WHAT IS STRUCTURAL EQUATION MODELING (SEM)?

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WHAT IS STRUCTURAL EQUATION MODELING (SEM)?
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LINEAR STRUCTURAL RELATIONS
3
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)

4
Computer programs

• LISREL
• EQS
• LISCOMP / Mplus
• COSAN
• MOMENTS
• CALIS
• AMOS
• RAMONA
• Mx

• Jöreskog and Sörbom
• Bentler
• Muthén
• McDonalds
• Schoenberg
• SAS
• Arbunckle
• Browne
• Neale

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Computer programs

• SEM software
• EQS http//www.mvsoft.com
• LISREL http//www.ssicentral.com
• MPLUS http//www.statmodel.com/index2.html
• AMOS http//smallwaters.com/amos/
• Mx http//www.vipbg.vcu.edu/vipbg/dr/MNEALE.shtm
  l

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... books

• Bollen (1989)
• Dwyer (1983)
• Hayduk (1987)
• Mueller (1996)
• Saris and Stronkhorst (1984)
• ....

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... 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.

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  Information on SEM bibliography, courses
.. General information on SEM
http//allserv.rug.ac.be/flievens/stat.htmStruc
tural   Jason Newsom's
Structural Equation Modeling Reference
List http//www.ioa.pdx.edu/newsom/semrefs.htm
David A. Kennys course http//users.rcn.com/d
akenny/causalm.htm Jouni Kuhas Model
Assessment and Model Choice An Annotated
Bibliography http//www.stat.psu.edu/jkuha/msbib
/biblio.html
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... 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/jcrawford/psych
  om.htm computing the scaling factor for the
  difference of chi squares

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Introduction to SEM

• Data
• Data matrix (raw data)
• Sufficient statistics (sample means, variances
  and covariances)

vars
Data Matrix (n x p)

• Sample Moments
• Vector of means
• Variance and covariance matrix (p x p)
• Fourth order moments
• G (p x p) p p(p1)/2, p20--gt p 210

Indiv.
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Moment Structure
S sample covariance matrix S population
covariance matrix
S S(q)
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Fitting S to S(q)
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Type of variables
Measurement Model
l32
e3
X3
x2
e4
X4
l42
Measurement error, disturbances ei , di
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The form of structural equation models
Latent constructs - Endogenous hi -
Exogenous xi
Structural Model - Regression of h1 on x2
g12 - Regression of h1 on h2 b12
Structural Error zi
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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)
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... path diagram (LISREL)
e1
e2
e3
Y1
Y2
Y3
X1
d1
g11
z1
x1
h1
b31
z2
X2
d2
e6
Y6
q21
h3
d3
X3
e7
Y7
g22
b32
d4
X4
x2
h2
z3
d5
X5
Y4
Y5
e4
e5
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SEM
i1,2, ...., 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
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El modelo general
donde
F var x
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... path diagram (EQS)
E6
E7
E8
V6
V7
V8
V1
E1
D3

F1
F3
D5
V2

E2
E11
V11

F5
E3
V3
E12
V12


E4
V4
F2
F4
D4
E5
V5
V9
V10
E9
E10
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Main virtues of SEM (ctd.)

• Flexibility on the type of data
• Continuous and ordinal variables
• multiple sample
• Informative missingness (MCA, MAR)
• Finite mixture distributions
• Multilevel models
• Samples with complex design
• General longitudinal type of data
• ...

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RESEARCH DESINGS
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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

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.. data collection designs

• Longitudinal
• a) Retrospective
• b) Prospective
• c) Repeated measures
• d) panel
• e) Rotating panel
• Experimental, quasi-experimental data
• Observational or non-experimental

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Type of Variables
VARIABLES
SCALE TYPE

• Continous
• Ordinal
• Nominal
• Censored, truncated

• Interval or ratio
• Ordinal
• Ordered categories
• Underordered caterogies

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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.

