LISREL matrices, LISREL programming ICPSR General Structural Equations Week 2 Class #4

Class Exercise (from previous class notes:)

Class exercise BETA 2 x 2 0 BE(1,2) BE(2,1) 0 PHI 2 X 2 PHI(1,1) GAMMA 2 X 2 GA(1,1) 0 0 GA(2,2) PSI 2 x 2 PS(1,1) PS(2,1) PS(2,2)

LAMBDA-X 1 0 LX(2,1) 0 LX(3,1) LX(3,2) 0 1 0 LX(5,2) LAMBDA-Y 1 0 LY(2,1) 0 LY(3,1) 0 0 1 0 LY(5,2) 0 LY(6,2)

MO NY=6 NX=5 NK=2 NE=2 LX=FU,FI LY=FU,FI C PH=SY BE=FU,FI GA=FU,FI TD=SY TE=SY PS=SY,FR VA 1.0 LX 1 1 LX 4 2 LY 1 1 LY 4 2 FR LX 2 1 LX 3 1 LX 3 2 LX 5 2 LY 2 1 LY 3 1 LY 5 2 LY 6 2 FR GA 1 1 GA 2 2 FR BE 2 1 BE 1 2

Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS

Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS Beta 2 x 2 0 0 BE(2,1) 0 PSI 2 x 2 PS(1,1) 0 PS(2,2) Not shown: zeta1 Theta-eps TE(1,1) 0 TE(2,2) 0 0 TE(3,3) TE(4,1) 0 0 TE(4,4) 0 TE(5,2) 0 0 TE(5,5) 0 0 0 0 0 TE(6,6)

Exercise 2:A (PANEL) MODEL WITH CORRELATED ERRORS MO NY=6 NE=2 LY=FU,FI PS=SY TE=SY BE=FU,FI VA 1.0 LY 1 1 LY 4 2 FR LY 2 1 LY 3 1 LY 5 2 LY 6 2 FR BE 2 1 FR TE 4 1 TE 5 2 Notes: PS=SY specification  free diagonals (PS(1,1) and PS(2,2), fixed off-diagonals [ps(2,1)=0 in this model].

Exercise 3

Exercise 3 BETA 2 X 2 0 0 BE(2,1) 0 LAMBDA-Y 1 0 LY(2,1) 0 Gamma 2 x 1 0 0 BE(2,1) 0 LAMBDA-Y 1 0 LY(2,1) 0 LY(3,1) LY(3,2) 0 1 0 LY(5,2) Gamma 2 x 1 GA(1,1) LAMBDA-X 1 X 1 1

Exercise 3 MO NX=1 NY=5 NK=1 NE=2 LX=ID LY=FU,FI C PS=SY PH=SY TD=ZE TE=SY BE=FU,FI GA=FU,FI VA 1.0 LY 1 1 LY 4 2 FR LY 2 1 LY 3 1 LY 3 2 LY 5 2 FR GA 1 1 BE 2 1

Exercise 4 This is a non-standard model.

Exercise 4 This parameter cannot be estimated in LISREL; must re-express the model (to an equivalent that CAN be estimated)

RE-EXPRESSED MODEL LAMBDA – Y BETA 1 0 0 BE(1,2) LY(2,1) 0 0 0 1 0 LY(2,1) 0 LY(3,1) 0 LY(4,1) 0 0 1 BETA 0 BE(1,2) 0 0

RE-EXPRESSED MODEL Now X1,X2 MO NY=5 NX=2 NK=1 NE=2 LY=FU,FI LX=FU,FR C GA=FU,FR PS=SY PH=SY TD=SY TE=SY VA 1.0 LX 1 1 LY 1 1 LY 5 2 FR LX 2 1 LY 2 1 LY 3 1 LY 4 1 FI TE 5 5  SINGLE INDICATOR, CANNOT ESTIMATE ERROR

Re-expressed as: e3 variance=0 Same variance as e3 in previous model Same as lambda parameter in previous model

The same sort of principle can be used for other purposes too: Imposing an inequality constraint. Example: We wish to impose a constraint such that VAR(e3) > 0 (don’t allow negative error variance).

