WHAT IS STRUCTURAL
EQUATION MODELING
(SEM)?
1
LINEAR STRUCTURAL
RELATIONS
2
Terminología
• LINEAR LATENT VARIABLE MODELS
• T.W. Anderson (1989), Journal of Econometrics
• MULTIVARIATE LINEAR RELATIONS
• T.W. Anderson (1987), 2nd International Temp.
Conference in Statistics
• LINEAR STATISTICAL RELATIONSHIPS
• T.W. Anderson (1984), Annals of Statistics, 12
• COVARIANCE STRUCTURES
•
•
•
•
•
Browne, Shapiro, Satorra, ...
Jöreskog (1973, 1977)
Wiley (1979)
Keesling (1972)
Koopmans and Hovel (1953)
3
Computer programs
•
•
•
•
•
•
•
•
•
LISREL
EQS
LISCOMP / Mplus
COSAN
MOMENTS
CALIS
AMOS
RAMONA
Mx
•
•
•
•
•
•
•
•
•
Jöreskog and Sörbom
Bentler
Muthén
McDonalds
Schoenberg
SAS
Arbunckle
Browne
Neale
4
Computer programs
• SEM software:
–
–
–
–
–
EQS
LISREL
MPLUS
AMOS
Mx
http://www.mvsoft.com
http://www.ssicentral.com
http://www.statmodel.com/index2.html
http://smallwaters.com/amos/
http://www.vipbg.vcu.edu/~vipbg/dr/MNEALE.shtml
5
... books
•
•
•
•
•
•
Bollen (1989)
Dwyer (1983)
Hayduk (1987)
Mueller (1996)
Saris and Stronkhorst (1984)
....
6
... many research papers
• Austin and Wolfle (1991): Annotated
bibliography of structural equation
modeling: Technical Works. BJMSP, 99,
pp. 85-152.
• Austin, J.T. and Calteron, R.F. (1996).
Theoretical and technical contributions to
structural equation modeling: An updated
annotated bibliography. SEM, pp. 105-175.
7
Information on SEM: bibliography, courses ..
General information on SEM:
http://allserv.rug.ac.be/~flievens/stat.htm#Structural
Jason Newsom's
Structural Equation Modeling Reference List
http://www.ioa.pdx.edu/newsom/semrefs.htm
David A. Kenny’s course
http://users.rcn.com/dakenny/causalm.htm
Jouni Kuha’s
Model Assessment and Model Choice: An Annotated Bibliography
http://www.stat.psu.edu/~jkuha/msbib/biblio.html
8
... web sites
• SEM webs:
– http://www.gsu.edu/~mkteer/semfaq.html
– http://www.ssicentral.com/lisrel/ref.htm
• http://www.psyc.abdn.ac.uk/homedir/jcrawf
ord/psychom.htm computing the scaling factor for the
difference of chi squares
9
Introduction to SEM:
• Data:
• Data matrix (“raw data”)
• Sufficient statistics (sample means, variances and
covariances)
vars
Sample Moments:
Indiv.
