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WHAT IS S TRUCTURAL E QUATION M ODELING ( SEM )?

WHAT IS S TRUCTURAL E QUATION M ODELING ( SEM )?. LI NEAR S TRUCTURAL REL ATIONS. 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

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WHAT IS S TRUCTURAL E QUATION M ODELING ( SEM )?

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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)

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

  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’s Model 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.htmcomputing 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) 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(p+1)/2, p=20--> p* =210 Indiv.

  11. Moment Structure S sample covariance matrix S population covariance matrix S = S(q)

  12. Min f(S,S) S = S(q) ^ ^ ^ S ≈ S ^ S – S≈ 0 Fitting S to S(q):

  13. Manifest Variables:Yi , Xi Type of variables Measurement Model: l32 e3 X3 x2 e4 X4 l42 Measurement error, disturbances:ei , di

  14. The form of structural equation models Latent constructs: - Endogenous hi - Exogenous xi Structural Model: - Regression of h1 on x2:g12 - Regression of h1 onh2: b12 Structural Error:zi

  15. 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)

  16. ... path diagram (LISREL) e1 e2 e3 Y1 Y2 Y3 d1 X1 g11 z1 x1 h1 b31 z2 d2 X2 e6 Y6 q21 h3 d3 X3 e7 Y7 g22 b32 d4 X4 x2 h2 z3 d5 X5 Y4 Y5 e4 e5

  17. SEM: i=1,2, ...., 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(xixi’): matrices de parámetros del modelo

  18. El modelo general: donde: F = var x

  19. ... path diagram (EQS) E6 E7 E8 V6 V7 V8 E1 V1 D3 * F1 F3 D5 E2 V2 * E11 V11 * F5 E3 V3 E12 V12 * * E4 V4 F2 F4 D4 E5 V5 V9 V10 E9 E10

  20. RESEARCH DESINGS

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

  23. 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.

  24. Polychorical correlation

  25. Polyserial correlation

  26. Threshold model

  27. Modelling the effect on behaviour Correla = .83 Affect Cognition .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

  28. Causal model with reciprocal effects W I U2 U1 P = price D = demand I = Income W = Wages + D P -

  29. Examples with Coupon data (Bagozzi, 1994)

  30. 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

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

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

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

  34. ... 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

  35. Hopkins and Hopkins (1997): “Strategic planning-financial performance relationships in banks: a causal examination”. Strategic Management Journal, Vol 18 (8), pp. (635-652)

  36. 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

  37. 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

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