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 1Time 2Time 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 (2 nd order)

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

12. 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. 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 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 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 V110 V211 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 Parm1 = mean intercept Parm2 = mean slope value 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

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) IntSlope V110 V211 V312 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 Non-linear growth Parm 3

45. 3 factor Growth Curve Model LY matrix INT LIN Quad V11 0 0 V21 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 parm3 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

52. Interactions Easiest case: X1 is 0/1 X2 ix 0/1 Options: 1. Manually construct X3=X1*X2 outside SEM software, estimate model with X1,X2,X3 exogenous. Test for interaction: fix regression coefficient for X3 to 0. 2. Create two groups: X1=0 and X1=1. In each group, X2 as exogenous variable. Test for interaction would be H0: gamma[1] = gamma[2]. Extensions for X1, X2 >2 categories straightfoward (more groups/dummy variables)

53. Interactions Option 3: Model as a 4-group problem. X1 10 X21gr1gr2 0gr3 gr4 AL[1]=0 al[2], al[3],al[4] parameters to be estimated. Main effects model (no interaction) would allow for al[2]≠al[3] ≠al[4] but pattern of differences would be constrained such that…..

54. Interactions Model as a 4-group problem. X1 10 X21gr1gr2 0gr3 gr4 AL[1]=0 al[2], al[3],al[4] parameters to be estimated. Main effects model (no interaction) would allow for al[2]≠al[3] ≠al[4] but pattern of differences would be constrained such that….. The group1 vs. group 2 difference = group 3 vs. group 4 difference (or group 1 vs. 3 difference = group 2 vs. group 4). Programming in LISREL would be: Al[1] – Al[2] = al[3]- al[4] 0 – al[2] = al[3] – al[4] Al[2] = al[4]-al[3] LISREL: CO al 2 1 = al 4 1 – al 3 1 Test for interaction: run another model removing this constraint (all AL completely free except group 1) … more examples provided in text

55. Interactions Interactions involving continuous variables. Case 1: One continuous (single or multiple indicator) and one categorical variable EASY: categorical variable becomes basis for grouping. Group 1 Eta = gamma[1] Ksi + zeta Group 2 Eta = gamma[2] Ksi + zeta Test for interaction: H0: gamma[1] = gamma[2] Case 2: Two continuous single indicator variables Also somewhat straightforward: Create single-indicator X3 = X2*X1 Case 3: Two continuous multiple indicator latent variables This is not so easy! Substantial literature on this question See course outline for extended list. (Schumacker and Mracoulides, eds., Interaction and Nonlinear Effects in Structural Equation Modeling). Case 3A, not talked about much: X1 single indicator Ksi1 (X2, X3,X4) Create: X1X2, X1X3, X1,X4

56. Latent variable interactions Major approaches: Kenny-Judd Simplified variants of Kenny-Judd, modifications, etc. (Joreskog & Yang, 1996; Ping) Two-stage least squares (get instrumental variables) Use SEM to estimate 2 factor model, save latent variable “scores” (analogous to factor scores), then use these

57. Latent variable interactions Use SEM to estimate 2 factor model, save latent variable “scores” (analogous to factor scores), then use these In LISREL: Mo nx=6 nk=2 lx=fu,fi ph-sy,fr td=sy Va 1.0 lx 1 1 lx 4 2 Fr lx 2 1 lx 3 1 lx 5 2 lx 6 2 PS=Newfile.psf OU

58. Latent variable interactions Use SEM to estimate 2 factor model, save latent variable “scores” (analogous to factor scores), then use these In LISREL: Mo nx=6 nk=2 lx=fu,fi ph-sy,fr td=sy Va 1.0 lx 1 1 lx 4 2 Fr lx 2 1 lx 3 1 lx 5 2 lx 6 2 PS=Newfile.psf OU LISREL documentation suggests that a simple regression can be estimated in PRELIS: Sy=newfile.psf ne inter=ksi1*ksi2 rg y on ksi1 ksi2 ksi1 ksi2 ou

59. Latent variable interactions LISREL documentation suggests that a simple regression can be estimated in PRELIS: Sy=newfile.psf ne inter=ksi1*ksi2 rg y on ksi1 ksi2 ksi1 ksi2 ou …. But it should also be possible to a) construct “inter” (=ksi1*ksi2) and read the 3 new “single indicator” variables back into LISREL for use with other variables (including those which form the basis of multiple-indicator endogenous variables. If all else fails, construct a LISREL model for Ksi1, Ksi2, and put FS (factor score regressions) on the OU line: OU ME=ML FS MI ND=4.. And use factor score regressions to compute estimated factor scores in any stat package (incl. PRELIS)

60. Example: INTERACTION MODEL WITH INTERACTION TERM CREATED EXTERNALLY SINGLE INDICATORS FOR EXOGENOUS LVS INVOLVED IN INTERACTION DA NO=1111 NI=10 MA=CM CM FI=G:\ICPSR\INTERACTIONS\INT5b.COV FU FO (10F10.7) LABELS lv1 lv2 interact sex race v217 v216 v125 v127 v130 se 8 9 10 1 2 3 4 5 6 7/ mo ny=3 ne=1 LY=FU,FI PS=SY,FR TE=SY c nx=7 nk=7 fixedx ga=fu,fr va 1.0 ly 1 1 fr ly 2 1 ly 3 1 ou me=ml se tv mi sc

