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The TCALIS Procedure

Example 88.1 Path Analysis: Stability of Alienation

The following covariance matrix from Wheaton et al. (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 Jöreskog (1978). You can also find a more detailed discussion of these models in the LISREL VI manual (Jöreskog and Sörbom 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 88.2. The same model is reanalyzed here by PROC TCALIS. However, for the analysis with the EQS implementation, the SEI variable 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.

   title "Stability of Alienation";                                    
   title2 "Data Matrix of WHEATON, MUTHEN, ALWIN & SUMMERS (1977)";    
   data Wheaton(TYPE=COV);                                             
      _type_ = 'cov'; 
      input _name_ $ 1-11 Anomie67 Powerless67 Anomie71 Powerless71 
                          Education SEI;
      label Anomie67='Anomie (1967)' Powerless67='Powerlessness (1967)' 
            Anomie71='Anomie (1971)' Powerless71='Powerlessness (1971)' 
            Education='Education'    SEI='Occupational Status Index';                            
   Anomie67       11.834     .        .        .       .        .                
   Powerless67     6.947    9.364     .        .       .        .                
   Anomie71        6.819    5.091   12.532     .       .        .                
   Powerless71     4.783    5.028    7.495    9.986    .        .                
   Education      -3.839   -3.889   -3.841   -3.625   9.610     .                
   SEI           -21.899  -18.831  -21.748  -18.775  35.522  450.288             
   ods graphics on;
   proc tcalis nobs=932 data=Wheaton plots=residuals;
         Anomie67     <-  Alien67   1.0,
         Powerless67  <-  Alien67   0.833,
         Anomie71     <-  Alien71   1.0,
         Powerless71  <-  Alien71   0.833,
         Education    <-  SES       1.0,
         SEI          <-  SES       lambda,
         Alien67      <-  SES       gamma1,
         Alien71      <-  SES       gamma2,
         Alien71      <-  Alien67   beta;
         Anomie67     = theta1,
         Powerless67  = theta2,
         Anomie71     = theta1,
         Powerless71  = theta2,
         Education    = theta3,
         SEI          = theta4,
         Alien67      = psi1,
         Alien71      = psi2,
         SES          = phi;
         Anomie67    Anomie71    = theta5,
         Powerless67 Powerless71 = theta5;
   ods graphics off;

Since no METHOD= option has been used, maximum likelihood estimates are computed by default. In this example, no global display options are specified. This means that PROC TCALIS prints the default set of results.

PROC TCALIS can produce a high-quality residual histogram that is useful for showing the distribution of residuals. To request the residual histogram, you must first enable ODS Graphics by specifying the ods graphics on statement, as shown in the preceding statements before the PROC TCALIS statement. Then, the residual histogram is requested by the plots=residuals option in the PROC TCALIS statement.

Output 88.1.1 displays the modeling information and variables in the analysis.

Output 88.1.1 Model Specification and Variables
Stability of Alienation
PATH Model Specification

The TCALIS Procedure
Covariance Structure Analysis: Model and Initial Values

Modeling Information
N Obs 932
Model Type PATH

Variables in the Model
Endogenous Manifest Anomie67 Anomie71 Education Powerless67 Powerless71 SEI
  Latent Alien67 Alien71
Exogenous Manifest  
  Latent SES
Number of Endogenous Variables = 8
Number of Exogenous Variables = 1

Output 88.1.1 shows that the data set Wheaton was used with observations. The model is specified with the PATH modeling language. Variables in the model are classified into different categories according to their roles. All manifest variables are endogenous in the model. Also, three latent variables are hypothesized in the model: Alien67, Alien71, and SES. While Alien67 and Alien71 are endogenous, SES is exogenous in the model.

Output 88.1.2 echoes the initial specification of the PATH model.

Output 88.1.2 Initial Estimates
Initial Estimates for PATH List
Path Parameter Estimate
Anomie67 <- Alien67   1.00000
Powerless67 <- Alien67   0.83300
Anomie71 <- Alien71   1.00000
Powerless71 <- Alien71   0.83300
Education <- SES   1.00000
SEI <- SES lambda .
Alien67 <- SES gamma1 .
Alien71 <- SES gamma2 .
Alien71 <- Alien67 beta .

Initial Estimates for Variance Parameters
Variable Parameter Estimate
Error Anomie67 theta1 .
  Powerless67 theta2 .
  Anomie71 theta1 .
  Powerless71 theta2 .
  Education theta3 .
  SEI theta4 .
  Alien67 psi1 .
  Alien71 psi2 .
Exogenous SES phi .

Initial Estimates for Covariances Among Errors
Error of Error of Parameter Estimate
Anomie67 Anomie71 theta5 .
Powerless67 Powerless71 theta5 .

In Output 88.1.2, numerical values for estimates are the initial values you input in the model specification. If the associated parameter name for a numerical estimate is blank, it means that the estimate is a fixed value, which would not be changed in the estimation. For example, the first five paths have fixed path coefficients with the fixed values given. For numerical estimates with parameter names given, the numerical values serve as initial values, which would be changed during the estimation. In Output 88.1.2, you actually do not have this kind of specification. All free parameters specified in the model are with missing initial values, denoted by '.'. For example, lambda, gamma1, theta1, and psi1, among others, are free parameters without initial values given. The initial values of these parameters will be generated by PROC TCALIS automatically.

You can examine this output to ensure that the desired model is being analyzed. PROC TCALIS outputs the initial specifications or the estimation results in the order you specify in the model, unless you use reordering options such as ORDERSPEC and ORDERALL. Therefore, the input order of specifications is important—it determines how your output would look.

Simple descriptive statistics are displayed in Output 88.1.3.

Output 88.1.3 Descriptive Statistics
Simple Statistics
Variable Mean Std Dev
Anomie67 Anomie (1967) 0 3.44006
Powerless67 Powerlessness (1967) 0 3.06007
Anomie71 Anomie (1971) 0 3.54006
Powerless71 Powerlessness (1971) 0 3.16006
Education Education 0 3.10000
SEI Occupational Status Index 0 21.21999

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.

Initial estimates are necessary in all kinds of optimization problems. They could be provided by users or automatically generated by PROC TCALIS. As shown in Output 88.1.2, you did not provide any initial estimates for all free parameters. PROC TCALIS uses a combination of well-behaved mathematical methods to complete the initial estimation. The initial estimation methods for the current analysis are shown in Output 88.1.4.

Output 88.1.4 Optimization Starting Point
Initial Estimation Methods
1 Instrumental Variables Method
2 McDonald Method
3 Two-Stage Least Squares

Optimization Start
Parameter Estimates
N Parameter Estimate Gradient
1 lambda 4.99508 -0.00206
2 gamma1 -0.62322 -0.04069
3 gamma2 -0.20437 -0.03816
4 beta 0.66589 0.03789
5 theta1 3.51433 -0.00409
6 theta2 3.65991 0.01182
7 theta3 2.49860 -0.00578
8 theta4 272.85274 0.0000194
9 psi1 5.57764 -0.00217
10 psi2 3.79636 -0.00935
11 phi 7.11140 0.00108
12 theta5 0.45298 -0.06463
Value of Objective Function = 0.0365979443

In this example, instrumental Variable Method, method by McDonald and Hartmann, and two-stage least squares method have been used for initial estimation. In the same output, the vector of initial parameter estimates and their gradients are also shown. The initial objective function value is .

Output 88.1.5 displays the optimization information, including technical details, iteration history and convergence status.

Output 88.1.5 Optimization
Parameter Estimates 12
Functions (Observations) 21

Optimization Start
Active Constraints 0 Objective Function 0.0365979443
Max Abs Gradient Element 0.0646338767 Radius 1

Iteration   Restarts Function
Max Abs
Lambda Ratio
1   0 4 0   0.01453 0.0221 0.00142 0 1.013
2   0 6 0   0.01448 0.000046 0.000249 0 1.001
3   0 8 0   0.01448 1.007E-7 4.717E-6 0 1.006

Optimization Results
Iterations 3 Function Calls 11
Jacobian Calls 5 Active Constraints 0
Objective Function 0.0144844814 Max Abs Gradient Element 4.7172823E-6
Lambda 0 Actual Over Pred Change 1.0060912391
Radius 0.001390392    

Convergence criterion (ABSGCONV=0.00001) satisfied.