Lambda 2, lambda 3: same parm’s Variance of ksi-2 fixed to 1.0 X3 = lambda3 KSI1 + lambda4 KSI2 VAR(X3) = lambda32*VAR(Ksi-1) + lambda42 *VAR(KSI-2) Since…..VAR(ksi-2) = 1.0 [expression lambda42 replaces VAR(e3) Regardless of estimate of lambda4, variance >0.

The LISREL PROGRAM: MO modelparameters statement FR free a parameter FI fix a parameter VA set a parameter to a value (if the parameter is free, this is the “start value” to override program default estimate; otherwise, it is the value to which a parameter is constrained

The LISREL PROGRAM: If reading in a “system” .dsf file created by prelis: Title SY= input file if LISREL .dsf DA - dataparameters SE selection of variables MO – modelparameters … various FI and FR statements OU – outputparameters (see handout)

The LISREL PROGRAM: ! Achievement Values Program #1 SY='z:\baer\Week2Examples\LISREL\Achieve1.dsf' SE REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT / MO NY=6 NE=1 LY=FU,FR PS=SY TE=SY FI LY 1 1 VA 1.0 LY 1 1 OU ME=ML SC MI SE statement lists variables to be used (always specify Y variables first) can change order on SE statement. Here, REDUCE is Y1, NEVHAPP is Y2, etc. LY 1 1 refers to REDUCE. OU : ME=ML (maximum likelihood) SC (standardized solution) MI (provide modification indices)

LISREL Output: Parameter Specifications LAMBDA-Y ETA 1 -------- REDUCE 0 NEVHAPP 1 NEW_GOAL 2 IMPROVE 3 ACHIEVE 4 CONTENT 5 PSI 6 THETA-EPS REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- 7 8 9 10 11 12 Reference indicator is “fixed” All fixed parameters represented by 0. Theta-eps is diagonal

LISREL Output LISREL Estimates (Maximum Likelihood) LAMBDA-Y ETA 1 -------- REDUCE 1.00 NEVHAPP 2.14 (0.37) 5.72 NEW_GOAL -2.76 (0.46) -6.00 IMPROVE -4.23 (0.70) -6.01 ACHIEVE -2.64 (0.45) -5.87 CONTENT 2.66 5.78

LISREL Output Covariance Matrix of ETA ETA 1 -------- 0.01 PSI (0.00) 3.08 THETA-EPS REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- 0.53 0.38 0.19 0.21 0.36 0.50 (0.01) (0.01) (0.01) (0.01) (0.01) (0.01) 38.84 36.44 28.79 18.92 34.53 35.92 Squared Multiple Correlations for Y - Variables 0.02 0.11 0.29 0.46 0.17 0.13

LISREL Output Modification Indices and Expected Change No Non-Zero Modification Indices for LAMBDA-Y No Non-Zero Modification Indices for PSI Modification Indices for THETA-EPS REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- REDUCE - - NEVHAPP 323.45 - - NEW_GOAL 24.46 4.29 - - IMPROVE 92.13 52.90 87.29 - - ACHIEVE 19.12 48.71 0.97 33.31 - - CONTENT 170.74 243.43 58.94 21.28 1.82 - - Expected Change for THETA-EPS NEVHAPP 0.15 - - NEW_GOAL 0.03 0.01 - - IMPROVE 0.08 0.06 0.10 - - ACHIEVE 0.04 0.05 0.01 0.06 - - CONTENT 0.13 0.14 0.06 0.05 0.01 - - Completely Standardized Expected Change for THETA-EPS NEVHAPP 0.32 - - NEW_GOAL 0.09 0.04 - - IMPROVE 0.18 0.15 0.29 - - ACHIEVE 0.08 0.12 0.02 0.14 - - CONTENT 0.23 0.27 0.14 0.10 0.02 - - Maximum Modification Index is 323.45 for Element ( 2, 1) of THETA-EPS

Lisrel program input SE 2 3 6 9 8 7 / If reading in a covariance matrix generated by PRELIS instead of a .dsf file: DA NO=# cases NI=# of input var’s MA=CM {MA = type of matrix to be analyzed; default = CM, or a covariance matrix} CM FI=‘c:\file1.cov’ input file specification(cov) SE 2 3 6 9 8 7 / Selection: corresponds to order in which variables located on input covariance matrix (3rd variable on the matrix is now Y2).

