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General Structural Equations (LISREL)

General Structural Equations (LISREL). Week 1 #4. Today:. Quick look at more AMOS examples… Extending the work with AMOS: Moving from factor model to causal model (construct equations among latent variables) adding single-indicator exogenous variables (assume no measurement error)

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General Structural Equations (LISREL)

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  1. General Structural Equations (LISREL) Week 1 #4

  2. Today: • Quick look at more AMOS examples… • Extending the work with AMOS: • Moving from factor model to causal model (construct equations among latent variables) • adding single-indicator exogenous variables (assume no measurement error) • adding single-indicator exogenous variables with assumed measurement error • Equality constraints in structural equation models • Dummy exogenous variables in structural equation models • SEM equivalents to contrasts • Block tests for dummy variables • AMOS example

  3. Also Today: • Model fit (an overview) • The SIMPLIS program (part of LISREL) • Moving from Standard Stats packages into SEM software • Conceptualizing SEM models in Matrix terms (some basics)

  4. The SIMPLIS interface for LISREL • Works in scalar, not matrix, terms • Fairly easy to use • Sometimes, output is provided in regular LISREL matrix form (can be a bit confusing) • Requires a lower-triangular covariance matrix (most stats packages produce “square” matrices) OR a special “.dsf” file (both can be created by the PRELIS program which accompanies LISREL).

  5. Two examples of SIMPLIS programs Example 1 SIMPLIS Example for Religion Sexual Morality Data System file from file f:\Classes\ICPSR2005\Week1Examples\ReligSexMoral-SIMPLIS\ReligSex1.dsf Latent Variables Relig Sexmor Relationships: V9 V175 V176 = Relig V147 = 1*Relig V304 V305 V307 V309 = Sexmor V308 = 1*Sexmor End of problem

  6. SIMPLIS Example for Religion Sexual Morality Data Covariance Matrix V9 V147 V175 V176 V304 V305 -------- -------- -------- -------- -------- -------- V9 0.82 V147 1.34 6.50 V175 0.31 0.75 0.48 V176 -1.64 -3.49 -1.09 6.77 V304 0.40 1.06 0.29 -1.52 2.90 V305 0.46 0.98 0.27 -1.45 1.34 3.52 V307 0.79 1.85 0.46 -2.57 1.70 1.69 V308 0.65 1.51 0.37 -1.93 1.59 1.61 V309 1.11 2.39 0.58 -3.12 1.61 1.83 Covariance Matrix V307 V308 V309 -------- -------- -------- V307 7.26 V308 3.13 4.61 V309 4.02 2.83 7.76 Output

  7. LISREL Estimates (Maximum Likelihood) Measurement Equations V9 = 0.44*Relig, Errorvar.= 0.28 , Rý = 0.66 (0.018) (0.015) 25.20 18.41 V147 = 1.00*Relig, Errorvar.= 3.73 , Rý = 0.43 (0.16) 23.93 V175 = 0.27*Relig, Errorvar.= 0.27 , Rý = 0.44 (0.013) (0.011) 21.54 23.78 V176 = - 1.35*Relig, Errorvar.= 1.74 , Rý = 0.74 (0.052) (0.12) -25.96 14.68 Output

  8. V304 = 0.63*Sexmor, Errorvar.= 1.96 , Rý = 0.33 (0.033) (0.082) 19.06 24.02 V305 = 0.66*Sexmor, Errorvar.= 2.49 , Rý = 0.29 (0.036) (0.10) 18.11 24.46 V307 = 1.25*Sexmor, Errorvar.= 3.57 , Rý = 0.51 (0.054) (0.17) 23.14 20.54 V308 = 1.00*Sexmor, Errorvar.= 2.23 , Rý = 0.52 (0.11) 20.33 V309 = 1.26*Sexmor, Errorvar.= 3.96 , Rý = 0.49 (0.055) (0.19) 22.81 21.01

  9. Covariance Matrix of Independent Variables Relig Sexmor -------- -------- Relig 2.77 (0.21) 13.18 Sexmor 1.59 2.38 (0.11) (0.17) 14.25 14.32

  10. Degrees of Freedom = 26 Minimum Fit Function Chi-Square = 213.09 (P = 0.0) Normal Theory Weighted Least Squares Chi-Square = 217.29 (P = 0.0) Estimated Non-centrality Parameter (NCP) = 191.29 90 Percent Confidence Interval for NCP = (147.95 ; 242.10) Minimum Fit Function Value = 0.15 Population Discrepancy Function Value (F0) = 0.13 90 Percent Confidence Interval for F0 = (0.10 ; 0.17) Root Mean Square Error of Approximation (RMSEA) = 0.071 90 Percent Confidence Interval for RMSEA = (0.063 ; 0.080) P-Value for Test of Close Fit (RMSEA < 0.05) = 0.00

