Download Quant_Chapter_09_cfa

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.
Mathematical
Marketing
Slide 9.1
Confirmatory
Factor Analysis
Regression with Multiple Dependent Variables
These matrices have only one column
in univariate regression analysis
 y11
y
 21


 y n1
y12
y 22

yn 2




y1p   1
y2p   1

  
 
y np   1
x11  x1k*   01 02
x 21  x 2 k*   11 12

     

x n1  x nk*  k*1 k*2
Y = XB +
Mathematical
Marketing
 0 p   e11
 1p  e 21

   
 
  k *p  e n 1
e12
e 22

en 2




e1p 
e2 p 



e np 

Slide 9.2
Confirmatory
Factor Analysis
Comparing Regression with Factor Analysis
Looking at a typical row corresponding to the data from subject i:
y
i1
y i 2  y ip   1 x i1  x ik* 
 01 02

12
 11
 

 k*1  k*2
 0 p 
 1p 
  e
i1
 

  k*p 
e i 2  e ip 
yi  xiB  ei
Mathematical
Marketing
Slide 9.3
Confirmatory
Factor Analysis
We Transpose It and Drop the Subscript i
From the previous slide we have
yi  xiB  ei
Transpose both sides to get
y i  Bxi  ei
Then dropping the subscript i altogether gets us to
y = Bx + e
Mathematical
Marketing
Slide 9.4
Confirmatory
Factor Analysis
The Factor Analysis Model
y1  λ11η1  λ12 η2    λ1m ηm  ε1
y 2  λ 21η1  λ 22 η2    λ 2 m ηm  ε 2
 
y p  λ p1η1  λ p 2 η2    λ pm ηm  ε p .
 y1   11 12
 y  
 22
 2    21
   
  
 y p   p 1  p 2
Observed variables
 1 
 1 
 
 
2
    2 .

 
 
 
 m 
p 
y  Λη  
Factor Loadings
Mathematical
Marketing
 1 m 
 2m 

 

  pm 
Unique Factors
Common Factors
Slide 9.5
Confirmatory
Factor Analysis
Assumptions of the Model
y  Λη  
Random inputs of the model:
 ~ N(0, )
 ~ N(0, )
Cov(, ) = 0
Mathematical
Marketing
Slide 9.6
Confirmatory
Factor Analysis
Now We Can Deduce the V(y)
V(y )  Σ  E(yy)
 E ( Λη  ε)( Λη  ε)
 Λ E( ηη) Λ  Λ E( ηε)  E(εη) Λ  E(εε)
Named 
Named 
Assumed 0
We end up with only components 1 and 4 from the
above equation
V(y) =  + 
Mathematical
Marketing
Slide 9.7
Confirmatory
Factor Analysis
A Simple Example to Get Us Going
Mathematical
Marketing
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
Slide 9.8
Confirmatory
Factor Analysis
The Pretend Example in Matrices
y  Λη ε
 y1 
11 0 
y 

0
2
21
 


 y3 
 31 0 
   

 y4 
 0  42 
 y5 
 0  52 
 


 y6 
 0  62 
 1 
 
 2
 3 
 1 

 
 
 2
 4 
 5 
 
 6 


Ψ   11

 21  22 
11 0
0 
22
Θ
 

0 0
Mathematical
Marketing
 0
 0
.
 

 66 
Slide 9.9
Confirmatory
Factor Analysis
Graphical Conventions of Factor Analysis
21
y1
y2
y3
42
11
21
1
31
2
52
62
y4
y5
y6
Note use of
 boxes
 circles
 single-headed arrows
 double-headed arrows
 unlabeled arrows
Mathematical
Marketing
Slide 9.10
Confirmatory
Factor Analysis
Two Alternative Models
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?
Mathematical
Marketing
Slide 9.11
Confirmatory
Factor Analysis
Ambiguity in the Model
My Model
Your Model
y =  + 
y = ** + 
V(y) = 2 + 
where * = a∙ and
* = /a
but
y  * *   

 a   and also
a
2 2
V( y)  * *    2  a   
a
2
Mathematical
Marketing
Slide 9.12
Confirmatory
Factor Analysis
Resolving the Ambiguity by Setting the Metric
Mathematical
Marketing
Plan A
0
1