Data
Matrix
(n x p)
• Vector of means
• Variance and covariance matrix (p x p)
• Fourth order moments:
G (p* x p*) p* = p(p+1)/2, p=20--> p* =210
10
Moment Structure
S sample covariance matrix
S population covariance matrix
S = S(q)
11
Fitting S to S(q):
Min f(S,S)
^
^
S = S(q)
^
S≈S
^
S–S≈0
12
Type of variables
Manifest Variables:
Yi , Xi
Measurement Model:
e3
X3
e4
X4
l32
x2
l42
Measurement error, disturbances:
ei , d i
13
The form of structural equation
models
Latent constructs:
- Endogenous
- Exogenous
hi
xi
Structural Model:
- Regression of h1 on x2: g12
- Regression of h1 on h2: b12
Structural Error:
zi
14
LISREL model:
h(m x 1) = B(m x m) h(m x 1) + G(m x n) x(n x 1) + z(m x 1)
y(p x 1) = Ly(p
x m)
h(m x 1) + e(p x 1)
x(q x 1) = Lx(q x n) x(n x 1) + d(q x 1)
15
... path diagram (LISREL)
d1
X1
d2
X2
d3
x1
e1
e2
e3
Y1
Y2
Y3
g11
z1
h1
b31
q21
h3
X3
d4
X4
d5
X5
x2
z2
g22
b32
h2
Y4
e4
Y6
e6
Y7
e7
z3
Y5
e5
16
SEM:
hi  B hi  G x i
zi 
U i
i=1,2, ...., ng,
donde:
zi: vector de variables observables,
hi : vector de variables endógenas
xi : vector de variables exógenas
vi = (hi’, xi’)’: vector de variables observables y latentes,
U(g): matriz de selección completamente especificada,
B, G y F = E(xi xi’): matrices de parámetros del modelo
17
El modelo general:
 (I  B ) G 
xi
zi  G 

I


1
 L xi
donde:
 ( I  B ) 1 G
L  G 
I

F  var x




18
... path diagram (EQS)
E1
E2
E3
E4
E5
V1
F1
E6
E7
E8
V6
V7
V8
*
D3
F3
*
V2
*
V5
F2
V11
E11
V12
E12
F5
V3
V4
D5
*
F4
V9
E9
*
D4
V10
E10
19
RESEARCH DESINGS
21
Data collection designs
• Cross-sectional
– N independent units observed or measured at
one time
• Time-series
– One unit observed or measured al T occasions
• Longitudinal
– N independent units observed or measured at
two or more occasions
22
Type of Variables
VARIABLES
• Continous
• Ordinal
• Nominal
SCALE TYPE
•
•
•
•
Interval or ratio
Ordinal
Ordered categories
Underordered
caterogies
• Censored, truncated …
24
Ordinal Variables
Is is assumed that there is a continuous
unobserved variable x* underlying the observed
ordinal variable x.
A threshold model is specified, as in ordinal probit
regression, but here we contemplate multivariate
regression.
It is the underlying variable x* that is acting in the
SEM model.
25
Polychorical correlation
26
Polyserial correlation
27
Threshold model
28
Modelling the effect on behaviour
Correla = .83
Affect
Cognition
.65
.23
U
Behaviour
Influence of affect on
Behaviour is almost
Three times stronger
(on a standardized scale)
Than the effect of Cognition.
A policy that changes
Affect will have more
influence on B
than one that
changes cognition
Bagozzi and Burnkrant (1979),
Attitude organization and the attitude behaviour relationship, Journal
Of Personality and Social Psychology, 37, 913-29
29
Causal model with reciprocal
effects
U1
P = price
D = demand
I = Income
W = Wages
W
I
D
+
-
U2
P
30
Examples with Coupon data
(Bagozzi, 1994)
31
Example: Data of Bagozzi, Baumgartner, and Yi (1992),
on “coupon usage” :
Sample A: Action oriented women (n = 85)
Intentions #1
4.389
Intentions #2
3.792 4.410
Behavior
1.935 1.855 2.385
Attitudes #1
1.454 1.453 0.989 1.914
Attitudes #2
1.087 1.309 0.841 0.961
Attitudes #3
1.623 1.701 1.175 1.279
Sample B: State oriented women (n = 64)
Intentions #1
3.730
Intentions #2
3.208 3.436
Behavior
1.687 1.675 2.171
Attitudes #1
0.621 0.616 0.605
Attitudes #2
1.063 0.864 0.428
Attitudes #3
0.895 0.818 0.595
1.373
0.671
0.912
1.480
1.220
1.397
0.663
1.971
1.498
32
Variables
/LABELS
V1 = Intentions1;
V2 = Intentions2;
V3 = Behavior;
V4 = Attitudes1;
V5 = Attitudes2;
V6 = Attitudes3;
F1 = Attitudes
F2 = Intentions
V3 = Behavior
33
SEM multiple indicators
E4
D2
V4
V1
E5
E6
V5
F1
F2
V2
E1
E2
E3
V6
V3
F1 = Attitudes
F2 = Intentions
V3 = Behavior
34
INTENTIO=V1
=
1.000 F2
+ 1.000 E1
INTENTIO=V2
=
1.014*F2
+ 1.000 E2
.088
CHI-SQUARE =
5.426, 7 DEGREES OF FREEDOM
PROBABILITY VALUE IS
0.60809
11.585
BEHAVIOR=V3
ATTITUDE=V4
=
=
.330*F2
+
.492*F1
.103
.204
3.203
2.411
1.020*F1
+ 1.000 E4
+ 1.000 E3
VARIANCES OF INDEPENDENT VARIABLES
---------------------------------E
--E1
-INTENTIO
E2
-INTENTIO
E3
-BEHAVIOR
E4
-ATTITUDE
E5
-ATTITUDE
E6
-ATTITUDE
.136
7.501
ATTITUDE=V5
=
.951*F1
+ 1.000 E5
.117
8.124
ATTITUDE=V6
=
1.269*F1
+ 1.000 E6
.127
10.005
INTENTIO=F2
=
1.311*F1
.214
6.116
D
--.649*I D2
.255 I
2.542 I
I
.565*I
.257 I
2.204 I
I
1.311*I
.213 I
6.166 I
I
.875*I
.161 I
5.424 I
I
.576*I
.115 I
5.023 I
I
.360*I
.132 I
2.729 I
-INTENTIO
2.020*I
.437 I
4.619 I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
+ 1.000 D2
35
... adding parameters ?
