Prof V Nallasivam

Regression Modeling Technique

Dependability of Regression Equation

Drawing Inferences

Optimising the independent variables in a Multiple Regression

Prof V Nallasivam

YearR & D

Expenditure(in Crs)

AnnualProfit

(in Crs)

x y2002 2 202003 3 252004 5 342005 4 302006 11 402007 5 31Total 30 180

Prof V Nallasivam

Prof V Nallasivam

ˆ 20 2y x= +

ANOVA - (Test the Model)

Scatter diagram

Standard Error of Estimate

Testing the significance of Regression Co-efficient (b) against Zero.

Co-efficient of determination

Prof V Nallasivam

Dependability of a Regression Equation

Prof V Nallasivam

2 Explained Variationr

Total Variation=

Prof V Nallasivam

2 2[ ]r Correlation=

Coefficient of Determination

x y

1 4 4 -14 196 -14 1962 8 8 -10 100 -10 1003 12 12 -6 36 -6 364 16 16 -2 4 -2 45 20 20 2 4 2 46 24 24 6 36 6 367 28 28 10 100 10 1008 32 32 14 196 14 196

144 144 0 672 0 672

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

Prof V Nallasivam

ˆ 4y x=

x y

1 4 4 -14 196 -14 1962 8 8 -10 100 -10 1003 12 12 -6 36 -6 364 16 16 -2 4 -2 45 20 20 2 4 2 46 24 24 6 36 6 367 28 28 10 100 10 1008 32 32 14 196 14 196

144 144 0 672 0 672

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

x y

1 4 4 -14 196 -14 1962 8 8 -10 100 -10 1003 12 12 -6 36 -6 364 16 16 -2 4 -2 45 20 20 2 4 2 46 24 24 6 36 6 367 28 28 10 100 10 1008 32 32 14 196 14 196

144 144 0 672 0 672

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

x y

1 4 4 -14 196 -14 1962 8 8 -10 100 -10 1003 12 12 -6 36 -6 364 16 16 -2 4 -2 45 20 20 2 4 2 46 24 24 6 36 6 367 28 28 10 100 10 1008 32 32 14 196 14 196

144 144 0 672 0 672

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

x y

1 4 4 -14 196 -14 1962 8 8 -10 100 -10 1003 12 12 -6 36 -6 364 16 16 -2 4 -2 45 20 20 2 4 2 46 24 24 6 36 6 367 28 28 10 100 10 1008 32 32 14 196 14 196

144 144 0 672 0 672

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

x y

1 4 4 -14 196 -14 1962 8 8 -10 100 -10 1003 12 12 -6 36 -6 364 16 16 -2 4 -2 45 20 20 2 4 2 46 24 24 6 36 6 367 28 28 10 100 10 1008 32 32 14 196 14 196

