Sociology 203B
Structural Equation Models with SAS PROC CALIS

This course is offered in the spring quarter of 2006-07  by Robert A. Hanneman of the Department of Sociology at the University of California, Riverside. When the course is in session, announcements, discussion groups, and other features may be found on the U.C.R. instructional web site. Your comments and suggestions are welcome by email to the instructor.
Introduction

"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.


Table of Contents

The materials below are rather long. Here are a set of jumps to go to the various parts of the output:


The problem:

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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The program:

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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Output step 1: Checking the model specification:

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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Output step 2: Initial and raw form parameter estimates:

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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Output step 3: Standardized estimates:

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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Output step 4: Global fit and modification indicies:

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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