Previous Page | Next Page

Introduction to Structural Equation Modeling with Latent Variables

A Combined Measurement-Structural Model with Reciprocal Influence and Correlated Residuals

To illustrate a more complex model, this example uses some well-known data from Haller and Butterworth (1960). Various models and analyses of these data are given by Duncan, Haller, and Portes (1968), Jöreskog and Sörbom (1988), and Loehlin (1987).

The study is concerned with the career aspirations of high school students and how these aspirations are affected by close friends. The data are collected from 442 seventeen-year-old boys in Michigan. There are 329 boys in the sample who named another boy in the sample as a best friend. The observations to be analyzed consist of the data from these 329 boys paired with the data from their best friends.

The method of data collection introduces two statistical problems. First, restricting the analysis to boys whose best friends are in the original sample causes the reduced sample to be biased. Second, since the data from a given boy might appear in two or more observations, the observations are not independent. Therefore, any statistical conclusions should be considered tentative. It is difficult to accurately assess the effects of the dependence of the observations on the analysis, but it could be argued on intuitive grounds that since each observation has data from two boys and since it seems likely that many of the boys will appear in the data set at least twice, the effective sample size might be as small as half of the reported 329 observations.

The correlation matrix, taken from Jöreskog and Sörbom (1988), is shown in the following DATA step:

   title 'Peer Influences on Aspiration: Haller & Butterworth (1960)';
   data aspire(type=corr);
      _type_='corr';
      input _name_ $ riq rpa rses roa rea fiq fpa fses foa fea;
      label riq='Respondent: Intelligence'
            rpa='Respondent: Parental Aspiration'
            rses='Respondent: Family SES'
            roa='Respondent: Occupational Aspiration'
            rea='Respondent: Educational Aspiration'
            fiq='Friend: Intelligence'
            fpa='Friend: Parental Aspiration'
            fses='Friend: Family SES'
            foa='Friend: Occupational Aspiration'
            fea='Friend: Educational Aspiration';
      datalines;
   riq   1.      .      .      .      .      .       .      .      .      .
   rpa   .1839  1.      .      .      .      .       .      .      .      .
   rses  .2220  .0489  1.      .      .      .       .      .      .      .
   roa   .4105  .2137  .3240  1.      .      .       .      .      .      .
   rea   .4043  .2742  .4047  .6247  1.      .       .      .      .      .
   fiq   .3355  .0782  .2302  .2995  .2863  1.       .      .      .      .
   fpa   .1021  .1147  .0931  .0760  .0702  .2087   1.      .      .      .
   fses  .1861  .0186  .2707  .2930  .2407  .2950  -.0438  1.      .      .
   foa   .2598  .0839  .2786  .4216  .3275  .5007   .1988  .3607  1.      .
   fea   .2903  .1124  .3054  .3269  .3669  .5191   .2784  .4105  .6404  1.
   ;

The model analyzed by Jöreskog and Sörbom (1988) is displayed in the path diagram in Figure 17.17.

Figure 17.17 Path Diagram: Career Aspiration – Jöreskog and Sörbom (1988)


Two latent variables, f_ramb and f_famb, represent the respondent’s level of ambition and his best friend’s level of ambition, respectively. The model states that the respondent’s ambition is determined by his intelligence (riq) and socioeconomic status (rses), his perception of his parents’ aspiration for him (rpa), and his friend’s socioeconomic status (fses) and ambition (f_famb). It is assumed that his friend’s intelligence (fiq) and parental aspiration (fpa) affect the respondent’s ambition only indirectly through the friend’s ambition (f_famb). Ambition is indexed by the manifest variables of occupational (roa) and educational aspiration (rea), which are assumed to have uncorrelated residuals. The path coefficient from ambition to occupational aspiration is set to 1.0 to determine the scale of the ambition latent variable.

This model can be analyzed with PROC TCALIS by using the PATH modeling language, as shown in the following statements:


   proc tcalis corr data=aspire nobs=329;
      path
         /* measurement model for aspiration */
         rea <-  f_ramb    lambda2,
         roa <-  f_ramb    1.,
         foa <-  f_famb    1.,
         fea <-  f_famb    lambda3,
         /* structural model of influences */
         f_ramb <- rpa     gam1,
         f_ramb <- riq     gam2,
         f_ramb <- rses    gam3, 
         f_ramb <- fses    gam4,
         f_famb <- rses    gam5,
         f_famb <- fses    gam6, 
         f_famb <- fiq     gam7,
         f_famb <- fpa     gam8,
         f_ramb <- f_famb  beta1,
         f_famb <- f_ramb  beta2;
      pvar
         f_ramb = psi11,
         f_famb = psi22,
         rpa riq rses fpa fiq fses = v1-v6,
         rea roa fea foa = theta1-theta4;
      pcov
         f_ramb f_famb = psi12,
         rpa riq rses fpa fiq fses = 15 * cov__;
   run;

In this specification, the names of the parameters correspond to those used by Jöreskog and Sörbom (1988). Since this TYPE=CORR data set does not contain an observation with _TYPE_=’N’ giving the sample size, it is necessary to specify the NOBS= option in the PROC TCALIS statement.

Specifying a name followed by double underscores is a quick way to generate unique parameter names. The double underscores are replaced with a unique number each time a new parameter name is generated. For example, in the COV statement, the specification

   rpa riq rses fpa fiq fses = 15 * cov__;

is equivalent to

   rpa riq rses fpa fiq fses = cov01-cov15;

In the PROC TCALIS statement, the CORR option is used to indicate that the correlation matrix is fitted by the model. Fitting correlation matrices by covariance structure modeling method is plagued with some statistical issues. For example, the chi-square statistic might not follow the theoretical distribution well, and the estimates of standard errors might not be accurate. Nonetheless, the correlation matrix is fitted here for illustration and comparison purposes.

The results from this analysis are displayed in Figure 17.18.

