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ICPSR General Structural Equation Models

ICPSR General Structural Equation Models

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ICPSR General Structural Equation Models

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  1. ICPSR General Structural Equation Models Week 4 # 3 Panel Data (including Growth Curve Models)

  2. Causal models: Cross-lagged panel coefficients [Reduced form of model on next slide]

  3. Causal models: Reciprocal effects, using lagged values to achieve model identification

  4. Causal models: A variant Issue: what does ga(1,1) mean given concern over causal direction?

  5. Lagged and contemporaneous effects This model is underidentified

  6. Lagged effects model Ksi-1 could be an “event” 1/0 dummy variable

  7. First order model for three wave data(univariate) Time 1 Time 2 Time 3

  8. First order model for three wave data(univariate) Tests: Equivalent of stability coefficients (b1) Mean differences (see earlier slide)

  9. Second order model for three wave data(univariate) No longer comparable to b1 (t1 t2)

  10. Second order model for three wave data(univariate) Issue: adding appropriate error terms (2nd order)

  11. Multivariate Model for Three-wave panel data: cross-lagged effects (first order)

  12. Multivariate Model for Three-wave panel data: cross-lagged effects (first order) Equivalence of parameters: T1  T2 T2  T3

  13. Multivariate Model for Three-wave panel data: cross-lagged effects (second order)

  14. Multivariate Model for Four-wave panel data: cross-lagged effects (second order)

  15. Lagged and contemporaneous effects Three wave model with constraints: Under many circumstances, there will be an empirical under-ident. problem, though in theory this model is identified

  16. Example: • Canada, Quality of Life data • In directory \Panel in Week4Examples

  17. Panel Data model Model for attitudes about labour unions, 1977-1979 Items: 5-pt. agree/disagree 199D QD6B Unions too much power Q156C QK16F Scabs (gov’t prohibit strikebreakers) Q156D QK16G Workers on Boards Q156B QK16E Teachers should not have right to strike

  18. Source: Cdn. Quality of life panel study, 1977-1979 waves

  19. Panel Data model LISREL Estimates (Maximum Likelihood) LAMBDA-Y LABOR77 LABOR79 -------- -------- Q199D 1.000 - - Q156C -1.803 - - (0.141) -12.796 Q156D -1.148 - - (0.101) -11.350 Q156B 0.789 - - (0.098) 8.040 QD7B - - 1.000 QK16F - - -1.352 (0.109) -12.355 QK16G - - -0.755 (0.072) -10.479 QK16E - - 0.709 (0.084) 8.427

  20. Panel Data model BETA LABOR77 LABOR79 -------- -------- LABOR77 - - - - LABOR79 1.420 - - (0.138) 10.318 PSI Note: This matrix is diagonal. LABOR77 LABOR79 -------- -------- 0.125 -0.066 (0.017) (0.018) 7.529 -3.611 Squared Multiple Correlations for Structural Equations LABOR77 LABOR79 -------- -------- - - 1.356 W_A_R_N_I_N_G: PSI is not positive definite

  21. Panel Data model Completely Standardized Solution LAMBDA-Y LABOR77 LABOR79 -------- -------- Q199D 0.425 - - Q156C -0.559 - - Q156D -0.436 - - Q156B 0.262 - - QD7B - - 0.409 QK16F - - -0.524 QK16G - - -0.382 QK16E - - 0.277 BETA LABOR77 LABOR79 -------- -------- LABOR77 - - - - LABOR79 1.165 - - What is the problem here?

  22. Panel Data model Theta-epsilon was specified as diagonal Modification Indices for THETA-EPS Q199D Q156C Q156D Q156B QD7B QK16F -------- -------- -------- -------- -------- -------- Q199D - - Q156C 2.845 - - Q156D 3.439 20.324 - - Q156B 17.009 5.334 13.004 - - QD7B 83.881 42.939 10.988 4.108 - - QK16F 10.361 108.940 28.775 23.541 2.034 - - QK16G 19.366 28.336 141.658 5.494 0.242 7.172 QK16E 0.158 7.133 14.031 169.430 25.246 6.019

  23. Panel Data model

  24. Panel Data model Added error covariances: FR TE 5 1 TE 6 2 TE 7 3 TE 8 4 BETA LABOR77 LABOR79 -------- -------- LABOR77 - - - - LABOR79 1.094 - - (0.115) 9.547 Covariance Matrix of ETA LABOR77 LABOR79 -------- -------- LABOR77 0.116 LABOR79 0.127 0.199

  25. Panel Data model Added error covariances: FR TE 5 1 TE 6 2 TE 7 3 TE 8 4 PSI Note: This matrix is diagonal. LABOR77 LABOR79 -------- -------- 0.116 0.060 (0.020) (0.016) 5.935 3.721 Squared Multiple Correlations for Structural Equations LABOR77 LABOR79 -------- -------- - - 0.698

