slide 10.1 structural equation models mathematicalmarketing chapter 10 structural equation models in...
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Slide 10.Slide 10.11Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Chapter 10 Structural Equation ModelsIn This Chapter We Will Cover
The theme of this chapter is causation. We have already looked at one class of causal models,confirmatory factor analysis. We will now look at a more general formulation including
Path analysis
All-y models
Nonrecursion
Path models with latent variables
Second order factor models
Models that include means
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Slide 10.Slide 10.22Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Key Terminology
Endogenous
Exogenous
Path analysis
Causal model
Structural equation model
Covariance structure model
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Slide 10.Slide 10.33Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
ζΓxByy
The Basic SEM Model
p
2
1
q
2
1
pq2p1p
q22221
q11211
p
2
1
2p1p
p221
p112
p
2
1
x
x
x
y
y
y
0
0
0
y
y
y
vector of pendogeneousvariables
regressioncoefficients for endogeneousvariables onother endogeneousvaribles
regression coefficients for endogeneousvariables onexogeneousvariables
specificationerror
exogeneousvariables
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Slide 10.Slide 10.44Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Some Basic Assumptions
E(y) = 0
E(x) = 0
Cov(x, ) = 0
0|| BI
V(x) = E(xx) =
V() = E() = .
Some Definitions
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Slide 10.Slide 10.55Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Reduced Form Coefficients vs. Structural Coefficients
ζBIΓxBIy
ζΓxyBI
ζΓxByy
ζΓxByy
11 )()(
)(
eGxy
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Slide 10.Slide 10.66Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
What Are the Model's Implications for V(y)?
))((E)(E eGxeGxyyΣ
.)(E)(E)(E)(E eeGxeeGxGxGx
ζBIGxeGx 1)(E)(E
1)()(E BIζxG
E(yy) = GE(xx)G + E(ee)
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Slide 10.Slide 10.77Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Proceeding With the V(y)
E(yy) = GE(xx)G + E(ee)
11
1111
)]([)(
)()()()(
BIΨΓΓΦBI
BIΨBIBIΓΓΦBI
G = (I – B)-1 E(xx) = V(x) =
e = (I – B)-1
V() =
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Slide 10.Slide 10.88Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Another Important Piece: Cov(x, y)
1))((E)(E
E
)(E)(E
BIζxGxx
exGxx
eGxxyx e = (I – B)-1
1)( BIΓΦ0GΦ
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Slide 10.Slide 10.99Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
The Structure of the Covariance Matrix for All the Variables
ΦBIΓΦ
BIΨΓΓΦBI
ΣΣ
ΣΣ
1
11
xxxy
yy
)(
)]([)(
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Slide 10.Slide 10.1010Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
A Simple Causal Model
Intention to Purchase
y1
Purchasing Behavior
y2
Perceived Costx2
Attractiveness of Product
x1
DescriptionVariable
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Slide 10.Slide 10.1111Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Graphical Representation of Path Models
x2
y1 y2
x1
21
11
12
Boxes are manifest variables Circles are latent variables Unlabeled arrows are error Labeled single headed arrows are causal paths Double headed arrows are covariances
21
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Slide 10.Slide 10.1212Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
The Equations for the Sample Model
21212
12121111
yy
xxy
ζΓxByy
2
1
2
11211
2
1
212
1
x
x
00y
y
0
00
y
y
ΦSx
ζ
xx
2221
11
22
11
)(V
0)(V
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Slide 10.Slide 10.1313Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
We Do Not Really Need x and y Variables
Rewriting the model slightly
y = By + x +
x = 0y + Ix + 0
Now define
x
yz
I0
ΓBG
0
ζe
eGzz
so the model is
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Slide 10.Slide 10.1414Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
We Can Also Consolidate the Covariance Matrices
eGzz
Now for our model
AΦ0
e
)(V
We have the following covariance matrices for the exogenous factors
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Slide 10.Slide 10.1515Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
So We Don't Need So Many Matrices
Instead of having x and y we can get by with one set of variables: z
Instead of having B and we can get by with just G
Instead of having and we can get by with just A
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Slide 10.Slide 10.1616Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
In What Sense Are These Causal Models?
y3y2 y1
y2
y1
y3
The "same" arrow is missing in both.