144 144 0 672 0 672

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

E V T V

Prof V Nallasivam

2ˆ( ) 672

1( ) 672

y yr

y y

−= = =

−∑∑

Y = 12

ˆ 4y x=

Y

X

ˆ 12y =

Example-1

Prof V Nallasivam

x y

1 6 9 0 0 -3 9

1 12 9 0 0 3 9

3 6 9 0 0 -3 9

3 12 9 0 0 3 9

5 6 9 0 0 -3 9

5 12 9 0 0 3 9

7 6 9 0 0 -3 9

7 12 9 0 0 3 90 72

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

Prof V Nallasivam

ˆ 9y =

x y

1 6 9 0 0 -3 9

1 12 9 0 0 3 9

3 6 9 0 0 -3 9

3 12 9 0 0 3 9

5 6 9 0 0 -3 9

5 12 9 0 0 3 9

7 6 9 0 0 -3 9

7 12 9 0 0 3 90 72

yy y− 2ˆ( )y y−y y− 2( )y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

E V T V

Prof V Nallasivam

2ˆ( ) 0

0( ) 72

y yr

y y

−= = =

−∑∑

Example-2

Prof V Nallasivam

Research & Development Expenditure - Profit

Prof V Nallasivam

x y

2 20 24 -6 36 -10 100

3 25 26 -4 16 -5 25

5 34 30 0 0 4 16

4 30 28 -2 4 0 0

11 40 42 12 144 10 100

5 31 30 0 0 1 1

30 180 180 200 241

yy y−y y−y y−y y−

y y y− 2ˆ( )y y− y y− 2( )y y−

Prof V Nallasivam

E V T V

Prof V Nallasivam

2ˆ( ) 200

0.829( ) 241

y yr

y y

−= = =

−∑∑

y

y

yTotal Variation

Unexplained Variation

Explained Variation

Prof V Nallasivam

x y

2 20 24 -4 16

3 25 26 -1 1

5 34 30 4 16

4 30 28 2 4

11 40 42 -2 4

5 31 30 1 1

30 180 180 0 42

y ˆy y− 2ˆ( )y y−

y ˆy y− 2ˆ( )y y−

Prof V Nallasivam

Standard Error of Estimate

Prof V Nallasivam

2ˆ( ) 423.24

2 4e

y ySE

n

−= = =

−∑

Formulation of HypothesisSignificance Level [

αα

Formulation of HypothesisSignificance Level [

α

Formulation of HypothesisSignificance Level [

α

Formation of Hypothesis

Significance Level

Probability Distribution

Find the Table Value

Find the Calculated Value

Prof V Nallasivam

Test Regression Coefficient ‘b’ against ZERO

-Statistic ParameterCV

Standard Error=

2 04.44

0.45r

b BCV

SE

− −= = =

Prof V Nallasivam

0- 2.78 2.784.44

Probability Curve t Distribution

Acceptance Region Rejected RegionRejected Region

Table ValueCalculated Value

P Value 0.025 0.025

Prof V Nallasivam

Acceptance Region Rejected Region

7.7119.048

Table ValueCalculated Value

P Value 0.05

Prof V Nallasivam

ANOVA

0.012

Prof V Nallasivam

I Hypothesis Testing

II Estimation of Population Parametersa) Point Estimateb) Interval Estimate

Prof V Nallasivam

r

b Bt

SE

−=

2.0 2.10.22

0.45t

−= = −

Population Growth Rate of Profit = 2.1

Prof V Nallasivam

0- 2.78 2.78- 0.22

Probability Curve t Distribution

Acceptance Region Rejected RegionRejected Region

Table ValueCalculated Value

P Value 0.025 0.025

Prof V Nallasivam

From Sample

statistic estimate, population Parameter

From Sample

y estimate, population Y

From Sample

estimate, population

From Sample

b estimate, population B

From Sample

a estimate, population A

yY

y Y

Prof V Nallasivam

Parameter = statistic ± [Standard Error × Critical Value]

Parameter = statistic + [Standard Error × Critical Value]

Parameter = statistic - [Standard Error × Critical Value]

General Formula to Calculate Interval Estimate

Upper Limit

Lower Limit

Prof V Nallasivam

Confidence Level Significance Level

90%(0.9)

10%(0.1)

95%(0.95)

5%(0.05)

99%(0.99)

1%(0.01)

Prof V Nallasivam

y

y

y

Upper Limit 3.25

5Prof V Nallasivam

Lower Limit 0.75

Point Estimate 2.00

34

31

y

From Sample b confidence Interval of B

b

y

y

y

Upper Limit 38.39

5Prof V Nallasivam

Lower Limit 21.11

Point Estimate 30

34

31

y

From Sample confidence Interval of y Y

y

y

y

Upper Limit 27.23

5Prof V Nallasivam

Lower Limit 12.77

Point Estimate 20

34

31

y

From Sample a confidence Interval of A

a

y

y

y

Upper Limit 39.45

5Prof V Nallasivam

Lower Limit 20.55

Point Estimate 30

34

31

y

From Sample y confidence Interval of Y

Prof V Nallasivam

Prof V Nallasivam

Dependent Variable Sales

Independent VariablesMarket PotentialNumber of dealersNumber of Sales PeopleCompetitors ActivitiesNumber of Service PeopleNumber of Existing Customers

Prof V Nallasivam

Regn SALES POTENTI DEALERS PEOPLE COMPT SERVICE CUSTOM

1 5.00 25.00 1.00 6.00 5.00 2.00 20.00

2 60.00 150.00 12.00 30.00 4.00 5.00 50.00

3 20.00 45.00 5.00 15.00 3.00 2.00 25.00

4 11.00 30.00 2.00 10.00 3.00 2.00 20.00

5 45.00 75.00 12.00 20.00 2.00 4.00 30.00

6 6.00 10.00 3.00 8.00 2.00 3.00 16.00

7 15.00 29.00 5.00 18.00 4.00 5.00 30.00

8 22.00 43.00 7.00 16.00 3.00 6.00 40.00

9 29.00 70.00 4.00 15.00 2.00 5.00 39.00

10 3.00 40.00 1.00 6.00 5.00 2.00 5.00

11 16.00 40.00 4.00 11.00 4.00 2.00 17.00

12 8.00 25.00 2.00 9.00 3.00 3.00 10.00

13 18.00 32.00 7.00 14.00 3.00 4.00 31.00

14 23.00 73.00 10.00 10.00 4.00 3.00 43.00

15 81.00 150.00 15.00 35.00 4.00 7.00 70.00

Prof V Nallasivam

1 1 2 2 3 3 4 4 5 5 6 6y a b x b x b x b x b x b x= + + + + + +

Sales = -3.17 + 0.227Pot + 0.819Dealers + 1.091People -1.893Compet – 0.549Service + 0.66Cust.