  26. Panel data model Cdn. Quality of Life 1977-81 ! Model for mean differences SY='H:\QOL3WAVE\imputed_data.dsf' SE Q199D Q156C Q156D Q156B QD7B QK16F QK16G QK16E / MO NY=8 NE=2 LY=FU,FI PS=SY,FR TE=SY BE=FU,FI TY=FR AL=FI LE LABOR77 LABOR79 VA 1.0 LY 1 1 LY 5 2 FR LY 2 1 LY 3 1 LY 4 1 FR LY 6 2 LY 7 2 LY 8 2 FR TE 5 1 TE 6 2 TE 7 3 TE 8 4 EQ TY 5 TY 1 EQ TY 6 TY 2 EQ TY 7 TY 3 EQ TY 8 TY 4 EQ LY 2 1 LY 6 2 EQ LY 3 1 LY 7 2 EQ LY 4 1 LY 8 2 FR AL 2 OU ME=ML MI SC ND=3 Panel Data model Alternative specification with stability coefficient: PS=SY BE=SD [or BE=FU,FI then FR BE 2 1]

  27. Panel Data ALPHA LABOR77 LABOR79 -------- -------- - - 0.043 (0.014) 3.051 Higher score = pro-union (ref. indicator: too much/too little power… too little=5 too much=1

  28. Panel Data Panel data model Cdn. Quality of Life 1977-81 ! Impact of TV newspapers on labor union attitudes SY='H:\QOL3WAVE\imputed_data.dsf' SE Q258 Q260 Q261 Q199D Q156C Q156D Q156B QD7B QK16F QK16G QK16E / MO NY=11 NE=4 LY=FU,FI PS=SY TE=SY BE=FU,FI LE NEWSP TV LABOR77 LABOR79 VA 1.0 LY 2 1 VA 1.0 LY 3 2 FR LY 1 1 FI TE 3 3 VA 1.0 LY 4 3 LY 8 4 FR LY 5 3 LY 6 3 LY 7 3 FR LY 9 4 LY 10 4 LY 11 4 FR BE 4 3 FR BE 3 2 BE 3 1 FR BE 4 2 BE 4 1 FR PS 2 1 FR TE 11 7 TE 10 6 TE 9 5 TE 8 4 OU ME=ML MI SC ND=3

  29. Panel Data LISREL Estimates (Maximum Likelihood) LAMBDA-Y NEWSP TV LABOR77 LABOR79 -------- -------- -------- -------- Q258 0.917 - - - - - - (0.176) 5.212 Q260 1.000 - - - - - - Q261 - - 1.000 - - - - Q199D - - - - 1.000 - - Q156C - - - - -1.891 - - (0.214) -8.819

  30. Panel Data BETA NEWSP TV LABOR77 LABOR79 -------- -------- -------- -------- NEWSP - - - - - - - - TV - - - - - - - - LABOR77 0.061 -0.005 - - - - (0.026) (0.011) 2.325 -0.406 LABOR79 0.047 -0.017 1.081 - - (0.030) (0.014) (0.113) 1.584 -1.216 9.564

  31. Panel Data Panel data model Cdn. Quality of Life 1977-81 ! Impact of TV newspapers on labor union attitudes ! Controls: education sex union membership SY='H:\QOL3WAVE\imputed_data.dsf' SE Q258 Q260 Q261 Q199D Q156C Q156D Q156B QD7B QK16F QK16G QK16E Q63 SEX Q201 RAGE Q157/ MO NY=11 NE=4 LY=FU,FI PS=SY TE=SY BE=FU,FI NX=5 NK=5 FIXEDX LE NEWSP TV LABOR77 LABOR79 LK MEMBER SEX EDUC AGE INCOME VA 1.0 LY 2 1 VA 1.0 LY 3 2 FR LY 1 1 FI TE 3 3 VA 1.0 LY 4 3 LY 8 4 FR LY 5 3 LY 6 3 LY 7 3 FR LY 9 4 LY 10 4 LY 11 4 FR BE 4 3 FR BE 3 2 BE 3 1 FR BE 4 2 BE 4 1 FR PS 2 1 FR TE 11 7 TE 10 6 TE 9 5 TE 8 4 OU ME=ML MI SC ND=3