Another LISREL example: ! Achievement Values Program #8: Adding One Extra Measurement Model Path SY='z:\baer\Week2Examples\LISREL\Achieve1.dsf' SE REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT GENDER AGE EDUC INCOME/ MO NX=4 NK=4 NY=6 NE=2 LX=ID PH=SY,FR TD=ZE LY=FU,FI C PS=SY,FR TE=SY GA=FU,FR FI LY 2 1 FI LY 3 2 VA 1.0 LY 2 1 LY 3 2 FR LY 1 1 LY 6 1 LY 4 2 LY 5 2 FR LY 1 2 PD OU ME=ML SE TV SC MI

(from output listing) Parameter Specifications LAMBDA-Y ETA 1 ETA 2 -------- -------- REDUCE 1 2 NEVHAPP 0 0 NEW_GOAL 0 0 IMPROVE 0 3 ACHIEVE 0 4 CONTENT 5 0 GAMMA GENDER AGE EDUC INCOME -------- -------- -------- -------- ETA 1 6 7 8 9 ETA 2 10 11 12 13 PHI GENDER 14 AGE 15 16 EDUC 17 18 19 INCOME 20 21 22 23 PSI ETA 1 24 ETA 2 25 26 THETA-EPS REDUCE NEVHAPP NEW_GOAL IMPROVE ACHIEVE CONTENT -------- -------- -------- -------- -------- -------- 27 28 29 30 31 32

(output) LISREL Estimates (Maximum Likelihood) LAMBDA-Y ETA 1 ETA 2 -------- -------- REDUCE 1.13 0.65 (0.07) (0.08) 17.32 8.53 NEVHAPP 1.00 - - NEW_GOAL - - 1.00 IMPROVE - - 1.85 (0.12) 16.00 ACHIEVE - - 0.99 (0.06) 15.95 CONTENT 1.16 - - 19.84 GAMMA GENDER AGE EDUC INCOME -------- -------- -------- -------- ETA 1 0.02 -0.01 0.03 0.01 (0.02) (0.00) (0.00) (0.00) 1.14 -10.40 10.04 5.67 ETA 2 0.07 0.00 0.01 0.00 (0.01) (0.00) (0.00) (0.00) 4.90 4.81 4.19 -0.79

Squared Multiple Correlations for Structural Equations Covariance Matrix of ETA and KSI ETA 1 ETA 2 GENDER AGE EDUC INCOME -------- -------- -------- -------- -------- -------- ETA 1 0.15 ETA 2 -0.04 0.07 GENDER -0.01 0.02 0.25 AGE -2.25 0.37 -0.08 269.69 EDUC 0.53 0.06 -0.07 -18.55 13.75 INCOME 0.47 -0.08 -0.98 -15.71 5.55 20.57 Squared Multiple Correlations for Structural Equations ETA 1 ETA 2 -------- -------- 0.22 0.03

(LISREL output) Modification Indices and Expected Change Modification Indices for LAMBDA-Y ETA 1 ETA 2 -------- -------- REDUCE - - - - NEVHAPP - - 3.55 NEW_GOAL 4.90 - - IMPROVE 0.84 - - ACHIEVE 2.18 - - CONTENT - - 3.55

Completely Standardized Solution LAMBDA-Y ETA 1 ETA 2 -------- -------- REDUCE 0.59 0.24 NEVHAPP 0.59 - - NEW_GOAL - - 0.52 IMPROVE - - 0.79 ACHIEVE - - 0.41 CONTENT 0.59 - - GAMMA GENDER AGE EDUC INCOME -------- -------- -------- -------- ETA 1 0.03 -0.25 0.25 0.15 ETA 2 0.12 0.11 0.10 -0.02 (could have used LA (labels) statement to provide labels for these latent variables)

Reproduced covariances in matrix form First examples are for SEM models that are “manifest variable only” – no latent variables.

Manifest variables only

Manifest variables only

Manifest variables only Previous example had no paths connecting endogenous y-variables (no “Beta” matrix). A bit more complicated with these included:

Manifest variables only With Beta matrix:

Manifest variables only

Manifest variables only

Manifest variables only

Manifest variables only

Latent variables included Measurement model only

Latent variables included

δ

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