WHAT IS STRUCTURAL EQUATION MODELING (SEM)?

Transcription

1 WHAT IS STRUCTURAL EQUATION MODELING (SEM)? 1

2 LINEAR STRUCTURAL RELATIONS 2

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) 3

4 Computer programs LISREL EQS LISCOMP / Mplus COSAN MOMENTS CALIS AMOS RAMONA Mx Jöreskog and Sörbom Bentler Muthén McDonalds Schoenberg SAS Arbunckle Browne Neale 4

5 Computer programs SEM software: EQS LISREL MPLUS AMOS Mx 5

6 ... books Bollen (1989) Dwyer (1983) Hayduk (1987) Mueller (1996) Saris and Stronkhorst (1984)... 6

7 ... many research papers Austin and Wolfle (1991): Annotated bibliography of structural equation modeling: Technical Works. BJMSP, 99, pp Austin, J.T. and Calteron, R.F. (1996). Theoretical and technical contributions to structural equation modeling: An updated annotated bibliography. SEM, pp

8 Information on SEM: bibliography, courses.. General information on SEM: Jason Newsom's Structural Equation Modeling Reference List David A. Kenny s course Jouni Kuha s Model Assessment and Model Choice: An Annotated Bibliography 8

9 ... web sites SEM webs: jcrawford/psychom.htm computing the scaling factor for the difference of chi squares 9

10 Introduction to SEM: Data: Data matrix ( raw data ) Sufficient statistics (sample means, variances and covariances) Indiv. vars Data Matrix (n x p) 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 10

11 Moment Structure S sample covariance matrix Σ population covariance matrix Σ = Σ(θ) 11

12 Fitting S to Σ(θ): Min f(s,σ) Σ ^ ^ = Σ(θ) ^ S Σ ^ S Σ 0 12

13 Type of variables Manifest Variables: Y i, X i Measurement Model: ε 3 X 3 λ 32 ξ 2 ε 4 X 4 λ 42 Measurement error, disturbances: ε i, δ i 13

14 The form of structural equation models Latent constructs: - Endogenous η i - Exogenous ξ i Structural Model: - Regression of η 1 on ξ 2 : γ 12 - Regression of η 1 on η 2 : β 12 Structural Error: ζ i 14

15 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) = Λ y(p x m) η (m x 1) + ε (p x 1) x (q x 1) = Λ x(q x n) ξ (n x 1) + δ (q x 1) 15

16 ... path diagram (LISREL) ε 1 ε 2 ε 3 Y 1 Y 2 Y 3 δ 1 X 1 γ 11 ζ 1 ζ 2 ξ 1 η 1 δ 2 X 2 β 31 δ 3 X 3 θ 21 η 3 Y 6 Y 7 ε 6 ε 7 δ 4 X 4 ξ 2 γ 22 η 2 β 32 ζ 3 δ 5 X 5 Y 4 Y 5 ε 4 ε 5 16

17 SEM: i=1,2,..., n g, donde: z i : vector de variables observables, η i : vector de variables endógenas ξ i : vector de variables exógenas v i = (η i, ξ i ) : vector de variables observables y latentes, U (g) : matriz de selección completamente especificada, B, Γ y Φ = E(ξ i ξ i ): matrices de parámetros del modelo 17

18 El modelo general: donde: Φ = var ξ 18

19 ... path diagram (EQS) Ε 6 Ε 7 Ε 8 V 6 V 7 V 8 Ε 1 Ε 2 V 1 V 2 F 1 F 3 D 3 D 5 V 11 Ε 11 Ε 3 Ε 4 V 3 V 4 F 2 F 4 F 5 V 12 Ε 12 D 4 Ε 5 V 5 V 9 V 10 Ε 9 Ε 10 19

20 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... 20

21 RESEARCH DESINGS 21

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 22

23 .. data collection designs Longitudinal a) Retrospective b) Prospective c) Repeated measures d) panel e) Rotating panel Experimental, quasi-experimental data Observational or non-experimental 23

24 Type of Variables VARIABLES Continous Ordinal Nominal SCALE TYPE Interval or ratio Ordinal Ordered categories Underordered caterogies Censored, truncated 24

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. 25

26 Polychorical correlation 26

27 Polyserial correlation 27

28 Threshold model 28

29 Modelling the effect on behaviour Correla =.83 Affect Cognition U.65 Behaviour A policy that changes Affect will have more influence on B than one that Bagozzi and Burnkrant (1979), changes cognition Attitude organization and the attitude behaviour relationship, Journal Of Personality and Social Psychology, 37, Influence of affect on Behaviour is almost Three times stronger (on a standardized scale) Than the effect of Cognition. 29

30 Causal model with reciprocal effects U 1 W I U 2 P = price D = demand I = Income W = Wages D + - P 30

31 Examples with Coupon data (Bagozzi, 1994) 31

32 Example: Data of Bagozzi, Baumgartner, and Yi (1992), on coupon usage : Sample A: Action oriented women (n = 85) Intentions # Intentions # Behavior Attitudes # Attitudes # Attitudes # Sample B: State oriented women (n = 64) Intentions # Intentions # Behavior Attitudes # Attitudes # Attitudes #

33 Variables /LABELS V1 = Intentions1; V2 = Intentions2; V3 = Behavior; V4 = Attitudes1; V5 = Attitudes2; V6 = Attitudes3; F1 = Attitudes F2 = Intentions V3 = Behavior 33

34 SEM multiple indicators E4 E5 E6 V4 V5 V6 F1 F2 D2 E3 V1 V2 E1 E2 V3 F1 = Attitudes F2 = Intentions V3 = Behavior 34

35 INTENTIO=V1 = F E1 INTENTIO=V2 = 1.014*F E CHI-SQUARE = 5.426, 7 DEGREES OF FREEDOM PROBABILITY VALUE IS BEHAVIOR=V3 =.330*F *F E VARIANCES OF INDEPENDENT VARIABLES ATTITUDE=V4 = 1.020*F E ATTITUDE=V5 =.951*F E ATTITUDE=V6 = 1.269*F E INTENTIO=F2 = 1.311*F D E1 -INTENTIO 2.020*I I I I E2 -INTENTIO I I I I E3 -BEHAVIOR I I I I E4 -ATTITUDE I I I I E5 -ATTITUDE I E D *I D2 -INTENTIO.255 I I I.565*I.257 I I I 1.311*I.213 I I I.875*I.161 I I I.576*I.115 I 35

36 ... adding parameters? LAGRANGE MULTIPLIER TEST (FOR ADDING PARAMETERS) ORDERED UNIVARIATE TEST STATISTICS: NO CODE PARAMETER CHI-SQUARE PROBABILITY PARAMETER CHANGE V2,F V1,F V4,F V5,F V6,F V3,F F1,F F2,D V1,F

37 Hopkins and Hopkins (1997): Strategic planningfinancial performance relationships in banks: a causal examination. Strategic Management Journal, Vol 18 (8), pp. ( ) 37

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 38

39 39

40 40

41 41

42 42

43 43

44 44

45 Covariance matrix:: Means: S.D.: