Regression Analysis

Lecture 9

Regression analysis establishes relationship between a dependent variable and independent variables

Relationship between “Cause” and “Effect”t

Relationship between variables

Usefulness of regression analysis

• Regression analysis is a vary widely used tool for research.

• It shows type and magnitude of relationship between two variables.

Example of Usefulness of Regression Analysis

:

1.Shows for example whether there is any relationship between an increase in household income (Y) land an increase in consumption (C ).

2.Whether there is positive or negative relationship between Y and C. Whether if :

Y C or reverse1.How much of an increase in income (Y) is spent on

consumption ( C ).

Example of Usefulness of Regression Analysis

• Regression is also used for prediction and forecasting,

• Regression analysis allows to measure confidence or significance level of the findings.

Example of Usefulness of Regression Analysis

• Increase in traffic jam (hours of non-movement) depends on Increase in number of cars in Dhaka City. (+ dependency)

• A decrease in number of School drop-out depends on an increase I income of parents.(-ve dependency)

• An increase in household income leads to an increase in household consumption.

Other Logical Examples of Positive and Negative Dependency

Forms of regression models

• A regression model relates dependent variable Y to be a function/relation of independent variable X.

• Symbolically, Y = f (Xi)

• Where i = 1,2,3,4,…

Diagrammatic Representation of Regression Model

Consumption Expenditure(,000Tk)

Income of the Household (,000 Tk)

0

Each dot represent sample data for Income and Expenditure for each sample household

120

100

130

90

•

Consumption Expenditure ( C )

Income of the Household (Y)

0

C = a + by

Regression analysis draw a mean /average line with equation C = a + b Y so that difference between sample data and estimated data is minimized.

Does dotted line minimize deviations?

Deviations between sample value and the mean value

Mean value line

Diagrammatic Representation of Regression Equation

• In mean or average line, square of the deviation ( C i) for each of the

sample from mean ( C )is minimized.

Why ?• Because simple sum of difference

from mean is always zero.

ExampleY 10 8 9

Av Y is 9

C 8 6 7

Av C is 7

Dependent variable

C - C

Sum is zero

1 -1 0

(C – C)**2 1 1 0

Sum of square is + number

Formula for Regression coefficient b when sum of square is minimized , b =

(Ci – C) (Yi –Y)

(Yi – Y) 2

i = 1,2, ….n

General Formula

• If Y is dependent variable and X is independent variable e.g. Y = f (x) then

• Regression coefficient =

Sum of (Xi –X) (Yi – Y)

Sum of (Xi –X)**2

Example : Given the following data C = f (Y), predict

Consumption level for a household with annual income of 500 thousand

TakaAnnual Income (Y)

(,000Tk)

100 150 200 250 300

Annual Expenditure

(,000Tk)

(C )

80 90 100 110 120

Example : Given the following data, predict Consumption level for a household with annual

income of 500 thousand Taka. (Fig in,000Tk)

Annual Income (Y)

Av Y = 200

Yi - Y

100

-100

150

-50

200

0

250

50

300

100

Annual Expenditure

(C )

Av C = 100

Ci - C

80

-20

90

-10

100

0

110

10

120

20

Example• (Ci – C) (Yi –Y) = 2000 +500 +

0 + 500 + 2000 = 5000• (Yi – Y) 2 = 10000+2500 + 0 + 2500+ 10000 = 25000

• Therefore b = (Ci – C) (Yi –Y) / (Yi – Y) 2 = 0.2

Calculated Regression Equation Example

C = a + b Y

Or C = a + 0.2 Y or C = a + 0.2 Y Or a = C -0.2 Y

Or a = 100-0.2 x 200 = 100 – 40 = 60

Therefore C = 60 + 0.2 Y

Calculated Regression Equation Example

C = 60 +0.2 Y What kind of relationship between

Y and C ? How much consumption increases

for Tk 1000 increase in income ?

C = 60 +0.2 Y

What is consumption, when income is zero?

What is predicted consumption, when income is Tk 500,000?

Correlation : A measure of simple relationship

• Correlation shows only associanship or relationship between two variables.

• Whereas Regression analysis shows dependency relationship

• Correlation between two variables ( for example Income and Expenditure) is measured by a formula shown as ;

Formula of Correlation coefficient r is

(Ci – C) (Yi –Y)

(Yi – Y) 2 (Ci – C)2

Formula of Correlation coefficient rin terms of regression

coefficient r

(Yi – Y)**2

(Ci – C )**2 r = b

The End

Given the following data, calculate correlation coefficient between Income and

Expenditure. Also predict how much Consumption will increase for a 1000 Tk

increase in household income?Annual Income (Y)

(,000Tk)

110 160 210 260 310

Annual Expenditure

(,000Tk)

(C )

75 85 95 105 115

Class Assignment

The End