Introduction to Structural Equation Modeling with LISREL.

1. Introduction to Structural Equation Modeling with LISREL. E. Kevin Kelloway, Ph.D. Professor of Management and Psychology Senior Research Fellow, CN Centre for Occupational Health and Safety Cover PageCover Page

2. Statement of causal relations Implies a pattern of covariances/correlations Necessary (but not sufficient) condition for validity is that the oberved pattern of correlations matches the implied pattern of correlations. Fundamental hypothesis of all SEM applications = ?(?) Content Page 2Content Page 2

3. Fishbein & Ajzen�s Theory of Reasoned Action Content Page 1Content Page 1

4. Model Specification Identification Estimation Testing Fit Respecification Content Page 2Content Page 2

5. Model Specification ? = ?(?)  Cover PageCover Page

6. Association Isolation Causal Direction Content Page 2Content Page 2

7. Causal flow is from left to right (top to bottom). Curved arrows represent bidirectional relationships (correlations). Straight arrows represent causal associations Relationships assumed to be linear What�s not in the model is just as important as what is in the model Causal Closure Content Page 2Content Page 2

8. Path Diagram Content Page 1Content Page 1

9. Factor AnalysisY = t + e Content Page 1Content Page 1

10.  IdentificationX + Y = 10Solve for X Cover PageCover Page

11. Just Identified (e.g., regression or multiple regression) Under Identified Over-Identified The t rule, given a k X k matrix there are k X (k-1)/2 elements that can be estimated Content Page 2Content Page 2

12. Direction � recursive models Assigning value � to parameters (often 0) Content Page 2Content Page 2

13.  Estimation Cover PageCover Page

14. Iterative estimation to a fitting criterion ML and GL allow for a fit test (N-1) * minimum of the fitting function is distributed as ?2 Partial vs Full information techniques Content Page 2Content Page 2

15.  Model Fit Cover PageCover Page

16. Absolute Comparative Parsimonious Content Page 2Content Page 2

17. Available for ML and GLS Tests the null that ?=?(?) Distributed with 1/2(q)(q+1)-k df where q is the number of variables and k is the number of estimated parameters Power Logical problem of accepting the null Content Page 2Content Page 2

18. Indicate degree of fit along a bounded continuum (normed) Be independent of sample size Have known distributional properties No fit indices (except possibly the RMSEA) meet these criteria Content Page 2Content Page 2

19. RMR & Standardized RMR RMSEA GFI AGFI ?2/df Content Page 2Content Page 2

20. Tests of individual parameters Called t values but are interpreted as Z scores Problems: Overall fit but parameters are not significant. Overall fit but parameters are in opposite direction. Lack of fit but all parameters as predicted Content Page 2Content Page 2

21. Null Model (Independence Model) Saturated Model Measures of absolute fit test the distance from the saturated model (i.e., are tests of identifying restrictions). Measures of comparative fit typically test the distance from the null model. Content Page 2Content Page 2

22. Normed Fit Index (NFI) Non-Normed Fit Index(NNFI) Incremental Fit Index(IFI) Comparative Fit Index(CFI) Relative Fit Index(RFI) Expected Cross-Validation Index(ECVI) Content Page 2Content Page 2

23. Fit (both absolute and comparative) increases with the number of parameters estimated. Rewards researcher for estimating trivial paths Parsimonious fit adjusts for the df in the model and penalizes accordingly Tend to reward the estimation of significant (and only significant) paths Content Page 2Content Page 2

24. Parsimonious Normed Fit Index (PNFI) Parsimonious Goodness of Fit Index(PGFI) Akaike Information Criterion(AIC) Consistent Akaike Information Criterion (CAIC) Content Page 2Content Page 2

25. Compare two (theoretically generated) plausible models of the data If the models stand in nested sequence (one model is completely contained in the other) then the difference may be tested with a ?2difference test Subtract the two ?2 values and the result is distribute as ?2 with df equal to the difference in model dfs Content Page 2Content Page 2

26. Compare competing and theoretically plausible models Identify sources of ambiguity a priori Using multiple indices/definitions of fit Recognize that fit does not equate to truth or validity Content Page 2Content Page 2

27.  Model Modification Cover PageCover Page

28. Theory trimming (significance tests) Theory Building (modification indices) Replication - holdout samples Simultaneous estimation �What percentage or researchers would find themselves unable to think up a theoretical justification for freeing a parameter? In the absence of empirical information, I assume that the answer� is near zero� (Steiger, 1990 p. 175) Content Page 2Content Page 2

29. LISRELThe beauty and the horror Cover PageCover Page

30. Run in batch (with limited interactivity) Written in the SIMPLIS language Three tasks Specify the data Specify the model Specify the output Content Page 2Content Page 2

31. Example 1: A regression Model Cover PageCover Page

32. Janes Safety Data (regression) Observed Variables Injury Training Tfl Passive Covariance Matrix 1.13 -.05 .096 -.279 -.092 1.973 .439 .067 -.807 2.406 Sample Size: 129 Equation: Injury = Training Tfl Passive End of Problem Content Page 2Content Page 2

