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The CALIS Procedure

Example 19.1: Path Analysis: Stability of Alienation

The following covariance matrix from Wheaton, Muthen, Alwin, and Summers (1977) has served to illustrate the performance of several implementations for the analysis of structural equation models. Two different models have been analyzed by an early implementation of LISREL and are mentioned in J\ddot{o}reskog (1978). You also can find a more detailed discussion of these models in the LISREL VI manual (J\ddot{o}reskog and S\ddot{o}rbom 1985). A slightly modified model for this covariance matrix is included in the EQS 2.0 manual (Bentler 1985, p. 28). The path diagram of this model is displayed in Figure 19.1. The same model is reanalyzed here by PROC CALIS. However, for the analysis with the EQS implementation, the last variable (V6) is rescaled by a factor of 0.1 to make the matrix less ill-conditioned. Since the Levenberg-Marquardt or Newton-Raphson optimization techniques are used with PROC CALIS, rescaling the data matrix is not necessary and, therefore, is not done here. The results reported here reflect the estimates based on the original covariance matrix.

   data Wheaton(TYPE=COV);                                             
   title "Stability of Alienation";                                    
   title2 "Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)";    
      _type_ = 'cov'; input _name_ $ v1-v6;                            
      label v1='Anomia (1967)' v2='Anomia (1971)' v3='Education'       
            v4='Powerlessness (1967)' v5='Powerlessness (1971)'        
            v6='Occupational Status Index';                            
      datalines;                                                       
   v1   11.834     .        .        .       .        .                
   v2    6.947    9.364     .        .       .        .                
   v3    6.819    5.091   12.532     .       .        .                
   v4    4.783    5.028    7.495    9.986    .        .                
   v5   -3.839   -3.889   -3.841   -3.625   9.610     .                
   v6  -21.899  -18.831  -21.748  -18.775  35.522  450.288             
   ;                                                                   
                                                                       
   proc calis cov data=Wheaton tech=nr edf=931 pall;                   
      Lineqs                                                             
         V1 =         F1                  + E1,                        
         V2 =    .833 F1                  + E2,                        
         V3 =         F2                  + E3,                        
         V4 =    .833 F2                  + E4,                        
         V5 =         F3                  + E5,                        
         V6 = Lamb (.5) F3                + E6,                        
         F1 = Gam1(-.5) F3                + D1,                        
         F2 = Beta (.5) F1 + Gam2(-.5) F3 + D2;                        
      Std                                                                
         E1-E6 = The1-The2 The1-The4 (6 * 3.),                         
         D1-D2 = Psi1-Psi2 (2 * 4.),                                   
         F3    = Phi (6.) ;                                            
      Cov                                                                
         E1 E3 = The5 (.2),                                            
         E4 E2 = The5 (.2);                                            
   run;

The COV option in the PROC CALIS statement requests the analysis of the covariance matrix. Without the COV option, the correlation matrix would be computed and analyzed. Since no METHOD= option has been used, maximum likelihood estimates are computed by default. The TECH=NR option requests the Newton-Raphson optimization method. The PALL option produces the almost complete set of displayed output, as displayed in Output 19.1.1 through Output 19.1.11. Note that, when you specify the PALL option, you can produce large amounts of output. The PALL option is used in this example to show how you can get a wide spectrum of useful information from PROC CALIS.

Output 19.1.1 displays the model specification in matrix terms, followed by the lists of endogenous and exogenous variables. Equations and initial parameter estimates are also displayed. You can use this information to ensure that the desired model is the model being analyzed.

Output 19.1.1: Model Specification
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values

LINEQS Model Statement
  Matrix Rows Columns Matrix Type
Term 1 1 _SEL_ 6 17 SELECTION  
  2 _BETA_ 17 17 EQSBETA IMINUSINV
  3 _GAMMA_ 17 9 EQSGAMMA  
  4 _PHI_ 9 9 SYMMETRIC  
 
The 8 Endogenous Variables
Manifest v1 v2 v3 v4 v5 v6
Latent F1 F2
 
The 9 Exogenous Variables
Manifest  
Latent F3
Error E1 E2 E3 E4 E5 E6 D1 D2

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values

v1 = 1.0000   F1 + 1.0000   E1
v2 = 0.8330   F1 + 1.0000   E2
v3 = 1.0000   F2 + 1.0000   E3
v4 = 0.8330   F2 + 1.0000   E4
v5 = 1.0000   F3 + 1.0000   E5
v6 = 0.5000 * F3 + 1.0000   E6
        Lamb        

