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# Modeling in regression

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Dependability of Regression Equation
Drawing Inferences
Optimising the independent variables in a Multiple Regression
Regression Modeling Technique

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### Modeling in regression

1. 1. Prof V Nallasivam
2. 2. Regression Modeling Technique Dependability of Regression Equation Drawing Inferences Optimising the independent variables in a Multiple Regression Prof V Nallasivam
3. 3. Year R & D Expenditure (in Crs) Annual Profit (in Crs) x y 2002 2 20 2003 3 25 2004 5 34 2005 4 30 2006 11 40 2007 5 31 Total 30 180 Prof V Nallasivam
4. 4. Prof V Nallasivam ˆ 20 2y x= +
5. 5. 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
6. 6. Prof V Nallasivam
7. 7. 2 Explained Variation r Total Variation = Prof V Nallasivam 2 2 [ ]r Correlation= Coefficient of Determination
8. 8. x y 1 4 4 -14 196 -14 196 2 8 8 -10 100 -10 100 3 12 12 -6 36 -6 36 4 16 16 -2 4 -2 4 5 20 20 2 4 2 4 6 24 24 6 36 6 36 7 28 28 10 100 10 100 8 32 32 14 196 14 196 144 144 0 672 0 672 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam
9. 9. Prof V Nallasivam ˆ 4y x=
10. 10. x y 1 4 4 -14 196 -14 196 2 8 8 -10 100 -10 100 3 12 12 -6 36 -6 36 4 16 16 -2 4 -2 4 5 20 20 2 4 2 4 6 24 24 6 36 6 36 7 28 28 10 100 10 100 8 32 32 14 196 14 196 144 144 0 672 0 672 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam
11. 11. x y 1 4 4 -14 196 -14 196 2 8 8 -10 100 -10 100 3 12 12 -6 36 -6 36 4 16 16 -2 4 -2 4 5 20 20 2 4 2 4 6 24 24 6 36 6 36 7 28 28 10 100 10 100 8 32 32 14 196 14 196 144 144 0 672 0 672 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam
12. 12. x y 1 4 4 -14 196 -14 196 2 8 8 -10 100 -10 100 3 12 12 -6 36 -6 36 4 16 16 -2 4 -2 4 5 20 20 2 4 2 4 6 24 24 6 36 6 36 7 28 28 10 100 10 100 8 32 32 14 196 14 196 144 144 0 672 0 672 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam
13. 13. x y 1 4 4 -14 196 -14 196 2 8 8 -10 100 -10 100 3 12 12 -6 36 -6 36 4 16 16 -2 4 -2 4 5 20 20 2 4 2 4 6 24 24 6 36 6 36 7 28 28 10 100 10 100 8 32 32 14 196 14 196 144 144 0 672 0 672 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam
14. 14. x y 1 4 4 -14 196 -14 196 2 8 8 -10 100 -10 100 3 12 12 -6 36 -6 36 4 16 16 -2 4 -2 4 5 20 20 2 4 2 4 6 24 24 6 36 6 36 7 28 28 10 100 10 100 8 32 32 14 196 14 196 144 144 0 672 0 672 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam E V T V
15. 15. Prof V Nallasivam 2 ˆ( ) 672 1 ( ) 672 y y r y y − = = = − ∑ ∑
16. 16. Y = 12 ˆ 4y x= Y X ˆ 12y = Example-1 Prof V Nallasivam
17. 17. 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 9 0 72 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam
18. 18. Prof V Nallasivam ˆ 9y =
19. 19. 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 9 0 72 ˆyˆy y− 2 ˆ( )y y−y y− 2 ( )y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam E V T V
20. 20. Prof V Nallasivam 2 ˆ( ) 0 0 ( ) 72 y y r y y − = = = − ∑ ∑
21. 21. Example-2 Prof V Nallasivam
22. 22. Research & Development Expenditure - Profit Prof V Nallasivam
23. 23. 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 ˆyˆy y−ˆy y−y y−y y− ˆy ˆy y− 2 ˆ( )y y− y y− 2 ( )y y− Prof V Nallasivam E V T V
24. 24. Prof V Nallasivam 2 ˆ( ) 200 0.829 ( ) 241 y y r y y − = = = − ∑ ∑
25. 25. y ˆy y Total Variation Unexplained Variation Explained Variation Prof V Nallasivam
26. 26. 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 ˆy ˆy y− 2 ˆ( )y y− ˆy ˆy y− 2 ˆ( )y y− Prof V Nallasivam Standard Error of Estimate
27. 27. Prof V Nallasivam 2 ˆ( ) 42 3.24 2 4 e y y SE n − = = = − ∑
28. 28. Formulation of Hypothesis Significance Level [ αα Formulation of Hypothesis Significance Level [ α Formulation of Hypothesis Significance 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
29. 29. -Statistic Parameter CV Standard Error = 2 0 4.44 0.45r b B CV SE − − = = = Prof V Nallasivam
30. 30. 0- 2.78 2.78 4.44 Probability Curve t Distribution Acceptance Region Rejected RegionRejected Region Table Value Calculated Value P Value 0.025 0.025 Prof V Nallasivam
