Regression Correlation Vs Regression Types of relationships Simple linear regression

1. REGRESSION
ANALYSIS
Dr.BalamuruganM
AssociateProfessor
AcharyaInstituteofGraduateStudies
Bangalore

2. Regression
• Regression analysis measures the nature and extent of the
relationship between two or more variables, thus enables
us to make predictions.
• Regression is the measure of the average relationship
between two or more variables.

3. Correlation vs. Regression
• Degree & Nature of Relationship
 Correlation is a measure of degree of relationship between X
& Y.
 Regression studies the nature of relationship between the
variables so that one may be able to predict the value of one
variable on the basis of another.
• Cause & Effect Relationship
 Correlation does not assume cause and effect relationship
between two variables.
 Regression clearly expresses the cause and effect relationship
between two variables. The independent variable is the cause
and dependent variable is effect.

4. Correlation vs. Regression
• Prediction
 Correlation does not help in making predictions.
 Regression enable us to make predictions using
regression line.
• Symmetric
 Correlation coefficients are symmetrical.
 Regression coefficients are not symmetrical.

5. Copyright © 2016 Pearson Education, Ltd. Chapter 12, Slide 5
Regression Models
Y
X
Y
X
Y
Y
X
X
Linear relationships Curvilinear relationships

6. Copyright © 2016 Pearson Education, Ltd. Chapter 12, Slide 6
Types of Relationships
Y
X
Y
X
Y
Y
X
X
Strong relationships Weak relationships

7. Copyright © 2016 Pearson Education, Ltd. Chapter 12, Slide 7
Types of Relationships
Y
X
Y
X
No relationship

8. Regression Analysis
 Regression analysis is used to:
 Predict the value of a dependent variable based on the value of at
least one independent variable.
 Explain the impact of changes in an independent variable on the
dependent variable.
 Dependent variable: The variable we wish to predict or
explain.
 Independent variable: The variable used to predict or explain
the dependent variable.

9. Simple Linear Regression
Model
• Only one independent variable, X
• Relationship between X and Y is described by a linear
function
• Changes in Y are assumed to be related to changes in X

10. Copyright © 2016 Pearson Education, Ltd. Chapter 12, Slide 10
i
i
1
0
i ε
X
β
β
Y 


Linear component
Simple Linear Regression Model
Population
Y intercept
Population
Slope
Coefficient
Random
Error
term
Dependent
Variable
Independent
Variable
Random Error
component

11. Copyright © 2016 Pearson Education, Ltd. Chapter 12, Slide 11
Simple Linear Regression Equation
(Prediction Line)
i
1
0
i X
b
b
Ŷ 

The simple linear regression equation provides an
estimate of the population regression line
Estimate of
the regression
intercept
Estimate of the
regression slope
Estimated
(or predicted)
Y value for
observation i
Value of X for
observation i

12. Interpretation of the Slope
and Intercept
• b0 is the estimated mean value of Y when the value of X
is zero.
• b1 is the estimated change in the mean value of Y as a
result of a one-unit increase in X.

13. Example
• A real estate agent wishes to examine the
relationship between the selling price of a home
and its size (measured in square feet)
• A random sample of 10 houses is selected
• Dependent variable (Y) = house price in $1000s
• Independent variable (X) = square feet

14. Data

15. 0
50
100
150
200
250
300
350
400
450
0 500 1000 1500 2000 2500 3000
House
Price
($1000s)
Square Feet
ScatterPlot
House price model: Scatter Plot

16. 0
50
100
150
200
250
300
350
400
450
0 500 1000 1500 2000 2500 3000
Square Feet
House
Price
($1000s)
GraphicalRepresentation
House price model: Scatter Plot and Prediction Line
feet)
(square
0.10977
98.24833
price
house 

Slope
= 0.10977
Intercept
= 98.248

17. Simple Linear Regression
Example: Interpretation
 b0 is the estimated mean value of Y when the value of X is zero
(if X = 0 is in the range of observed X values)
 Because a house cannot have a square footage of 0, b0 has no
practical application.
 b1 estimates the change in the mean value of Y as a result of a
one-unit increase in X.
 Here, b1 = 0.10977 tells us that the mean value of a house
increases by .10977($1000) = $109.77, on average, for each
additional one square foot of size.
feet)
(square
0.10977
98.24833
price
house 


18. 317.85
0)
0.1098(200
98.25
(sq.ft.)
0.1098
98.25
price
house





Predict the price for a house
with 2000 square feet:
The predicted price for a house with 2000
square feet is 317.85($1,000s) = $317,850
Simple Linear Regression Example:
Making Predictions

19. THANK YOU