 


 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values

F1 = -0.5000 * F3 + 1.0000   D1        
        Gam1                
F2 = 0.5000 * F1 + -0.5000 * F3 + 1.0000   D2
        Beta       Gam2        

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Pattern and Initial Values

Variances of Exogenous
Variables
Variable Parameter Estimate
F3 Phi 6.00000
E1 The1 3.00000
E2 The2 3.00000
E3 The1 3.00000
E4 The2 3.00000
E5 The3 3.00000
E6 The4 3.00000
D1 Psi1 4.00000
D2 Psi2 4.00000
 
Covariances Among Exogenous
Variables
Var1 Var2 Parameter Estimate
E1 E3 The5 0.20000
E2 E4 The5 0.20000

General modeling information and simple descriptive statistics are displayed in Output 19.1.2. Because the input data set contains only the covariance matrix, the means of the manifest variables are assumed to be zero. Note that this has no impact on the estimation, unless a mean structure model is being analyzed. The twelve parameter estimates in the model and their respective locations in the parameter matrices are also displayed. Each of the parameters, The1, The2, and The5, is specified for two elements in the parameter matrix _PHI_.

Output 19.1.2: Modeling Information, Simple Statistics and Parameter Vector
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Observations 932 Model Terms 1
Variables 6 Model Matrices 4
Informations 21 Parameters 12
 
Variable Mean Std Dev
v1 Anomia (1967) 0 3.44006
v2 Anomia (1971) 0 3.06007
v3 Education 0 3.54006
v4 Powerlessness (1967) 0 3.16006
v5 Powerlessness (1971) 0 3.10000
v6 Occupational Status Index 0 21.21999

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Covariances
  v1 v2 v3 v4 v5 v6
v1 Anomia (1967) 11.83400000 6.94700000 6.81900000 4.78300000 -3.83900000 -21.8990000
v2 Anomia (1971) 6.94700000 9.36400000 5.09100000 5.02800000 -3.88900000 -18.8310000
v3 Education 6.81900000 5.09100000 12.53200000 7.49500000 -3.84100000 -21.7480000
v4 Powerlessness (1967) 4.78300000 5.02800000 7.49500000 9.98600000 -3.62500000 -18.7750000
v5 Powerlessness (1971) -3.83900000 -3.88900000 -3.84100000 -3.62500000 9.61000000 35.5220000
v6 Occupational Status Index -21.89900000 -18.83100000 -21.74800000 -18.77500000 35.52200000 450.2880000
 
Determinant 6080570 Ln 15.620609

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Vector of Initial Estimates
  Parameter Estimate Type
1 Beta 0.50000 Matrix Entry: _BETA_[8:7]
2 Lamb 0.50000 Matrix Entry: _GAMMA_[6:1]
3 Gam1 -0.50000 Matrix Entry: _GAMMA_[7:1]
4 Gam2 -0.50000 Matrix Entry: _GAMMA_[8:1]
5 Phi 6.00000 Matrix Entry: _PHI_[1:1]
6 The1 3.00000 Matrix Entry: _PHI_[2:2] _PHI_[4:4]
7 The2 3.00000 Matrix Entry: _PHI_[3:3] _PHI_[5:5]
8 The5 0.20000 Matrix Entry: _PHI_[4:2] _PHI_[5:3]
9 The3 3.00000 Matrix Entry: _PHI_[6:6]
10 The4 3.00000 Matrix Entry: _PHI_[7:7]
11 Psi1 4.00000 Matrix Entry: _PHI_[8:8]
12 Psi2 4.00000 Matrix Entry: _PHI_[9:9]

PROC CALIS examines whether each element in the moment matrix is modeled by the parameters defined in the model. If an element is not structured by the model parameters, it is predetermined by its observed value. This occurs, for example, when there are exogenous manifest variables in the model. If present, the predetermined values of the elements will be displayed. In the current example, the `.' displayed for all elements in the predicted moment matrix (Output 19.1.3) indicates that there are no predetermined elements in the model.

Output 19.1.3: Predetermined Elements
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Predetermined Elements of the Predicted Moment Matrix
  v1 v2 v3 v4 v5 v6
v1 Anomia (1967) . . . . . .
v2 Anomia (1971) . . . . . .
v3 Education . . . . . .
v4 Powerlessness (1967) . . . . . .
v5 Powerlessness (1971) . . . . . .
v6 Occupational Status Index . . . . . .
 
Sum of Squared Differences 0

Output 19.1.4 displays the optimization information. You can check this table to determine whether the convergence criterion is satisfied. PROC CALIS displays an error message when problematic solutions are encountered.

Output 19.1.4: Optimization
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Parameter Estimates 12
Functions (Observations) 21
 
Optimization Start
Active Constraints 0 Objective Function 119.33282242
Max Abs Gradient Element 74.016932345    
 
Iteration   Restarts Function
Calls
Active
Constraints
  Objective
Function
Objective
Function
Change
Max Abs
Gradient
Element
Ridge Ratio
Between
Actual
and
Predicted
Change
1   0 2 0   0.82689 118.5 1.3507 0 0.0154
2   0 3 0   0.09859 0.7283 0.2330 0 0.716
3   0 4 0   0.01581 0.0828 0.00684 0 1.285
4   0 5 0   0.01449 0.00132 0.000286 0 1.042
5   0 6 0   0.01448 9.936E-7 0.000045 0 1.053
6   0 7 0   0.01448 4.227E-9 1.685E-6 0 1.056
 
Optimization Results
Iterations 6 Function Calls 8
Jacobian Calls 7 Active Constraints 0
Objective Function 0.0144844811 Max Abs Gradient Element 1.6847829E-6
Ridge 0 Actual Over Pred Change 1.0563187228
 
ABSGCONV convergence criterion satisfied.

The predicted model matrix is displayed next, followed by a list of model test statistics or fit indices (Output 19.1.5). Depending on your modeling philosophy, some indices may be preferred to others. In this example, all indices and test statistics point to a good fit of the model.

Output 19.1.5: Predicted Model Matrix and Fit Statistics
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Predicted Model Matrix
  v1 v2 v3 v4 v5 v6
v1 Anomia (1967) 11.90390632 6.91059048 6.83016211 4.93499582 -4.16791157 -22.3768816
v2 Anomia (1971) 6.91059048 9.35145064 4.93499582 5.01664889 -3.47187034 -18.6399424
v3 Education 6.83016211 4.93499582 12.61574998 7.50355625 -4.06565606 -21.8278873
v4 Powerlessness (1967) 4.93499582 5.01664889 7.50355625 9.84539112 -3.38669150 -18.1826302
v5 Powerlessness (1971) -4.16791157 -3.47187034 -4.06565606 -3.38669150 9.61000000 35.5219999
v6 Occupational Status Index -22.37688158 -18.63994236 -21.82788734 -18.18263015 35.52199986 450.2879993
 
Determinant 6169285 Ln 15.635094

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0145
Goodness of Fit Index (GFI) 0.9953
GFI Adjusted for Degrees of Freedom (AGFI) 0.9890
Root Mean Square Residual (RMR) 0.2281
Parsimonious GFI (Mulaik, 1989) 0.5972
Chi-Square 13.4851
Chi-Square DF 9
Pr > Chi-Square 0.1419
Independence Model Chi-Square 2131.4
Independence Model Chi-Square DF 15
RMSEA Estimate 0.0231
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0470
ECVI Estimate 0.0405
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.0556
Probability of Close Fit 0.9705
Bentler's Comparative Fit Index 0.9979
Normal Theory Reweighted LS Chi-Square 13.2804
Akaike's Information Criterion -4.5149
Bozdogan's (1987) CAIC -57.0509
Schwarz's Bayesian Criterion -48.0509
McDonald's (1989) Centrality 0.9976
Bentler & Bonett's (1980) Non-normed Index 0.9965
Bentler & Bonett's (1980) NFI 0.9937
James, Mulaik, & Brett (1982) Parsimonious NFI 0.5962
Z-Test of Wilson & Hilferty (1931) 1.0754
Bollen (1986) Normed Index Rho1 0.9895
Bollen (1988) Non-normed Index Delta2 0.9979
Hoelter's (1983) Critical N 1170

PROC CALIS can perform a detailed residual analysis. Large residuals may indicate misspecification of the model. In Output 19.1.6 for example, note the table for the 10 largest asymptotically standardized residuals. As the table shows, the specified model performs the poorest concerning the variable V5 and its covariance with V2, V1, and V3. This may be the result of a misspecification of the model equation for V5. However, because the model fit is quite good, such a possible misspecification may have no practical significance and is not a serious concern in the analysis.

Output 19.1.6: Residual Analysis
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Raw Residual Matrix
  v1 v2 v3 v4 v5 v6
v1 Anomia (1967) -.0699063150 0.0364095216 -.0111621061 -.1519958205 0.3289115712 0.4778815840
v2 Anomia (1971) 0.0364095216 0.0125493646 0.1560041795 0.0113511059 -.4171296612 -.1910576405
v3 Education -.0111621061 0.1560041795 -.0837499788 -.0085562504 0.2246560598 0.0798873380
v4 Powerlessness (1967) -.1519958205 0.0113511059 -.0085562504 0.1406088766 -.2383085022 -.5923698474
v5 Powerlessness (1971) 0.3289115712 -.4171296612 0.2246560598 -.2383085022 0.0000000000 0.0000000000
v6 Occupational Status Index 0.4778815840 -.1910576405 0.0798873380 -.5923698474 0.0000000000 0.0000000000
 
Average Absolute Residual 0.153928
Average Off-diagonal Absolute Residual 0.195045

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Rank Order of the 10 Largest Raw Residuals
Row Column Residual
v6 v4 -0.59237
v6 v1 0.47788
v5 v2 -0.41713
v5 v1 0.32891
v5 v4 -0.23831
v5 v3 0.22466
v6 v2 -0.19106
v3 v2 0.15600
v4 v1 -0.15200
v4 v4 0.14061

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Asymptotically Standardized Residual Matrix
  v1 v2 v3 v4 v5 v6
v1 Anomia (1967) -0.308548787 0.526654452 -0.056188826 -0.865070455 2.553366366 0.464866661
v2 Anomia (1971) 0.526654452 0.054363484 0.876120855 0.057354415 -2.763708659 -0.170127806
v3 Education -0.056188826 0.876120855 -0.354347092 -0.121874301 1.697931678 0.070202664
v4 Powerlessness (1967) -0.865070455 0.057354415 -0.121874301 0.584930625 -1.557412695 -0.495982427
v5 Powerlessness (1971) 2.553366366 -2.763708659 1.697931678 -1.557412695 0.000000000 0.000000000
v6 Occupational Status Index 0.464866661 -0.170127806 0.070202664 -0.495982427 0.000000000 0.000000000
 
Average Standardized Residual 0.646622
Average Off-diagonal Standardized Residual 0.818457

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Rank Order of the 10 Largest Asymptotically Standardized Residuals
Row Column Residual
v5 v2 -2.76371
v5 v1 2.55337
v5 v3 1.69793
v5 v4 -1.55741
v3 v2 0.87612
v4 v1 -0.86507
v4 v4 0.58493
v2 v1 0.52665
v6 v4 -0.49598
v6 v1 0.46487

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Distribution of Asymptotically Standardized Residuals
Each * Represents 1 Residuals
Range Freq Percent  
-3.00000 -2.75000 1 4.76 *
-2.75000 -2.50000 0 0.00  
-2.50000 -2.25000 0 0.00  
-2.25000 -2.00000 0 0.00  
-2.00000 -1.75000 0 0.00  
-1.75000 -1.50000 1 4.76 *
-1.50000 -1.25000 0 0.00  
-1.25000 -1.00000 0 0.00  
-1.00000 -0.75000 1 4.76 *
-0.75000 -0.50000 0 0.00  
-0.50000 -0.25000 3 14.29 ***
-0.25000 0 3 14.29 ***
0 0.25000 6 28.57 ******
0.25000 0.50000 1 4.76 *
0.50000 0.75000 2 9.52 **
0.75000 1.00000 1 4.76 *
1.00000 1.25000 0 0.00  
1.25000 1.50000 0 0.00  
1.50000 1.75000 1 4.76 *
1.75000 2.00000 0 0.00  
2.00000 2.25000 0 0.00  
2.25000 2.50000 0 0.00  
2.50000 2.75000 1 4.76 *

Output 19.1.7 displays the equations and parameter estimates. Each parameter estimate is displayed with its standard error and the corresponding t ratio. As a general rule, a t ratio larger than 2 represents a statistically significant departure from 0. From these results, it is observed that both F1 (Alienation 1967) and F2 (Alienation 1971) are regressed negatively on F3 (Socioeconomic Status), and F1 has a positive effect on F2. The estimates and significance tests for the variance and covariance of the exogenous variables are also displayed.

Output 19.1.7: Equations and Parameter Estimates
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

v1 = 1.0000   F1 + 1.0000   E1
v2 = 0.8330   F1 + 1.0000   E2
v3 = 1.0000   F2 + 1.0000   E3
v4 = 0.8330   F2 + 1.0000   E4
v5 = 1.0000   F3 + 1.0000   E5
v6 = 5.3688 * F3 + 1.0000   E6
Std Err   0.4337   Lamb        
t Value   12.3788            

 


 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

F1 = -0.6299 * F3 + 1.0000   D1        
Std Err   0.0563   Gam1                
t Value   -11.1809                    
F2 = 0.5931 * F1 + -0.2409 * F3 + 1.0000   D2
Std Err   0.0468   Beta   0.0549   Gam2        
t Value   12.6788       -4.3885            

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable Parameter Estimate Standard
Error
t Value
F3 Phi 6.61632 0.63914 10.35
E1 The1 3.60788 0.20092 17.96
E2 The2 3.59493 0.16448 21.86
E3 The1 3.60788 0.20092 17.96
E4 The2 3.59493 0.16448 21.86
E5 The3 2.99368 0.49861 6.00
E6 The4 259.57580 18.31150 14.18
D1 Psi1 5.67047 0.42301 13.41
D2 Psi2 4.51480 0.33532 13.46
 
Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
E1 E3 The5 0.90580 0.12167 7.44
E2 E4 The5 0.90580 0.12167 7.44

The measurement scale of variables is often arbitrary. Therefore, it can be useful to look at the standardized equations produced by PROC CALIS. Output 19.1.8 displays the standardized equations and predicted moments. From the standardized structural equations for F1 and F2, you can conclude that SES (F3) has a larger impact on earlier Alienation (F1) than on later Alienation (F3).

The squared multiple correlation for each equation, the correlation among the exogenous variables, and the covariance matrices among the latent variables and between the observed and the latent variables help to describe the relationships among all variables.

Output 19.1.8: Standardized Equations and Predicted Moments
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

v1 = 0.8348   F1 + 0.5505   E1
v2 = 0.7846   F1 + 0.6200   E2
v3 = 0.8450   F2 + 0.5348   E3
v4 = 0.7968   F2 + 0.6043   E4
v5 = 0.8297   F3 + 0.5581   E5
v6 = 0.6508 * F3 + 0.7593   E6
        Lamb        

 


 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

F1 = -0.5626 * F3 + 0.8268   D1        
        Gam1                
F2 = 0.5692 * F1 + -0.2064 * F3 + 0.7080   D2
        Beta       Gam2        

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Squared Multiple Correlations
  Variable Error Variance Total Variance R-Square
1 v1 3.60788 11.90391 0.6969
2 v2 3.59493 9.35145 0.6156
3 v3 3.60788 12.61575 0.7140
4 v4 3.59493 9.84539 0.6349
5 v5 2.99368 9.61000 0.6885
6 v6 259.57580 450.28800 0.4235
7 F1 5.67047 8.29603 0.3165
8 F2 4.51480 9.00787 0.4988
 
Correlations Among Exogenous
Variables
Var1 Var2 Parameter Estimate
E1 E3 The5 0.25106
E2 E4 The5 0.25197

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Predicted Moments of Latent Variables
  F1 F2 F3
F1 8.296026985 5.924364730 -4.167911571
F2 5.924364730 9.007870649 -4.065656060
F3 -4.167911571 -4.065656060 6.616317547
 
Predicted Moments between Manifest and Latent
Variables
  F1 F2 F3
v1 8.29602698 5.92436473 -4.16791157
v2 6.91059048 4.93499582 -3.47187034
v3 5.92436473 9.00787065 -4.06565606
v4 4.93499582 7.50355625 -3.38669150
v5 -4.16791157 -4.06565606 6.61631755
v6 -22.37688158 -21.82788734 35.52199986

Output 19.1.9 displays the latent variable score regression coefficients that produce the latent variable scores. Each latent variable is expressed as a linear combination of the observed variables. See Chapter 57, "The SCORE Procedure," for more information on the creation of latent variable scores. Note that the total effects and indirect effects of the exogenous variables are also displayed.

Output 19.1.9: Latent Variable Score Regression, Direct and Indirect Effects
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Latent Variable Score Regression Coefficients
  F1 F2 F3
v1 Anomia (1967) 0.4131113567 0.0482681051 -.0521264408
v2 Anomia (1971) 0.3454029627 0.0400143300 -.0435560637
v3 Education 0.0526632293 0.4306175653 -.0399927539
v4 Powerlessness (1967) 0.0437036855 0.3600452776 -.0334000265
v5 Powerlessness (1971) -.0749215200 -.0639697183 0.5057060770
v6 Occupational Status Index -.0046390513 -.0039609288 0.0313127184
 
