1

Chapter 17

Data Analysis:

Investigation of Association

© 2005 Thomson/South-Western

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Figure 1: Scatter Diagrams of Sales vs. Marketing-Mix Variables

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Sales-Y

($000’s)

TV Spots-X1

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10

Number of Salespersons-X2

Sales-Y

($000’s)

3

Figure 1 continued

0

100

200

300

400

500

600

700

800

0 1 2 3 4 5

Wholesaler Efficiency Index-X3

Sales-Y

($000’s)

4

Figure 2: Relationship between Y and X1 in the Probabilistic Model

Y

Yi

Yi^

ei

Yi = α1 + β1 X1i^ ^ ^

X1i

X1

5

Figure 3: Plot of Equation Relating Sales to TV Spots

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Sales

($000’s)

TV Spots

Y=135.4+25.3X1

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Figure 4: Rectangular Distribution of Error Term

Frequency

Y

X

7

Figure 5: Scatter of Points for Sample of n Observations

X

Y

y

x

yi

xi

P

8

Figure 6: Sample Scatter Diagrams and Their Correlation Coefficients

A: r = .95 B: r = .60

E: r = -.40

C: r = 1.00

F: r = -1.00D: r = -.60

I: r = .00H: r = .00G: r = .00

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Figure 7: Hypothetical Relationship between Sales and TV Spotsand between TV Spots and Number of Sales Representatives

TV Spots

200

175

150

125

100

75

50

251 2 3 4 5 6 7 8 9 10

Sales

9

8

7

6

5

4

3

2

1

TV Spots

Number of

Salespersons

1 2 3 4 5 6 7 8 9 10

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Territory Data for Click Ballpoint Pens

Territory

Sales(In Thousands)Y

Advertising(TV Spots/Month)

X1

Number ofSalespersons

X2

WholesalerEfficiency Index

X3

005

019

033

039

061

082

091

101

115

118

133

149

162

164

178

187

189

205

260.3

286.1

279.4

410.8

438.2

315.3

565.1

570.0

426.1

315.0

403.6

220.5

343.6

644.6

520.4

329.5

426.0

343.2

5

7

6

9

12

8

11

16

13

7

10

4

9

17

19

9

11

8

3

5

3

4

6

3

7

8

4

3

6

4

4

8

7

3

6

3

4

2

3

4

1

4

3

2

3

4

1

1

3

4

2

2

4

3

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Territory Data for Click Ballpoint Pens

Territory

Sales(In Thousands)Y

Advertising(TV Spots/Month)

X1

Number ofSalespersons

X2

WholesalerEfficiency Index

X3

222237242251260266279298306332347358362370391408412430442467471488

450.4421.8245.6503.3375.7265.5620.6450.5270.1368.0556.1570.0318.5260.2667.0618.3525.3332.2393.2283.5376.2481.8

1314 716 9 51818 5 71213 8 61619171012 81012

5546536536764388745355

4243334322143222433342

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Sales vs. TV Spots

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Sales-Y

ThousandsofDollars

TV Spots-X1

13

Sales vs. Number of Salespersons

0

100

200

300

400

500

600

700

800

0 2 4 6 8 10

Sales-Y

ThousandsofDollars

Number of Salespersons-X2

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Sales vs. Wholesaler Efficiency Index

0

100

200

300

400

500

600

700

800

0 1 2 3 4 5

Sales-Y

ThousandsofDollars

Wholesaler Efficiency Index-X3

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Computer Output of Regression of Sales Versus TV Spots

Coefficient of Multiple Determination

Coefficient of Multiple Correlation

Standard Error of Estimate

Constant 135.433

TV Spots in 25.307

2.214 11.430

130.644 .880 .880

Analysis of Variance Summary Table

Due to Regression

Due to Residuals

Total

463451.00

134802.01

598253.02

1

38

39

463451.01

3547.42

130.644

Sum ofSquares

Degrees of Freedom

MeanSquare

FRatio

Sales

.775

.880

59.560

Dependent Variable

Variable Regression Standard T- F- Partial StandardizedStatus Coefficient Error Value Level Correlation Coefficient

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Plot of Equation Relating Sales to TV Spots

0

100

200

300

400

500

600

700

800

0 5 10 15 20

Sales

ThousandsofDollars

TV Spots

Y=135.4+25.3X1

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Computer Output of Sales Versus TV Spots and Number of Salespersons

Coefficient of Multiple Determination

Coefficient of Multiple Correlation

Standard Error of Estimate

Variable Regression Standard T- F- Partial StandardizedStatus Coefficient Error Value Level Correlation CoefficientConstant 69.328TV Spots 14.15

62.664 5.315 28.246 .658 .492

Analysis of Variance Summary Table

Due to Regression

Due to Residuals

Total

522778.45

75474.56

598253.02

2

37

39

261389.23

2039.85

128.410

.141

Sum ofSquares

Degrees of Freedom

MeanSquare

FRatio

Sales .874

.935

45.165

Salespersons 37.531

6.959 5.393 29.084 .663 .500

Dependent Variable

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Computer Output of Sales Versus TV Spots, Number of Salespersons and Wholesaler Efficiency

Coefficient of Multiple Determination

Coefficient of Multiple Correlation

Standard Error of Estimate

Variable Regression Standard T- F- Partial StandardizedStatus Coefficient Error Value Level Correlation CoefficientConstant 31.150TV Spots 12.96

82.737 4.738 22.446 .620 .451

Analysis of Variance Summary Table

Due to Regression

Due to Residuals

Total

527209.08

71043.94

598253.02

3

36

39

175736.36

1973.44

89.050

Sum ofSquares

Degrees of Freedom

MeanSquare

FRatio

.881

.939

44.423

Salespersons 41.246

7.280 5.666 32.098 .687 .549

Wholeeff 11.524

7.691 1.498 2.245 .242 .092

Sales

Dependent Variable

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Computer Output of Sales Versus TV Spots, # of Salespersons and Wholesaler Efficiency with Wholesaler Efficiency Expressed as a Dummy Variable

