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1
WHAT IS STRUCTURAL
EQUATION MODELING
(SEM)?
2
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)
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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’sModel 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: (p* x p*) p* = p(p+1)/2, p=20--> p* =210
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Moment Structure
= ()
S sample covariance matrix population covariance matrix
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Fitting S to ():
Min f(S,)
= ()^^ S ≈ ̂
S – ≈ 0^
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Type of variables
Manifest Variables: Yi , Xi
Measurement Model:
2
X3
X4
32
42
Measurement error, disturbances: i , i
3
4
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The form of structural equation models
Latent constructs:
- Endogenous i
- Exogenous i
Structural Model:- Regression of 1 on 2 12
- Regression of 1 on 2: 12
Structural Error: i
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LISREL model:
(m x 1) = (m x m) (m x 1) + (m x n) (n x 1) + (m x 1)
y(p x 1) = y(p x m) (m x 1) + (p x 1)
x(q x 1) = x(q x n) (n x 1) + (q x 1)
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... path diagram (LISREL)
X1
X2
X3
X4
X5
1
2
1
2
3
Y6
Y7
Y1 Y2 Y3
Y4 Y5
11
22
31
32
1
2
3
21
1
2
3
4
5
1 2 3
6
7
4 5
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SEM:
ii
iii
Uz
B
i=1,2, ...., ng,
donde: zi: vector de variables observables, i
: vector de variables endógenasi
: vector de variables exógenas vi = (i’, i’)’: vector de variables observables y latentes, U(g): matriz de selección completamente especificada, B, y = E(i i’): matrices de parámetros del modelo
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El modelo general:
i
iiI
BIGz
1)(
I
BIG
1)(
donde:
var
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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
1
2
3
4
5
11
12
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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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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
CognitionAffect
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 changesAffect will have more influence on B than one thatchanges cognition
U
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Causal model with reciprocal effects
D P
U1WI U2
+
-
P = priceD = demandI = IncomeW = 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 4.389Intentions #2 3.792 4.410Behavior 1.935 1.855 2.385Attitudes #1 1.454 1.453 0.989 1.914Attitudes #2 1.087 1.309 0.841 0.961 1.480Attitudes #3 1.623 1.701 1.175 1.279 1.220 1.971
Sample B: State oriented women (n = 64)Intentions #1 3.730Intentions #2 3.208 3.436Behavior 1.687 1.675 2.171Attitudes #1 0.621 0.616 0.605 1.373Attitudes #2 1.063 0.864 0.428 0.671 1.397Attitudes #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 = AttitudesF2 = IntentionsV3 = Behavior
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F1 F2
V3
D2
E3
SEM multiple indicators
V4
V5
V6
V1
V2
E4
E5
E6
E1
E2
F1 = AttitudesF2 = IntentionsV3 = 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 I 5.023 I I I I E6 -ATTITUDE .360*I I .132 I I 2.729 I I
CHI-SQUARE = 5.426, 7 DEGREES OF FREEDOM PROBABILITY VALUE IS 0.60809
36
... 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)
38
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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42
43
44
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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 0.810.12 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
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