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General Structural Equation (LISREL) Models

General Structural Equation (LISREL) Models. Week #2 Class #2. Today’s class. Latent variable structural equations in matrix form (from yesterday) Fit measures SEM assumptions What to write up LISREL matrices. From yesterday’s lab :. Reference indicator: REDUCE .

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General Structural Equation (LISREL) Models

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  1. General Structural Equation (LISREL) Models Week #2 Class #2

  2. Today’s class • Latent variable structural equations in matrix form (from yesterday) • Fit measures • SEM assumptions • What to write up • LISREL matrices

  3. From yesterday’s lab: Reference indicator: REDUCE Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- ------- REDUCE <---------- Ach1 1.000 NEVHAPP <--------- Ach1 2.142 0.374 5.721 NEW_GOAL <-------- Ach1 -2.759 0.460 -5.995 IMPROVE <--------- Ach1 -4.226 0.703 -6.009 ACHIEVE <--------- Ach1 -2.642 0.450 -5.874 CONTENT <--------- Ach1 2.657 0.460 5.779

  4. From yesterday’s lab: Reference indicator: REDUCE Standardized Regression Weights: Estimate -------------------------------- -------- REDUCE <---------- Ach1 0.138 NEVHAPP <--------- Ach1 0.332 NEW_GOAL <-------- Ach1 -0.541 IMPROVE <--------- Ach1 -0.682 ACHIEVE <--------- Ach1 -0.410 CONTENT <--------- Ach1 0.357

  5. From yesterday’s lab: Reference indicator: REDUCE

  6. Regression Weights: Estimate S.E. C.R. Label ------------------- -------- ------- ------- ------- REDUCE <---------- Ach1 1.000 NEVHAPP <--------- Ach1 -113.975 1441.597 -0.079 NEW_GOAL <-------- Ach1 215.393 2717.178 0.079 IMPROVE <--------- Ach1 373.497 4711.675 0.079 ACHIEVE <--------- Ach1 211.419 2667.067 0.079 CONTENT <--------- Ach1 -155.262 1961.974 -0.079 Standardized Regression Weights: Estimate -------------------------------- -------- REDUCE <---------- Ach1 0.002 NEVHAPP <--------- Ach1 -0.223 NEW_GOAL <-------- Ach1 0.534 IMPROVE <--------- Ach1 0.762 ACHIEVE <--------- Ach1 0.415 CONTENT <--------- Ach1 -0.264

  7. Solution: • Use a different reference indicator • (Note: REDUCE can be used as a reference indicator in a 2-factor model, though other reference indicators might be better because REDUCE is factorally complex)

  8. When to add, when not to add parameters

  9. Modification Indices Covariances: M.I. Par Change e1 <--> Ach1 63.668 0.032 e1 <--> Cont1 6.692 0.016 e6 <--> Ach1 32.540 -0.023 e5 <--> Cont1 4.370 0.012 e5 <--> e6 13.033 -0.028 e4 <--> e1 28.242 0.036 e4 <--> e6 24.104 -0.034 e3 <--> e1 4.500 0.012 e2 <--> e1 5.440 0.016 e2 <--> e6 5.290 -0.016 e2 <--> e5 14.681 0.025 e2 <--> e3 12.410 -0.017 Discrepancy 125.260 0.000 Degrees of freedom 8 P 0.000 0.000

  10. Regression Weights: M.I. Par Change REDUCE <-- Ach1 52.853 0.406 REDUCE <-- ACHIEVE 16.291 0.076 REDUCE <-- IMPROVE 50.413 0.140 REDUCE <-- NEW_GOAL 23.780 0.117 CONTENT <-- Ach1 27.051 -0.293 CONTENT <-- ACHIEVE 24.336 -0.094 CONTENT <-- IMPROVE 31.694 -0.112 ACHIEVE <-- REDUCE 4.791 0.033 ACHIEVE <-- NEVHAPP 11.086 0.056 IMPROVE <-- REDUCE 18.169 0.058 IMPROVE <-- CONTENT 16.219 -0.053 NEW_GOAL <-- NEVHAPP 6.137 -0.032 NEVHAPP <-- REDUCE 4.031 0.029 NEVHAPP <-- ACHIEVE 9.687 0.050 NEVHAPP <-- NEW_GOAL 9.452 -0.063

  11. Choice to add or not to add parameter from Ach1  REDUCE a matter of theoretical judgement. (Note changes in other parameters)

  12. Goodness of Fit Measures in Structural Equation Models A Good Reference: Bollen and Long, TESTING STRUCTURAL EQUATION MODELS, Sage, 1993.

