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Slide 9. Slide 9.1 Confirmatory Confirmatory Factor Analysis Factor Analysis Mathematical Mathematical Marketing Marketing In This Chapter We Will Cover Models with multiple dependent variables, where the independent variables are not observed. This is called Factor Analysis. We cover The factor analysis model A factor analysis example Measurement properties of the unobserved variables Maximum Likelihood estimation of the model Some interesting special cases When statistical reasoning is applied to factor analysis, as it will be in this chapter, we often call this Confirmatory Factor Analysis.

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In This Chapter We Will Cover. Models with multiple dependent variables, where the independent variables are not observed. This is called Factor Analysis. We cover The factor analysis model A factor analysis example Measurement properties of the unobserved variables - PowerPoint PPT Presentation

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Page 1: In This Chapter We Will Cover

Slide 9.Slide 9.11

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

In This Chapter We Will Cover

Models with multiple dependent variables, where the independent variablesare not observed. This is called Factor Analysis. We cover

The factor analysis model

A factor analysis example

Measurement properties of the unobserved variables

Maximum Likelihood estimation of the model

Some interesting special cases

When statistical reasoning is applied to factor analysis, as it will be inthis chapter, we often call this Confirmatory Factor Analysis.

Page 2: In This Chapter We Will Cover

Slide 9.Slide 9.22

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Regression with Multiple Dependent Variables

Y = XB +

np2n1n

p22221

p11211

p*k2*k1*k

p11211

p00201

*nk1n

*k221

*k111

np2n1n

p22221

p11211

eee

eeeeee

xx1

xx1xx1

yyy

yyyyyy

These matrices have only one columnin univariate regression analysis

Page 3: In This Chapter We Will Cover

Slide 9.Slide 9.33

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Comparing Regression with Factor Analysis

ip2i1i

p*k2*k1*k

p11211

p00201

*ik1iip2i1i eeexx1yyy

Looking at a typical row corresponding to the data from subject i:

iii eBxy

Page 4: In This Chapter We Will Cover

Slide 9.Slide 9.44

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

We Transpose It and Drop the Subscript i

iii eBxy

y = Bx + e

Then dropping the subscript i altogether gets us to

iii exBy

From the previous slide we have

Transpose both sides to get

Page 5: In This Chapter We Will Cover

Slide 9.Slide 9.55

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

The Factor Analysis Model

.εηληληλy

εηληληλyεηληληλy

pmpm22p11pp

2mm22221212

1mm12121111

.

y

yy

p

2

1

m

2

1

pm2p1p

m22221

m11211

p

2

1

ΛηyObserved variables

Factor Loadings Common Factors

Unique Factors

Page 6: In This Chapter We Will Cover

Slide 9.Slide 9.66

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Assumptions of the Model

Ληy

Random inputs of the model:

~ N(0, ) 

~ N(0, )  

Cov(, ) = 0

Page 7: In This Chapter We Will Cover

Slide 9.Slide 9.77

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Now We Can Deduce the V(y)

)(E)(E)(E)(E

))((E

)(E)(V

εεΛηεεηΛΛηηΛ

εΛηεΛη

yyΣy

Named

Assumed 0

Named

We end up with only components 1 and 4 from the above equation

V(y) = +

Page 8: In This Chapter We Will Cover

Slide 9.Slide 9.88

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

A Simple Example to Get Us Going

Variables Description

y1 Measurement 1 of B

y2 Measurement 2 of B

y3 Measurement 3 of B

y4 Measurement 1 of C

y5 Measurement 2 of C

y6 Measurement 3 of C

Page 9: In This Chapter We Will Cover

Slide 9.Slide 9.99

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

The Pretend Example in Matrices

6

5

4

3

2

1

2

1

62

52

42

31

21

11

6

5

4

3

2

1

000

000

yyyyyy

εΛηy

.

00

0000

66

22

11

2221

11

Θ

Ψ

Page 10: In This Chapter We Will Cover

Slide 9.Slide 9.1010

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Graphical Conventions of Factor Analysis

y2

y3

y4

y5

y6

y1

1 2

11

21

31

42

52

62

21

Note use of

boxes circles single-headed arrows double-headed arrows unlabeled arrows

Page 11: In This Chapter We Will Cover

Slide 9.Slide 9.1111

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Assume I have a model with just one y and one .

My model is then

y = +

Now assume you have a model y = ** +

where * = a∙ and

* = /a

Whose model is right?

