"Structural equation models," "covariance structure models," "confirmatory factor analysis models," "EQS," and "LISREL," are among the more common names of approaches to modeling with multiple simultaneous equations in observed and latent continuous variables. Programs for estimating models of this type directly from observations, or from moments, covariance, or correlation matrices have now become commonplace. The most common programs offer similar ranges of estimation methods and statistical output, but differ considerably in how models are specified.
In sociology, the two most common approaches to specifying models of this type are the conventions of LISREL (Jorskog and Sorbom), and the conventions of EQS (Bentler and Bonnett). The LISREL conventions have the disadvantage of being rather more difficult to understand and implement as program control statements; LISREL conventions, however, are more orderly and elegant than EQS conventions. SAS's PROC CALIS does not implement LISREL directly (although it is not difficult to translate from LISREL to CALIS); CALIS does offer (among other things) the EQS approach. The example below follows these conventions.
The materials below are rather long. Here are a set of jumps to go to the various parts of the output:
We believe that the educational aspirations of youth are positively associated with their academic ability. We are interested in using six items to assess the convergent validity and internal consistency of measures of ability and of educational aspiration. We are also interested in assessing the strength of the association between the two latent variables.
Using the confirmatory logic, we will propose that the correlation matrix of four measures of ability (SCABIL, PPEVAL, PTEVAL, and PFEVAL) and two measures of educational aspiration (EDASP and COLPLAN) can be accounted for by an oblique two factor model, where the former four items load on F1 (the latent construct supposed to be "ability") and the latter two items load on F2 (the latent construct supposed to represent "educational aspiration"). We will estimate this model by maximum likelihood, assess it's goodness of fit, and examine possible modifications of the model.
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options ls=80 ps=60 nocenter compress=yes ;
data aspire (type=corr) ; input _type_ $ _name_ $ scabil ppeval pteval pfeval edasp colplan ; cards; std . 1.0 1.0 1.0 1.0 1.0 1.0 corr scabil 1.0 .73 .70 .58 .46 .56 corr ppeval .73 1.0 .68 .61 .43 .52 corr pteval .70 .68 1.0 .57 .40 .48 corr pfeval .58 .61 .57 1.0 .37 .41 corr edasp .46 .43 .40 .37 1.0 .72 corr colplan .56 .52 .48 .41 .72 1.0 ; proc calis corr modification pestim privec toteff se ; lineqs scabil=1.0 f1 +e1, ppeval=lambx2 (.5) f1 +e2, pteval=lambx3 (.5) f1 +e3, pfeval=lambx4 (.5) f1 +e4, edasp=1.0 f2 +e5, colplan=lamby2 (.5) f2 + e6 ; std f1 = phi1 (.6), f2 = phi2 (.6), e1 = te1 (.5), e2 = te2 (.5), e3 = te3 (.5), e4 = te4 (.5), e5 = te5 (.5), e6 = te6 (.5) ; cov f1 f2 = oblique (.3) ; run;
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Covariance Structure Analysis: Pattern and Initial Values
LINEQS Model Statement