Figure 17.18 Career Aspiration Data: Fit Summary of Jöreskog and Sörbom (1988) Analysis 1
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 40
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0814
  Chi-Square 26.6972
  Chi-Square DF 15
  Pr > Chi-Square 0.0313
  Z-Test of Wilson & Hilferty 1.8625
  Hoelter Critical N 309
  Root Mean Square Residual (RMSR) 0.0202
  Standardized RMSR (SRMSR) 0.0202
  Goodness of Fit Index (GFI) 0.9844
Parsimony Index Adjusted GFI (AGFI) 0.9428
  Parsimonious GFI 0.3281
  RMSEA Estimate 0.0488
  RMSEA Lower 90% Confidence Limit 0.0145
  RMSEA Upper 90% Confidence Limit 0.0783
  Probability of Close Fit 0.4876
  ECVI Estimate 0.3338
  ECVI Lower 90% Confidence Limit 0.3012
  ECVI Upper 90% Confidence Limit 0.3910
  Akaike Information Criterion -3.3028
  Bozdogan CAIC -75.2437
  Schwarz Bayesian Criterion -60.2437
  McDonald Centrality 0.9824
Incremental Index Bentler Comparative Fit Index 0.9859
  Bentler-Bonett NFI 0.9694
  Bentler-Bonett Non-normed Index 0.9576
  Bollen Normed Index Rho1 0.9082
  Bollen Non-normed Index Delta2 0.9864
  James et al. Parsimonious NFI 0.3231

Jöreskog and Sörbom (1988) present more detailed results from a second analysis in which two constraints are imposed:

  • The coefficients connecting the latent ambition variables are equal (that is, beta1 beta2).

  • The covariance of the disturbances of the ambition variables is zero (that is, psi12 ).

This analysis can be performed by changing the names beta1 and beta2 to beta and omitting the line from the COV statement for psi12, as shown in the following statements:


   proc tcalis corr data=aspire nobs=329;
      path
         /* measurement model for aspiration */
         rea <-  f_ramb    lambda2,
         roa <-  f_ramb    1.,
         foa <-  f_famb    1.,
         fea <-  f_famb    lambda3,
         /* structural model of influences */
         f_ramb <- rpa     gam1,
         f_ramb <- riq     gam2,
         f_ramb <- rses    gam3, 
         f_ramb <- fses    gam4,
         f_famb <- rses    gam5,
         f_famb <- fses    gam6, 
         f_famb <- fiq     gam7,
         f_famb <- fpa     gam8,
         f_ramb <- f_famb  beta,
         f_famb <- f_ramb  beta;
      pvar
         f_ramb = psi11,
         f_famb = psi22,
         rpa riq rses fpa fiq fses = v1-v6,
         rea roa fea foa = theta1-theta4;
      pcov
         rpa riq rses fpa fiq fses = 15 * cov__;
    run;

The fit summary is displayed in Figure 17.19, and the estimation results are displayed in Figure 17.20.

Figure 17.19 Career Aspiration Data: Fit Summary of Jöreskog and Sörbom (1988) Analysis 2
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 38
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0820
  Chi-Square 26.8987
  Chi-Square DF 17
  Pr > Chi-Square 0.0596
  Z-Test of Wilson & Hilferty 1.5599
  Hoelter Critical N 338
  Root Mean Square Residual (RMSR) 0.0203
  Standardized RMSR (SRMSR) 0.0203
  Goodness of Fit Index (GFI) 0.9843
Parsimony Index Adjusted GFI (AGFI) 0.9492
  Parsimonious GFI 0.3718
  RMSEA Estimate 0.0421
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0710
  Probability of Close Fit 0.6367
  ECVI Estimate 0.3218
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.3781
  Akaike Information Criterion -7.1013
  Bozdogan CAIC -88.6343
  Schwarz Bayesian Criterion -71.6343
  McDonald Centrality 0.9851
Incremental Index Bentler Comparative Fit Index 0.9880
  Bentler-Bonett NFI 0.9692
  Bentler-Bonett Non-normed Index 0.9683
  Bollen Normed Index Rho1 0.9183
  Bollen Non-normed Index Delta2 0.9884
  James et al. Parsimonious NFI 0.3661

Figure 17.20 Career Aspiration Data: Estimation Results of Jöreskog and Sörbom (1988) Analysis 2
PATH List
Path Parameter Estimate Standard
Error
t Value
rea <- f_ramb lambda2 1.06097 0.08921 11.89233
roa <- f_ramb   1.00000    
foa <- f_famb   1.00000    
fea <- f_famb lambda3 1.07359 0.08063 13.31498
f_ramb <- rpa gam1 0.16367 0.03872 4.22740
f_ramb <- riq gam2 0.25395 0.04186 6.06725
f_ramb <- rses gam3 0.22115 0.04187 5.28219
f_ramb <- fses gam4 0.07728 0.04149 1.86264
f_famb <- rses gam5 0.06840 0.03868 1.76809
f_famb <- fses gam6 0.21839 0.03948 5.53198
f_famb <- fiq gam7 0.33063 0.04116 8.03314
f_famb <- fpa gam8 0.15204 0.03636 4.18169
f_ramb <- f_famb beta 0.18007 0.03912 4.60305
f_famb <- f_ramb beta 0.18007 0.03912 4.60305

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value
Error f_ramb psi11 0.28113 0.04640 6.05867
  f_famb psi22 0.22924 0.03889 5.89393
Exogenous rpa v1 1.00000 0.07809 12.80625
  riq v2 1.00000 0.07809 12.80625
  rses v3 1.00000 0.07809 12.80625
  fpa v4 1.00000 0.07809 12.80625
  fiq v5 1.00000 0.07809 12.80625
  fses v6 1.00000 0.07809 12.80625
Error rea theta1 0.33764 0.05178 6.52039
  roa theta2 0.41205 0.05103 8.07403
  fea theta3 0.31337 0.04574 6.85165
  foa theta4 0.40381 0.04608 8.76428

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
rpa riq cov01 0.18390 0.05614 3.27564
rpa rses cov02 0.04890 0.05528 0.88456
riq rses cov03 0.22200 0.05656 3.92503
rpa fpa cov04 0.11470 0.05558 2.06377
riq fpa cov05 0.10210 0.05550 1.83955
rses fpa cov06 0.09310 0.05545 1.67885
rpa fiq cov07 0.07820 0.05538 1.41195
riq fiq cov08 0.33550 0.05824 5.76060
rses fiq cov09 0.23020 0.05666 4.06284
fpa fiq cov10 0.20870 0.05641 3.70000
rpa fses cov11 0.01860 0.05523 0.33680
riq fses cov12 0.18610 0.05616 3.31352
rses fses cov13 0.27070 0.05720 4.73226
fpa fses cov14 -0.04380 0.05527 -0.79249
fiq fses cov15 0.29500 0.05757 5.12435

The difference between the chi-square values for the two preceding models is with degrees of freedom, which is far from significant. This indicates that the restricted model (analysis 2) fits as well as the unrestricted model (analysis 1). However, the chi-square test of the restricted model against the alternative of a completely unrestricted covariance matrix yields a p-value of 0.0596, which indicates that the model might not be entirely satisfactory (p-values from these data are probably too small because of the dependence of the observations).