61. Example: LISREL Estimates (Maximum Likelihood) LAMBDA-Y ETA 1 -------- v125 1.00 v127 1.34 (0.24) 5.59 v130 0.65 (0.11) 5.74 GAMMA lv1 lv2 interact sex race v217 -------- -------- -------- -------- -------- -------- ETA 1 -0.04 -0.21 0.85 0.22 -0.30 0.05 (0.06) (0.08) (0.45) (0.11) (0.13) (0.03) -0.65 -2.57 1.89 2.10 -2.27 1.75 GAMMA v216 -------- ETA 1 0.09 (0.03) 2.92 Dep var = inequality att’s (high score  “more individual effort”) Lv1=relig. Lv2=econ. status

62. Kenny-Judd model Typically, literature (e.g., Kenny-Judd, 1984; Hayduk, 1987) starts with 2- indicator example (2 LV’s each with 2 indicators). Ksi1 Ksi2Ksi1*Ksi2 (interaction term) Indicators:Ksi1:x1 x2 Ksi2:x3 x4 Possible product terms: x1*x3x1*x4 x2*X3X2*x4 Kenny-Judd model use 4 product terms but Joreskog and Yang show that the model can be constructed with 1 product term.

63. Kenny-Judd model Typically, literature (e.g., Kenny-Judd, 1984; Hayduk, 1987) starts with 2-indicator example (2 LV’s each with 2 indicators). Ksi1 Ksi2Ksi1*Ksi2 (interaction term) Indicators:Ksi1:x1 x2 Ksi2:x3 x4 Possible product terms: x1*x3x1*x4 x2*X3X2*x4 Kenny-Judd model use 4 product terms but Joreskog and Yang show that the model can be constructed with 1 product term. Kenny-Judd do not include constant intercept terms (alpha, tau).. But even if dependent variable, Ksi1, Ksi2 and zeta have zero means, alpha will still be nonzero. - means of observed variables functions of other parameters in the model and therefore intercept terms have to be included. - Nonnormality even if x’s are normal (ADF estimation often recommended if sample size acceptable)

64. Kenny-Judd model

65. alpha=1 term

66. Kenny-Judd model, mod. INTERACTION MODEL KENNY JUDD MODIFICATION (JORESKOG AND YANG) ONE INTERACTION INDICATOR 3 INDICATORS PER L.V. DA NO=1111 NI=22 CM FI=G:\ICPSR2000\INTERACTIONS\INT5c.COV FU FO (22F20.11) ME FI=G:\ICPSR2000\INTERACTIONS\INT5C.MN FO (22F20.11) LABELS v181 v9 v190 v221 v226 v227 relinc1 relinc2 relinc3 relinc4 relinc5 relinc6 relinc7 relinc8 reling9 sex race v217 v216 v125 v127 v130 se 20 21 22 1 2 3 4 5 6 9 16 17 18 19/ mo ny=3 ne=1 NX=11 NK=7 LY=FU,FI PS=SY,FR C TE=SY TX=FR KA=FI C LX=FU,FI GA=FU,FR PH=SY,FR TD=SY AL=FI TY=FR va 1.0 ly 1 1 fr ly 2 1 ly 3 1 FI PH 3 1 PH 3 2 FR KA 3 VA 1.0 LX 1 1 LX 4 2 LX 7 3 LX 8 4 LX 9 5 LX 10 6 LX 11 7 FR TD 1 1 TD 2 2 TD 3 3 TD 4 4 TD 5 5 TD 6 6 TD 7 7 FR LX 2 1 LX 3 1 LX 5 2 LX 6 2 LX 7 1 LX 7 2 CO LX(7,1)=TX(1) CO LX(7,2)=TX(4) CO KA(3) = PH(2,1) FI PH 3 1 PH 3 2 CO PH(3,3) = PH(1,1)*PH(2,2) + PH(2,1)**2 CO TX(6) = TX(1)*TX(4) FI TD(8,8) TD(9,9) TD(10,10) TD(11,11) CO TD(7,7) = TX(1)**2*TD(3,3) + TX(4)**2*TD(1,1) + PH(1,1)*TX(4) + C PH(2,2)*TX(1) + TD(1,1)*TD(4,4) OU ME=ML SE TV ND=3 AD=off

67. Kenny-Judd model, modified Joreskog/Yang Parameter Specifications LAMBDA-Y ETA 1 -------- v125 0 v127 1 v130 2 LAMBDA-X KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 KSI 6 -------- -------- -------- -------- -------- -------- v181 0 0 0 0 0 0 v9 3 0 0 0 0 0 v190 4 0 0 0 0 0 v221 0 0 0 0 0 0 v226 0 5 0 0 0 0 v227 0 6 0 0 0 0 relinc3 Constr'd Constr'd 0 0 0 0 sex 0 0 0 0 0 0 race 0 0 0 0 0 0 v217 0 0 0 0 0 0 v216 0 0 0 0 0 0

68. Kenny-Judd model, modified Joreskog/Yang GAMMA KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 KSI 6 -------- -------- -------- -------- -------- -------- ETA 1 -0.023 -0.003 -0.008 0.209 -0.324 0.051 (0.009) (0.015) (0.004) (0.098) (0.125) (0.024) -2.557 -0.198 -1.984 2.130 -2.593 2.094 GAMMA KSI 7 -------- ETA 1 0.080 (0.029) 2.735