assignment 3.1

Assignment 3: Multiple Regression

This data set consists of a sample of over eight hundred used cars in this country. The retail price of these cars was calculated from the tables provided by the association of car manufacturer. You are provided with a data set containing the following variables:

· Price: suggested retail price of the used car in excellent condition. The condition of a car can greatly affect price. All cars in this data set were less than one year old when priced and considered to be in excellent condition.

· Mileage: number of miles the car has been driven · Make: manufacturer of the car. · Model: specific models for each car manufacturer.· Trim (of car): specific type of car model such as SE Sedan 4D, Quad Coupe 2D· Type: body type such as sedan, coupe, etc.· Cylinder: number of cylinders in the engine· Liter: a more specific measure of engine size · Doors: number of doors · Cruise: indicator variable representing whether the car has cruise control (1 = cruise)· Sound: indicator variable representing whether the car has upgraded speakers (1 = upgraded)· Leather: indicator variable representing whether the car has leather seats (1 = leather)

Perform the following tasks on this data set:

1. Use simple linear regression to explore the intuitive relationship between miles traveled and retail price.From the simple regression results, answer the following questions:a. In general, what happens to price when there is one more mile on the car?b. Does mileage help you predict price? What does the p-value tell you?c. Does mileage help you predict price? What does the R-Sq value tell you?

Answers

Variables Entered/Removeda

Model

Variables

Entered

Variables

Removed Method

1 Mileageb . Enter

a. Dependent Variable: Price

b. All requested variables entered.

Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

t

Sig.

B Std. Error Beta

1 (Constant) 24764.559 904.363 27.383 .000

Mileage -.173 .042 -.143 -4.093 .000


a. The price will be reduced by 1.73 cents with each added mile on the car.

Coefficientsa

Model


Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) 24764.559 904.363 27.383 .000

Mileage -.173 .042 -.143 -4.093 .000


b. Yes.p-value explains that the relationship between mileage and price is negatively significant corelated.It is significant but in negative direction.

Model Summary

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate

1 .143a .020 .019 9789.288

a. Predictors: (Constant), Mileage

c. Yes. R-Sq(R2) is the correlation coefficient squared(.1432 = .020) referred to as the coefficent of determination. This values indicates the percentage of total variation of Y ( Price) explained by the regression model consisting of miles. Only 2% can be influenced by mileage and the rest (98%) by other factors.

2. Taking price as the dependent variable, perform stepwise multiple regression on this data set.What is your final model? How many variable/variables was/were dropped from the model. Explain why?


Model

Variables

Entered

Variables

Removed Method

1

Cylinder .

Stepwise

(Criteria:

Probability-of-F-

to-enter <= .050,

Probability-of-F-

to-remove

>= .100).

2

Cruise .

Stepwise

(Criteria:

Probability-of-F-

to-enter <= .050,

Probability-of-F-

to-remove

>= .100).

3

Leather .

Stepwise

(Criteria:

Probability-of-F-

to-enter <= .050,

Probability-of-F-

to-remove

>= .100).

4

Mileage .

Stepwise

(Criteria:

Probability-of-F-

to-enter <= .050,

Probability-of-F-

to-remove

>= .100).

5

Doors .

Stepwise

(Criteria:

Probability-of-F-

to-enter <= .050,

Probability-of-F-

to-remove

>= .100).

6

Sound .

Stepwise

(Criteria:

Probability-of-F-

to-enter <= .050,

Probability-of-F-

to-remove

>= .100).


Coefficientsa

Model


Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) -17.057 1126.944 -.015 .988

Cylinder 4054.203 206.852 .569 19.600 .000

2 (Constant) -1046.431 1082.655 -.967 .334

Cylinder 3392.587 211.273 .476 16.058 .000

Cruise 6000.366 678.841 .262 8.839 .000

3 (Constant) -2978.398 1129.554 -2.637 .009

Cylinder 3276.233 209.189 .460 15.662 .000

Cruise 6362.343 671.901 .278 9.469 .000

Leather 3139.484 608.259 .142 5.161 .000

4 (Constant) 412.562 1296.815 .318 .750

Cylinder 3232.656 206.188 .454 15.678 .000

Cruise 6492.035 662.181 .284 9.804 .000

Leather 3161.569 599.032 .143 5.278 .000

Mileage -.165 .032 -.137 -5.087 .000

5 (Constant) 5530.335 1709.446 3.235 .001

Cylinder 3257.643 203.798 .457 15.985 .000

Cruise 6319.636 655.373 .276 9.643 .000

Leather 2978.887 593.246 .135 5.021 .000

Mileage -.167 .032 -.139 -5.214 .000

Doors -1402.112 310.015 -.121 -4.523 .000

6 (Constant) 7323.164 1770.837 4.135 .000

Cylinder 3200.125 202.983 .449 15.765 .000

Cruise 6205.511 651.463 .271 9.525 .000

Leather 3327.143 597.114 .151 5.572 .000

Mileage -.171 .032 -.141 -5.352 .000

Doors -1463.399 308.274 -.126 -4.747 .000

Sound -2024.401 570.718 -.096 -3.547 .000


Model Summaryg

Model R R Square

Adjusted R

Square

Std. Error of the

Estimate Durbin-Watson

1 .569a .324 .323 8133.162

2 .620b .384 .382 7768.193

3 .635c .404 .402 7646.769

4 .650d .423 .420 7530.569

5 .661e .437 .433 7440.529

6 .668f .446 .442 7387.114 .304

a. Predictors: (Constant), Cylinder

b. Predictors: (Constant), Cylinder, Cruise

c. Predictors: (Constant), Cylinder, Cruise, Leather

d. Predictors: (Constant), Cylinder, Cruise, Leather, Mileage

e. Predictors: (Constant), Cylinder, Cruise, Leather, Mileage, Doors

f. Predictors: (Constant), Cylinder, Cruise, Leather, Mileage, Doors, Sound

g. Dependent Variable: Price

In the Model Summary, we can see that litre is deleted.

Excluded Variablesa

Model Beta In t Sig.

Partial

Correlation

Collinearity

Statistics

Tolerance

1 Mileage -.126b -4.401 .000 -.154 .999

Liter .158b 1.563 .118 .055 .082

Doors -.140b -4.890 .000 -.170 1.000

Cruise .262b 8.839 .000 .298 .874

Sound -.074b -2.543 .011 -.090 .992

Leather .115b 3.981 .000 .139 .994

2 Mileage -.136c -4.966 .000 -.173 .998

Liter .037c .383 .702 .014 .081

Doors -.128c -4.655 .000 -.162 .997

Sound -.058c -2.094 .037 -.074 .988

Leather .142c 5.161 .000 .180 .983

3 Mileage -.137d -5.087 .000 -.177 .998

Liter .004d .037 .970 .001 .080

Doors -.119d -4.377 .000 -.153 .993

Sound -.084d -3.048 .002 -.107 .960

4 Liter .017e .180 .858 .006 .080

Doors -.121e -4.523 .000 -.158 .992

Sound -.088e -3.243 .001 -.114 .959

5 Liter -.108f -1.108 .268 -.039 .074

Sound -.096f -3.547 .000 -.125 .956

6 Liter -.088g -.908 .364 -.032 .074


b. Predictors in the Model: (Constant), Cylinder

c. Predictors in the Model: (Constant), Cylinder, Cruise

d. Predictors in the Model: (Constant), Cylinder, Cruise, Leather

e. Predictors in the Model: (Constant), Cylinder, Cruise, Leather, Mileage

f. Predictors in the Model: (Constant), Cylinder, Cruise, Leather, Mileage, Doors

g. Predictors in the Model: (Constant), Cylinder, Cruise, Leather, Mileage, Doors, Sound

Litre is excluded where the P value is high. For each model ( 1 – 6 ) the p values are more than 0.05 significant.

Only one variable is dropped. Because the P value of Litre is more than the significance

value; p < 0.05.

3. Transform price to log price and take this new variable as your dependent variable. Perform multiple regression by including variables in (2) as independent variables. Discuss the results.