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34. Relationships Injury = Training - Passive Paths Training � Passive -> Injury Content Page 2Content Page 2

35. Add the words �Path Diagram� just before the End of Problem Statement Theory Trimming Theory Building Content Page 2Content Page 2

36. Compare competing and theoretically plausible models Identify sources of ambiguity a priori Using multiple indices/definitions of fit Recognize that fit does not equate to truth or validity Content Page 2Content Page 2

37. Example 2: A Path Analysis(observed variable) Cover PageCover Page

38. Basic hypotheses - [a] leadership affects wellbeing [b] effects are indirect being mediated by trust and self efficacy A FULLY MEDIATED Model Content Page 2Content Page 2

39. t rule is met Null B rule (no relationships among the endogenous variables) - e.g., a multiple regression equation; Recursive rule - Recursive models are identified Rank and Order conditions - essentially allows for non-recursive models, need a unique predictor for one of the variables in a non-recursive relationship Content Page 2Content Page 2

40. Leadership Data Fully mediated model Observed Variables = Wellbeing Trust Efficacy Leadership Means 22.3035294 4.9588235 3.9641765 10.4242353 Standard Deviations 3.9405502 .8590221 .6941727 3.1419617 Correlations 1.0000000 -.2361636 1.0000000 -.1746880 .1860385 1.0000000 -.1441248 .4604753 .1907934 1.0000000 sample size = 425 Paths Trust Efficacy ->Wellbeing Leadership ->Trust Efficacy path diagram end of problem Content Page 2Content Page 2

41. Does the model fit? Are the paths significant? Do the data suggest changing the model? Content Page 2Content Page 2

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43. Both the fully mediated and the non-mediated are nested within the partially mediated (but are not directly comparable) Mediation exists if: [a] Fully mediated Fit is not significantly different than Partially mediated Fit and [b] Non-mediated Fit is significantly worse than Partially mediated fit Content Page 2Content Page 2

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45. Leadership Data Fully mediated model Observed Variables = Wellbeing Trust Efficacy Leadership Means 22.3035294 4.9588235 3.9641765 10.4242353 Standard Deviations 3.9405502 .8590221 .6941727 3.1419617 Correlations 1.0000000 -.2361636 1.0000000 -.1746880 .1860385 1.0000000 -.1441248 .4604753 .1907934 1.0000000 sample size = 425 Y variables = Wellbeing Trust Efficacy path diagram end of problem Content Page 2Content Page 2

46. Example 3: Confirmatory Factor Analysis Cover PageCover Page

47. Union commitment literature identifies 3 components of union commitment (loyalty, responsibility, willingness) Does the same structure hold for commitment to other representative groups (student union). Content Page 2Content Page 2

48. One factor model is always a reasonable alternative Orthogonal models are always nested within oblique models (but may be trivial) If one generates an alternative model by combining factors (i.e., by fixing the interfactor correlation to 1) a nested sequence is obtained In this case the literature suggests both a 3 factor (loyalty, responsibility, willingness) and a 2 factor (attitudes and behavior) model Estimate a 1 factor, two factor and three factor model Content Page 2Content Page 2

49. CFA models are recursive t rule (estimate less parameters than the number of non-redundant elements in the covariance matrix) 3 indicator rule - 3 observed variables for each latent variable 2 indicator rule - 2 observed variables for each latent variable and latent variables are allowed to correlate Both 3 indicator and 2 indicator rule assume that unique factor loadings (error terms) are uncorrelated Monte Carlo research supports the use of 3 indicators with sample sizes greater than 200 Content Page 2Content Page 2

50. Example_3.spl (Three Factor) Content Page 1Content Page 1

51. Example_3.spl (Two Factor) Content Page 1Content Page 1

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53. INTERACTIVE VERSION Content Page 1Content Page 1

54. Example 4: Latent Variable Path Analysis Cover PageCover Page

55. CFA and Path Analysis at the same time Corrects structural parameters for measurement - modeling with �true� as opposed to �observed� scores Increased complexity - only real advantage is when you care about both questions of measurement and structural relations Content Page 2Content Page 2

56. Lack of fit may result from [a] the measurement model, [b] the structural model, or [c] both Establish the fit of the measurement model (provides a baseline for the full model), then move to testing structural parameters Content Page 2Content Page 2

57. Virtually all of the organizational literature treats gossip as a �bad� thing We hypothesize that gossip can be a good thing It enhances individual control It may enhance organizational citizenship behaviors Content Page 2Content Page 2

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59. OCB = Ocb1 ocb2 ocb3 = item parcels each made up by summing 2 items CTRL = 3 single indicators (items) GOSSIP = 4 scale scores (toldsup, hearsup, toldcow, hearcow) Need to assign a scale for each latent variable (fix a factor loading to 1 � shouldn�t matter which one) Content Page 2Content Page 2

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61. Does the model fit? Can it be fixed? (If so, how?) � Identifying the problem, resolving the problem (Hopefully) Number of indicators Single indicator latent variables Content Page 2Content Page 2

62. Start with a correlation matrix (Kevin�s preference) Reading from a file Import SPSS data Content Page 2Content Page 2

63. THANK YOU Content Page 2Content Page 2