-------------------------------
Matrix Rows & Cols Matrix Type
TERM 1----------------------------------------------------
1 _SEL_ 6 14 SELECTION
2 _BETA_ 14 14 EQSBETA IMINUSINV
3 _GAMMA_ 14 8 EQSGAMMA
4 _PHI_ 8 8 SYMMETRIC
Number of endogenous variables = 6
Manifest: SCABIL PPEVAL PTEVAL PFEVAL EDASP COLPLAN
Number of exogenous variables = 8
Latent: F1 F2
Error: E1 E2 E3 E4 E5 E6
Covariance Structure Analysis: Pattern and Initial Values
Manifest Variable Equations
Initial Estimates
SCABIL = 1.0000 F1 + 1.0000 E1
PPEVAL = 0.5000*F1 + 1.0000 E2
LAMBX2
PTEVAL = 0.5000*F1 + 1.0000 E3
LAMBX3
PFEVAL = 0.5000*F1 + 1.0000 E4
LAMBX4
EDASP = 1.0000 F2 + 1.0000 E5
COLPLAN = 0.5000*F2 + 1.0000 E6
LAMBY2
Variances of Exogenous Variables
-------------------------------------
Variable Parameter Estimate
-------------------------------------
F1 PHI1 0.600000
F2 PHI2 0.600000
E1 TE1 0.500000
E2 TE2 0.500000
E3 TE3 0.500000
E4 TE4 0.500000
E5 TE5 0.500000
E6 TE6 0.500000
Covariances among Exogenous Variables
---------------------------------------------
Parameter Estimate
---------------------------------------------
F2 F1 OBLIQUE 0.300000
10000 Observations Model Terms 1
6 Variables Model Matrices 4
21 Informations Parameters 13
VARIABLE Mean Std Dev
SCABIL 0 1.000000000
PPEVAL 0 1.000000000
PTEVAL 0 1.000000000
PFEVAL 0 1.000000000
EDASP 0 1.000000000
COLPLAN 0 1.000000000
Correlations
SCABIL PPEVAL PTEVAL PFEVAL EDASP COLPLAN
SCABIL 1.0000 0.7300 0.7000 0.5800 0.4600 0.5600
PPEVAL 0.7300 1.0000 0.6800 0.6100 0.4300 0.5200
PTEVAL 0.7000 0.6800 1.0000 0.5700 0.4000 0.4800
PFEVAL 0.5800 0.6100 0.5700 1.0000 0.3700 0.4100
EDASP 0.4600 0.4300 0.4000 0.3700 1.0000 0.7200
COLPLAN 0.5600 0.5200 0.4800 0.4100 0.7200 1.0000
Determinant = 0.03685 (Ln = -3.301)
Vector of Initial Estimates
LAMBX2 1 0.50000 Matrix Entry: _GAMMA_[2:1]
LAMBX3 2 0.50000 Matrix Entry: _GAMMA_[3:1]
LAMBX4 3 0.50000 Matrix Entry: _GAMMA_[4:1]
LAMBY2 4 0.50000 Matrix Entry: _GAMMA_[6:2]
PHI1 5 0.60000 Matrix Entry: _PHI_[1:1]
OBLIQUE 6 0.30000 Matrix Entry: _PHI_[2:1]
PHI2 7 0.60000 Matrix Entry: _PHI_[2:2]
TE1 8 0.50000 Matrix Entry: _PHI_[3:3]
TE2 9 0.50000 Matrix Entry: _PHI_[4:4]
TE3 10 0.50000 Matrix Entry: _PHI_[5:5]
TE4 11 0.50000 Matrix Entry: _PHI_[6:6]
TE5 12 0.50000 Matrix Entry: _PHI_[7:7]
TE6 13 0.50000 Matrix Entry: _PHI_[8:8]
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Covariance Structure Analysis: Maximum Likelihood Estimation
Levenberg-Marquardt Minimization
Algorithm for Hessian= 11
Maximum Iterations= 50
Maximum Function Calls= 125
Maximum Absolute Gradient Criterion= 0.001
Number of Estimates= 13 Lower Bounds= 0 Upper Bounds= 0
Minimization Start: Active Constraints= 0 Criterion= 1.315
Maximum Gradient Element= 1.117 Radius= 3.055
Iter nfun act mincrit maxgrad difcrit lambda rhoratio
1 3 0 0.36202 0.5850 0.9531 1.0946 0.8486
2 4 0 0.03544 0.2458 0.3266 0 1.2003
3 5 0 0.01672 0.0129 0.0187 0 1.0889
4 6 0 0.01668 0.000336 0.000039 0 0.9950
Minimization Results: Iterations= 4 Function Calls= 6 Derivative Calls= 5
Active Constraints= 0 Criterion= 0.0167 Maximum Gradient Element= 0.000336
Radius= 0.00237
NOTE: Convergence criterion satisfied.