Model

Variables

Entered

Variables

Removed Method

1 Leather,

Mileage, Doors,

Cylinder, Sound,

Cruiseb

. Enter

a. Dependent Variable: LgPrice

b. All requested variables entered.

Model Summaryb

Mode

l R

R

Square

Adjusted R

Square

Std. Error

of the

Estimate

Change Statistics

Durbin-

Watson

R Square

Change

F

Change df1 df2

Sig. F

Change

1 .695a .484 .480 .12847 .484 124.410 6 797 .000 .376

a. Predictors: (Constant), Leather, Mileage, Doors, Cylinder, Sound, Cruise

b. Dependent Variable: LgPrice

After running the Log Price as the dependent variable, we can see that Litre is excluded but cylinder is included.

Coefficientsa

Model Unstandardized

Coefficients

Standardized

Coefficients

t Sig. 95.0%

Confidence

Interval for B

Correlations Collinearity

Statistics

B

Std.

Error Beta

Lower

Bound

Upper

Bound

Zero-

order Partial Part Tolerance VIF

1 (Constant) 3.996 .031 129.744 .000 3.935 4.056

Mileage -

3.206E-

6

.000 -.148 -5.786 .000 .000 .000 -.148 -.201-.14

7.997 1.003

Cylinder .057 .004 .440 16.018 .000 .050 .063 .583 .493 .408 .857 1.167

Doors-.016 .005 -.077 -3.007 .003 -.027 -.006 -.092 -.106

-.07

7.989 1.011

Cruise .139 .011 .338 12.298 .000 .117 .162 .494 .399 .313 .859 1.165

Sound-.038 .010 -.099 -3.816 .000 -.057 -.018 -.139 -.134

-.09

7.956 1.046

Leather .053 .010 .132 5.078 .000 .032 .073 .130 .177 .129 .952 1.050


Excluded Variablesa

Model Beta In t Sig.

Partial

Correlation

Collinearity

Statistics

Tolerance

1 Mileage -.137b -4.876 .000 -.170 1.000

Cylinder .215b 2.170 .030 .076 .082

Doors -.045b -1.578 .115 -.056 .994

Cruise .316b 10.999 .000 .362 .857

Sound -.101b -3.561 .000 -.125 .996

Leather .079b 2.777 .006 .098 .992

2 Mileage -.147c -5.646 .000 -.196 .998

Cylinder .243c 2.636 .009 .093 .082

Doors -.039c -1.480 .139 -.052 .993

Sound -.080c -3.017 .003 -.106 .990

Leather .113c 4.271 .000 .149 .980

3 Cylinder .223d 2.463 .014 .087 .082

Doors -.042d -1.610 .108 -.057 .993

Sound -.084d -3.220 .001 -.113 .990

Leather .114d 4.393 .000 .154 .980

4 Cylinder .236e 2.632 .009 .093 .082

Doors -.036e -1.375 .169 -.049 .990

Sound -.106e -4.060 .000 -.142 .963

5 Cylinder .204f 2.284 .023 .081 .081

Doors -.042f -1.642 .101 -.058 .986

6 Doors -.062g -2.346 .019 -.083 .916

a. Dependent Variable: TrPrice

b. Predictors in the Model: (Constant), Liter

c. Predictors in the Model: (Constant), Liter, Cruise

d. Predictors in the Model: (Constant), Liter, Cruise, Mileage

e. Predictors in the Model: (Constant), Liter, Cruise, Mileage, Leather

f. Predictors in the Model: (Constant), Liter, Cruise, Mileage, Leather, Sound

g. Predictors in the Model: (Constant), Liter, Cruise, Mileage, Leather, Sound, Cylinder

Doors is excluded. Where the P values are more than 0.05 for each model.

Only one variable is dropped. Because the P value of Doors is more than the significance

value; p < 0.05.

4. Since Type (Sedan, Hatchback, Convertible or Coupe) and Make (A,B,C,D,E or F) are also criterias considered by many car buyers, perform another regression by considering these two variables. Discuss the results.