The convergence status is important for the validity of your solution. In most cases, you should interpret your results only when the solution is converged. In this example, you obtain a converged solution, as shown in the message at the bottom of the table. The final objective function value is , which is the minimized function value during the optimization. If problematic solutions such as nonconvergence are encountered, PROC TCALIS issues an error message.

The fit summary statistics are displayed in Output 88.1.6. By default, PROC TCALIS displays all available fit indices and modeling information.

Output 88.1.6 Fit Summary
Fit Summary
Modeling Info N Observations 932
  N Variables 6
  N Moments 21
  N Parameters 12
  N Active Constraints 0
  Independence Model Chi-Square 2131.4327
  Independence Model Chi-Square DF 15
Absolute Index Fit Function 0.0145
  Chi-Square 13.4851
  Chi-Square DF 9
  Pr > Chi-Square 0.1419
  Z-Test of Wilson & Hilferty 1.0754
  Hoelter Critical N 1170
  Root Mean Square Residual (RMSR) 0.2281
  Standardized RMSR (SRMSR) 0.0150
  Goodness of Fit Index (GFI) 0.9953
Parsimony Index Adjusted GFI (AGFI) 0.9890
  Parsimonious GFI 0.5972
  RMSEA Estimate 0.0231
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0470
  Probability of Close Fit 0.9705
  ECVI Estimate 0.0405
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.0556
  Akaike Information Criterion -4.5149
  Bozdogan CAIC -57.0509
  Schwarz Bayesian Criterion -48.0509
  McDonald Centrality 0.9976
Incremental Index Bentler Comparative Fit Index 0.9979
  Bentler-Bonett NFI 0.9937
  Bentler-Bonett Non-normed Index 0.9965
  Bollen Normed Index Rho1 0.9895
  Bollen Non-normed Index Delta2 0.9979
  James et al. Parsimonious NFI 0.5962

First, the fit summary table starts with some basic modeling information, as shown in Output 88.1.6. You can check the number of observations, number of variables, number of moments being fitted, number of parameters, number of active constraints in the solution, and the independent model chi-square and its degrees of freedom in this modeling information category. Next, three types of fit indices are shown: absolute, parsimony, and incremental.

The absolute indices are fit measures that are interpreted without referring to any baseline model and without adjusting model parsimony. The chi-square test statistic is the best-known absolute index in this category. In this example, the -value of the chi-square is , which is larger than the conventional value. From a statistical hypothesis testing point of view, this model cannot be rejected. The Z-test of Wilson and Hilferty is also insignificant at , which echoes the result of the chi-square test. Other absolute indices can also be consulted. Although it seems that there are no clear conventional levels for these indices to indicate an acceptable model fit, you can always use these indices to compare the relative fit between different models.

Next, the parsimony fit indices take the model parsimony into account. These indices adjust the model fit by the degrees of freedom (or the number of the parameters) of the model in certain ways. The advantage of these indices is that merely increasing the number of parameters in the model might not necessarily improve the model fit indicated by these indices. Models with large numbers of parameters are penalized. There is no uniform way to interpret all these indices. However, for the relatively well-known RMSEA estimate, by convention values under indicate good fit. The RMSEA value for this example is , and so this is a very good fit. For interpretations of other parsimony indices, you can consult the original articles for these indices.

Last, the incremental fit indices are computed based on comparison with a baseline model, usually the independence model where all mainfest variables are assumed to be uncorrelated. The independence model fit statistic is shown under the Modeling Info category of the same fit summary table. In this example, the model fit chi-square of the independence model is , with degrees of freedom. The incremental indices show how well the hypothesized model improves over the independence model for the data. Various incremental fit indices have been proposed. In the fit summary table, there are six of such fit indices. Large values for these indices are desired. It has been suggested that values larger than for these indices indicate acceptable model fit. In this example, all incremental indices but James et al. parsimonious NFI show that the hypothesized model fits well.

