Regression & Correlation Class 2011-2013 L17

7/31/2019 Regression & Correlation Class 2011-2013 L17

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Regression Models


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Introduction

Regression analysisis a very valuabletool for a manager

Regression can be used to

Understand the relationship betweenvariables

Predict the value of one variable based onanother variable

Simple linear regression models haveonly two variables

Multiple regression models have morevariables


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Coefficient of Correlation

Measures the relative strength of the linear relationshipbetween two variables

1. Price and Quantity demand (Simple Regression)

2. Height and Weight (Simple Regression)

3. Advertisement expenditure and Sales(Simple Regression)

4. Family income and expenditure on luxury items

(Simple Regression)

5. Sales revenue of the product is influenced by the Adv. exp., Quality

of the product, and price. (Multiple Regression)6. Employer-employee relationship in any organization may be

examined with reference to training and development facilities,medical, housing, salary structure etc (Multiple Regression)


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Introduction

The variable to be predicted is calledthe dependent variable Sometimes called the response variable

The value of this variable depends onthe value of the independent variable Sometimes called the explanatoryor

predictor variable

Independentvariable

Dependentvariable

Independentvariable

= +


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Scatter Diagram

Graphing is a helpful way to investigatethe relationship between variables

A scatter diagramor scatter plotis

often used The independent variable is normally

plotted on theXaxis

The dependent variable is normallyplotted on the Yaxis


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Triple A Construction

Triple A Construction renovates old homes

They have found that the dollar volume ofrenovation work is dependent on the areapayroll

TRIPLE AS SALES(Rs100,000s)

LOCAL PAYROLL(Rs100,000,000s)

6 3

8 4

9 65 4

4.5 2

9.5 5


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12

10

8

6

4

2

0

Sales(Rs100,0

00)

Payroll (Rs100 million)

| | | | | | | |

0 1 2 3 4 5 6 7 8

Fig. 1


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Simple Linear Regression

where

Y = dependent variable (response)

X = independent variable (predictor or explanatory)

0 = constant (value of YwhenX= 0)

1 = slope of the regression line

= random error

Regression models are used to test if there is arelationship between variables

There is some random error that cannot bepredicted

XY10


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Simple Linear Regression

True values for the slope and constant are notknown so they are estimated using sample data

XbbY 10

where

Y = dependent variable (response)

X

= independent variable (predictor or explanatory)b0 = constant (value of YwhenX= 0)

b1 = slope of the regression line

^


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Triple A Construction is trying to predict salesbased on area payroll

Y= Sales

X= Area payroll

The line chosen in Figure 1 is the one thatminimizes the errors

Error = (Actual value) (Predicted value)

YYe


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For the simple linear regression model, the values of theconstant and slope can be calculated using the formulaebelow

XbbY 10

0 1

1 1

N N

n n

Y nb b X

2

0 1

1 1 1

N N N

n n n

YX b X b X

Step 1.

Step 2.


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Regression calculations (Step 3)

Y X XY X2

6 3 18 9

8 4 32 16

9 6 54 36

5 4 20 16

4.5 2 9 4

9.5 5 47.5 25

Y= 42Y= 42/6 = 7

X= 24X= 24/6 = 4

5.180XY 1062

X


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Slop, constant and equation

25.1;2 10 bb

XY 2512 .Therefore

Sales = 2 + 1.25(Payroll)

If the payroll next year is Rs600 million

000,9505.9)6(25.12 orY

Put values from step 3 to Step 1; 2 equations

Results are:


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Measuring the Fitof the Regression Model

Regression models can be developedfor any variablesXand Y

How do we know the model is actually

helpful in predicting Ybased onX? We could just take the average error, but

the positive and negative errors wouldcancel each other out

Three measures of variability are SST Total variability about the mean

SSE Variability about the regression line

SSR Total variability that is explained bythe model


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Sum of the squares total2)( YYSST

Sum of the squared error22)( YYeSSE

Sum of squares due to regression

2

)

( YYSSR

An important relationship

SSESSRSST


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Y X (YY)2 Y (YY)2 (YY)2

6 3 (6 7)2 = 1 2 + 1.25(3) = 5.75 0.0625 1.563

8 4 (8 7)2 = 1 2 + 1.25(4) = 7.00 1 0

9 6 (9 7)2 = 4 2 + 1.25(6) = 9.50 0.25 6.25

5 4 (5 7)2 = 4 2 + 1.25(4) = 7.00 4 0

4.5 2 (4.5 7)2 = 6.25 2 + 1.25(2) = 4.50 0 6.25

9.5 5 (9.5 7)2 = 6.25 2 + 1.25(5) = 8.25 1.5625 1.563

(YY)2 = 22.5 (YY)2 = 6.875 (YY)2 = 15.625

Y= 7 SST= 22.5 SSE = 6.875 SSR = 15.625

^

^^

^^


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Sum of the squares total2)( YYSST