A Dummy variable or Indicator Variable is an artificial variable created to represent an attribute with two or more distinct categories/levels. Regression analysis treats all independent (X) variables in the analysis as numerical. Numerical variables are interval or ratio scale variables whose values are directly comparable. For multiple regression analysis, all but one of the dummy variables is entered as independent variables for each of the original categorical variables. With dummy variables, the regression coefficients indicate the difference in the dependent variable between the category specified by the dummy variable and the category omitted from the analysis.

After Type and Make are changed into dummy variables. The data is analysed.

Descriptive Statistics

Mean Std. Deviation N

LgPrice 4.2904 .17811 804

Mileage 19831.93 8196.320 804

Cylinder 5.27 1.388 804

Doors 3.53 .850 804

Cruise .75 .432 804

Sound .68 .467 804

Leather .72 .447 804

A .10 .300 804

B .10 .300 804

C .40 .490 804

D .14 .349 804

E .07 .263 804

Sedan .42 .494 804

Convertible .06 .242 804

Hatchback .05 .218 804

Coupe .17 .379 804


Model

Variables

Entered

Variables

Removed Method

1 Coupe, Mileage,

Cruise, Leather,

Convertible,

Sound,

Hatchback, E, A,

B, D, C,

Cylinder, Sedanb

. Enter


b. Tolerance = .000 limit reached.

Model Summaryb

Mod

el R

R

Square

Adjusted R

Square

Std. Error

of the

Estimate

Change Statistics

Durbin-

Watson

R Square

Change

F

Change df1 df2

Sig. F

Change

1 .960a .922 .921 .05008 .922 669.112 14 789 .000 .274

a. Predictors: (Constant), Coupe, Mileage, Cruise, Leather, Convertible, Sound, Hatchback, E, A, B, D, C, Cylinder,

Sedan

b. Dependent Variable: LgPrice

Coefficientsa

Model

Unstandardized

Coefficients

Standardized

Coefficients

t Sig.

95.0%

Confidence

Interval for B Correlations

Collinearity

Statistics

B

Std.

