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

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

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

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

Lagged and contemporaneous effects This model is underidentified

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

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

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

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

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

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

Equivalence of parameters: T1  T2 T2  T3

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

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

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

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

Panel Data model Model for attitudes about labour unions, 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

Source: Cdn. Quality of life panel study, waves

Panel Data model LISREL Estimates (Maximum Likelihood) LAMBDA-Y LABOR77 LABOR Q199D Q156C (0.141) Q156D (0.101) Q156B (0.098) QD7B QK16F (0.109) QK16G (0.072) QK16E (0.084) 8.427

Panel Data model BETA LABOR77 LABOR LABOR LABOR (0.138) PSI Note: This matrix is diagonal. LABOR77 LABOR (0.017) (0.018) Squared Multiple Correlations for Structural Equations LABOR77 LABOR W_A_R_N_I_N_G: PSI is not positive definite

Panel Data model Completely Standardized Solution LAMBDA-Y LABOR77 LABOR Q199D Q156C Q156D Q156B QD7B QK16F QK16G QK16E BETA LABOR77 LABOR LABOR LABOR What is the problem here?

Panel Data model Theta-epsilon was specified as diagonal Modification Indices for THETA-EPS Q199D Q156C Q156D Q156B QD7B QK16F Q199D - - Q156C Q156D Q156B QD7B QK16F QK16G QK16E

Panel Data model

Added error covariances: FR TE 5 1 TE 6 2 TE 7 3 TE 8 4 BETA LABOR77 LABOR LABOR LABOR (0.115) Covariance Matrix of ETA LABOR77 LABOR LABOR LABOR

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 LABOR (0.020) (0.016) Squared Multiple Correlations for Structural Equations LABOR77 LABOR

Panel Data model Panel data model Cdn. Quality of Life ! 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]

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

Panel Data Panel data model Cdn. Quality of Life ! 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

Panel Data LISREL Estimates (Maximum Likelihood) LAMBDA-Y NEWSP TV LABOR77 LABOR Q (0.176) Q Q Q199D Q156C (0.214)

Panel Data BETA NEWSP TV LABOR77 LABOR NEWSP TV LABOR (0.026) (0.011) LABOR (0.030) (0.014) (0.113)

Panel Data Panel data model Cdn. Quality of Life ! 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

Panel Data BETA NEWSP TV LABOR77 LABOR NEWSP TV LABOR (0.034) (0.011) LABOR (0.042) (0.013) (0.115) GAMMA MEMBER SEX EDUC AGE INCOME NEWSP (0.039) (0.035) (0.009) (0.001) (0.005) TV (0.070) (0.062) (0.015) (0.002) (0.009) LABOR (0.036) (0.026) (0.008) (0.001) (0.004) LABOR (0.042) (0.033) (0.009) (0.001) (0.004)

Another model (panel7) BETA INEQ77 LABOR77 INEQ79 LABOR INEQ LABOR INEQ (0.069) (0.110) LABOR (0.044) (0.124)

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

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)

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

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

Some growth curve trajectories: Parallel stability

Some growth curve trajectories: Strict stability

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) )

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

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”

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

3 factor Growth Curve Model Non-linear growth Parm 3

3 factor Growth Curve Model LY matrix INT LIN Quad V V V 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)

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

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

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

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)

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

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 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)

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…..

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

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

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

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

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

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)

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 / 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

Example: LISREL Estimates (Maximum Likelihood) LAMBDA-Y ETA v v (0.24) 5.59 v (0.11) 5.74 GAMMA lv1 lv2 interact sex race v ETA (0.06) (0.08) (0.45) (0.11) (0.13) (0.03) GAMMA v ETA (0.03) 2.92 Dep var = inequality att’s (high score  “more individual effort”) Lv1=relig. Lv2=econ. status

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 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)

Kenny-Judd model

alpha=1 term

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 / 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

Kenny-Judd model, modified Joreskog/Yang Parameter Specifications LAMBDA-Y ETA v125 0 v127 1 v130 2 LAMBDA-X KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 KSI v v v v v v relinc3 Constr'd Constr'd sex race v v

Kenny-Judd model, modified Joreskog/Yang GAMMA KSI 1 KSI 2 KSI 3 KSI 4 KSI 5 KSI ETA (0.009) (0.015) (0.004) (0.098) (0.125) (0.024) GAMMA KSI ETA (0.029) 2.735