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Slide 10.Slide 10.1717Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Both Models Have 1 DF and That DF Implies the Same Restriction
Both causal diagrams require only that the partial covariance 23·1 = 0
where 23·1 is the Cov(e2, e3) and e2 and e3 are defined as the errors in
y2 = y1 + e2 and
y3 = y1 + e3
Assuming that this constraint is met, that just means we have failed to reject Ho. Of course that doesn't mean we have proven it.
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Slide 10.Slide 10.1818Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Regression Is a Special Case of a Causal Model
x2 y1
x1
21
11
12
x3
1332
31
What are the degrees of freedom for this model?
Can we reject the causal hypothesis?
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Slide 10.Slide 10.1919Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Multivariate Regression Is Also a Special Case
x2
y1
x1
21
11
12
x3 23
32
31
y2
22
21
13
12
The same questions apply here
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Slide 10.Slide 10.2020Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Recursive Systems
A recursive system is characterized by V() = diagonal, and by the fact that it is possible to arrange the y variables so that B is lower (or upper) triangular
x2
y1x1
y2 x2
y1x1
y2
Two Nonrecursive Examples
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Slide 10.Slide 10.2121Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Structural Equation Models with Latent Variables
y = y +
x = x +
= B + +
Measurement models
Structural equation models
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Slide 10.Slide 10.2222Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Some of the Assumptions of the Model
Cov (, ) = 0
Cov (, ) = 0
Cov (, ) = 0
Cov (, , ) = 0
Diag (B) = 0
| I – B| 0
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Slide 10.Slide 10.2323Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Naming Some More Matrices
y = y +
x = x +
= B + +
V() =
V() =
V() =
V() =
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Slide 10.Slide 10.2424Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
An Example Longitudinal Model with Latent Variables
y1
21
1
121
y2
y3
42
1
232
y4
y5
63
1
343
y6
y7
84
1
4
y8
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Slide 10.Slide 10.2525Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
The Matrices for the Model
8
7
6
5
4
3
2
1
4
3
2
1
84
63
42
21
8
7
6
5
4
3
2
1
000
1000
000
0100
000
0010
000
0001
y
y
y
y
y
y
y
y
4
3
2
1
4
3
2
1
43
32
21
4
3
2
1
000
000
000
0000
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Slide 10.Slide 10.2626Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Variance Matrices for Exogenous Factors
88
22
11
00
00
00
Θ
44
33
22
11
000
000
000
000
Ψ
The Matrices for the Model
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Slide 10.Slide 10.2727Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
The Second Order Factor Analysis Model
y = y + = +
εζΓξΛy y
V() =
V() =
V() =
ΘΛΨΓΓΦΛyyy
)(V
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Slide 10.Slide 10.2828Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Path Diagram of Second Order Factor Analysis
y2y1 y3
211
1 2
y5y4 y6
3 4
y7 y8
1
1 1 142 63 84
1 413121
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Slide 10.Slide 10.2929Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Models with Structured Means
x = x + x +
y = y + y +
= + B + +
E(x) = x + x
E(y) = y + y (I – B)-1 ( + )
E() = (I - B)-1( + )
x measurement model. Includes x0
y measurement model
SEM
= E()
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Slide 10.Slide 10.3030Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
Details on the Model
1 = 10 + 0
δ
1
ξ
η
νΛ0
ν0Λ
x
y
xx
yyε
0
1
1101
κξ
ζ
κ
α
ξ
η
00
000
0ΓB
ξ
η
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Slide 10.Slide 10.3131Structural Structural
Equation ModelsEquation ModelsMathematicalMathematicalMarketingMarketing
A Sequence of Hypotheses for Multiple Group Analysis
)2(
y
)1(
y0:H ΛΛ
)2(
y
)1(
y0:H νν
)2()1(
0:H ΘΘ
)2()1(
0:H αα
)2()1(
0:H ΨΨ