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)

4

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

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

5

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

6

... books

• Bollen (1989)

• Dwyer (1983)

• Hayduk (1987)

• Mueller (1996)

• Saris and Stronkhorst (1984)

• ....

7

... 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.

8

 

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

9

... 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

10

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

11

Moment Structure

= ()

S sample covariance matrix population covariance matrix

12

Fitting S to ():

Min f(S,)

= ()^^ S ≈ ̂

S – ≈ 0^

13

Type of variables

Manifest Variables: Yi , Xi

Measurement Model:

2

X3

X4

32

42

Measurement error, disturbances: i , i

3

4

14

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

15

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)

16

... 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

17

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

18

El modelo general:

i

iiI

BIGz

1)(

I

BIG

1)(

donde:

var

19

... 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

21

RESEARCH DESINGS

22

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

24

Type of Variables

• Continous

• Ordinal

• Nominal

• Censored, truncated …

• Interval or ratio• Ordinal• Ordered categories• Underordered

caterogies

VARIABLES SCALE TYPE

25

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.

26

Polychorical correlation

27

Polyserial correlation

28

Threshold model

29

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

30

Causal model with reciprocal effects

D P

U1WI U2

+

-

P = priceD = demandI = IncomeW = Wages

31

Examples with Coupon data (Bagozzi, 1994)

32

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

33

Variables

/LABELS V1 = Intentions1; V2 = Intentions2; V3 = Behavior; V4 = Attitudes1; V5 = Attitudes2; V6 = Attitudes3;

F1 = AttitudesF2 = IntentionsV3 = Behavior

34

F1 F2

V3

D2

E3

SEM multiple indicators

V4

V5

V6

V1

V2

E4

E5

E6

E1

E2

F1 = AttitudesF2 = IntentionsV3 = Behavior

35

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

37

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

39

40

41

42

43

44

45

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