Predicted Model Matrix
SCABIL PPEVAL PTEVAL PFEVAL EDASP COLPLAN
SCABIL 1.0000 0.7332 0.6949 0.6001 0.4458 0.5343
PPEVAL 0.7332 1.0000 0.6838 0.5905 0.4387 0.5258
PTEVAL 0.6949 0.6838 1.0000 0.5597 0.4158 0.4983
PFEVAL 0.6001 0.5905 0.5597 1.0000 0.3591 0.4304
EDASP 0.4458 0.4387 0.4158 0.3591 1.0000 0.7200
COLPLAN 0.5343 0.5258 0.4983 0.4304 0.7200 1.0000
Determinant = 0.03747 (Ln = -3.284)
The SAS System 13:45 Friday, April 19, 1996 21
Covariance Structure Analysis: Maximum Likelihood Estimation
Fit criterion . . . . . . . . . . . . . . . . . . 0.0167
Goodness of Fit Index (GFI) . . . . . . . . . . . 0.9944
GFI Adjusted for Degrees of Freedom (AGFI). . . . 0.9854
Root Mean Square Residual (RMR) . . . . . . . . . 0.0120
Parsimonious GFI (Mulaik, 1989) . . . . . . . . . 0.5304
Chi-square = 166.7535 df = 8 Prob>chi**2 = 0.0001
Null Model Chi-square: df = 15 33005.3170
RMSEA Estimate . . . . . . 0.0445 90%C.I.[0.0388, 0.0506]
Probability of Close Fit . . . . . . . . . . . . 0.9315
ECVI Estimate . . . . . . . 0.0193 90%C.I.[0.0154, 0.0239]
Bentler's Comparative Fit Index . . . . . . . . . 0.9952
Normal Theory Reweighted LS Chi-square . . . . . 168.1534
Akaike's Information Criterion. . . . . . . . . . 150.7535
Bozdogan's (1987) CAIC. . . . . . . . . . . . . . 85.0708
Schwarz's Bayesian Criterion. . . . . . . . . . . 93.0708
McDonald's (1989) Centrality. . . . . . . . . . . 0.9921
Bentler & Bonett's (1980) Non-normed Index. . . . 0.9910
Bentler & Bonett's (1980) NFI . . . . . . . . . . 0.9949
James, Mulaik, & Brett (1982) Parsimonious NFI. . 0.5306
Z-Test of Wilson & Hilferty (1931). . . . . . . . 10.6792
Bollen (1986) Normed Index Rho1 . . . . . . . . . 0.9905
Bollen (1988) Non-normed Index Delta2 . . . . . . 0.9952
Hoelter's (1983) Critical N . . . . . . . . . . . 931
Vector of Estimates
LAMBX2 1 0.983980 Matrix Entry: _GAMMA_[2:1]
LAMBX3 2 0.932675 Matrix Entry: _GAMMA_[3:1]
LAMBX4 3 0.805434 Matrix Entry: _GAMMA_[4:1]
LAMBY2 4 1.198434 Matrix Entry: _GAMMA_[6:2]
PHI1 5 0.745096 Matrix Entry: _PHI_[1:1]
OBLIQUE 6 0.445848 Matrix Entry: _PHI_[2:1]
PHI2 7 0.600784 Matrix Entry: _PHI_[2:2]
TE1 8 0.254904 Matrix Entry: _PHI_[3:3]
TE2 9 0.278586 Matrix Entry: _PHI_[4:4]
TE3 10 0.351855 Matrix Entry: _PHI_[5:5]
TE4 11 0.516639 Matrix Entry: _PHI_[6:6]
TE5 12 0.399216 Matrix Entry: _PHI_[7:7]
TE6 13 0.137127 Matrix Entry: _PHI_[8:8]
Vector of Standard Errors
LAMBX2 1 0.009543 Matrix Entry: _GAMMA_[2:1]
LAMBX3 2 0.009753 Matrix Entry: _GAMMA_[3:1]
LAMBX4 3 0.010385 Matrix Entry: _GAMMA_[4:1]
LAMBY2 4 0.017276 Matrix Entry: _GAMMA_[6:2]
PHI1 5 0.014295 Matrix Entry: _PHI_[1:1]
OBLIQUE 6 0.010441 Matrix Entry: _PHI_[2:1]
PHI2 7 0.014739 Matrix Entry: _PHI_[2:2]
TE1 8 0.005505 Matrix Entry: _PHI_[3:3]
TE2 9 0.005684 Matrix Entry: _PHI_[4:4]
TE3 10 0.006342 Matrix Entry: _PHI_[5:5]
TE4 11 0.008182 Matrix Entry: _PHI_[6:6]
TE5 12 0.008999 Matrix Entry: _PHI_[7:7]
TE6 13 0.010249 Matrix Entry: _PHI_[8:8]
Vector of t Values
LAMBX2 1 103.107045 Matrix Entry: _GAMMA_[2:1]
LAMBX3 2 95.631070 Matrix Entry: _GAMMA_[3:1]
