Ken Black QA ch15

8/3/2019 Ken Black QA ch15

1/24

Business Statistics, 5th ed.

by Ken Black

Chapter 15

Multiple RegressionAnalysis

Discrete Distributions

PowerPoint presentations prepared by Lloyd Jaisingh,Morehead State University


2/24

Learning Objectives

Develop a multiple regression model.

Understand and apply significance tests of the regressionmodel and its coefficients.

Compute and interpret residuals, the standard error of theestimate, and the coefficient of determination.

Interpret multiple regression computer output.


3/24

Regression Models

Probabilistic Multiple Regression Model

Y = 0 + 1X1+ 2X2+ 3X3 + . . . + kXk+

Y= the value of the dependent (response) variable

0 = the regression constant1 = the partial regression coefficient of independent variable 12 = the partial regression coefficient of independent variable 2k = the partial regression coefficient of independent variable kk = the number of independent variables

= the error of prediction


4/24

Estimated Regression Model

st variableindependenofnumber=

tcoefficienregressionofestimate

3tcoefficienregressionofestimate



constantregressionofestimate

ofvaluepredicted:

3

2

1

0

3322110

k

k

YYwhere

Y

b

b

b

b

b

XbXbXbXbb

k

kk


5/24

Multiple Regression Model with Two

Independent Variables (First-Order)

Y

where

Y

where Y

X X

b b X b X

b

b

b

0 1 1 2 2

0

1

2

0 1 1 2 2

0

1

2

:

:

= the regression constant

the partial regression coefficient for independent variable 1

the partial regression coefficient for independent variable 2

= the error of prediction

predicted value of Yestimate of regression constant

estimate of regression coefficient 1

estimate of regression coefficient 2

PopulationModel

Estimated

Model


6/24

Response Plane for First-Order Two-

Predictor Multiple Regression Model

X1X2

Response Plane

Y1

Vertical InterceptY


7/24

Least Squares Equations for k = 2

0 1 1 2 2

0 1 1 1

2

2 1 2 1

0 2 1 1 2 2 2

2

2

b b X b X Y

b X b X b X X X Y

b X b X X b X X Y

n


8/24

Real Estate Data

Observation Y X1 X2 Observation Y X1 X21 63.0

65.1

1,605 35 13 79.7 2,121 14

2 2,489 45 14 84.5 2,485 93 69.9

7

1,553 20 15 96.0 2,300 19

4 76.8 2,404 32 16 109.5 2,714 4

5 73.9 1,884 25 17 102.5 2,463 5

6 77.9 1,558 14 18 121.0 3,076 7

7 74.9 1,748 8 19 104.9 3,048 3

8 78.0 3,105 10 20 128.0 3,267 6

9 79.0 1,682 28 21 129.0 3,069 1010 63.4 2,470 30 22 117.9 4,765 11

11 79.5 1,820 2 23 140.0 4,540 8

12 83.9 2,143 6

Market

Price

($1,000)

Square

Feet

Age

(Years)

Market

Price

($1,000)

Square

Feet

Age

(Years)


9/24

Real Estate Data

5000

4000

Price

60

90

120

3000

150

SquareFeet0200015

3045Age

Real Estate Data


10/24

MINITAB Output

for the Real Estate Example

The regression equation is

Price = 57.4 + 0.0177 Sq.Feet - 0.666 Age

Predictor Coef StDev T P

Constant 57.35 10.01 5.73 0.000

Sq.Feet 0.017718 0.003146 5.63 0.000Age -0.6663 0.2280 -2.92 0.008

S = 11.96 R-Sq = 74.1% R-Sq(adj) = 71.5%

Analysis of Variance

Source DF SS MS F PRegression 2 8189.7 4094.9 28.63 0.000

Residual Error 20 2861.0 143.1

Total 22 11050.7


11/24

Predicting the Price of Home

dollarsthousand658.93

12666.025000177.04.57,12and2500

666.00177.04.57

21

21

Y

XXFor

XXY


12/24

Evaluating the Multiple

Regression Model

H

H

k

a

01 2 3

0:

:

At least one of the regression coefficients is 0

H

H

H

H

H

H

H

H

a a

a

k

ak

01

1

03

3

02

2

0

0

0

0

0

0

0

0

0

:

:

:

:

:

:

:

:

Significance

Tests for

IndividualRegression

Coefficients

Testing

the

Overall

Model


13/24

Testing the Overall Model for the

Real Estate Example

0istscoefficienregressiontheofoneleastAt:

0: 21

0

aH

H

MSRSSR

kMSE

SSE

n kF

MSR

MSE

1

ANOVAdf SS MS F p

Regression 2 8189.723 4094.86 28.63 .000

Residual (Error) 20 2861.017 143.1

Total 22 11050.74

. , ,.

. . ,

01 2 20585

28 63 585

F

FCal

reject H .0


14/24

Significance Test

of the Regression

Coefficients forthe Real Estate

Example

H

H

H

H

a

a

01

1

02

2

0

0

0

0

:

:

:

:

tCal = 5.63 > 2.086, reject H0.