  11. Normed Fit Index (NFI) = 0.97 Non-Normed Fit Index (NNFI) = 0.97 Parsimony Normed Fit Index (PNFI) = 0.70 Comparative Fit Index (CFI) = 0.98 Incremental Fit Index (IFI) = 0.98 Relative Fit Index (RFI) = 0.96 Critical N (CN) = 312.87 Root Mean Square Residual (RMR) = 0.15 Standardized RMR = 0.035 Goodness of Fit Index (GFI) = 0.97 Adjusted Goodness of Fit Index (AGFI) = 0.94 Parsimony Goodness of Fit Index (PGFI) = 0.56

  12. Another SIMPLIS Example(Same 2 latent variables with single-indicator exogenous variables added) SIMPLIS Example for Religion Sexual Morality Data Observed variables: V9 V147 V175 V176 V304 V305 V307 V308 V309 V310 V355 V356 SEX OCC1 OCC2 OCC3 OCC4 OCC5 Covariance matrix from file e:\ICPSR2005\RSM1.COV Sample size = 1457 Latent Variables: Relig Sexmor Relationships: V9 V175 V176 = Relig V147 = 1*Relig V304 V305 V307 V309 = Sexmor V308 = 1*Sexmor Equations: Relig = V355 V356 SEX Sexmor = V355 V356 SEX Let the error covariance of Relig and Sexmor be free Let the error covariance of V175 and V176 be free Options MI ND=3 SC End of problem

  13. Output Covariance Matrix (portion) V307 V308 V309 V355 V356 SEX -------- -------- -------- -------- -------- -------- V307 7.264 V308 3.132 4.606 V309 4.023 2.832 7.758 V355 -7.317 -5.385 -4.860 305.580 V356 1.447 0.656 1.455 -8.744 4.869 SEX -0.101 0.123 0.019 -0.107 0.090 0.250

  14. Error Covariance for V176 and V175 = -0.210 (0.0327) -6.417 Structural Equations Relig = - 0.0139*V355 + 0.0801*V356 + 0.443*SEX, Errorvar.= 2.735 , R2 = 0.0575 (0.00281) (0.0222) (0.0958) (0.204) -4.962 3.607 4.626 13.425 Sexmor = - 0.0148*V355 + 0.155*V356 + 0.0413*SEX, Errorvar.= 2.089 , R2 = 0.0973 (0.00255) (0.0204) (0.0860) (0.149) -5.795 7.587 0.480 13.995 Error Covariance for Sexmor and Relig = 1.454 (0.104) 13.954

  15. Standardized Completely Standardized Solution LAMBDA-Y Relig Sexmor -------- -------- V9 0.848 - - V147 0.668 - - V175 0.598 - - V176 -0.821 - - V304 - - 0.560 V305 - - 0.540 V307 - - 0.721 V308 - - 0.709 V309 - - 0.706 • In Simplis: OPTIONS SC

  16. Standardized GAMMA V355 V356 SEX -------- -------- -------- Relig -0.143 0.104 0.130 Sexmor -0.170 0.225 0.014

  17. Use PRELIS SPSS SYSTEM FILE SPSS SYSTEM FILE A ‘DSF’ file created by PRELIS AMOS (reads Directly from SPSS system files) LISREL reads DSF files A raw covariancematrix (lower triangle) created by PRELIS SAS, Stata, etc. SYSTEM FILE LISREL reads lower triangular matrices Moving from Stat Package System files to SEM Software AMOS LISREL

  18. Fit of a model • How far apart are Σ and S? • Test of significance for H0: Σ=S • chi-square test • Note: “Independence chi-square” is a different test! It tests H0: S=0 • Test is a simple function of N: • Χ2 = F*(N-1) • “Perfect fit” (non-significant chi-square) much easier to obtain in small samples

  19. Fit of a model • Search for “fit indices” that are not a function of N • Desirable properties of fit indices: • Not a direct, linear function of N • Not affected by N (expect wider sampling distribution with smaller Ns.. this might imply that some types of fit indices yield “better” values for the same model in larger samples • Easily interpretable metric (e.g., 0  1) • Consistent across estimation methods • Not affected by metric of variables (e.g., same results whether variables standardized or not)

  20. Fit of a model • Desirable properties of fit indices (more): • Do not reward data dredging (vs. construction of parsimonious models) • So-called “parsimony” measures include a penalty function for adding parameters to a model • Commonly-used fit measures: • Joreskog & Sorbom’s GFI (affected by N though) • Bentler’s Normed Fit Index (and NNFI) • Incremental, Comparative fit indices • Root Mean Square Error of Approximation (RMSEA) (for this index, low values are good)

  21. Improving the fit of a model: diagnostics • Residuals: • Matrix of differences between sigma and S • Would need to standardize before we could determine where a model should be improved • A residual is not necessarily connected to one single parameter: • A high residual might imply any one of 3 or 4 parameters could/should be added to the model

  22. Improving the fit of a model: diagnostics • Modification indices • Based on 2nd order derivative matrix • Estimate the improvement in model fit if a particular parameter is added • Metric: chi-square (difference) • Any value greater than 3.84 is “significant” at p<.05 BUT criteria other than straight significance can/have been employed • Reason: otherwise, sensitive to N; in large samples will never get parsimonious model, etc.