Coefficient of Multiple Determination

Coefficient of Multiple Correlation

Standard Error of Estimate

Constant 44.211TV Spots 13.06

32.940 4.443 19.738 .606

Analysis of Variance Summary Table

Due to Regression

Due to Residuals

Total

527235.24

71017.77

598253.02

5

34

39

105447.05

2088.76

50.483

Sum ofSquares

Degrees of Freedom

MeanSquare

FRatio

Sales

.881

.939

45.703

Salespersons 40.948

8.009 5.113 26.143 .659Fairdist

8.39925.378

.331 .110 .057

Excldist 32.924

27.671

1.190 1.416 .200Gooddist 20.03

128.770

.696 .485 .119

Dependent Variable

Variable Regression Standard T- F- Partial StandardizedStatus Coefficient Error Value Level Correlation Coefficient

.454

.545

.033

.123

.077

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Hypothetical Relationship Between Sales and TV Spots, & Between TV Spots and #of Salespersons

200

175

150

125

100

75

50

25

9

8

7

6

5

4

3

2

1

1 2 3 4 5 6 7 8 9 10

TV Spots

1 2 3 4 5 6 7 8 9 10

TV Spots

Number of

Salespersons

Sales

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Measure of Linear Association between 2 Variables Range: -1.00 < r xy < 1.00

n

i yx

iixy SnS

yyxxr

1

))((

Correlation Coefficient

22

y

xrxy=-1.00

y

xrxy=-.20

y

xrxy=-.70

Relationship Between Scatterplots and Correlation Coefficients

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y

xrxy=.00

Nonlinear Relationship in the Data? “r” will be an approximation.

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Effect of Multicollinearity

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Predictions

“Experts are sure the Dow will either rise or decline.” ‘Boy, business forecasting is an exact science, isn’t it?’

(Headlines, compiled by Jay Leno)

A safe prediction for the market is the time

of the closing bell. (101 Corporate Haiku, W. Warriner)

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Representation of Categorical Variables for Regressions Ex/ Favorability rating of Auto prototype as function of: age,

gender, nationality

– X1=age

– X2=gender: M=1, W=0

– X3=nationality: 1=Asian, 2=European, 3=U.S.

#dummy variables required = # categories -1

Use X3 and define:

– If X3=1 then DV1=1, else DV1=0

– If X3=2 then DV2=1, else DV2=0

Dummy Variables

Nationality X3 DV1 DV2Asian 1 1 0European 2 0 1U.S. 3 0 0

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Source:

Conjoint Measurement

Understand how consumers make trade-offsDiscover attributes most valued by consumers

Implications of attribute values, combinations for product design

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Respondent’s Ordering of Various Product Descriptions

Source:

Capacity

Price

4 Cup

$28 $32 $38

8 Cup

$28 $32 $38

22 Cup

$28 $32 $38

Brewing Time

3 Minutes

6 Minutes

9 Minutes

12 Minutes

17 15 6

16 12 5

9 8 3

4 2 1

30 26 24

29 25 22

21 20 8

14 13 7

36 34 28

35 33 27

32 31 23

19 18 11

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Some Attribute Utility Values & the Resulting Utilities for the Alternatives Under an Additive Rule

Capacity

Price

4 Cup

$28 $32 $38

8 Cup

$28 $32 $38

22 Cup

$28 $32 $38

Brewing Time

3 Minutes

6 Minutes

9 Minutes

12 Minutes

1.3 1.0 0.8

1.1 0.3 0.6

0.9 0.6 0.4

0.9 0.6 0.4

1.4 1.1 0.9

1.2 0.9 0.7

1.0 0.7 0.5

1.0 0.7 0.5

1.6 1.3 1.1

1.4 1.1 0.9

1.2 0.9 0.7

1.2 0.9 0.7

Capacity Brewing Time

Price

4 Cup .2

8 Cup .3

10 Cup .5

$28 .6

$32 .3

$38 .1

3 Minutes .5

6 Minutes .3

9 Minutes .1

12 Minutes .1

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Plot of Input Ranks Versus Derived Cell Values

Input

Ranks

Derived Cell Values

40

30

20

10

0.5 1.0 1.5 2.0

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Figure 1: Key Decisions when Conducting a Conjoint Analysis

Select Attributes

Determine Attribute Levels

Select Form of Presentation of Stimuli and Natureof Judgments to Be Secured from Subjects

Decide on Whether, and If Yes How, Judgments Will Be Aggregated

Determine Attribute Combinationsto Be Used

Select Analysis Technique

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Coke store brand

$1.99$2.99

Coke store

$1.99$2.99

1

2 4

3

Coke store

$1.99$2.99

1

3 4

2

Which consumer is price sensitive, and which values quality or brand names?:

Using Conjoint to Determine Price Sensitivity and Brand Equity

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Figure 2: Pair-wise Approach to Data Collection in Conjoint Analysis

$28 $32 $38

4

8

10

Price

Capacity(cups)

3 6 9 12

4

8

10

Brewing Time (mins)

Capacity(cups)

3 6 9 12

$28

$32

$38

Brewing Time (mins)

Price

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Figure 3: Computer Administered Paired Comparison Choice

Which would you prefer?Use the scale below to indicate your preference.

4-cup capacity 8-cup capacity9-min.brewing time 3-min.brewing time

$28 $38

Strongly StronglyPrefer 1 2 3 4 5 6 7 8 9 PreferLeft Right