Error Beta

Lower

Bound

Upper

Bound

Zero-

order Partial Part Tolerance VIF

1 (Constant) 3.903 .013 306.064 .000 3.877 3.928

Cylinder .072 .002 .560 36.209 .000 .068 .076 .583 .790 .359 .412 2.429

Cruise .010 .005 .024 1.963 .050 .000 .020 .494 .070 .019 .663 1.507

Sound .002 .004 .004 .425 .671 -.006 .010 -.139 .015 .004 .884 1.131

Leather .017 .004 .042 3.927 .000 .008 .025 .130 .138 .039 .845 1.183

A .067 .009 .113 7.316 .000 .049 .085 .044 .252 .073 .414 2.416

B .213 .010 .359 21.339 .000 .194 .233 .580 .605 .212 .348 2.870

C-.007 .006 -.019 -1.167 .243 -.018 .005 -.467 -.042

-.01

2.377 2.654

D .310 .008 .608 38.839 .000 .294 .326 .402 .810 .385 .402 2.486

E .008 .009 .012 .896 .371 -.010 .026 -.237 .032 .009 .561 1.781

Sedan-.032 .006 -.089 -5.275 .000 -.044 -.020 .029 -.185

-.05

2.349 2.862

Convertible .113 .009 .154 13.072 .000 .096 .130 .440 .422 .130 .712 1.404

Hatchback-.083 .010 -.102 -8.737 .000 -.102 -.065 -.263 -.297

-.08

7.727 1.375

Coupe .002 .006 .005 .364 .716 -.010 .014 -.178 .013 .004 .613 1.630


Coefficient Correlationsa

Model

Cou

pe

Mile

age

Crui

se

Leat

her

Conve

rtible

Sou

nd

Hatch

back E A B D C

Cylin

der

Sed

an

1 Correla

tions

Coupe 1.00

0.021

-.07

3

-.07

1.235

-.06

6.337

-.30

6

-.21

5

-.20

7.009

-.30

8.128 .513

Mileag

e.021

1.00

0

-.01

2

-.02

1.007 .023 .038

-.04

7

-.06

0

-.01

7

-.04

7

-.03

9.017 .051

Cruise -.07

3

-.01

2

1.00

0.102 .028 .005 .106 .073

-.13

3

-.04

7

-.32

2.022

-.33

4.041

Leathe

r

-.07

1

-.02

1.102

1.00

0.030

-.14

2-.015 .079 .095

-.14

7

-.10

5

-.08

7

-.06

8

-.02

0

Conve

rtible.235 .007 .028 .030 1.000

-.03

3.167

-.20

4

-.21

1

-.26

1

-.38

4

-.24

1

-.07

0.374

Sound -.06

6.023 .005

-.14

2-.033

1.00

0.042 .115

-.01

4.044 .065

-.09

8.076

-.06

2

Hatchb

ack.337 .038 .106

-.01

5.167 .042 1.000

-.14

5

-.25

0

-.25

9

-.05

6

-.33

7.161 .398

E -.30

6

-.04

7.073 .079 -.204 .115 -.145

1.00

0.431 .341 .387 .534 .098

-.42

4

A -.21

5

-.06

0

-.13

3.095 -.211

-.01

4-.250 .431

1.00

0.586 .361 .541

-.20

1

-.62

2

B -.20

7

-.01

7

-.04

7

-.14

7-.261 .044 -.259 .341 .586

1.00

0.257 .498

-.43

5

-.60

8

D.009

-.04

7

-.32

2

-.10

5-.384 .065 -.056 .387 .361 .257

1.00

0.504 .425

-.20

4

C -.30

8

-.03

9.022

-.08

7-.241

-.09

8-.337 .534 .541 .498 .504

1.00

0.025

-.42

6

Cylind

er.128 .017

-.33

4

-.06

8-.070 .076 .161 .098

-.20

1

-.43

5.425 .025

1.00

0.253

Sedan.513 .051 .041

-.02

0.374

-.06

2.398

-.42

4

-.62

2

-.60

8

-.20

4

-.42

6.253

1.00

0

Covari

ances

Coupe 3.53

6E-5

2.76

6E-

11

-

2.18

4E-6

-

1.81

4E-6

1.210

E-5

-

1.57

6E-6

1.906

E-5

-

1.63

2E-5

-

1.17

5E-5

-

1.23

3E-5

4.44

6E-7

-

1.07

6E-5

1.51

1E-6

1.84

4E-5

Mileag

e2.76

6E-

11

4.69

7E-

14

-

1.34

5E-

11

-

1.97

9E-

11

1.223

E-11

2.00

7E-

11

7.819

E-11

-

9.10

0E-

11

-

1.18

3E-

10

-

3.75

6E-

11

-

8.12

4E-

11

-

5.00

7E-

11

7.23

6E-

12

6.71

5E-

11

Cruise-

2.18

4E-6

-

1.34

5E-

11

2.52

5E-5

2.19

8E-6