There is no consensus as to which fit index is the best to judge model fit. Probably, with artificial data and model, all fit indices can be shown defective in some aspects of measuring model fit. Conventional wisdom is to look at all fit indices and determine whether the majority of them are close to the desirable ranges of values. In this example, almost all fit indices are good, and so it is safe to conclude that the model fits well.

Nowadays, most researchers do pay less attention to the model fit chi-square statistic because it tends to reject all meaningful models with minimum departures from the truth. Although the model fit chi-square test statistic is an impeccable statistical inference tool when the underlying statistical assumptions are satisfied, for practical purposes it is just too powerful to accept any useful and reasonable models with only tiny imperfections. Some fit indices are more popular than others. Standardized RMSR, RMSEA estimate, adjusted AGFI, and Bentler’s comparative fit index are frequently reported in empirical research for judging model fit. In this example, all these measures show good model fit of the hypothesized model. While there are certainly legitimate reasons why these fit indices are more popular than others, they are out of the current scope of discussion.

PROC TCALIS can perform a detailed residual analysis. Large residuals might indicate misspecification of the model. In Output 88.1.7, raw residuals are reported and ranked.

Output 88.1.7 Raw Residuals and Ranking
Raw Residual Matrix
  Anomie67 Powerless67 Anomie71 Powerless71 Education SEI
Anomie67 Anomie (1967) -0.06997 0.03642 -0.01116 -0.15200 0.32892 0.47786
Powerless67 Powerlessness (1967) 0.03642 0.01261 0.15600 0.01135 -0.41712 -0.19108
Anomie71 Anomie (1971) -0.01116 0.15600 -0.08381 -0.00854 0.22464 0.07976
Powerless71 Powerlessness (1971) -0.15200 0.01135 -0.00854 0.14067 -0.23832 -0.59248
Education Education 0.32892 -0.41712 0.22464 -0.23832 0 4.03148E-6
SEI Occupational Status Index 0.47786 -0.19108 0.07976 -0.59248 4.03148E-6 0.0000221

Average Absolute Residual 0.153940
Average Off-diagonal Absolute Residual 0.195044

Rank Order of the 10 Largest Raw Residuals
Var1 Var2 Residual
SEI Powerless71 -0.59248
SEI Anomie67 0.47786
Education Powerless67 -0.41712
Education Anomie67 0.32892
Education Powerless71 -0.23832
Education Anomie71 0.22464
SEI Powerless67 -0.19108
Anomie71 Powerless67 0.15600
Powerless71 Anomie67 -0.15200
Powerless71 Powerless71 0.14067

Because of the differential scaling of the variables, it is usually more useful to examine the standardized residuals instead. In Output 88.1.8, for example, the table for the 10 largest asymptotically standardized residuals is displayed.

Output 88.1.8 Asymptotically Standardized Residuals and Ranking
Asymptotically Standardized Residual Matrix
  Anomie67 Powerless67 Anomie71 Powerless71 Education SEI
Anomie67 Anomie (1967) -0.30882 0.52686 -0.05619 -0.86507 2.55338 0.46484
Powerless67 Powerlessness (1967) 0.52686 0.05464 0.87613 0.05735 -2.76371 -0.17015
Anomie71 Anomie (1971) -0.05619 0.87613 -0.35460 -0.12169 1.69781 0.07009
Powerless71 Powerlessness (1971) -0.86507 0.05735 -0.12169 0.58521 -1.55750 -0.49608
Education Education 2.55338 -2.76371 1.69781 -1.55750 0 0
SEI Occupational Status Index 0.46484 -0.17015 0.07009 -0.49608 0 0

Average Standardized Residual 0.646672
Average Off-diagonal Standardized Residual 0.818456

Rank Order of the 10 Largest Asymptotically Standardized Residuals
Var1 Var2 Residual
Education Powerless67 -2.76371
Education Anomie67 2.55338
Education Anomie71 1.69781
Education Powerless71 -1.55750
Anomie71 Powerless67 0.87613
Powerless71 Anomie67 -0.86507
Powerless71 Powerless71 0.58521
Powerless67 Anomie67 0.52686
SEI Powerless71 -0.49608
SEI Anomie67 0.46484

The model performs the poorest concerning the covariances of Education with all measures of Powerless and Anomie. This might suggest a misspecification of the functional relationships of Education with other variables in the model. However, because the model fit is quite good, such a possible misspecification should not be a serious concern in the analysis.