LAMBX4 3 77.553979 Matrix Entry: _GAMMA_[4:1]
LAMBY2 4 69.368459 Matrix Entry: _GAMMA_[6:2]
PHI1 5 52.124216 Matrix Entry: _PHI_[1:1]
OBLIQUE 6 42.700519 Matrix Entry: _PHI_[2:1]
PHI2 7 40.761360 Matrix Entry: _PHI_[2:2]
TE1 8 46.300873 Matrix Entry: _PHI_[3:3]
TE2 9 49.013982 Matrix Entry: _PHI_[4:4]
TE3 10 55.479198 Matrix Entry: _PHI_[5:5]
TE4 11 63.146765 Matrix Entry: _PHI_[6:6]
TE5 12 44.364128 Matrix Entry: _PHI_[7:7]
TE6 13 13.379712 Matrix Entry: _PHI_[8:8]
First Derivatives (Gradient)
LAMBX2 1 0.000095566 Matrix Entry: _GAMMA_[2:1]
LAMBX3 2 -0.000019265 Matrix Entry: _GAMMA_[3:1]
LAMBX4 3 -0.000065882 Matrix Entry: _GAMMA_[4:1]
LAMBY2 4 -0.000010529 Matrix Entry: _GAMMA_[6:2]
PHI1 5 -0.000012008 Matrix Entry: _PHI_[1:1]
OBLIQUE 6 0.000029413 Matrix Entry: _PHI_[2:1]
PHI2 7 -0.000015056 Matrix Entry: _PHI_[2:2]
TE1 8 0.000107 Matrix Entry: _PHI_[3:3]
TE2 9 -0.000336 Matrix Entry: _PHI_[4:4]
TE3 10 0.000063012 Matrix Entry: _PHI_[5:5]
TE4 11 0.000105 Matrix Entry: _PHI_[6:6]
TE5 12 -0.000019497 Matrix Entry: _PHI_[7:7]
TE6 13 0.000048396 Matrix Entry: _PHI_[8:8]
Manifest Variable Equations
SCABIL = 1.0000 F1 + 1.0000 E1
PPEVAL = 0.9840*F1 + 1.0000 E2
Std Err 0.0095 LAMBX2
t Value 103.1070
PTEVAL = 0.9327*F1 + 1.0000 E3
Std Err 0.0098 LAMBX3
t Value 95.6311
PFEVAL = 0.8054*F1 + 1.0000 E4
Std Err 0.0104 LAMBX4
t Value 77.5540
EDASP = 1.0000 F2 + 1.0000 E5
COLPLAN = 1.1984*F2 + 1.0000 E6
Std Err 0.0173 LAMBY2
t Value 69.3685
Variances of Exogenous Variables
---------------------------------------------------------------------
Standard
Variable Parameter Estimate Error t Value
---------------------------------------------------------------------
F1 PHI1 0.745096 0.014295 52.124
F2 PHI2 0.600784 0.014739 40.761
E1 TE1 0.254904 0.005505 46.301
E2 TE2 0.278586 0.005684 49.014
E3 TE3 0.351855 0.006342 55.479
E4 TE4 0.516639 0.008182 63.147
E5 TE5 0.399216 0.008999 44.364
E6 TE6 0.137127 0.010249 13.380
Covariances among Exogenous Variables
-----------------------------------------------------------------------------
Standard
Parameter Estimate Error t Value
-----------------------------------------------------------------------------
F2 F1 OBLIQUE 0.445848 0.010441 42.701
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Equations with Standardized Coefficients
SCABIL = 0.8632 F1 + 0.5049 E1
PPEVAL = 0.8494*F1 + 0.5278 E2
LAMBX2
PTEVAL = 0.8051*F1 + 0.5932 E3
LAMBX3
PFEVAL = 0.6952*F1 + 0.7188 E4
LAMBX4
EDASP = 0.7751 F2 + 0.6318 E5
COLPLAN = 0.9289*F2 + 0.3703 E6
LAMBY2
Variances of Endogenous Variables
-----------------------------------------------
Variable Estimate R-squared
-----------------------------------------------
1 SCABIL 1.000000 0.745096
2 PPEVAL 1.000000 0.721414
3 PTEVAL 1.000002 0.648145
4 PFEVAL 1.000001 0.483361
5 EDASP 1.000000 0.600784
6 COLPLAN 0.999999 0.862873
Correlations among Exogenous Variables
---------------------------------------------
Parameter Estimate
---------------------------------------------
F2 F1 OBLIQUE 0.666380