Two Alternative Models

Page 12: In This Chapter We Will Cover

Slide 9.Slide 9.1212

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Ambiguity in the Model

alsoandaa

**y

2

2

22 a

a**)y(V

My Model

y = +

V(y) = 2 +

Your Model

y = ** +

where * = a∙ and* = /a

but

Page 13: In This Chapter We Will Cover

Slide 9.Slide 9.1313

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Resolving the Ambiguity by Setting the Metric

2221

11

62

52

31

21

and,

00

100001

ΨΛ

1

1and,

000

000

21

62

52

42

31

21

11

ΨΛ

Plan A

Plan B

Page 14: In This Chapter We Will Cover

Slide 9.Slide 9.1414

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Degrees of Freedom

2)1p(p

4 ’s3 ’s6 ’s

13 parameters

The General Alternative The Model

H0: = + HA: = S

Page 15: In This Chapter We Will Cover

Slide 9.Slide 9.1515

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

ML Estimation of the Factor Analysis Model

i1

i2/12/pi 21exp

||)2(1)Pr( yΣy

Σy

n

ii

1

i2/n2/np

n

ii0 2

1exp||)2(

1)yPr( yΣyl

]n[TrTrTr 11n

iii

1

i

n

ii

1

i

SΣΣyyyΣyyΣy i

The likelihood of observation i is

The likelihood of the sample is

Because eaeb = ea+b

Page 16: In This Chapter We Will Cover

Slide 9.Slide 9.1616

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

The Log of the Likelihood

)(tr||lnn21ttancons

)(trn21||lnn

21)2(lnpn

21Lln

1

1

00

SΣΣ

SΣΣl

n

ii

1

i2/n2/np

n

ii0 2

1exp||)2(

1)yPr( yΣyl

Page 17: In This Chapter We Will Cover

Slide 9.Slide 9.1717

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

The Log of the Likelihood Under HA

p||lnn21

SLA = constant -

)(tr||lnn21ttanconsln 1

A

SSSl

)(tr||lnn21ttanconsln 1

0

SΣΣl

Page 18: In This Chapter We Will Cover

Slide 9.Slide 9.1818

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

The Likelihood Ratio

)df(~LL2ln2 2A0

A

0

l

l

p)(tr||ln||lnnˆ 12 SΣSΣ

From L0 From LA

S, 0 2ˆ

n , 2ˆ

Page 19: In This Chapter We Will Cover

Slide 9.Slide 9.1919

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

The Single Factor Model

p

2

1

p

2

1

p

2

1

y

yy

= +

if V() = 11 = 1

The latent variable is called a true score

The model is called congeneric tests

Page 20: In This Chapter We Will Cover

Slide 9.Slide 9.2020

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Even More Restrictive Models with More Degrees of Freedom

p21

pp2211

-equivalent tests

Parallel tests

00

0000

Σ

Page 21: In This Chapter We Will Cover

Slide 9.Slide 9.2121

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Multi-Trait Multi-Method Models

y21 y31y11 y22 y32y12 y32 y33y13

1 2 2

4 5 6

Page 22: In This Chapter We Will Cover

Slide 9.Slide 9.2222

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

The MTMM Model in Equations

33

23

13

32

22

12

31

21

11

6

5

4

3

2

1

9693

8583

7473

6662

5552

4442

3631

2521

1411

33

23

13

32

22

12

31

21

11

000000000000

000000000000

000000000000

yyyyyyyyy

10001000

10001

11

)(V

3231

21

3231

21

Ψη

β0α

Page 23: In This Chapter We Will Cover

Slide 9.Slide 9.2323

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Goodness of Fit According to Bentler and Bonett (1980)

j

2

j dfˆ

Q

2

s

2

0

2

s0s ˆ

ˆˆ

1Q

QQ

S

0S0s

Define

HA: = S

H0: = +

HS: = (with diagonal)

Then we could have

Perfect Fit (1)

No Fit (0)(for off-diagonal)

Page 24: In This Chapter We Will Cover

Slide 9.Slide 9.2424

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Goodness of Fit

)(tr

tr1GFI1

21

SΣISΣ

)GFI1(df2

)1p(p1AGFI0

2/)1p(p

)s(RMSE

p

1i

i

1j

2

ijij

Page 25: In This Chapter We Will Cover

Slide 9.Slide 9.2525

Confirmatory Confirmatory Factor AnalysisFactor Analysis

MathematicalMathematicalMarketingMarketing

Modification Indices

22

22

)ˆ(

ˆ2n

MI