LAGRANGE MULTIPLIER TEST (FOR ADDING PARAMETERS)
ORDERED UNIVARIATE TEST STATISTICS:
NO
-1
2
3
4
5
6
7
8
9
CODE
---2
2
2
2
2
2
2
2
2
12
12
20
20
20
20
0
0
0
PARAMETER
--------V2,F1
V1,F1
V4,F2
V5,F2
V6,F2
V3,F2
F1,F1
F2,D2
V1,F2
CHI-SQUARE
---------1.427
1.427
0.720
0.289
0.059
0.000
0.000
0.000
0.000
PROBABILITY
----------0.232
0.232
0.396
0.591
0.808
1.000
1.000
1.000
1.000
PARAMETER CHANGE
---------------0.410
-0.404
0.080
-0.045
-0.025
0.000
0.000
0.000
0.000
36
Hopkins and Hopkins (1997): “Strategic planningfinancial performance relationships in banks: a
causal examination”. Strategic Management Journal,
Vol 18 (8), pp. (635-652)
37
Data to be analyzed
• Sample: 112 comercial bancs
• Data obtained by survey
• Dependent variable:
• Intensity of strategic plannification
• Finance results
• Independent variables:
• Directive factors
• Contour factors
• Organizative factors
38
39
40
41
42
43
44
Covariance matrix::
0.48
0.76 0.60
0.51 0.46 0.54
-0.06 -0.09 0.01 0.31
-0.17 -0.21 -0.16 0.04
-0.26 -0.06 -0.16 -0.19
0.52 0.32 0.44 0.66
0.52 0.40 0.51 0.76
0.49 0.27 0.43 0.64
0.12 0.16 0.09 0.28
0.34 0.24 0.27 0.64
0.23 0.08 0.16 0.07
0.03 0.02 0.04 -0.07
0.20 0.32 0.22 0.09
0.15 0.06 0.11 -0.03
0.44
0.16
0.23
0.26
0.17
0.18
0.31
0.09
-0.05
-0.24
0.10
0.27
0.07
0.19
0.10
0.24
0.23
0.16
-0.03
-0.33
0.13
-0.24
-0.15
-0.21
0.07
-0.01
-0.01
-0.05
0.05
0.16
0.76
0.77
0.36
0.56
0.28
0.06
-0.02
0.13
0.81
0.41
0.67
0.30
-0.06
-0.07
0.07
0.35
0.57
0.27
0.03
-0.08
0.06
0.45
0.29
0.01
0.02
0.16
0.30
-0.07 0.03
0.05 -0.23 -0.03
0.19 0.21 0.13 0.16
Means:
34.30 12.75 3.50 6.70 7.10 7.00 7.10 7.00 7.05 7.20 7.20 7.30 7.45 21.50 3.54 2.35
S.D.:
58.58 4.10 1.61 1.95 1.65 1.62 1.55 1.52 1.64 1.96 1.88 1.78 1.54 12.87 0.56 0.67
45