Coefficients Std Dev t Stat p

x1 (Sq.Feet) 0.0177 0.003146 5.63 .000

x2 (Age) -0.666 0.2280 -2.92 .008

t.025,20 = 2.086


15/24

Residuals and Sum of Squares Error

for the Real Estate Example

SSE

Observation Y Observation Y

1 43.0 42.466 0.534 0.285 13 59.7 65.602 -5.902 34.832

2 45.1 51.465 -6.365 40.517 14 64.5 75.383 -10.883 118.438

3 49.9 51.540 -1.640 2.689 15 76.0 65.442 10.558 111.479

4 56.8 58.622 -1.822 3.319 16 89.5 82.772 6.728 45.265

5 53.9 54.073 -0.173 0.030 17 82.5 77.659 4.841 23.440

6 57.9 55.627 2.273 5.168 18 101.0 87.187 13.813 190.799

7 54.9 62.991 -8.091 65.466 19 84.9 89.356 -4.456 19.858

8 58.0 85.702 -27.702 767.388 20 108.0 91.237 16.763 280.982

9 59.0 48.495 10.505 110.360 21 109.0 85.064 23.936 572.936

10 63.4 61.124 2.276 5.181 22 97.9 114.447 -16.547 273.81511 59.5 68.265 -8.765 76.823 23 120.0 112.460 7.540 56.854

12 63.9 71.322 -7.422 55.092 2861.017

Y Y Y 2

Y Y Y Y Y 2

Y Y


16/24

MINITAB Residual Diagnostics for

the Real Estate Problem

Residual

Pe

rcent

30150-15-30

99

90

50

10

1

Fitted Value

Residual

1401201008060

30

15

0

-15

-30

Residual

Frequenc

y

24120-12-24

6.0

4.5

3.0

1.5

0.0

Observation Order

Residual

222018161412108642

30

15

0

-15

-30

Normal Probability Plot of the Residuals Residuals Versus the Fitted Values

Histogram of the Residuals Residuals Versus the Order o f the Data

Residual Plots for Price


17/24

SSE and Standard Error

of the Estimate

eSSSE

n k

where

1

2861

23 2 1

1196.

: n = number of observations

k = number of independent variables

SSE

ANOVA

df SS MS F PRegression 2 8189.7 4094.9 28.63 .000

Residual (Error) 20 2861.0 143.1

Total 22 11050.7


18/24

Coefficient of Multiple Determination (R2)

2

2

8189 723

11050 74741

1 1

2861017

11050 74 741

R

R

SSR

SSY

SSE

SSY

.

..

.

. .

SSEANOVA

df SS MS F p

Regression 2 8189.7 4094.89 28.63 .000


Total 22 11050.7

SSYY SSR


19/24

Adjusted R2

adj

SSE

n kSSY

n

R.

.

.. .

21

1

1

1

2861017

23 2 111050 74

23 1

1 285 715

ANOVA

df SS MS F p

Regression 2 8189.7 4094.9 28.63 .000


Total 22 11050.7

SSYYSSEn-k-1n-1


20/24

Demonstration Problem 15.1: Freight

Data

Country

Freight Cargo Shipped by

Road

(Million Short-Ton

Miles)

Length 0f Roads

(Miles)

Number of Commercial

Vehicles

China 278,806 673,239 5,010,000

Brazil 178,359 1,031,693 1,371,127

India 144,000 1,342,000 1,980,000

Germany 138,975 395,367 2,923,000

Italy 125,171 188,597 2,745,500

Spain 105,824 206,271 2,859,438

Mexico 96,049 157,036 3,758,034


21/24

Demonstration Problem 15.1: Excel

Output

Regression Statistics

Multiple R 0.812

R Square 0.659

Adjusted R Square 0.488

Standard Error 44273.86677

Observations 7

ANOVA

df SS MS F Sig. F

Regression 2 15148592381 7.57E+09 3.86 0.116

Residual 4 7840701114 1.96E+09

Total 6 22989293495

Coefficients Standard Error t Stat P-value

Intercept -26425.45085 67624.93769 -0.39 0.716

Length 0f Roads 0.101820862 0.043495015 2.34 0.079

Commercial Vehicles 0.04094856 0.017121018 2.39 0.075


22/24

Demonstration Problem 15.1: MINITAB

Output


23/24

MINITAB Residual Diagnostics for

Demonstration Problem 15.1

Residual

Pe

rcent

100000500000-50000-100000

99

90

50

10

1

Fitted Value

Re

sidual

250000200000150000100000

50000

25000

0

-25000

-50000

Residual

Frequency

50000250000-25000-50000

2.0

1.5

1.0

0.5

0.0

Observation Order

Residual

7654321

50000

25000

0

-25000

-50000

Normal Probability Plot of the Residuals Residuals Versus the Fitted Values

Hist ogram of t he Residuals Residuals Versus t he Order of t he Dat a

Residual Plots for Freight


24/24

Copyright 2008 John Wiley & Sons, Inc.All rights reserved. Reproduction or translation

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Ken Black QA ch15

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