Modeling in regression
description
Transcript of Modeling in regression
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
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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
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Dependability of a Regression Equation
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2 Explained Variationr
Total Variation=
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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−
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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−
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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−
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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−
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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−
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E V T V
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2ˆ( ) 672
1( ) 672
y yr
y y
−= = =
−∑∑
Y = 12
ˆ 4y x=
Y
X
ˆ 12y =
Example-1
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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−
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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−
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E V T V
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2ˆ( ) 0
0( ) 72
y yr
y y
−= = =
−∑∑
Example-2
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Research & Development Expenditure - Profit
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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−
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E V T V
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2ˆ( ) 200
0.829( ) 241
y yr
y y
−= = =
−∑∑
y
y
yTotal Variation
Unexplained Variation
Explained Variation
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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−
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Standard Error of Estimate
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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
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Test Regression Coefficient ‘b’ against ZERO
-Statistic ParameterCV
Standard Error=
2 04.44
0.45r
b BCV
SE
− −= = =
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0- 2.78 2.784.44
Probability Curve t Distribution
Acceptance Region Rejected RegionRejected Region
Table ValueCalculated Value
P Value 0.025 0.025
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Acceptance Region Rejected Region
7.7119.048
Table ValueCalculated Value
P Value 0.05
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ANOVA
0.012
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I Hypothesis Testing
II Estimation of Population Parametersa) Point Estimateb) Interval Estimate
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r
b Bt
SE
−=
2.0 2.10.22
0.45t
−= = −
Population Growth Rate of Profit = 2.1
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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
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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
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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
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Confidence Level Significance Level
90%(0.9)
10%(0.1)
95%(0.95)
5%(0.05)
99%(0.99)
1%(0.01)
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y
y
y
Upper Limit 3.25
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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
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Lower Limit 21.11
Point Estimate 30
34
31
y
From Sample confidence Interval of y Y
y
y
y
Upper Limit 27.23
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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
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Lower Limit 20.55
Point Estimate 30
34
31
y
From Sample y confidence Interval of Y
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Dependent Variable Sales
Independent VariablesMarket PotentialNumber of dealersNumber of Sales PeopleCompetitors ActivitiesNumber of Service PeopleNumber of Existing Customers
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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
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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
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1 1 2 2y a b x b x= + +
Sales = - 10.616 + 0.234 Pot + 1.424People
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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
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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
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MonthsEmployed
BaseSalary
6 7.5010 8.6012 9.1018 10.3030 13.005 6.213 8.715 9.421 9.8
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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
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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
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Ho: There is no difference in the base Salary between Male and Female
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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
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0.7893.31
0.238r
b BCV
SE
− −= = =
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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
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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
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y y ˆy y−
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