0
21


 31 0 
Λ
, and
0
1


 0  52 


 0  62 


Ψ   11

 21  22 
Plan B
11 0 

0
 21


0
Λ   31
, and
0

42 

 0  52 


 0  62 
1
Ψ
 21

1 
Slide 9.13
Confirmatory
Factor Analysis
Degrees of Freedom
The General Alternative
HA:  = S
p(p  1)
2
The Model
H0:  =  + 
4 ’s
3 ’s
6 ’s
13 parameters
Mathematical
Marketing
Slide 9.14
Confirmatory
Factor Analysis
ML Estimation of the Factor Analysis Model
The likelihood of observation i is
Pr(y i ) 
 1

exp  yi Σ 1y i 
(2) |Σ|
 2

1
p/2
1
n
The likelihood of the sample is
l 0   Pr( yi ) 
i
1/ 2
(2) np / 2 ||n / 2
 1 n

exp   yi Σ 1y i 
 2 i

 yΣ 1y   Tr n y yΣ 1  Tr [nSΣ 1 ]
1

y
Σ
y

Tr
i i i
i i i
i
 i

n
Because eaeb = ea+b
Mathematical
Marketing
Slide 9.15
Confirmatory
Factor Analysis
The Log of the Likelihood
n
l 0   Pr( yi ) 
i
1
(2) np / 2 ||n / 2
 1 n

exp   yi Σ 1y i 
 2 i

1
1
1
ln l0  L 0   n p ln (2)  n ln |Σ |  n tr (SΣ 1 )
2
2
2
1
 constant  nln |Σ |  tr (SΣ 1 )
2
Mathematical
Marketing
Slide 9.16
Confirmatory
Factor Analysis
The Log of the Likelihood Under HA
1
ln l0  constant  nln |Σ |  tr (SΣ1 )
2
1
ln lA  constant  nln |S |  tr (SS1 )
2
LA = constant -
Mathematical
Marketing
1
n ln | S | p
2
Slide 9.17
Confirmatory
Factor Analysis
The Likelihood Ratio
l 
 2 ln  0   2L 0  L A  ~  2 (df )
 lA 
From LA
From L0
ˆ 2  n ln |Σ |  ln |S |  tr(SΣ 1 )  p
  S,
ˆ 2  0
2
n  , ˆ  
Mathematical
Marketing
Slide 9.18
Confirmatory
Factor Analysis
The Single Factor Model
 y1    1 
 1 
 y   
 
2
2
       2
 

   
 
y

p
p
   
 p 
if V() = 11 = 1
 =  + 
The latent variable  is called a true score
The model is called congeneric tests
Mathematical
Marketing
Slide 9.19
Confirmatory
Factor Analysis
Even More Restrictive Models with More Degrees of Freedom
1  2    p  
-equivalent tests
11  22    pp  
Parallel tests

 0  0

0   0

Σ          

   
 



0 0  
Mathematical
Marketing
Slide 9.20
Confirmatory
Factor Analysis
Multi-Trait Multi-Method Models
1
y11
y21
y31
y12
4
Mathematical
Marketing
2
y22
5
2
y32
y13
y32
y33
6
Slide 9.21
Confirmatory
Factor Analysis
The MTMM Model in Equations
 y11 
11 0
y 

0
 21 
 21
 y31 
 31 0
 

 y12 
 0  42
 y 22    0  52
 

 y32 
 0  62
 y13 
0
0
 

0
 y 23 
0
y 
0
0
 33 

0
14
0
0
0
 25
0
0
0
0
 44
0
0
0
 55
0
0
0
 73
 74
0
 83
0
 85
 93
0
0
1

1
 21
  32
V( η)  Ψ   31
0
0
0
0

0
0
Mathematical
Marketing
0
0

 36 

0
0

 66 
0

0
 96 

1
0 1
0 21 1
0 31 32
 1 
 
 2
3 
 
4 
5 
 
6 








1
 11 
 
 21 
31 
 
12 
  22 
 
32 
 13 
 
 23 
 
 33 
α 


0 β
Slide 9.22
Confirmatory
Factor Analysis
Goodness of Fit According to Bentler and Bonett (1980)
Define
Perfect Fit (1)
HA:  = S
H0:  =  + 
HS:  =  (with  diagonal)
No Fit (0)
(for off-diagonal)
ˆ 2
Qj 
df j
Then we could have
s 0 
Mathematical
Marketing
ˆ s2  ˆ 02
ˆ s2
s 0 
QS  Q 0
QS  1
Slide 9.23
Confirmatory
Factor Analysis
Goodness of Fit
tr Σ 1S  I 
GFI  1 
tr ( Σ 1S)
2
AGFI  1 
p(p  1)
(1  GFI)
2  df0
p
RMSE 
Mathematical
Marketing
i
 (s
i 1
j1
ij
 ij ) 2
p(p  1) / 2
Slide 9.24
Confirmatory
Factor Analysis
Modification Indices
n  ˆ 2 


2   
MI 
(ˆ 2 ) 2
 
Mathematical
Marketing
2
Slide 9.25
Confirmatory
Factor Analysis