Total Effects
  F3 F1 F2
v1 -0.629944307 1.000000000 0.000000000
v2 -0.524743608 0.833000000 0.000000000
v3 -0.614489258 0.593112208 1.000000000
v4 -0.511869552 0.494062469 0.833000000
v5 1.000000000 0.000000000 0.000000000
v6 5.368847492 0.000000000 0.000000000
F1 -0.629944307 0.000000000 0.000000000
F2 -0.614489258 0.593112208 0.000000000
 
Indirect Effects
  F3 F1 F2
v1 -.6299443069 0.0000000000 0
v2 -.5247436076 0.0000000000 0
v3 -.6144892580 0.5931122083 0
v4 -.5118695519 0.4940624695 0
v5 0.0000000000 0.0000000000 0
v6 0.0000000000 0.0000000000 0
F1 0.0000000000 0.0000000000 0
F2 -.3736276589 0.0000000000 0

PROC CALIS can display Lagrange multiplier and Wald statistics for model modifications. Modification indices are displayed for each parameter matrix. Only the Lagrange multiplier statistics have significance levels and approximate changes of values displayed. The significance level of the Wald statistic for a given parameter is the same as that shown in the equation output. An insignificant p-value for a Wald statistic means that the corresponding parameter can be dropped from the model without significantly worsening the fit of the model.

A significant p-value for a Lagrange multiplier test indicates that the model would achieve a better fit if the corresponding parameter is free. To aid in determining significant results, PROC CALIS displays the rank order of the ten largest Lagrange multiplier statistics. For example, [E5:E2] in the _PHI_ matrix is associated with the largest Lagrange multiplier statistic; the associated p-value is 0.0067. This means that adding a parameter for the covariance between E5 and E2 will lead to a significantly better fit of the model. However, adding parameters indiscriminately can result in specification errors. An over-fitted model may not perform well with future samples. As always, the decision to add parameters should be accompanied with consideration and knowledge of the application area.

Output 19.1.10: Lagrange Multiplier and Wald Tests
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Lagrange Multiplier and Wald Test Indices _PHI_[9:9]
Symmetric Matrix
Univariate Tests for Constant Constraints
Lagrange Multiplier or Wald Index / Probability / Approx Change of Value
  F3 E1 E2 E3 E4 E5 E6 D1 D2
F3 107.1619
.
.
[Phi]
3.3903
0.0656
0.5079
 