1.233

E-6

1.10

5E-7

5.078

E-6

3.28

2E-6

-

6.11

9E-6

-

2.36

8E-6

-

1.29

2E-5

6.38

4E-7

-

3.33

0E-6

1.23

9E-6

Leathe

r-

1.81

4E-6

-

1.97

9E-

11

2.19

8E-6

1.84

7E-5

1.122

E-6

-

2.45

8E-6

-

6.332

E-7

3.05

2E-6

3.74

1E-6

-

6.29

6E-6

-

3.60

3E-6

-

2.21

0E-6

-

5.75

9E-7

-

5.16

4E-7

Conve

rtible1.21

0E-5

1.22

3E-

11

1.23

3E-6

1.12

2E-6

7.507

E-5

-

1.16

8E-6

1.379

E-5

-

1.58

6E-5

-

1.67

5E-5

-

2.26

3E-5

-

2.65

9E-5

-

1.22

6E-5

-

1.19

7E-6

1.96

2E-5

Sound -

1.57

6E-6

2.00

7E-

11

1.10

5E-7

-

2.45

8E-6

-

1.168

E-6

1.62

0E-5

1.599

E-6

4.14

9E-6

-

5.01

8E-7

1.78

1E-6

2.09

7E-6

-

2.31

3E-6

6.06

5E-7

-

1.50

7E-6

Hatchb

ack1.90

6E-5

7.81

9E-

11

5.07

8E-6

-

6.33

2E-7

1.379

E-5

1.59

9E-6

9.075

E-5

-

1.23

6E-5

-

2.18

8E-5

-

2.46

4E-5

-

4.27

8E-6

-

1.88

6E-5

3.03

9E-6

2.29

2E-5

E-

1.63

2E-5

-

9.10

0E-

11

3.28

2E-6

3.05

2E-6

-

1.586

E-5

4.14

9E-6

-

1.236

E-5

8.04

7E-5

3.54

4E-5

3.05

3E-5

2.76

9E-5

2.81

5E-5

1.75

0E-6

-

2.30

1E-5

A-

1.17

5E-5

-

1.18

3E-

10

-

6.11

9E-6

3.74

1E-6

-

1.675

E-5

-

5.01

8E-7

-

2.188

E-5

3.54

4E-5

8.41

3E-5

5.36

9E-5

2.64

1E-5

2.91

7E-5

-

3.65

3E-6

-

3.45

2E-5

B-

1.23

3E-5

-

3.75

6E-

11

-

2.36

8E-6

-

6.29

6E-6

-

2.263

E-5

1.78

1E-6

-

2.464

E-5

3.05

3E-5

5.36

9E-5

9.99

1E-5

2.05

4E-5

2.92

7E-5

-

8.63

2E-6

-

3.67

4E-5

D

4.44

6E-7

-

8.12

4E-

11

-

1.29

2E-5

-

3.60

3E-6

-

2.659

E-5

2.09

7E-6

-

4.278

E-6

2.76

9E-5

2.64

1E-5

2.05

4E-5

6.37

4E-5

2.36

3E-5

6.73

3E-6

-

9.85

8E-6

C -

1.07

6E-5

-

5.00

7E-

11

6.38

4E-7

-

2.21

0E-6

-

1.226

E-5

-

2.31

3E-6

-

1.886

E-5

2.81

5E-5

2.91

7E-5

2.92

7E-5

2.36

3E-5

3.45

6E-5

2.93

7E-7

-

1.51

4E-5

Cylind

er1.51

1E-6

7.23

6E-

12

-

3.33

0E-6

-

5.75

9E-7

-

1.197

E-6

6.06

5E-7

3.039

E-6

1.75

0E-6

-

3.65

3E-6

-

8.63

2E-6

6.73

3E-6

2.93

7E-7

3.94

1E-6

3.04

2E-6

Sedan1.84

4E-5

6.71

5E-

11

1.23

9E-6

-

5.16

4E-7

1.962

E-5

-

1.50

7E-6

2.292

E-5

-

2.30

1E-5

-

3.45

2E-5

-

3.67

4E-5

-

9.85

8E-6

-

1.51

4E-5

3.04

2E-6

3.65

9E-5


After including type and make (dummy variables), other variables are excluded from the

model as their partial correlation was significant. This suggest that if we maintain them in the

model, it will not have significant influence on the ability of the model to predict retail price

of the car.

The prediction model contained 11 variables in total and 9 dummies. All 11 predictors were

gathered in 11 steps with 5 variables removed. The model was statistically significant and

counted for 97.3% of the variance of the retail price. Only litre and mileage have the highest

influence on retail price of the car.

GRADUATE SCHOOL OF BUSINESS (UKM – GSB)

ZCZA6043

MULTIVARIATE ANALYSIS

ASSIGNMENT 3

MULTIPLE REGRESSION

PREPARED FOR :

PROF. MADYA DR. RASIDAH MOHAMAD SAID

PREPARED BY:

AL AZMI BIN ABDUL RAHMAN

(ZP 02311)

assignment 3.1

Documents