The histogram of the asymptotically standardized residuals is displayed in Output 88.1.9, which also shows the normal and kernel approximations.

Output 88.1.9 Distribution of Asymptotically Standardized Residuals
Distribution of Asymptotically Standardized Residuals

The residual distribution looks quite symmetrical. It shows a small to medium departure from the normal distribution, as evidenced by the discrepancies between the kernel and the normal distribution curves.

Estimation results are shown in Output 88.1.10.

Output 88.1.10 Estimation Results
Path Parameter Estimate Standard
t Value
Anomie67 <- Alien67   1.00000    
Powerless67 <- Alien67   0.83300    
Anomie71 <- Alien71   1.00000    
Powerless71 <- Alien71   0.83300    
Education <- SES   1.00000    
SEI <- SES lambda 5.36883 0.43371 12.37880
Alien67 <- SES gamma1 -0.62994 0.05634 -11.18092
Alien71 <- SES gamma2 -0.24086 0.05489 -4.38836
Alien71 <- Alien67 beta 0.59312 0.04678 12.67884

Variance Parameters
Variable Parameter Estimate Standard
t Value
Error Anomie67 theta1 3.60796 0.20092 17.95717
  Powerless67 theta2 3.59488 0.16448 21.85563
  Anomie71 theta1 3.60796 0.20092 17.95717
  Powerless71 theta2 3.59488 0.16448 21.85563
  Education theta3 2.99366 0.49861 6.00398
  SEI theta4 259.57639 18.31151 14.17559
  Alien67 psi1 5.67046 0.42301 13.40500
  Alien71 psi2 4.51479 0.33532 13.46394
Exogenous SES phi 6.61634 0.63914 10.35190

Covariances Among Errors
Error of Error of Parameter Estimate Standard
t Value
Anomie67 Anomie71 theta5 0.90580 0.12167 7.44472
Powerless67 Powerless71 theta5 0.90580 0.12167 7.44472

The paths, variances and partial (or error) variances, and covariances and partial covariances are shown. When you have fixed parameters such as the first five path coefficients in the output, the standard errors and values are all blanks. For free or constrained estimates, standard errors and values are computed. Researchers in structural equation modeling usually use the value as an approximate critical value for the observed values. The reason is that the estimates are asymptotically normal, and so the two-sided critical point with is , which is close to . Using this criterion, all estimates shown in Output 88.1.10 are significantly different from zero, supporting the presence of these parameters in the model.

Squared multiple correlations are shown in Output 88.1.11.

Output 88.1.11 Squared Multiple Correlations
Squared Multiple Correlations
Variable Error Variance Total Variance R-Square
Anomie67 3.60796 11.90397 0.6969
Anomie71 3.60796 12.61581 0.7140
Education 2.99366 9.61000 0.6885
Powerless67 3.59488 9.35139 0.6156
Powerless71 3.59488 9.84533 0.6349
SEI 259.57639 450.28798 0.4235
Alien67 5.67046 8.29601 0.3165
Alien71 4.51479 9.00786 0.4988

For each endogenous variable in the model, the corresponding squared multiple correlation is computed by:


In regression analysis, this is the percentage of explained variance of the endogenous variable by the predictors. However, this interpretation is complicated or even uninterpretable when your structural equation model has correlated errors or reciprocal casual relations. In these situations, it is not uncommon to see negative R-squares. Negative R-squares do not necessarily mean that your model is wrong or the model prediction is weak. Rather, the R-square interpretation is questionable in these situations.