3.3901
0.0656
-0.4231
 
0.5752
0.4482
0.2090
 
0.5753
0.4482
-0.1741
 
.
.
.
Sing
.
.
.
Sing
.
.
.
Sing
.
.
.
Sing
E1 3.3903
0.0656
0.5079
 
322.4501
.
.
[The1]
0.1529
0.6958
0.0900
 
55.4237
.
.
[The5]
1.2037
0.2726
-0.3262
 
5.8025
0.0160
0.5193
 
0.7398
0.3897
-1.2587
 
0.4840
0.4866
0.2276
 
0.0000
0.9961
0.0014
 
E2 3.3901
0.0656
-0.4231
 
0.1529
0.6958
0.0900
 
477.6768
.
.
[The2]
0.5946
0.4406
0.2328
 
55.4237
.
.
[The5]
7.3649
0.0067
-0.5060
 
1.4168
0.2339
1.5431
 
0.4840
0.4866
-0.1896
 
0.0000
0.9961
-0.0011
 
E3 0.5752
0.4482
0.2090
 
55.4237
.
.
[The5]
0.5946
0.4406
0.2328
 
322.4501
.
.
[The1]
0.1528
0.6958
-0.0900
 
1.5982
0.2062
0.2709
 
0.0991
0.7529
-0.4579
 
1.1825
0.2768
0.2984
 
0.5942
0.4408
-0.2806
 
E4 0.5753
0.4482
-0.1741
 
1.2037
0.2726
-0.3262
 
55.4237
.
.
[The5]
0.1528
0.6958
-0.0900
 
477.6768
.
.
[The2]
1.2044
0.2724
-0.2037
 
0.0029
0.9568
0.0700
 
1.1825
0.2768
-0.2486
 
0.5942
0.4408
0.2338
 
E5 .
.
.
Sing
5.8025
0.0160
0.5193
 
7.3649
0.0067
-0.5060
 
1.5982
0.2062
0.2709
 
1.2044
0.2724
-0.2037
 
36.0486
.
.
[The3]
.
.
.
Sing
0.1033
0.7479
-0.2776
 
0.1035
0.7477
0.1062
 
E6 .
.
.
Sing
0.7398
0.3897
-1.2587
 
1.4168
0.2339
1.5431
 
0.0991
0.7529
-0.4579
 
0.0029
0.9568
0.0700
 
.
.
.
Sing
200.9466
.
.
[The4]
0.1034
0.7478
1.4906
 
0.1035
0.7477
-0.5700
 
D1 .
.
.
Sing
0.4840
0.4866
0.2276
 
0.4840
0.4866
-0.1896
 
1.1825
0.2768
0.2984
 
1.1825
0.2768
-0.2486
 
0.1033
0.7479
-0.2776
 
0.1034
0.7478
1.4906
 
179.6950
.
.
[Psi1]
.
.
.
Sing
D2 .
.
.
Sing
0.0000
0.9961
0.0014
 
0.0000
0.9961
-0.0011
 
0.5942
0.4408
-0.2806
 
0.5942
0.4408
0.2338
 
0.1035
0.7477
0.1062
 
0.1035
0.7477
-0.5700
 
.
.
.
Sing
181.2787
.
.
[Psi2]

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Rank Order of the 10 Largest Lagrange Multipliers in _PHI_
Row Column Chi-Square Pr > ChiSq
E5 E2 7.36486 0.0067
E5 E1 5.80246 0.0160
E1 F3 3.39030 0.0656
E2 F3 3.39013 0.0656
E5 E3 1.59820 0.2062
E6 E2 1.41677 0.2339
E5 E4 1.20437 0.2724
E4 E1 1.20367 0.2726
D1 E3 1.18251 0.2768
D1 E4 1.18249 0.2768

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Lagrange Multiplier
and Wald Test Indices
_GAMMA_[8:1]
General Matrix
Univariate Tests
for Constant Constraints
Lagrange Multiplier
or Wald Index /
Probability / Approx
Change of Value
  F3
v1 3.3903
0.0656
0.0768
 
v2 3.3901
0.0656
-0.0639
 
v3 0.5752
0.4482
0.0316
 
v4 0.5753
0.4482
-0.0263
 
v5 .
.
.
Sing
v6 153.2354
.
.
[Lamb]
F1 125.0132
.
.
[Gam1]
F2 19.2585
.
.
[Gam2]

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Rank Order of the 4 Largest Lagrange Multipliers in _GAMMA_
Row Column Chi-Square Pr > ChiSq
v1 F3 3.39030 0.0656
v2 F3 3.39013 0.0656
v4 F3 0.57526 0.4482
v3 F3 0.57523 0.4482

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Lagrange Multiplier and Wald Test Indices _BETA_[8:8]
General Matrix
Identity-Minus-Inverse Model Matrix
Univariate Tests for Constant Constraints
Lagrange Multiplier or Wald Index / Probability / Approx Change of Value
  v1 v2 v3 v4 v5 v6 F1 F2
v1 .
.
.
Sing
0.1647
0.6849
-0.0159
 