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Polychorical correlation
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Polyserial correlation
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Threshold model
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Modelling the effect on behaviour
Correla .83
Cognition
Affect
.65
Influence of affect on Behaviour is almost
Three times stronger (on a standardized
scale) Than the effect of Cognition.
.23
U
Behaviour
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
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Causal model with reciprocal effects
W
I
U2
U1
P price D demand I Income W Wages

D
P
-
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Examples with Coupon data (Bagozzi, 1994)
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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 A
ttitudes 1 1.454 1.453 0.989 1.914 Attitudes
2 1.087 1.309 0.841 0.961 1.480 Attitudes
3 1.623 1.701 1.175 1.279 1.220 1.971
Sample B State oriented women (n
64) Intentions 1 3.730 Intentions
2 3.208 3.436 Behavior 1.687 1.675 2.171 A
ttitudes 1 0.621 0.616 0.605 1.373 Attitudes
2 1.063 0.864 0.428 0.671 1.397 Attitudes
3 0.895 0.818 0.595 0.912 0.663 1.498
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Variables
/LABELS V1 Intentions1 V2 Intentions2
V3 Behavior V4 Attitudes1 V5
Attitudes2 V6 Attitudes3
F1 Attitudes F2 Intentions V3 Behavior
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SEM multiple indicators
D2
E4
V4
E1
V1
F1
F2
E5
V5
E2
V2
E6
E3
V6
V3
F1 Attitudes F2 Intentions V3 Behavior
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INTENTIOV1 1.000 F2 1.000 E1


INTENTIOV2 1.014F2 1.000 E2

.088
11.585

BEHAVIORV3 .330F2 .492F1
1.000 E3
.103 .204
3.203
2.411
ATTITUDEV4 1.020F1 1.000 E4

.136
7.501

ATTITUDEV5 .951F1 1.000 E5

.117
8.124

ATTITUDEV6 1.269F1 1.000 E6

.127
10.005
INTENTIOF2 1.311F1 1.000 D2

.214
6.116
CHI-SQUARE 5.426, 7 DEGREES OF
FREEDOM PROBABILITY VALUE IS 0.60809
VARIANCES OF INDEPENDENT VARIABLES
----------------------------------
E D
--- --- E1
-INTENTIO .649I D2 -INTENTIO
2.020I
.255 I .437 I
2.542 I
4.619 I
I
I E2 -INTENTIO
.565I I
.257 I
I
2.204 I
I I
I
E3 -BEHAVIOR 1.311I
I
.213 I I
6.166 I
I
I
I E4 -ATTITUDE
.875I I
.161 I
I
5.424 I
I I
I
E5 -ATTITUDE .576I
I
.115 I I
5.023 I
I
I
I E6 -ATTITUDE
.360I I
.132 I
I
2.729 I
I
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... adding parameters ?
LAGRANGE MULTIPLIER TEST (FOR ADDING
PARAMETERS)   ORDERED UNIVARIATE
TEST STATISTICS   NO CODE PARAMETER
CHI-SQUARE PROBABILITY PARAMETER CHANGE --
---- --------- ---------- -----------
----------------   1 2 12 V2,F1
1.427 0.232 0.410 2 2
12 V1,F1 1.427 0.232
-0.404 3 2 20 V4,F2 0.720
0.396 0.080 4 2 20
V5,F2 0.289 0.591
-0.045 5 2 20 V6,F2 0.059
0.808 -0.025 6 2 20
V3,F2 0.000 1.000
0.000 7 2 0 F1,F1 0.000
1.000 0.000 8 2 0 F2,D2
0.000 1.000 0.000 9
2 0 V1,F2 0.000 1.000
0.000
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Hopkins and Hopkins (1997) Strategic
planning-financial performance relationships in
banks a causal examination. Strategic
Management Journal, Vol 18 (8), pp. (635-652)
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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

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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 0.44
-0.26 -0.06 -0.16 -0.19 0.16 0.27
0.52 0.32 0.44 0.66 0.23 0.07
-0.24 0.52 0.40 0.51 0.76 0.26
0.19 -0.15 0.76 0.49 0.27 0.43
0.64 0.17 0.10 -0.21 0.77 0.81 0.12
0.16 0.09 0.28 0.18 0.24 0.07
0.36 0.41 0.35 0.34 0.24 0.27
0.64 0.31 0.23 -0.01 0.56 0.67
0.57 0.45 0.23 0.08 0.16 0.07
0.09 0.16 -0.01 0.28 0.30 0.27
0.29 0.30 0.03 0.02 0.04 -0.07
-0.05 -0.03 -0.05 0.06 -0.06 0.03
0.01 -0.07 0.03 0.20 0.32 0.22
0.09 -0.24 -0.33 0.05 -0.02 -0.07
-0.08 0.02 0.05 -0.23 -0.03 0.15
0.06 0.11 -0.03 0.10 0.13 0.16
0.13 0.07 0.06 0.16 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