CH5 Quadratic Regression notes PDF

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Chapter 15

15-1

Statistics for Managers Using Microsoft Excel

Chapter 5 Multiple Regression Model Building

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Chap 15-1

Nonlinear Relationships DCOVA 





The relationship between the dependent variable and an independent variable may not be linear Can review the scatter plot to check for nonlinear relationships Example: Quadratic model

Yi  β0  β1X1i  β2 X21i  εi 

The second independent variable is the square of the first variable

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Basic Business Statistics, 10/e

Chap 15-2

© 2006 Prentice Hall, Inc.

Chapter 15

15-2

Quadratic Regression Model DCOVA Model form:

Yi  β0  β1X1i  β2 X21i  εi 

where: β0 = Y intercept β1 = regression coefficient for linear effect of X on Y β2 = regression coefficient for quadratic effect on Y εi = random error in Y for observation i

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Chap 15-3

Linear vs. Nonlinear Fit DCOVA

Y

Y

X

X Linear fit does not give random residuals

residuals

residuals

X

X



Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Basic Business Statistics, 10/e

Nonlinear fit gives random residuals Chap 15-4

© 2006 Prentice Hall, Inc.

Chapter 15

15-3

Quadratic Regression Model DCOVA

Yi  β0  β1X1i  β2 X21i  εi

Quadratic models may be considered when the scatter plot takes on one of the following shapes: Y

Y

β1 < 0

X1

Y

β1 > 0

β2 > 0

β2 > 0

X1

Y

β1 < 0

X1

β2 < 0

β1 > 0

X1

β2 < 0

β1 = the coefficient of the linear term β2 = the coefficient of the squared term Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Chap 15-5

Testing the Overall Quadratic Model DCOVA 

Estimate the quadratic model to obtain the regression equation: ˆ i  b0  b1X1i  b2 X1i2 Y



Test for Overall Relationship H0: β1 = β2 = 0 (no overall relationship between X and Y) H1: β1 and/or β2 ≠ 0 (there is a relationship between X and Y) 

FSTAT =

MSR MSE

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Basic Business Statistics, 10/e

Chap 15-6

© 2006 Prentice Hall, Inc.

Chapter 15

15-4

Testing for Significance: Quadratic Effect DCOVA 

Testing the Quadratic Effect 

Compare quadratic regression equation

Yi  b0  b1X1i  b2 X1i2 with the linear regression equation

Yi  b0  b1X1i

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Chap 15-7

Testing for Significance: Quadratic Effect (continued) 

Testing the Quadratic Effect 

DCOVA

Consider the quadratic regression equation

Yi  b0  b1X1i  b2 X1i2

Hypotheses H0: β2 = 0 (The quadratic term does not improve the model) H1: β2  0 (The quadratic term improves the model)

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Basic Business Statistics, 10/e

Chap 15-8

© 2006 Prentice Hall, Inc.

Chapter 15

15-5

Testing for Significance: Quadratic Effect (continued) 

Testing the Quadratic Effect

DCOVA

Hypotheses H0: β2 = 0 (The quadratic term does not improve the model) H1: β2  0 (The quadratic term improves the model) 

The test statistic is

b β2 t STAT  2 Sb 2

d.f.  n  3

where: b2 = squared term slope coefficient β2 = hypothesized slope (zero) Sb = standard error of the slope 2

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Chap 15-9

Testing for Significance: Quadratic Effect (continued) 

Testing the Quadratic Effect

DCOVA

Compare r2 from simple regression to adjusted r2 from the quadratic model 

If adj. r2 from the quadratic model is larger than the r2 from the simple model, then the quadratic model is likely a better model

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Basic Business Statistics, 10/e

Chap 15-10

© 2006 Prentice Hall, Inc.

Chapter 15

15-6

Example: Quadratic Model DCOVA 3

1

7

2

8

3

15

5

22

7

33

8

40

10

54

12

67

13

70

14

78

15

85

15

87

16

99

17

Purity increases as filter time increases: Purity vs. Time 100 80 60

Purity

Purity

Filter Time

40 20 0 0

5

10

15

20

Time

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Chap 15-11

Example: Quadratic Model (continued) 

Simple regression results:

DCOVA

Y^ = -11.283 + 5.985 Time Coefficients Intercept Time

Standard Error

t Stat

t statistic, F statistic, and r2 are all high, but the residuals are not random:

P-value

-11.28267

3.46805

-3.25332

0.00691

5.98520

0.30966

19.32819

2.078E-10

Regression Statistics 0.96888

Adjusted R Square

0.96628

Standard Error

6.15997

F 373.57904

Time Residual Plot

Significance F

10

2.0778E-10 Residuals

R Square

5 0 -5 0

5

10

15

20

-10 Time Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Basic Business Statistics, 10/e

Chap 15-12

© 2006 Prentice Hall, Inc.

Chapter 15

15-7

Example: Quadratic Model in Excel (continued) 

Quadratic regression results:

DCOVA

Y^ = 1.5387 + 1.5650 Time + 0.2452 (Time)2 Coefficients

Standard Error

Intercept

1.5387

2.2447

0.6855

0.5072

Time

1.5650

0.6018

2.6005

0.0247

Time-squared

0.2452

0.0326

7.5241

0.0000

Time Residual Plot

P-value

10 Resi duals

t Stat

5 0 -5

F

R Square

0.9949

Adjusted R Square

0.9940

Standard Error

2.5951

1080.7330

5

Significance F 2.368E-13

The quadratic term is significant and improves the model: adj. r2 is higher and SYX is lower, residuals are now random

10

15

20

Time-squared Residual Plot 10

Statistics for Managers Using Microsoft Excel® 7e Copyright ©2014 Pearson Education, Inc.

Basic Business Statistics, 10/e

0

Time

Resi duals

Regression Statistics

5 0 -5

0

100

200

300

400

Time-squared

Chap 15-13

© 2006 Prentice Hall, Inc....