Loehlin (1987) points out that the models considered are unrealistic in at least two respects. First, the variables of parental aspiration, intelligence, and socioeconomic status are assumed to be measured without error. Loehlin adds uncorrelated measurement errors to the model and assumes, for illustrative purposes, that the reliabilities of these variables are known to be 0.7, 0.8, and 0.9, respectively. In practice, these reliabilities would need to be obtained from a separate study of the same or a very similar population. If these constraints are omitted, the model is not identified. However, constraining parameters to a constant in an analysis of a correlation matrix might make the chi-square goodness-of-fit test inaccurate, so there is more reason to be skeptical of the p-values. Second, the error terms for the respondent’s aspiration are assumed to be uncorrelated with the corresponding terms for his friend. Loehlin introduces a correlation between the two educational aspiration error terms and between the two occupational aspiration error terms. These additions produce the path diagram for Loehlin’s model 1 shown in Figure 17.21.

Figure 17.21 Path Diagram: Career Aspiration – Loehlin (1987)


In Figure 17.21, the observed variables rpa, riq, rses, fses, fiq, and fpa are all treated as measurements with errors. Their purified counterparts f_rpa, f_riq, f_rses, f_fses, f_fiq, and f_fpa are latent variables created in the model to represent measurements without errors. Path coefficients from these latent variables to the observed variables are fixed coefficients, indicating the square roots of the theoretical reliabilities in the model. These latent variables, rather than the observed counterparts, serve as predictors of the ambition variables f_ramb and f_famb. Correlated errors for the occupational aspiration variables and the educational aspiration variables are also shown in Figure 17.21. The error covariance for the educational aspiration variables rea and fea is indicated by the parameter covea, and the error covariance for the occupational aspiration variables roa and foa is indicated by the parameter covoa.


The statements for fitting this model by using the PATH modeling language are as follows:

   proc tcalis corr data=aspire nobs=329;
      path
         /* measurement model for aspiration */
         rea  <-  f_ramb    lambda2,
         roa  <-  f_ramb    1.,
         foa  <-  f_famb    1.,
         fea  <-  f_famb    lambda3,
   
         /* measurement model for intelligence and environment */
         rpa  <-  f_rpa     0.837,
         riq  <-  f_riq     0.894,
         rses <-  f_rses    0.949,
         fses <-  f_fses    0.949,
         fiq  <-  f_fiq     0.894,
         fpa  <-  f_fpa     0.837,
   
         /* structural model of influences */
         f_ramb <- f_rpa     gam1,
         f_ramb <- f_riq     gam2,
         f_ramb <- f_rses    gam3, 
         f_ramb <- f_fses    gam4,
         f_famb <- f_rses    gam5,
         f_famb <- f_fses    gam6, 
         f_famb <- f_fiq     gam7,
         f_famb <- f_fpa     gam8,
         f_ramb <- f_famb    beta1,
         f_famb <- f_ramb    beta2;
      pvar
         f_ramb = psi11,
         f_famb = psi22,
         f_rpa f_riq f_rses f_fses f_fiq f_fpa = 6 * 1.0,
         rea roa fea foa                       = theta1-theta4,
         rpa riq rses fpa fiq fses             = err1-err6;
      pcov
         f_ramb f_famb   = psi12,
         rea  fea        = covea,
         roa  foa        = covoa,
         f_rpa f_riq f_rses f_fses f_fiq f_fpa = 15 * cov__;
   run;

The fit summary is displayed in Figure 17.22, and the estimation results are displayed in Figure 17.23.

Figure 17.22 Career Aspiration Data: Fit Summary of Loehlin (1987) Model 1
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 42
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0366
  Chi-Square 12.0132
  Chi-Square DF 13
  Pr > Chi-Square 0.5266
  Z-Test of Wilson & Hilferty -0.0679
  Hoelter Critical N 612
  Root Mean Square Residual (RMSR) 0.0149
  Standardized RMSR (SRMSR) 0.0149
  Goodness of Fit Index (GFI) 0.9927
Parsimony Index Adjusted GFI (AGFI) 0.9692
  Parsimonious GFI 0.2868
  RMSEA Estimate 0.0000
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0512
  Probability of Close Fit 0.9435
  ECVI Estimate 0.3016
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.3392
  Akaike Information Criterion -13.9868
  Bozdogan CAIC -76.3356
  Schwarz Bayesian Criterion -63.3356
  McDonald Centrality 1.0015
Incremental Index Bentler Comparative Fit Index 1.0000
  Bentler-Bonett NFI 0.9862
  Bentler-Bonett Non-normed Index 1.0041
  Bollen Normed Index Rho1 0.9523
  Bollen Non-normed Index Delta2 1.0011
  James et al. Parsimonious NFI 0.2849

Figure 17.23 Career Aspiration Data: Estimation Results of Loehlin (1987) Model 1
PATH List
Path Parameter Estimate Standard
Error
t Value
rea <- f_ramb lambda2 1.08398 0.09417 11.51054
roa <- f_ramb   1.00000    
foa <- f_famb   1.00000    
fea <- f_famb lambda3 1.11631 0.08627 12.93937
rpa <- f_rpa   0.83700    
riq <- f_riq   0.89400    
rses <- f_rses   0.94900    
fses <- f_fses   0.94900    
fiq <- f_fiq   0.89400    
fpa <- f_fpa   0.83700    
f_ramb <- f_rpa gam1 0.18370 0.05044 3.64201
f_ramb <- f_riq gam2 0.28004 0.06139 4.56183
f_ramb <- f_rses gam3 0.22616 0.05223 4.33004
f_ramb <- f_fses gam4 0.08698 0.05476 1.58836
f_famb <- f_rses gam5 0.06327 0.05219 1.21240
f_famb <- f_fses gam6 0.21539 0.05121 4.20600
f_famb <- f_fiq gam7 0.35386 0.06741 5.24971
f_famb <- f_fpa gam8 0.16877 0.04934 3.42051
f_ramb <- f_famb beta1 0.11897 0.11396 1.04397
f_famb <- f_ramb beta2 0.13022 0.12067 1.07921

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value
Error f_ramb psi11 0.25418 0.04469 5.68738
  f_famb psi22 0.19698 0.03814 5.16533
Exogenous f_rpa   1.00000    
  f_riq   1.00000    
  f_rses   1.00000    
  f_fses   1.00000    
  f_fiq   1.00000    
  f_fpa   1.00000    
Error rea theta1 0.32707 0.05452 5.99883
  roa theta2 0.42307 0.05243 8.06948
  fea theta3 0.28715 0.04804 5.97748
  foa theta4 0.42240 0.04730 8.93103
  rpa err1 0.29584 0.07774 3.80573
  riq err2 0.20874 0.07832 2.66519
  rses err3 0.09887 0.07803 1.26715
  fpa err4 0.29987 0.07807 3.84089
  fiq err5 0.19988 0.07674 2.60475
  fses err6 0.10324 0.07824 1.31950

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov01 0.24677 0.07519 3.28204
f_rpa f_rses cov02 0.06184 0.06945 0.89034
f_riq f_rses cov03 0.26351 0.06687 3.94075
f_rpa f_fses cov04 0.02383 0.06952 0.34272
f_riq f_fses cov05 0.22135 0.06648 3.32976
f_rses f_fses cov06 0.30156 0.06359 4.74205
f_rpa f_fiq cov07 0.10853 0.07362 1.47419
f_riq f_fiq cov08 0.42476 0.07219 5.88373
f_rses f_fiq cov09 0.27250 0.06660 4.09143
f_fses f_fiq cov10 0.34922 0.06771 5.15755
f_rpa f_fpa cov11 0.15789 0.07873 2.00553
f_riq f_fpa cov12 0.13085 0.07418 1.76393
f_rses f_fpa cov13 0.11517 0.06978 1.65053
f_fses f_fpa cov14 -0.05623 0.06971 -0.80655
f_fiq f_fpa cov15 0.27867 0.07530 3.70083

Covariances Among Errors
Error of Error of Parameter Estimate Standard
Error
t Value
f_ramb f_famb psi12 -0.00935 0.05010 -0.18669
rea fea covea 0.02308 0.03139 0.73543
roa foa covoa 0.11206 0.03258 3.43993

Since the p-value for the chi-square test is , this model clearly cannot be rejected. However, Schwarz’s Bayesian criterion for this model (SBC ) is somewhat larger than for Jöreskog and Sörbom’s (1988) analysis 2 (SBC ), suggesting that a more parsimonious model would be desirable.

Since it is assumed that the same model applies to all the boys in the sample, the path diagram should be symmetric with respect to the respondent and his friend. In particular, the corresponding coefficients should be equal. By imposing equality constraints on the pairs of corresponding coefficients, this example obtains Loehlin’s (1987) model 2. The PATH model is as follows, where an OUTMODEL= data set is created to facilitate subsequent hypothesis tests:


   proc tcalis corr data=aspire nobs=329 outmodel=model2;
      path
         /* measurement model for aspiration */
         rea  <-  f_ramb    lambda,              /* 1 ec! */
         roa  <-  f_ramb    1.,
         foa  <-  f_famb    1.,
         fea  <-  f_famb    lambda,
   
         /* measurement model for intelligence and environment */
         rpa  <-  f_rpa     0.837,
         riq  <-  f_riq     0.894,
         rses <-  f_rses    0.949,
         fses <-  f_fses    0.949,
         fiq  <-  f_fiq     0.894,
         fpa  <-  f_fpa     0.837,
   
         /* structural model of influences */
         f_ramb <- f_rpa     gam1,               /* 5 ec! */
         f_ramb <- f_riq     gam2,
         f_ramb <- f_rses    gam3, 
         f_ramb <- f_fses    gam4,
         f_famb <- f_rses    gam4,
         f_famb <- f_fses    gam3, 
         f_famb <- f_fiq     gam2,
         f_famb <- f_fpa     gam1,
         f_ramb <- f_famb    beta,
         f_famb <- f_ramb    beta;
      pvar
         f_ramb = psi,                           /* 1 ec! */
         f_famb = psi,
         f_rpa f_riq f_rses f_fpa f_fiq f_fses = 6 * 1.0,
         rea fea                 = 2 * thetaea,  /* 2 ec! */
         roa foa                 = 2 * thetaoa,
         rpa fpa                 = errpa1 errpa2,
         riq fiq                 = erriq1 erriq2,
         rses fses               = errses1 errses2;
      pcov
         f_ramb f_famb           = psi12,
         rea  fea                = covea,
         roa  foa                = covoa,
         f_rpa f_riq f_rses      = cov1-cov3,          /* 3 ec! */
         f_fpa f_fiq f_fses      = cov1-cov3,
         f_rpa f_riq f_rses * f_fpa f_fiq f_fses =     /* 3 ec! */
             cov4 cov5 cov6
             cov5 cov7 cov8
             cov6 cov8 cov9;
   run;

The fit summary is displayed in Figure 17.24, and the estimation results are displayed in Figure 17.25.

Figure 17.24 Career Aspiration Data: Fit Summary of Loehlin (1987) Model 2
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 27
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0581
  Chi-Square 19.0697
  Chi-Square DF 28
  Pr > Chi-Square 0.8960
  Z-Test of Wilson & Hilferty -1.2599
  Hoelter Critical N 713
  Root Mean Square Residual (RMSR) 0.0276
  Standardized RMSR (SRMSR) 0.0276
  Goodness of Fit Index (GFI) 0.9884
Parsimony Index Adjusted GFI (AGFI) 0.9772
  Parsimonious GFI 0.6150
  RMSEA Estimate 0.0000
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0194
  Probability of Close Fit 0.9996
  ECVI Estimate 0.2285
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.2664
  Akaike Information Criterion -36.9303
  Bozdogan CAIC -171.2200
  Schwarz Bayesian Criterion -143.2200
  McDonald Centrality 1.0137
Incremental Index Bentler Comparative Fit Index 1.0000
  Bentler-Bonett NFI 0.9781
  Bentler-Bonett Non-normed Index 1.0174
  Bollen Normed Index Rho1 0.9649
  Bollen Non-normed Index Delta2 1.0106
  James et al. Parsimonious NFI 0.6086

Figure 17.25 Career Aspiration Data: Estimation Results of Loehlin (1987) Model 2
PATH List
Path Parameter Estimate Standard
Error
t Value
rea <- f_ramb lambda 1.10067 0.06842 16.08795
roa <- f_ramb   1.00000    
foa <- f_famb   1.00000    
fea <- f_famb lambda 1.10067 0.06842 16.08795
rpa <- f_rpa   0.83700    
riq <- f_riq   0.89400    
rses <- f_rses   0.94900    
fses <- f_fses   0.94900    
fiq <- f_fiq   0.89400    
fpa <- f_fpa   0.83700    
f_ramb <- f_rpa gam1 0.17585 0.03508 5.01299
f_ramb <- f_riq gam2 0.32234 0.04702 6.85568
f_ramb <- f_rses gam3 0.22273 0.03629 6.13725
f_ramb <- f_fses gam4 0.07564 0.03750 2.01699
f_famb <- f_rses gam4 0.07564 0.03750 2.01699
f_famb <- f_fses gam3 0.22273 0.03629 6.13725
f_famb <- f_fiq gam2 0.32234 0.04702 6.85568
f_famb <- f_fpa gam1 0.17585 0.03508 5.01299
f_ramb <- f_famb beta 0.11578 0.08390 1.38007
f_famb <- f_ramb beta 0.11578 0.08390 1.38007

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value
Error f_ramb psi 0.22456 0.02971 7.55930
  f_famb psi 0.22456 0.02971 7.55930
Exogenous f_rpa   1.00000    
  f_riq   1.00000    
  f_rses   1.00000    
  f_fpa   1.00000    
  f_fiq   1.00000    
  f_fses   1.00000    
Error rea thetaea 0.30662 0.03726 8.22956
  fea thetaea 0.30662 0.03726 8.22956
  roa thetaoa 0.42295 0.03651 11.58311
  foa thetaoa 0.42295 0.03651 11.58311
  rpa errpa1 0.30758 0.07511 4.09498
  fpa errpa2 0.28834 0.07369 3.91289
  riq erriq1 0.26656 0.07389 3.60730
  fiq erriq2 0.15573 0.06700 2.32445
  rses errses1 0.11467 0.07267 1.57800
  fses errses2 0.08814 0.07089 1.24333

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.26470 0.05442 4.86370
f_rpa f_rses cov2 0.00177 0.04996 0.03533
f_riq f_rses cov3 0.31129 0.05057 6.15553
f_fpa f_fiq cov1 0.26470 0.05442 4.86370
f_fpa f_fses cov2 0.00177 0.04996 0.03533
f_fiq f_fses cov3 0.31129 0.05057 6.15553
f_rpa f_fpa cov4 0.15784 0.07872 2.00521
f_rpa f_fiq cov5 0.11837 0.05447 2.17325
f_rpa f_fses cov6 0.06910 0.04996 1.38303
f_riq f_fpa cov5 0.11837 0.05447 2.17325
f_riq f_fiq cov7 0.43061 0.07258 5.93255
f_riq f_fses cov8 0.24967 0.05060 4.93420
f_rses f_fpa cov6 0.06910 0.04996 1.38303
f_rses f_fiq cov8 0.24967 0.05060 4.93420
f_rses f_fses cov9 0.30190 0.06362 4.74578

Covariances Among Errors
Error of Error of Parameter Estimate Standard
Error
t Value
f_ramb f_famb psi12 -0.00344 0.04931 -0.06981
rea fea covea 0.02160 0.03144 0.68686
roa foa covoa 0.11208 0.03257 3.44076

The test of Loehlin’s (1987) model 2 against model 1 yields a chi-square of with degrees of freedom, which is clearly not significant. This indicates the restricted model 2 fits at least as well as model 1. Schwarz’s Bayesian criterion (SBC) is also much lower for model 2 () than for model 1 (). Hence, model 2 seems preferable on both substantive and statistical grounds.

A question of substantive interest is whether the friend’s socioeconomic status (SES) has a significant direct influence on a boy’s ambition. This can be addressed by omitting the paths from f_fses to f_ramb and from f_rses to f_famb designated by the parameter name gam4, yielding Loehlin’s (1987) model 3:

   title2 'Loehlin (1987) analysis: Model 3';
   data model3(type=calismdl);
      set model2;
      if _name_='gam4' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;
   proc tcalis corr data=aspire nobs=329 inmodel=model3;
   run;

The fit summary is displayed in Figure 17.26.

Figure 17.26 Career Aspiration Data: Fit Summary of Loehlin (1987) Model 3
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 26
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0702
  Chi-Square 23.0365
  Chi-Square DF 29
  Pr > Chi-Square 0.7749
  Z-Test of Wilson & Hilferty -0.7563
  Hoelter Critical N 607
  Root Mean Square Residual (RMSR) 0.0304
  Standardized RMSR (SRMSR) 0.0304
  Goodness of Fit Index (GFI) 0.9858
Parsimony Index Adjusted GFI (AGFI) 0.9731
  Parsimonious GFI 0.6353
  RMSEA Estimate 0.0000
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0295
  Probability of Close Fit 0.9984
  ECVI Estimate 0.2343
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.2780
  Akaike Information Criterion -34.9635
  Bozdogan CAIC -174.0492
  Schwarz Bayesian Criterion -145.0492
  McDonald Centrality 1.0091
Incremental Index Bentler Comparative Fit Index 1.0000
  Bentler-Bonett NFI 0.9736
  Bentler-Bonett Non-normed Index 1.0112
  Bollen Normed Index Rho1 0.9590
  Bollen Non-normed Index Delta2 1.0071
  James et al. Parsimonious NFI 0.6274

The chi-square value for testing model 3 versus model 2 is with degree of freedom and a p-value of . Although the parameter is of marginal significance, the estimate in model 2 () is small compared to the other coefficients, and SBC indicates that model 3 is preferable to model 2.

Another important question is whether the reciprocal influences between the respondent’s and friend’s ambitions are needed in the model. To test whether these paths are zero, set the parameter beta for the paths linking f_ramb and f_famb to zero to obtain Loehlin’s (1987) model 4:


   title2 'Loehlin (1987) analysis: Model 4';
   data model4(type=calismdl);
      set model2;
      if _name_='beta' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;
   proc tcalis corr data=aspire nobs=329 inmodel=model4;
   run;

The fit summary is displayed in Figure 17.27, and the estimation results are displayed in Figure 17.28.

