŷ 


By using the least squares method (a procedure
By using the least squares method (a procedure
that minimizes the vertical deviations of plotted
that minimizes the vertical deviations of plotted
points surrounding a straight line) we are
points surrounding a straight line) we are
able to construct a best fitting straight line to the
able to construct a best fitting straight line to the
scatter diagram points and then formulate a
scatter diagram points and then formulate a
regression equation in the form of:
regression equation in the form of:



 



n
x)
(
x
n
y
x
xy
b 2
2
1
b
bX
a
ŷ 

Y mean X-Xmean

Hours studying and grades
Hours studying and grades

Regressing grades on hours
grades on hours
Linear Regression
2.00 4.00 6.00 8.00 10.00
Number of hours spent studying
70.00
80.00
90.00












Final grade in course = 59.95 + 3.17 * study
R-Square = 0.88
Predicted final grade in class =
59.95 + 3.17*(number of hours you study per week)

Predict the final grade of
Predict the final grade of
…
…
 Someone who studies for 12 hours
 Final grade = 59.95 + (3.17*12)
 Final grade = 97.99
 Someone who studies for 1 hour:
 Final grade = 59.95 + (3.17*1)
 Final grade = 63.12
Predicted final grade in class = 59.95 + 3.17*(hours of study)

Exercise
Exercise
A sample of 6 persons was selected the
A sample of 6 persons was selected the
value of their age ( x variable) and their
value of their age ( x variable) and their
weight is demonstrated in the following
weight is demonstrated in the following
table. Find the regression equation and
table. Find the regression equation and
what is the predicted weight when age is
what is the predicted weight when age is
8.5 years
8.5 years.
.

Serial no
. Age (x) Weight (y)
1
2
3
4
5
6
7
6
8
5
6
9
12
8
12
10
11
13

Answer
Answer
Serial no
. Age (x) Weight (y) xy X2
Y2
1
2
3
4
5
6
7
6
8
5
6
9
12
8
12
10
11
13
84
48
96
50
66
117
49
36
64
25
36
81
144
64
144
100
121
169
Total 41 66 461 291 742

6.83
6
41
x 
 11
6
66


y
92
.
0
6
)
41
(
291
6
66
41
461
2





b
Regression equation
6.83)
0.9(x
11
ŷ(x) 



0.92x
4.675
ŷ(x) 

12.50Kg
8.5
*
0.92
4.675
ŷ(8.5) 


Kg
58
.
11
7.5
*
0.92
4.675
ŷ(7.5) 



11.4
11.6
11.8
12
12.2
12.4
12.6
7 7.5 8 8.5 9
Age (in years)
Weight
(in
Kg)
we create a regression line by plotting two
estimated values for y against their X component,
then extending the line right and left.

Exercise 2
Exercise 2
The following are the
The following are the
age (in years) and
age (in years) and
systolic blood
systolic blood
pressure of 20
pressure of 20
apparently healthy
apparently healthy
adults.
adults.
Age
(x)
B.P
(y)
Age
(x)
B.P
(y)
20
43
63
26
53
31
58
46
58
70
120
128
141
126
134
128
136
132
140
144
46
53
60
20
63
43
26
19
31
23
128
136
146
124
143
130
124
121
126
123

Find the correlation between age
Find the correlation between age
and blood pressure using simple
and blood pressure using simple
and Spearman's correlation
and Spearman's correlation
coefficients, and comment.
coefficients, and comment.
Find the regression equation?
Find the regression equation?
What is the predicted blood
What is the predicted blood
pressure for a man aging 25 years?
pressure for a man aging 25 years?

Serial x y xy x2
1 20 120 2400 400
2 43 128 5504 1849
3 63 141 8883 3969
4 26 126 3276 676
5 53 134 7102 2809
6 31 128 3968 961
7 58 136 7888 3364
8 46 132 6072 2116
9 58 140 8120 3364
10 70 144 10080 4900

Serial x y xy x2
11 46 128 5888 2116
12 53 136 7208 2809
13 60 146 8760 3600
14 20 124 2480 400
15 63 143 9009 3969
16 43 130 5590 1849
17 26 124 3224 676
18 19 121 2299 361
19 31 126 3906 961
20 23 123 2829 529
Total 852 2630 114486 41678




 



n
x)
(
x
n
y
x
xy
b 2
2
1 4547
.
0
20
852
41678
20
2630
852
114486
2




=
=112.13 + 0.4547 x
for age 25
B.P = 112.13 + 0.4547 * 25=123.49 = 123.5 mm hg
ŷ

Multiple Regression
Multiple regression analysis is a
straightforward extension of simple
regression analysis which allows more
than one independent variable.

"Understanding Correlation and Regression: Key Concepts for Data Analysis"

"Understanding Correlation and Regression: Key Concepts for Data Analysis"

Recommended

More Related Content

Similar to "Understanding Correlation and Regression: Key Concepts for Data Analysis" (20)

Recently uploaded (20)

"Understanding Correlation and Regression: Key Concepts for Data Analysis"