When your variables are measured on different scales, comparison of path coefficients cannot be made directly. For example, in Output 88.1.10, the path coefficient for path Education <– SES is fixed at one, while the path coefficient for path SEI <– SES is . It would be very simple-minded to conclude that the effect of SES on SEI is larger than that SES on Education. Because SEI and Education are measured on different scales, direct comparison of the corresponding path coefficients is simply inappropriate.

In alleviating this problem, some might resort to the standardized solution for a better comparison. In a standardized solution, because the variances of manifest variables and systematic predictors are all standardized to ones, you hope the path coefficients are more comparable. In this example, PROC TCALIS standardizes your results in Output 88.1.12.

Output 88.1.12 Standardized Results
Standardized Results for PATH List
Path Parameter Estimate Standard
t Value
Anomie67 <- Alien67   0.83481 0.01093 76.35313
Powerless67 <- Alien67   0.78459 0.01163 67.47756
Anomie71 <- Alien71   0.84499 0.01031 81.97956
Powerless71 <- Alien71   0.79678 0.01107 71.96263
Education <- SES   0.82975 0.03172 26.15990
SEI <- SES lambda 0.65079 0.03019 21.55331
Alien67 <- SES gamma1 -0.56257 0.03456 -16.27961
Alien71 <- SES gamma2 -0.20642 0.04483 -4.60430
Alien71 <- Alien67 beta 0.56920 0.04066 14.00001

Standardized Results for Variance Parameters
Variable Parameter Estimate Standard
t Value
Error Anomie67 theta1 0.30309 0.01825 16.60309
  Powerless67 theta2 0.38442 0.01825 21.06948
  Anomie71 theta1 0.28599 0.01742 16.41782
  Powerless71 theta2 0.36514 0.01764 20.69424
  Education theta3 0.31152 0.05264 5.91822
  SEI theta4 0.57647 0.03930 14.66804
  Alien67 psi1 0.68352 0.03888 17.57968
  Alien71 psi2 0.50121 0.03321 15.08974
Exogenous SES phi 1.00000    

Standardized Results for Covariances Among Errors
Error of Error of Parameter Estimate Standard
t Value
Anomie67 Anomie71 theta5 0.07391 0.01013 7.29574
Powerless67 Powerless71 theta5 0.09440 0.01274 7.40922

Now, the standardized path coefficient for path Education <– SES is , while the standardized path coefficient for path SEI <– SES is . So the standardized effect of SES on SEI is actually smaller than that of SES on Education.

Furthermore, in PROC TCALIS the standardized estimates are computed with standard error estimates and values so that you can make statistical inferences on the standardized estimates as well.

PROC TCALIS might differ from other software in its standardization scheme. Unlike other software that might standardize the path coefficients that attach to the error terms (unsystematic sources), PROC TCALIS keeps these path coefficients at ones (not shown in the output). Unlike other software that might also standardize the corresponding error variances to ones, the error variances in the standardized solution of PROC TCALIS are rescaled so as to keep the mathematical consistency of the model.

Essentially, in PROC TCALIS only variances of mainfest and non-error-type latent variables are standardized to ones. The error variances are rescaled, but not standardized. For example, in the standardized solution shown in Output 88.1.12, the error variances for all endogenous variables are not ones (see the middle portion of the output). Only the variance for the latent variable SES is standardized to one. See the section Standardized Solutions for the logic of the standardization scheme adopted by PROC TCALIS.

In appearance, the standardized solution is like a correlational analysis on the standardized manifest variables with standardized exogenous latent factors. Unfortunately, this statement is over-simplified, if not totally inappropriate. In standardizing a solution, the implicit equality constraints are likely destroyed. In this example, the unstandardized error variances for Anomie67 and Anomie71 are both , represented by a common parameter theta1. However, after standardization, these error variances have different values at and , respectively. In addition, fixed parameter values are no longer fixed in a standardized solution (for example, the first five paths in the current example). The issue of standardization is common to all other SEM software and beyond the current discussion. PROC TCALIS provides the standardized solution so that users can interpret the standardized estimates whenever they find them appropriate.

Note: This procedure is experimental.

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