Figure 17.27 Career Aspiration Data: Fit Summary of Loehlin (1987) Model 4
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 26
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0640
  Chi-Square 20.9981
  Chi-Square DF 29
  Pr > Chi-Square 0.8592
  Z-Test of Wilson & Hilferty -1.0780
  Hoelter Critical N 666
  Root Mean Square Residual (RMSR) 0.0304
  Standardized RMSR (SRMSR) 0.0304
  Goodness of Fit Index (GFI) 0.9873
Parsimony Index Adjusted GFI (AGFI) 0.9760
  Parsimonious GFI 0.6363
  RMSEA Estimate 0.0000
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0234
  Probability of Close Fit 0.9994
  ECVI Estimate 0.2281
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.2685
  Akaike Information Criterion -37.0019
  Bozdogan CAIC -176.0876
  Schwarz Bayesian Criterion -147.0876
  McDonald Centrality 1.0122
Incremental Index Bentler Comparative Fit Index 1.0000
  Bentler-Bonett NFI 0.9759
  Bentler-Bonett Non-normed Index 1.0150
  Bollen Normed Index Rho1 0.9626
  Bollen Non-normed Index Delta2 1.0095
  James et al. Parsimonious NFI 0.6289

Figure 17.28 Career Aspiration Data: Estimation Results of Loehlin (1987) Model 4
PATH List
Path Parameter Estimate Standard
Error
t Value
rea <- f_ramb lambda 1.10505 0.06804 16.24157
roa <- f_ramb   1.00000    
foa <- f_famb   1.00000    
fea <- f_famb lambda 1.10505 0.06804 16.24157
rpa <- f_rpa   0.83700    
riq <- f_riq   0.89400    
rses <- f_rses   0.94900    
fses <- f_fses   0.94900    
fiq <- f_fiq   0.89400    
fpa <- f_fpa   0.83700    
f_ramb <- f_rpa gam1 0.17760 0.03610 4.91945
f_ramb <- f_riq gam2 0.34856 0.04625 7.53618
f_ramb <- f_rses gam3 0.23834 0.03549 6.71576
f_ramb <- f_fses gam4 0.10810 0.02992 3.61340
f_famb <- f_rses gam4 0.10810 0.02992 3.61340
f_famb <- f_fses gam3 0.23834 0.03549 6.71576
f_famb <- f_fiq gam2 0.34856 0.04625 7.53618
f_famb <- f_fpa gam1 0.17760 0.03610 4.91945
f_ramb <- f_famb   0    
f_famb <- f_ramb   0    

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value
Error f_ramb psi 0.22738 0.03140 7.24263
  f_famb psi 0.22738 0.03140 7.24263
Exogenous f_rpa   1.00000    
  f_riq   1.00000    
  f_rses   1.00000    
  f_fpa   1.00000    
  f_fiq   1.00000    
  f_fses   1.00000    
Error rea thetaea 0.30502 0.03728 8.18091
  fea thetaea 0.30502 0.03728 8.18091
  roa thetaoa 0.42429 0.03645 11.64071
  foa thetaoa 0.42429 0.03645 11.64071
  rpa errpa1 0.31354 0.07543 4.15664
  fpa errpa2 0.29051 0.07374 3.93945
  riq erriq1 0.29611 0.07299 4.05703
  fiq erriq2 0.18181 0.06611 2.75034
  rses errses1 0.12320 0.07273 1.69400
  fses errses2 0.09873 0.07109 1.38881

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.27241 0.05520 4.93523
f_rpa f_rses cov2 0.00476 0.05032 0.09455
f_riq f_rses cov3 0.32463 0.05089 6.37870
f_fpa f_fiq cov1 0.27241 0.05520 4.93523
f_fpa f_fses cov2 0.00476 0.05032 0.09455
f_fiq f_fses cov3 0.32463 0.05089 6.37870
f_rpa f_fpa cov4 0.16949 0.07863 2.15559
f_rpa f_fiq cov5 0.13539 0.05407 2.50384
f_rpa f_fses cov6 0.07362 0.05027 1.46453
f_riq f_fpa cov5 0.13539 0.05407 2.50384
f_riq f_fiq cov7 0.46893 0.06980 6.71822
f_riq f_fses cov8 0.26289 0.05093 5.16164
f_rses f_fpa cov6 0.07362 0.05027 1.46453
f_rses f_fiq cov8 0.26289 0.05093 5.16164
f_rses f_fses cov9 0.30880 0.06409 4.81849

Covariances Among Errors
Error of Error of Parameter Estimate Standard
Error
t Value
f_ramb f_famb psi12 0.05479 0.02699 2.03009
rea fea covea 0.02127 0.03150 0.67534
roa foa covoa 0.11245 0.03258 3.45136

The chi-square value for testing model 4 versus model 2 is with degree of freedom and a p-value of . Hence, there is little evidence of reciprocal influence.

Loehlin’s (1987) model 2 has not only the direct paths connecting the latent ambition variables f_ramb and f_famb but also a covariance between the disturbance terms d_ramb and d_famb to allow for other variables omitted from the model that might jointly influence the respondent and his friend. To test the hypothesis that this covariance is zero, set the parameter psi12 to zero, yielding Loehlin’s (1987) model 5:

   title2 'Loehlin (1987) analysis: Model 5';
   data model5(type=calismdl);
      set model2;
      if _name_='psi12' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;
   proc tcalis corr data=aspire nobs=329 inmodel=model5;
   run;

The fit summary is displayed in Figure 17.29, and the estimation results are displayed in Figure 17.30.