0.0511
0.8212
-0.0063
 
0.8029
0.3702
-0.0284
 
5.4083
0.0200
0.0697
 
0.1233
0.7255
0.0015
 
0.4047
0.5247
-0.0257
 
0.4750
0.4907
-0.0239
 
v2 0.5957
0.4402
0.0218
 
.
.
.
Sing
0.6406
0.4235
0.0185
 
0.0135
0.9076
0.0032
 
5.8858
0.0153
-0.0609
 
0.0274
0.8686
-0.0006
 
0.4047
0.5247
0.0214
 
0.4750
0.4907
0.0199
 
v3 0.3839
0.5355
0.0178
 
0.3027
0.5822
0.0180
 
.
.
.
Sing
0.1446
0.7038
-0.0145
 
1.1537
0.2828
0.0322
 
0.0296
0.8634
0.0007
 
0.1588
0.6902
0.0144
 
0.0817
0.7750
-0.0110
 
v4 0.4487
0.5030
-0.0160
 
0.2519
0.6157
-0.0144
 
0.0002
0.9877
-0.0004
 
.
.
.
Sing
0.9867
0.3206
-0.0249
 
0.1442
0.7041
-0.0014
 
0.1588
0.6903
-0.0120
 
0.0817
0.7750
0.0092
 
v5 5.4085
0.0200
0.1242
 
8.6455
0.0033
-0.1454
 
2.7123
0.0996
0.0785
 
2.1457
0.1430
-0.0674
 
.
.
.
Sing
.
.
.
Sing
0.1033
0.7479
-0.0490
 
0.1035
0.7476
0.0329
 
v6 0.4209
0.5165
-0.2189
 
1.4387
0.2304
0.3924
 
0.3044
0.5811
-0.1602
 
0.0213
0.8841
0.0431
 
.
.
.
Sing
.
.
.
Sing
0.1034
0.7478
0.2629
 
0.1035
0.7477
-0.1765
 
F1 1.0998
0.2943
0.0977
 
1.1021
0.2938
-0.0817
 
1.6114
0.2043
0.0993
 
1.6128
0.2041
-0.0831
 
0.1032
0.7480
-0.0927
 
0.1035
0.7477
0.0057
 
.
.
.
Sing
.
.
.
Sing
F2 0.0193
0.8896
-0.0104
 
0.0194
0.8892
0.0087
 
0.4765
0.4900
-0.0625
 
0.4760
0.4902
0.0522
 
0.1034
0.7477
0.0355
 
0.1035
0.7477
-0.0022
 
160.7520
.
.
[Beta]
.
.
.
Sing

 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Rank Order of the 10 Largest Lagrange Multipliers in _BETA_
Row Column Chi-Square Pr > ChiSq
v5 v2 8.64546 0.0033
v2 v5 5.88576 0.0153
v5 v1 5.40848 0.0200
v1 v5 5.40832 0.0200
v5 v3 2.71233 0.0996
v5 v4 2.14572 0.1430
F1 v4 1.61279 0.2041
F1 v3 1.61137 0.2043
v6 v2 1.43867 0.2304
v3 v5 1.15372 0.2828

When you specify equality constraints, PROC CALIS displays Lagrange multiplier tests for releasing the constraints. In the current example, none of the three constraints achieve a p-value smaller than 0.05. This means that releasing the constraints may not lead to a significantly better fit of the model. Therefore, all constraints are retained in the model.

Output 19.1.11: Tests for Equality Constraints
 
Stability of Alienation
Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Univariate Lagrange Multiplier Test for Releasing Equality
Constraints
Equality Constraint Changes Chi-Square Pr > ChiSq
[E1:E1] = [E3:E3] 0.0293 -0.0308 0.02106 0.8846
[E2:E2] = [E4:E4] -0.1342 0.1388 0.69488 0.4045
[E3:E1] = [E4:E2] 0.2468 -0.1710 1.29124 0.2558

The model is specified using the LINEQS, STD, and COV statements. The section "Getting Started" also contains the COSAN and RAM specifications of this model. These model specifications would give essentially the same results.

   proc calis cov data=Wheaton tech=nr edf=931;                
      Cosan J(9, Ide) * A(9, Gen, Imi) * P(9, Sym);              
      Matrix A                                                   
              [ ,7] = 1. .833  5 * 0. Beta (.5) ,              
              [ ,8] = 2 * 0.  1.  .833 ,                       
              [ ,9] = 4 * 0.  1.  Lamb Gam1-Gam2 (.5 2 * -.5); 
      Matrix P                                                   
              [1,1] = The1-The2 The1-The4 (6 * 3.) ,           
              [7,7] = Psi1-Psi2 Phi (2 * 4. 6.) ,              
              [3,1] = The5 (.2) ,                              
              [4,2] = The5 (.2) ;                              
      Vnames J V1-V6 F1-F3 ,                                     
             A = J ,                                             
             P E1-E6 D1-D3 ;                                     
   run;                                                        
                                                               
   proc calis cov data=Wheaton tech=nr edf=931;                
      Ram                                                        
         1   1  7  1.       ,                                  
         1   2  7  .833     ,                                  
         1   3  8  1.       ,                                  
         1   4  8  .833     ,                                  
         1   5  9  1.       ,                                  
         1   6  9  .5    Lamb ,                                
         1   7  9  -.5   Gam1 ,                                
         1   8  7  .5    Beta ,                                
         1   8  9  -.5   Gam2 ,                                
         2   1  1  3.    The1 ,                                
         2   2  2  3.    The2 ,                                
         2   3  3  3.    The1 ,                                
         2   4  4  3.    The2 ,                                
         2   5  5  3.    The3 ,                                
         2   6  6  3.    The4 ,                                
         2   1  3  .2    The5 ,                                
         2   2  4  .2    The5 ,                                
         2   7  7  4.    Psi1 ,                                
         2   8  8  4.    Psi2 ,                                
         2   9  9  6.    Phi ;                                 
      Vnames 1 F1-F3,                                            
             2 E1-E6 D1-D3;                                      
   run;

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