  23. Modification Indices • In AMOS, click “modification indices” under output options • In SIMPLIS, Options MI Modification Indices and Expected Change (SIMPLIS model discussed ealrier) The Modification Indices Suggest to Add the Path to from Decrease in Chi-Square New Estimate V9 Sexmor 8.4 -0.06 V176 Sexmor 11.6 -0.18 V307 Relig 14.2 -0.21 V309 Relig 34.0 0.33

  24. Important note on modification indices • It is not always the case that the parameter with the highest MI should be added to a model • Some MIs will not make substantive sense (e.g., in a causal model, an MI suggesting a path from respondent’s social status to parent’s social status).

  25. Improving the fit of a model: diagnostics • Estimated parameter change values • Estimated value of a parameter that is currently fixed (if this parameter is “freed” [included in the model]). • Standardized values can be helpful in determining whether adding a parameter is substantively important

  26. Equality Constraints in Structural Equation Models • We can set “equality constraints” on any two (or more) parameters in a model • E.g.: b1=b2 • E.g.: VAR(e1) = VAR(e2)

  27. Equality Constraints in Structural Equation Models • We can set “equality constraints” on any two (or more) parameters in a model • In AMOS we do this by giving parameters names, and then using the same name in the locations where we want to impose equality constraints

  28. Equality Constraints in Structural Equation Models • We can set “equality constraints” on any two (or more) parameters in a model • In SIMPLIS, we do this by adding statements: Let the path from Relig to V176 be equal to the path from Relig to V167.

  29. Equality Constraints in Structural Equation Models • The b1=b2 constraint may not make sense if the metric of the 2 latent variables is not the same (makes most sense if variances are the same – would work if the variables were standardized]

  30. Equality Constraints in Structural Equation Models • In this model, we could test b1=b2, b2=b3, b1=b3 or b1=b2=b3 by setting the parameter names to be the same • Equality constraints only make sense if variances of the 3 exogenous manifest variables are the same, though

  31. Equality Constraints in Structural Equation Models • Formal tests: • Model 1 b1, b2 estimated separately • Model 2 b1=b2 (i.e., labels “b1” in each of 2 locations) • Model 2 has 1 more degree of freedom than model 1 • A df=1 test for the equality constraint is obtained by subtracting the model 1 chi-square from the model 2 chi-square

  32. Dummy Variables in Structural Equation Models • Dummy variables can be included in structural equation models if they are completely exogenous Sex: 0/1 variable

  33. Dummy Variables in Structural Equation Models • Dummy variables can be included in structural equation models if they are completely exogenous

  34. Dummy Variables in Structural Equation Models • Dummy variables cannot be included in structural equation models as indicators of latent constructs VOTED = 0/1 voted/did not vote last election TRUST = 5 pt. trust in government item POL COR = 5 pt. agree/disagree politicians corrupt  This model is NOT appropriate

  35. Dummy Variables in Structural Equation Models • Dummy variables can be included in structural equation models if they are completely exogenous • For categorical independent variables with more than 2 categories, sets of dummy variables can be included (just like in regression models)

  36. Dummy Variables in Structural Equation Models • For categorical independent variables with more than 2 categories, sets of dummy variables can be included (just like in regression models) • Design matrix as with Regression (could use effects or indicator coding; example below uses indicator coding): D1 D2 D3 Catholic 1 0 0 Protestant 0 1 0 Jewish 0 0 1 Atheist 0 0 0

  37. DUMMY VARIABLES (add curved arrow D1 D2 )

  38. DUMMY VARIABLES Test H0 for entire religion variable: estimate model with parameters b1, b2 and b3 all set to 0 (add curved arrow D1 D2 )

  39. DUMMY VARIABLES Test H0 for entire religion variable: estimate model with parameters b1, b2 and b3 all set to 0

  40. DUMMY VARIABLES • Test religion variable : b1=b2=b3=0 Model 1 (3 separate parameters) vs. Model 2 (all parameters = 0) df=3 test • Test Protestant (category 1) vs. Atheist (reference group): • Model 1 (3 separate parameters) • Model 2 (fix b1=0) df=1 • OR: look at t-test for b1 parameter

  41. DUMMY VARIABLES • Test Protestant (category 1) vs. Catholic (category2): • Model 1 (3 separate parameters) • Model 2 (fix b1=b2) df=1

  42. LV Structural Equation Models in Matrix terms Thus far, our work has involved “scalar” equations. • one equation at a time • Specify a model (e.g, with software) by writing these equations out, one line per equation

  43. Matrix form We can represent the previous 2 equations in matrix form: Matrix Form (single, double subscript)

  44. There are other matrices in this model Variance-covariance matrix of error terms (e’s)

  45. (other matrices, continued) Variance covariance matrix of exogenous (manifest) variables

  46. Two scalar equations re-written scalar Matrix Contents of matrices

  47. More generic form (combines all exogenous variables into single matrix) More generic: Where E1 Ξ X1, E2 Ξ X2 and E3 Ξ X3

  48. More generic form: All exogenous variables part of a single variance-covariance matrix

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