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WHAT IS STRUCTURAL EQUATION MODELING

(SEM)?

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LINEAR STRUCTURAL RELATIONS

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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)

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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.shtml

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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.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

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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/psychom.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)

Data Matrix

(n x p)

Indiv.

vars

Sample Moments: •  Vector of means •  Variance and covariance matrix (p x p) •  Fourth order moments: Γ (p* x p*) p* = p(p+1)/2, p=20--> p* =210

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Moment Structure

Σ = Σ(θ)

S sample covariance matrix Σ population covariance matrix

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Fitting S to Σ(θ):

Min f(S,Σ)

Σ = Σ(θ) ^ ^ S ≈ Σ ^

S – Σ ≈ 0 ^

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Type of variables

Manifest Variables: Yi , Xi

Measurement Model:

ξ2 X3

X4

λ32

λ42

Measurement error, disturbances: εi , δi

ε3

ε4

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The form of structural equation models

Latent constructs: - Endogenous ηi - Exogenous ξi

Structural Model: - Regression of η1 on ξ2: γ12

- Regression of η1 on η2: β12

Structural Error: ζi

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LISREL model:

η(m x 1) = Β(m x m) η(m x 1) + Γ(m x n) ξ(n x 1) + ζ(m x 1)

y(p x 1) = Λy(p x m) η(m x 1) + ε(p x 1)

x(q x 1) = Λx(q x n) ξ(n x 1) + δ(q x 1)

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... path diagram (LISREL)

X1

X2

X3

X4

X5

ξ1

ξ2

η1

η2

η3 Y6

Y7

Y1 Y2 Y3

Y4 Y5

γ11

γ22

β31

β32

ζ1

ζ2

ζ3

θ21

δ1

δ2

δ3

δ4

δ5

ε1 ε2 ε3

ε6

ε7

ε4 ε5

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SEM:

i=1,2, ...., ng, donde: zi: vector de variables observables, ηi

: vector de variables endógenas ξi

: vector de variables exógenas vi = (ηi’, ξi’)’: vector de variables observables y latentes, U(g): matriz de selección completamente especificada, B, Γ y Φ = E(ξi ξi’): matrices de parámetros del modelo

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El modelo general:

donde:

Φ = var ξ

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... path diagram (EQS)

V1

V2

V3

V4

V5

F1

F2

F3

F4

F5 V11

V12

V6 V7 V8

V9 V10

*

*

*

*

D3

D5

D4

*

Ε1

Ε2

Ε3

Ε4

Ε5

Ε6 Ε7 Ε8

Ε11

Ε12

Ε9 Ε10

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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

•  Continous •  Ordinal

•  Nominal

•  Censored, truncated …

•  Interval or ratio •  Ordinal •  Ordered categories •  Underordered

caterogies

VARIABLES SCALE TYPE

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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

Behaviour

Cognition Affect

Bagozzi and Burnkrant (1979), Attitude organization and the attitude behaviour relationship, Journal Of Personality and Social Psychology, 37, 913-29

Correla = .83

.65 .23

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

U

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Causal model with reciprocal effects

D P

U1 W I U2

+

-

P = price D = demand I = Income W = Wages

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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 Attitudes #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 Attitudes #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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F1 F2

V3

D2

E3

SEM multiple indicators

V4

V5

V6

V1

V2

E4

E5

E6

E1

E2

F1 = Attitudes F2 = Intentions V3 = Behavior

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INTENTIO=V1 = 1.000 F2 + 1.000 E1

INTENTIO=V2 = 1.014*F2 + 1.000 E2

.088

11.585

BEHAVIOR=V3 = .330*F2 + .492*F1 + 1.000 E3

.103 .204

3.203 2.411

ATTITUDE=V4 = 1.020*F1 + 1.000 E4

.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 + 1.000 D2

.214

6.116

VARIANCES OF INDEPENDENT VARIABLES ----------------------------------

E D --- --- E1 -INTENTIO .649*I D2 -INTENTIO 2.020*I .255 I .437 I 2.542 I 4.619 I I I E2 -INTENTIO .565*I I .257 I I 2.204 I I I I E3 -BEHAVIOR 1.311*I I .213 I I 6.166 I I I I E4 -ATTITUDE .875*I I .161 I I 5.424 I I I I E5 -ATTITUDE .576*I I .115 I

CHI-SQUARE = 5.426, 7 DEGREES OF FREEDOM PROBABILITY VALUE IS 0.60809

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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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40

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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