  13. Goodness of Fit Measures in Structural Equation Models • A fit measure expresses the difference between Σ(θ) and S. Using whatever metric it employs, it should register “perfect” whenever Σ(θ) = S exactly. • This occurs trivially when df=0 • 0 to 1 usually thought of as best metric (see Tanaka in Bollen & Long, 1993)

  14. Goodness of Fit Measures in Structural Equation Models • Early fit measures: • Model Χ2 : • Asks the question, is there a statistically significant difference between S and Σ ? • If the answer to this question is “no”, we should definitely NOT try to add parameters to the model (capitalizing on change) • If the answer to this question is “yes”, we can cautiously add parameters • Contemporary thinking is that we need some other measure that is not sample-size dependent

  15. Goodness of Fit Measures in Structural Equation Models • Model Χ2 : • X2 = (N-1) * Fml • Contemporary thinking is that we need some other measure that is not sample-size dependent • An issue in fit measures: “sample size dependency” (not considered a good thing) • Chi-square is very much sample size dependent (a direct function of N)

  16. Goodness of Fit Measures in Structural Equation Models • Model Χ2 : • X2 = (N-1) * Fml • Contemporary thinking is that we need some other measure that is not sample-size dependent • An issue in fit measures: “sample size dependency” (not considered a good thing) • Chi-square is very much sample size dependent (a direct function of N)

  17. Goodness of Fit Measures in Structural Equation Models • Problem with Χ2 itself as a measure (aside from the fact that it is a direct function of N): • Logic of trying to “embrace” the null hypothesis. • Even if chi-square not used, it IS important as a “cut off” (never add parameters to a model when chi-square is non-signif. • Many measures are based on Χ2

  18. Goodness of Fit Measures in Structural Equation Models • The “first generation” fit measures: • Jöreskog and Sörbom’s Goodness of Fit Index (GFI) [LISREL] • Bentler’s Normed Fit Index (NFI) [EQS] • These have now been supplemented in most software packages with a wide variety of fit measures

  19. Fit Measures GFI = 1 – tr[Σ-1S – I]2 tr (Σ-1S)2 Takes on value from 0 to 1 Conventional wisdom: .90 cutoff GFI tends to yield higher values than other coefficients GFI is affected by sample size, since in small samples, we would expect larger differences between Σ and S even if the model is correct (sampling variation is larger)

  20. Fit Measures GFI is an “absolute” fit measure There are “incremental” fit measures that compare the model against some baseline. - one such baseline is the “Independence Model - Independence Model: models only the variances of manifest variables (no covariances) [=assumpt. all MVs independent] “Independence Model chi-square” (usually very large) - Σ will have 0’s in the off-diagonals

  21. Fit Measures NFI = (Χ2b-Χ2m)/ Χ2b Normed Fit Index (Bentler) (subscript b = baseline m=model) Both NFI and GFI will increase as the number of model parameters increases and are affected by N (though not as a simple *N or *N-1 function). GFI = widely used in earlier literature since it was the only measure (along with AGFI) available in LISREL NFI (along with NNFI) only measure available in early versions of EQs

  22. Fit Measures Thinking about fit indices: Desirable properties: • Normed (esp. to 0  1) Some measures only approx: TLI Arbitrary metric: AIC (Tanaka: AIC could be normed) • Not affected by sample size (GFI, NFI are) • “Penalty function” for extra parameters (no inherent advantage to complex models) – “Parsimony” indices deal with this • Consistent across estimation techniques (ML, GLS, other methods)

  23. Fit Measures Bollens delta-2 (Χ2b – Χ2m )/ Χ2b – dfm RMR – root mean residual (only works with standardized residuals) SRMR - standardized RMR Parsimony GFI 2df/p * (p+1) * GFI AGFI = 1 – [1(q+1) / 2df ] [1 – GFI] RNI (Relative Noncentrality Index) = [(Χ2b – dfb) – (X2m- dfm)] / (Χ2b – dfb) CFI = 1 – max[(X2m- dfm),0] / max[(X2m- dfm), (X2 b- dfb),0] RMSEA = sqrt (MAX[(X2m- dfm),(n-1),0) / dfm

  24. Fit Measures Some debate on conventional .90 criterion for most of these measures Hu & Bentler, SEM 6(1), 1999 suggest: • Use at least 2 measures • Use criterion of >.95 for 0-1 measure, <.06 for RMSEA or SRMR

  25. SEM Assumptions Fml estimator: 1. No Kurtosis 2. Covariance matrix analysed * 3. Large sample 4. H0: S = Σ(θ) holds exactly

  26. SEM Assumptions Fml estimator: 1. Consistent 2. Asymptotically efficient 3. Scale invariant 4. Distribution approximately normal as N increases

  27. SEM Assumptions Fml estimator: Small Samples 1980s simulations: - Not accurate N<50 - 100 + highly recommended - “large sample” usually 200+ - in small samples, chi-square tends ot be too large

  28. Writing up results from Structural Equation Models What to Report, What to Omit

  29. Writing up results from Structural Equation Models • Reference: Hoyle and Panter chapter in Hoyle. • Important to note that there is a wide variety of reporting styles (no one “standard”).