CORRELATION

COMPUTER OUTPUT [SPSS]

Model R R Square Adjusted R Square Std. Error of the Estimate

1 .989 .977 .960 4.39102

Model Summarya Predictors: (Constant), CUSTOMER, COMPT, SERVICE, POTENTIA, DEALERS, PEOPLE

Model Sum of Squares df Mean Square F Sig.

1 Regression 6609.485 6 1101.581 57.133 .000

Residual 154.249 8 19.281

Total 6763.733 14

ANOVAa Predictors: (Constant), CUSTOMER, COMPT, SERVICE, POTENTIA, DEALERS, PEOPLE b Dependent Variable: SALES

Unstandardized

Coefficients

Standardized Coefficients

t Sig.

Model B Std. Error Beta

1 (Constant) -3.173 5.813 -.546 .600

POTENTIA .227 .075 .439 3.040 .016

DEALERS .819 .631 .164 1.298 .230

PEOPLE 1.091 .418 .414 2.609 .031

COMPT -1.893 1.340 -.085 -1.413 .195

SERVICE -.549 1.568 -.041 -.350 .735

CUSTOMER 6.594E-02 .195 .050 .338 .744

Coefficientsa Dependent Variable: SALES

Prof V Nallasivam

1 1 2 2y a b x b x= + +

Sales = - 10.616 + 0.234 Pot + 1.424People

Prof V Nallasivam

Men Women

MonthsEmployed

BaseSalary

MonthsEmployed

BaseSalary

6 7.50 5 6.2

10 8.60 13 8.7

12 9.10 15 9.4

18 10.30 21 9.8

30 13.00

Prof V Nallasivam

Ho: There is no difference in the base Salary between Male and Female

1 2:oH x x=Prof V Nallasivam

1

1

21

5

9.7

4.415

n

x

s

==

=

2

2

22

4

8.525

2.609

n

x

s

==

=

Men Women

( =0.01; =7)

Calculated t Value = 0.92

Table Value t 2.998α γ =Prof V Nallasivam

Rejected Region

0- 2.365 2.3560.92

Acceptance Region Rejected Region

0.0250.025 P - Value

Prof V Nallasivam

Prof V Nallasivam

MonthsEmployed

BaseSalary

6 7.5010 8.6012 9.1018 10.3030 13.005 6.213 8.715 9.421 9.8

Prof V Nallasivam

OBS ACTUAL PREDICTEDVALUE

RESIDUAL

1 7.5000 7.2085 0.2915

2 8.6000 8.1413 0.4587

3 9.1000 8.6077 0.4923

4 10.3000 10.0069 0.2913

5 13.0000 12.8054 0.1946

6 6.2000 6.9753 -0.7753

7 8.7000 8.8409 -0.1407

8 9.4000 9.3073 0.0927

9 9.8000 10.7066 -0.9066

Prof V Nallasivam

MonthsEmployed Sex Base

SalaryM 6 0 7.50M 10 0 8.60M 12 0 9.10M 18 0 10.30M 30 0 13.00F 5 1 6.2F 13 1 8.7F 15 1 9.4F 21 1 9.8

Prof V Nallasivam

Ho: There is no difference in the base Salary between Male and Female

Prof V Nallasivam

Prof V Nallasivam

Unstandardized Coefficients

Standardized Coefficients

t Sig.

Model B Std. Error Beta

1 (Constant) 6.248 .291 21.439 .000

MONEMP

.227 .016 .937 14.089 .000

SEX -.789 .238 -.220 -3.309 .016

Prof V Nallasivam

0.7893.31

0.238r

b BCV

SE

− −= = =

Prof V Nallasivam

Unstandardized Coefficients

Standardized Coefficients

t Sig.

Model B Std. Error Beta

1 (Constant) 6.248 .291 21.439 .000

MONEMP

.227 .016 .937 14.089 .000

SEX -.789 .238 -.220 -3.309 .016

0.025

Rejected Region

0- 2.45 2.45

- 3.309

Acceptance Region Rejected Region

0.016 0.0250.025 P - Value

Prof V Nallasivam

7.5000 7.6109 -0.1109

8.6000 8.5192 0.0808

9.1000 8.9734 0.1266

10.3000 10.3358 -0.0358

13.0000 13.0607 -0.0607

6.2000 6.5949 -0.3949

8.7000 8.4115 0.2885

9.4000 8.8656 0.5344

9.8000 10.2281 -0.4281

Prof V Nallasivam

y y ˆy y−

Prof V Nallasivam