  32. Panel Data BETA NEWSP TV LABOR77 LABOR79 -------- -------- -------- -------- NEWSP - - - - - - - - TV - - - - - - - - LABOR77 -0.025 -0.012 - - - - (0.034) (0.011) -0.738 -1.157 LABOR79 0.068 -0.010 1.033 - - (0.042) (0.013) (0.115) 1.622 -0.751 8.970 GAMMA MEMBER SEX EDUC AGE INCOME -------- -------- -------- -------- -------- NEWSP -0.017 0.011 -0.097 -0.014 -0.014 (0.039) (0.035) (0.009) (0.001) (0.005) -0.422 0.311 -11.303 -13.496 -2.898 TV -0.013 -0.150 0.025 -0.017 0.001 (0.070) (0.062) (0.015) (0.002) (0.009) -0.182 -2.408 1.685 -9.807 0.113 LABOR77 0.286 -0.056 -0.039 -0.005 -0.010 (0.036) (0.026) (0.008) (0.001) (0.004) 7.880 -2.131 -5.158 -5.331 -2.557 LABOR79 0.045 0.114 0.001 0.001 -0.006 (0.042) (0.033) (0.009) (0.001) (0.004) 1.082 3.487 0.069 0.966 -1.436

  33. Another model (panel7) BETA INEQ77 LABOR77 INEQ79 LABOR79 -------- -------- -------- -------- INEQ77 - - - - - - - - LABOR77 - - - - - - - - INEQ79 0.704 0.012 - - - - (0.069) (0.110) 10.214 0.105 LABOR79 -0.106 0.819 - - - - (0.044) (0.124) -2.400 6.622

  34. Re-expressing parameters:GROWTH CURVE MODELS Intercept & linear (& sometimes quadratic) terms • Suitable for panel models with >2 waves • Best for panel models with >3 waves

  35. Linear Growth Model LISREL: 2 manifest variable, 2 latent variable model LY matrix INT Slope V1 1 0 V2 1 1 TE matrix = elements equal PS matrix = SY,FR (parm1 in model = variance of INT, parm2 = variance of Slope) TY zero AL free (“parm1” and “parm2” above)

  36. Linear Growth Model • Interpretation: • intercept factor represents initial status • Slope factor represents difference scores (V2-V1) With single indicators, cannot estimate error variances (as with any single indicator SEM model) Parm1 = mean intercept Parm2 = mean slope value

  37. Linear Growth Model E.g., TV use, adolescents, hours/day Parm1 = 2.5 Parm2 = 1.0 Increase of 1 hour/day from t1 to t2 We will also get variances for the Intercept and the Slope factors Parm1 = mean intercept Parm2 = mean slope value

  38. Some growth curve trajectories: • Parallel stability

  39. Some growth curve trajectories: • Strict stability

  40. Single-factor LGM • Actually nested within 2 factor model • take 2 factor model, intercept with 0 mean and 0 variance or strictly proportional to slope Not generally the best model unless assumptions met: (cf. Duncan et al. p. 31: when rank ordering of individuals does not vary across time despite mean level changes) (can estimate var(e1),(e2),(e3) if we impose constraint v(e1)=v(e2)=v(e3) )

  41. Linear Growth Model A bit more complicated with latent variables instead of single manifest variables … but the same basic principle.

  42. Linear Growth Model LY matrix (LISREL) Int Slope V1 1 0 V2 1 1 V3 1 2 Same principle would apply to k time points where k>3 More time points: test of linearity of “growth” (changes in mean)* *general test: vs. “unspecified growth model”

  43. Unspecified 2 factor Growth Curve Model 1 free lambda parameter in LY matrix In k time-point model, all but first 2 time points are represented by free parameters

  44. 3 factor Growth Curve Model Parm 3 Non-linear growth

  45. 3 factor Growth Curve Model parm3 LY matrix INT LIN Quad V1 1 0 0 V2 1 1 2 V3 1 2 4 TE is constrained to equality across t’s PS is free AL is free (parm1-3) All TY elements 0 This is a “saturated” model (perfect fit by definition)

  46. Examples: Z:\baer\Week4Examples\LatentGrowth Single variable models: LGMProg1.ls8 (output=.out) intercept model LGMProg2.ls8 - single factor curve model LGMProg3.ls8 - intercept + slope LGMProg4.ls8 – intercept + slope + quadratic

  47. Where do “growth factors” fit into models? • Examination of predictors (antecedents) and consequences of change Note: Intercept-slope covariance now disturbance covariance PROGRAM LGMProg5

  48. Consequences Model LGMProg6.ls8 Dependent variable: job satisfaction, wave 8.

  49. Multiple indicators for the variable(s) involved in growth curves • “factor of curves” LGM • Intercept term and slope term (e.g.) constructed for each indicator • if there are 3 variables & 4 waves, we will have an intercept term based on 4 manifest variables representing time x 3 manifest variables per time (3 intercept terms) • “common intercept” variable will have 3 indicators (intercept terms) • “common slope” will have 3 indicators (slope terms)

  50. Error variances now estimated (not constrained to equality).. Could include corr. Errors too