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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 Jreskog (1978). You also can find a more detailed discussion of these models in the LISREL VI manual (Jreskog and Srbom 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.

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.

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.

   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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