Figure 17.29 Career Aspiration Data: Fit Summary of Loehlin (1987) Model 5
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 26
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0582
  Chi-Square 19.0745
  Chi-Square DF 29
  Pr > Chi-Square 0.9194
  Z-Test of Wilson & Hilferty -1.4014
  Hoelter Critical N 733
  Root Mean Square Residual (RMSR) 0.0276
  Standardized RMSR (SRMSR) 0.0276
  Goodness of Fit Index (GFI) 0.9884
Parsimony Index Adjusted GFI (AGFI) 0.9780
  Parsimonious GFI 0.6370
  RMSEA Estimate 0.0000
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0152
  Probability of Close Fit 0.9998
  ECVI Estimate 0.2222
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.2592
  Akaike Information Criterion -38.9255
  Bozdogan CAIC -178.0111
  Schwarz Bayesian Criterion -149.0111
  McDonald Centrality 1.0152
Incremental Index Bentler Comparative Fit Index 1.0000
  Bentler-Bonett NFI 0.9781
  Bentler-Bonett Non-normed Index 1.0186
  Bollen Normed Index Rho1 0.9661
  Bollen Non-normed Index Delta2 1.0118
  James et al. Parsimonious NFI 0.6303

Figure 17.30 Career Aspiration Data: Estimation Results of Loehlin (1987) Model 5
PATH List
Path Parameter Estimate Standard
Error
t Value
rea <- f_ramb lambda 1.10086 0.06836 16.10408
roa <- f_ramb   1.00000    
foa <- f_famb   1.00000    
fea <- f_famb lambda 1.10086 0.06836 16.10408
rpa <- f_rpa   0.83700    
riq <- f_riq   0.89400    
rses <- f_rses   0.94900    
fses <- f_fses   0.94900    
fiq <- f_fiq   0.89400    
fpa <- f_fpa   0.83700    
f_ramb <- f_rpa gam1 0.17618 0.03502 5.03081
f_ramb <- f_riq gam2 0.32351 0.04346 7.44351
f_ramb <- f_rses gam3 0.22334 0.03533 6.32150
f_ramb <- f_fses gam4 0.07698 0.03225 2.38702
f_famb <- f_rses gam4 0.07698 0.03225 2.38702
f_famb <- f_fses gam3 0.22334 0.03533 6.32150
f_famb <- f_fiq gam2 0.32351 0.04346 7.44351
f_famb <- f_fpa gam1 0.17618 0.03502 5.03081
f_ramb <- f_famb beta 0.11074 0.04283 2.58539
f_famb <- f_ramb beta 0.11074 0.04283 2.58539

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value
Error f_ramb psi 0.22453 0.02973 7.55201
  f_famb psi 0.22453 0.02973 7.55201
Exogenous f_rpa   1.00000    
  f_riq   1.00000    
  f_rses   1.00000    
  f_fpa   1.00000    
  f_fiq   1.00000    
  f_fses   1.00000    
Error rea thetaea 0.30645 0.03721 8.23647
  fea thetaea 0.30645 0.03721 8.23647
  roa thetaoa 0.42304 0.03650 11.58877
  foa thetaoa 0.42304 0.03650 11.58877
  rpa errpa1 0.30781 0.07510 4.09880
  fpa errpa2 0.28837 0.07366 3.91467
  riq erriq1 0.26748 0.07295 3.66672
  fiq erriq2 0.15653 0.06614 2.36682
  rses errses1 0.11477 0.07265 1.57975
  fses errses2 0.08832 0.07088 1.24608

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.26494 0.05436 4.87395
f_rpa f_rses cov2 0.00185 0.04995 0.03696
f_riq f_rses cov3 0.31164 0.05039 6.18460
f_fpa f_fiq cov1 0.26494 0.05436 4.87395
f_fpa f_fses cov2 0.00185 0.04995 0.03696
f_fiq f_fses cov3 0.31164 0.05039 6.18460
f_rpa f_fpa cov4 0.15828 0.07846 2.01729
f_rpa f_fiq cov5 0.11895 0.05383 2.20978
f_rpa f_fses cov6 0.06924 0.04993 1.38664
f_riq f_fpa cov5 0.11895 0.05383 2.20978
f_riq f_fiq cov7 0.43180 0.07084 6.09540
f_riq f_fses cov8 0.25004 0.05039 4.96207
f_rses f_fpa cov6 0.06924 0.04993 1.38664
f_rses f_fiq cov8 0.25004 0.05039 4.96207
f_rses f_fses cov9 0.30203 0.06360 4.74852

Covariances Among Errors
Error of Error of Parameter Estimate Standard
Error
t Value
f_ramb f_famb   0    
rea fea covea 0.02120 0.03094 0.68516
roa foa covoa 0.11197 0.03254 3.44068

The chi-square value for testing model 5 versus model 2 is with degree of freedom. This test statistic is insignificant. Omitting the covariance between the disturbance terms, therefore, causes hardly any deterioration in the fit of the model.

These data fail to provide evidence of direct reciprocal influence between the respondent’s and friend’s ambitions or of a covariance between the disturbance terms when these hypotheses are considered separately. Notice, however, that the covariance psi12 between the disturbance terms increases from for model 2 to for model 4. Before you conclude that all of these paths can be omitted from the model, it is important to test both hypotheses together by setting both beta and psi12 to zero as in Loehlin’s (1987) model 7:


   title2 'Loehlin (1987) analysis: Model 7';
   data model7(type=calismdl);
      set model2;
      if _name_='psi12'|_name_='beta' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;
   proc tcalis corr data=aspire nobs=329 inmodel=model7;
   run;

The fit summary is displayed in Figure 17.31, and the estimation results are displayed in Figure 17.32.

Figure 17.31 Career Aspiration Data: Fit Summary of Loehlin (1987) Model 7
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 25
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.0773
  Chi-Square 25.3466
  Chi-Square DF 30
  Pr > Chi-Square 0.7080
  Z-Test of Wilson & Hilferty -0.5487
  Hoelter Critical N 568
  Root Mean Square Residual (RMSR) 0.0363
  Standardized RMSR (SRMSR) 0.0363
  Goodness of Fit Index (GFI) 0.9846
Parsimony Index Adjusted GFI (AGFI) 0.9718
  Parsimonious GFI 0.6564
  RMSEA Estimate 0.0000
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0326
  Probability of Close Fit 0.9975
  ECVI Estimate 0.2350
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.2815
  Akaike Information Criterion -34.6534
  Bozdogan CAIC -178.5351
  Schwarz Bayesian Criterion -148.5351
  McDonald Centrality 1.0071
Incremental Index Bentler Comparative Fit Index 1.0000
  Bentler-Bonett NFI 0.9709
  Bentler-Bonett Non-normed Index 1.0084
  Bollen Normed Index Rho1 0.9564
  Bollen Non-normed Index Delta2 1.0055
  James et al. Parsimonious NFI 0.6473