Total Effects of Exogenous on Endogenous Variables
F1 F2
SCABIL 1.0000 0.0000
PPEVAL 0.9840 0.0000
PTEVAL 0.9327 0.0000
PFEVAL 0.8054 0.0000
EDASP 0.0000 1.0000
COLPLAN 0.0000 1.1984
Indirect Effects of Exogenous on Endogenous Variables
F1 F2
SCABIL 0 0
PPEVAL 0 0
PTEVAL 0 0
PFEVAL 0 0
EDASP 0 0
COLPLAN 0 0
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Lagrange Multiplier and Wald Test Indices _PHI_[8:8]
Symmetric Matrix
Univariate Tests for Constant Constraints
------------------------------------------
| Lagrange Multiplier or Wald Index |
------------------------------------------
| Probability | Approx Change of Value |
------------------------------------------
F1 F2 E1
F1 2716.934[PHI1 ] 1823.334[OBLIQUE] 72.185
0.000 -0.063
F2 1823.334[OBLIQUE] 1661.488[PHI2 ] 72.209
0.000 0.038
E1 72.185 72.209 2143.771[TE1 ]
0.000 -0.063 0.000 0.038
E2 4.687 4.708 6.770
0.030 0.016 0.030 -0.010 0.009 -0.015
E3 24.372 24.362 9.783
0.000 0.038 0.000 -0.023 0.002 0.017
E4 7.964 7.942 65.771
0.005 0.024 0.005 -0.015 0.000 -0.043
E5 SING SING 0.804
. . . . 0.370 -0.004
E6 SING SING 53.217
. . . . 0.000 0.030
Lagrange Multiplier and Wald Test Indices _PHI_[8:8]
Symmetric Matrix
Univariate Tests for Constant Constraints (Contd.)
E2 E3 E4
F1 4.687 24.372 7.964
0.030 0.016 0.000 0.038 0.005 0.024
F2 4.708 24.362 7.942
0.030 -0.010 0.000 -0.023 0.005 -0.015
E1 6.770 9.783 65.771
0.009 -0.015 0.002 0.017 0.000 -0.043
E2 2402.370[TE2 ] 4.476 51.660
0.034 -0.012 0.000 0.038
E3 4.476 3077.941[TE3 ] 9.328
0.034 -0.012 0.002 0.017
E4 51.660 9.328 3987.514[TE4 ]
0.000 0.038 0.002 0.017
E5 2.721 1.187 26.924
0.099 -0.007 0.276 -0.005 0.000 0.027
E6 0.145 9.197 39.671
0.703 -0.002 0.002 -0.013 0.000 -0.031
E5 E6
F1 SING SING
. . . .
F2 SING SING
. . . .
E1 0.804 53.217
0.370 -0.004 0.000 0.030
E2 2.721 0.145
0.099 -0.007 0.703 -0.002
E3 1.187 9.197
0.276 -0.005 0.002 -0.013
E4 26.924 39.671
0.000 0.027 0.000 -0.031
E5 1968.176[TE5 ] SING
. .
E6 SING 179.017[TE6 ]
. .
Rank order of 10 largest Lagrange multipliers in _PHI_
E1 : F2 E1 : F1 E4 : E1
72.2088 : 0.000 72.1852 : 0.000 65.7715 : 0.000
E6 : E1 E4 : E2 E6 : E4
53.2172 : 0.000 51.6598 : 0.000 39.6706 : 0.000
E5 : E4 E3 : F1 E3 : F2
26.9236 : 0.000 24.3718 : 0.000 24.3620 : 0.000
E3 : E1
9.7831 : 0.002
Lagrange Multiplier and Wald Test Indices _GAMMA_[6:2]
General Matrix
Univariate Tests for Constant Constraints
------------------------------------------
| Lagrange Multiplier or Wald Index |
------------------------------------------
| Probability | Approx Change of Value |
------------------------------------------
F1 F2
SCABIL SING 72.228
. . 0.000 0.113
PPEVAL 10631.063[LAMBX2] 4.725
0.030 -0.029
PTEVAL 9145.301[LAMBX3] 24.354
0.000 -0.068
PFEVAL 6014.620[LAMBX4] 7.925
0.005 -0.043
EDASP SING SING
. . . .
COLPLAN SING 4811.983[LAMBY2]
. .
Rank order of 4 largest Lagrange multipliers in _GAMMA_
SCABIL : F2 PTEVAL : F2 PFEVAL : F2
72.2276 : 0.000 24.3541 : 0.000 7.9251 : 0.005
PPEVAL : F2
4.7247 : 0.030
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