Sum of the squared error22)( YYeSSE

Sum of squares due to regression

2

)

( YYSSR

An important relationship

SSESSRSST


For Triple A Construction

SST= 22.5

SSE = 6.875

SSR = 15.625


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12

10

8

6

4

2

0

Sales(Rs100,0

00)

Payroll (Rs100 million)

| | | | | | | |

0 1 2 3 4 5 6 7 8

Y= 2 + 1.25X^

YYYY

^

YYY^


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Coefficient of Determination

The proportion of the variability in Yexplained byregression equation is called the coefficient ofdetermination

The coefficient of determination isr2

SST

SSE

SST

SSRr 12


69440522

625152 ..

.r

About 69% of the variability in Yis explained bythe equation based on payroll (X)

r2= 0 to 1


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Correlation Coefficient

Thecorrelation coefficientis an expression of thestrength of the linear relationship

It will always be between +1 and1

The correlation coefficient isr

2rr


8333069440 ..r

r will be ve if slope isve and +ve if slope is +ve


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Correlation Coefficient

*

**

*(a) Perfect Positive

Correlation:r = +1

X

Y

*

**

*

(c) No Correlation:r = 0

X

Y

* *

**

* *

* **

*

(d) Perfect NegativeCorrelation:

r =1

X

Y

* **

*

* ***

*(b) Positive

Correlation:0


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Using Excel for Regression


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Using Excel for RegressionCorrelation coefficient is

called Multiple R in Excel


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Multiple Regression Analysis

Multiple regression modelsareextensions to the simple linear modeland allow the creation of models withseveral independent variables

Y= 0 + 1X1 + 2X2+ + kXk +

where

Y= dependent variable (response variable)

Xi = ith independent variable (predictor or explanatoryvariable)

0 = constant (value of Ywhen allXi= 0)I = coefficient of the ith independent variable

k = number of independent variables

= random error


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Multiple Regression Analysis

To estimate these values, a sample is takenthe following equation developed

kkXbXbXbbY ...

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where

= predicted value of Y

b0 = sample constant (and is an estimate of 0)

bi= sample coefficient of the ith variable (and isan estimate of i)

Y


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Jenny Wilson Realty

Jenny Wilson wants to develop a model todetermine the suggested listing price for housesbased on the size and age of the house

kkXbXbXbbY ...

22110

where

= predicted value of dependent variable (sellingprice)

b0 = Yconstant

X1

andX2

= value of the two independent variables (squarefootage and age) respectively

b1 andb2 = slopes forX1 andX2 respectively

Y

She selects a sample of houses that have soldrecently and records the data shown in followingTable


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Jenny Wilson Realty

SELLINGPRICE (Rs)

SQUAREFOOTAGE

AGE OFHOUSE

CONDITION

95,000 1,926 30 Good

119,000 2,069 40 Excellent

124,800 1,720 30 Excellent

135,000 1,396 15 Good142,000 1,706 32 Mint

145,000 1,847 38 Mint

159,000 1,950 27 Mint

165,000 2,323 30 Excellent

182,000 2,285 26 Mint

183,000 3,752 35 Good

200,000 2,300 18 Good

211,000 2,525 17 Good

215,000 3,800 40 Excellent

219,000 1,740 12 Mint


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Press OK


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One Dep. variable Selling price; Two indep. Variables

Square Footage & Age. So Input X Range will containBoth dataset (Square footage & Age together i.e. B2:C15).

And remaining process is same.


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Constant i.e. b0

Proportion of Ist Indep. Var.i.e. Square footage

Proportion of IInd Indep. Var.i.e. Age


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Jenny Wilson Realty

21289944146631 XXY


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Assumptions of the Regression Model

1. Errors are independent

2. Errors are normally distributed

3. Errors have a mean of zero

4. Errors have a constant variance

If we make certain assumptions about the errorsin a regression model, we can perform statisticaltests to determine if the model is useful

A plot of the residuals (errors) will often highlightany glaring violations of the assumption


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Example

Bus and Subway ridership in Delhi during winter months is believed to be

heavily tied to the number of tourists visiting the city. During the past 12

years, the following data have been obtained:

Develop a regression model.

What is expected ridership if 10 thousand tourists visit the city?

If there are no tourists at all, explain the predicted ridership.

Year 1 2 3 4 5 6 7 8 9 10 11 12No. of Tourists

(10,000s)

7 2 6 4 14 15 16 12 14 20 15 7

Ridership

(1000s)

15 10 13 15 25 27 24 20 27 44 34 17

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Regression & Correlation Class 2011-2013 L17