  30. Writing up results from Structural Equation Models • A Diagram • Construct Equation Model • Measurement Equation model Some simplification may be required. Adding parameter estimates may clutter (but for simple models helps with reporting). Alternatives exist (present matrices).

  31. Reporting Structural Equation Models • “Written explanation justifying each path and each absence of a path” (Hoyle and Panter) (just how much journal space is available here? ) It might make more sense to try to identify potential controversies (with respect to inclusion, exclusion).

  32. Controversial paths?

  33. What to report and what not to report….. • Present the details of the statistical model • Clear indication of all free parameters • Clear indication of all fixed parameters • It should be possible for the reader to reproduce the model • Describe the data • Correlations and standard errors (or covariances) for all variables ?? Round to 3-4 digits and not just 2 if you do this

  34. What to report and what not to report… 4. Describing the data (continued) • Distributions of the data • Any variable highly skewed? • Any variable only nominally continuous (i.e., 5-6 discrete values or less)? • Report Mardia’s Kurtosis coefficient (multivariate statistic) • Dummy exogenous variables, if any 5. Estimation Method If the estimation method is not ML, report ML results.

  35. What to report and what not to report… 6. Treatment of Missing Data • How big is the problem? • Treatment method used? • Pretend there are no missing data • Listwise deletion • Pairwise deletion • FIML estimation (AMOS, LISREL >=8.5) • Nearest neighbor imputation (LISREL >=8.1) • EM algorithm (covariance matrix imputation ) (LISREL >=8.5)

  36. What to report and what not to report… 7. Fit criterion • Hoyle and Panter suggest “.90; justify if lower”. • Choice of indices also an issue. There appears to be “little consensus on the best index” (H & P recommend using multiple indices in presentations) Standards: Bollen’s delta 2 (IFI) Comparative Fit Index RMSEA

  37. Fit indices • Older measures: • GFI (Joreskog & Sorbom) • Bentler’s Normed Fit index • Model Chi-Square

  38. What to report & what not to report…. 8. Alternative Models used for Nested Comparisons (if appropriate)

  39. 9. Plausible explanation for correlated errors [“these things were just too darned big to ignore”] • Generally assumed when working with panel model with equivalent indicators across time:

  40. What to report 10. Interpretation of regression-based model • Present standardized and unstandardized coefficients (usually) • Standard errors? (* significance test indicators?) • R-square for equations • Measurement model too? • (expect higher R-squares)

  41. What to report. • Problems and issues • Negative error variances or other reasons for non-singular parameter covariance matrices • How dealt with? Does the final model entail any “improper estimates”? • Convergence difficulties, if any • LISREL: can look at Fml across values of given parameter, holding other parameters constant • Collinearity among exogenous variables • Factorially complex items

  42. What to report & what not to report…. • General Model Limitations, Future Research issues: • Where the number of available indicators compromised the model • 2-indicator variables? (any constraints required?) • Single-indicator variables? (what assumptions made about error variances?) • Indicators not broadly representative of the construct being measured? • Where the distribution of data presented problems • Larger sample sizes can help

  43. What to report & what not to report…. • General Model Limitations, Future Research issues: • Missing data (extent of, etc.) • Cause-effect issues, if any (what constraints went into non-recursive model? How reasonable are these?)

  44. Matrix form: LISREL MEASUREMENT MODEL MATRICES Manifest variables: X’s Measurement errors: DELTA ( δ) Coefficients in measurement equations: LAMBDA ( λ ) Sample equation: X1 = λ1ξ1+ δ1 MATRICES: LAMBDA-x THETA-DELTA PHI

  45. Matrix form: LISREL MEASUREMENT MODEL MATRICES A slightly more complex example:

  46. Matrix form: LISREL MEASUREMENT MODEL MATRICES Labeling shown here applies ONLY if this matrix is specified as “diagonal” Otherwise, the elements would be: Theta-delta 1, 2, 5, 9, 15. OR, using double-subscript notation: Theta-delta 1,1 Theta-delta 2,2 Theta-delta 3,3 Etc.

  47. Matrix form: LISREL MEASUREMENT MODEL MATRICES While this numbering is common in some journal articles, the LISREL program itself does not use it. Two subscript notations possible: Single subscript Double subscript

  48. Matrix form: LISREL MEASUREMENT MODEL MATRICES Models with correlated measurement errors:

  49. Matrix form: LISREL MEASUREMENT MODEL MATRICES Measurement models for endogenous latent variables (ETA) are similar: • Manifest variables are Ys • Measurement error terms: EPSILON ( ε ) • Coefficients in measurement equations: LAMBDA (λ) • same as KSI/X side • to differentiate, will sometimes refer to LAMBDAs as Lambda-Y (vs. Lambda-X) • Equations • Y1 = λ1η1+ ε1

  50. Matrix form: LISREL MEASUREMENT MODEL MATRICES Measurement models for endogenous latent variables (ETA) are similar:

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