31. 31. Acceptance Region Rejected Region 7.71 19.048 Table Value Calculated Value P Value 0.05 Prof V Nallasivam ANOVA 0.012
32. 32. Prof V Nallasivam
33. 33. I Hypothesis Testing II Estimation of Population Parameters a) Point Estimate b) Interval Estimate Prof V Nallasivam
34. 34. r b B t SE − = 2.0 2.1 0.22 0.45 t − = = − Population Growth Rate of Profit = 2.1 Prof V Nallasivam
35. 35. 0- 2.78 2.78 - 0.22 Probability Curve t Distribution Acceptance Region Rejected RegionRejected Region Table Value Calculated Value P Value 0.025 0.025 Prof V Nallasivam
36. 36. 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 ˆyˆY ˆy ˆY Prof V Nallasivam
37. 37. 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
38. 38. 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
39. 39. y ˆy y Upper Limit 3.25 5 Prof V Nallasivam Lower Limit 0.75 Point Estimate 2.00 34 31 y From Sample b confidence Interval of B b
40. 40. y ˆy y Upper Limit 38.39 5 Prof V Nallasivam Lower Limit 21.11 Point Estimate 30 34 31 y From Sample confidence Interval ofˆy ˆY
41. 41. y ˆy y Upper Limit 27.23 5 Prof V Nallasivam Lower Limit 12.77 Point Estimate 20 34 31 y From Sample a confidence Interval of A a
42. 42. y ˆy y Upper Limit 39.45 5 Prof V Nallasivam Lower Limit 20.55 Point Estimate 30 34 31 y From Sample y confidence Interval of Y
43. 43. Prof V Nallasivam
44. 44. Prof V Nallasivam Dependent Variable Sales Independent Variables Market Potential Number of dealers Number of Sales People Competitors Activities Number of Service People Number of Existing Customers
45. 45. Prof V Nallasivam Reg n 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
46. 46. 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.
47. 47. CORRELATION
48. 48. COMPUTER OUTPUT [SPSS]
49. 49. Model R R Square Adjusted R Square Std. Error of the Estimate 1 .989 .977 .960 4.39102 Model Summary a Predictors: (Constant), CUSTOMER, COMPT, SERVICE, POTENTIA, DEALERS, PEOPLE
50. 50. 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 ANOVA a Predictors: (Constant), CUSTOMER, COMPT, SERVICE, POTENTIA, DEALERS, PEOPLE b Dependent Variable: SALES
51. 51. 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 Coefficients a Dependent Variable: SALES
52. 52. Prof V Nallasivam 1 1 2 2y a b x b x= + + Sales = - 10.616 + 0.234 Pot + 1.424People
53. 53. Prof V Nallasivam
54. 54. Men Women Months Employed Base Salary Months Employed Base Salary 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
55. 55. Ho: There is no difference in the base Salary between Male and Female 1 2:oH x x= Prof V Nallasivam
56. 56. 1 1 2 1 5 9.7 4.415 n x s = = = 2 2 2 2 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
57. 57. Rejected Region 0- 2.365 2.356 0.92 Acceptance Region Rejected Region 0.0250.025 P - Value Prof V Nallasivam
58. 58. Prof V Nallasivam
59. 59. Months Employed Base Salary 6 7.50 10 8.60 12 9.10 18 10.30 30 13.00 5 6.2 13 8.7 15 9.4 21 9.8 Prof V Nallasivam
60. 60. OBS ACTUAL PREDICTED VALUE 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
61. 61. Months Employed Sex Base Salary M 6 0 7.50 M 10 0 8.60 M 12 0 9.10 M 18 0 10.30 M 30 0 13.00 F 5 1 6.2 F 13 1 8.7 F 15 1 9.4 F 21 1 9.8Prof V Nallasivam
62. 62. Ho: There is no difference in the base Salary between Male and Female Prof V Nallasivam
63. 63. Prof V Nallasivam Unstandardized Coefficients Standardized Coefficients t Sig. Model B Std. Error Beta 1 (Constant) 6.248 .291 21.439 .000 MONEM P .227 .016 .937 14.089 .000 SEX -.789 .238 -.220 -3.309 .016
64. 64. Prof V Nallasivam 0.789 3.31 0.238r b B CV SE − − = = =
65. 65. Prof V Nallasivam Unstandardized Coefficients Standardized Coefficients t Sig. Model B Std. Error Beta 1 (Constant) 6.248 .291 21.439 .000 MONEM P .227 .016 .937 14.089 .000 SEX -.789 .238 -.220 -3.309 .016 0.025
66. 66. Rejected Region 0- 2.45 2.45 - 3.309 Acceptance Region Rejected Region 0.016 0.0250.025 P - Value Prof V Nallasivam
67. 67. 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 ˆy ˆy y−
68. 68. Prof V Nallasivam