Figure 17.32 Career Aspiration Data: Estimation Results of Loehlin (1987) Model 7
PATH List
Path Parameter Estimate Standard
Error
t Value
rea <- f_ramb lambda 1.10371 0.06784 16.27015
roa <- f_ramb   1.00000    
foa <- f_famb   1.00000    
fea <- f_famb lambda 1.10371 0.06784 16.27015
rpa <- f_rpa   0.83700    
riq <- f_riq   0.89400    
rses <- f_rses   0.94900    
fses <- f_fses   0.94900    
fiq <- f_fiq   0.89400    
fpa <- f_fpa   0.83700    
f_ramb <- f_rpa gam1 0.17653 0.03604 4.89810
f_ramb <- f_riq gam2 0.35727 0.04609 7.75204
f_ramb <- f_rses gam3 0.24187 0.03628 6.66710
f_ramb <- f_fses gam4 0.11087 0.03056 3.62795
f_famb <- f_rses gam4 0.11087 0.03056 3.62795
f_famb <- f_fses gam3 0.24187 0.03628 6.66710
f_famb <- f_fiq gam2 0.35727 0.04609 7.75204
f_famb <- f_fpa gam1 0.17653 0.03604 4.89810
f_ramb <- f_famb   0    
f_famb <- f_ramb   0    

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value
Error f_ramb psi 0.21011 0.02940 7.14704
  f_famb psi 0.21011 0.02940 7.14704
Exogenous f_rpa   1.00000    
  f_riq   1.00000    
  f_rses   1.00000    
  f_fpa   1.00000    
  f_fiq   1.00000    
  f_fses   1.00000    
Error rea thetaea 0.31633 0.03648 8.67106
  fea thetaea 0.31633 0.03648 8.67106
  roa thetaoa 0.42656 0.03610 11.81508
  foa thetaoa 0.42656 0.03610 11.81508
  rpa errpa1 0.31329 0.07538 4.15589
  fpa errpa2 0.29286 0.07389 3.96366
  riq erriq1 0.30776 0.07307 4.21157
  fiq erriq2 0.19193 0.06613 2.90250
  rses errses1 0.14303 0.07313 1.95574
  fses errses2 0.11804 0.07147 1.65171

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.27533 0.05552 4.95900
f_rpa f_rses cov2 0.00611 0.05085 0.12020
f_riq f_rses cov3 0.33510 0.05150 6.50648
f_fpa f_fiq cov1 0.27533 0.05552 4.95900
f_fpa f_fses cov2 0.00611 0.05085 0.12020
f_fiq f_fses cov3 0.33510 0.05150 6.50648
f_rpa f_fpa cov4 0.17099 0.07872 2.17210
f_rpa f_fiq cov5 0.13859 0.05431 2.55174
f_rpa f_fses cov6 0.07563 0.05077 1.48956
f_riq f_fpa cov5 0.13859 0.05431 2.55174
f_riq f_fiq cov7 0.48105 0.06993 6.87858
f_riq f_fses cov8 0.27235 0.05157 5.28154
f_rses f_fpa cov6 0.07563 0.05077 1.48956
f_rses f_fiq cov8 0.27235 0.05157 5.28154
f_rses f_fses cov9 0.32046 0.06517 4.91719

Covariances Among Errors
Error of Error of Parameter Estimate Standard
Error
t Value
f_ramb f_famb   0    
rea fea covea 0.04535 0.02918 1.55444
roa foa covoa 0.12085 0.03214 3.75976

When model 7 is tested against models 2, 4, and 5, the p-values are respectively 0.0433, 0.0370, and 0.0123, indicating that the combined effect of the reciprocal influence and the covariance of the disturbance terms is statistically significant. Thus, the hypothesis tests indicate that it is acceptable to omit either the reciprocal influences or the covariance of the disturbances, but not both.

It is also of interest to test the covariances between the error terms for educational (covea) and occupational aspiration (covoa), since these terms are omitted from Jöreskog and Sörbom’s (1988) models. Constraining covea and covoa to zero produces Loehlin’s (1987) model 6:

   title2 'Loehlin (1987) analysis: Model 6';
   data model6(type=calismdl);
      set model2;
      if _name_='covea'|_name_='covoa' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;
   proc tcalis corr data=aspire nobs=329 inmodel=model6;
   run;

The fit summary is displayed in Figure 17.33.

Figure 17.33 Career Aspiration Data: Loehlin (1987) Model 6
Fit Summary
Modeling Info N Observations 329
  N Variables 10
  N Moments 55
  N Parameters 25
  N Active Constraints 0
  Independence Model Chi-Square 872.0008
  Independence Model Chi-Square DF 45
Absolute Index Fit Function 0.1020
  Chi-Square 33.4475
  Chi-Square DF 30
  Pr > Chi-Square 0.3035
  Z-Test of Wilson & Hilferty 0.5151
  Hoelter Critical N 431
  Root Mean Square Residual (RMSR) 0.0306
  Standardized RMSR (SRMSR) 0.0306
  Goodness of Fit Index (GFI) 0.9802
Parsimony Index Adjusted GFI (AGFI) 0.9638
  Parsimonious GFI 0.6535
  RMSEA Estimate 0.0187
  RMSEA Lower 90% Confidence Limit .
  RMSEA Upper 90% Confidence Limit 0.0471
  Probability of Close Fit 0.9686
  ECVI Estimate 0.2597
  ECVI Lower 90% Confidence Limit .
  ECVI Upper 90% Confidence Limit 0.3164
  Akaike Information Criterion -26.5525
  Bozdogan CAIC -170.4342
  Schwarz Bayesian Criterion -140.4342
  McDonald Centrality 0.9948
Incremental Index Bentler Comparative Fit Index 0.9958
  Bentler-Bonett NFI 0.9616
  Bentler-Bonett Non-normed Index 0.9937
  Bollen Normed Index Rho1 0.9425
  Bollen Non-normed Index Delta2 0.9959
  James et al. Parsimonious NFI 0.6411

The chi-square value for testing model 6 versus model 2 is with degrees of freedom and a p-value of , indicating that there is considerable evidence of correlation between the error terms.

The following table summarizes the results from Loehlin’s (1987) seven models.

Model

df

p-value

SBC

1. Full model

2. Equality constraints

3. No SES path

4. No reciprocal influence

5. No disturbance correlation

6. No error correlation

7. Constraints from both 4 and 5

For comparing models, you can use a DATA step to compute the differences of the chi-square statistics and p-values:

   data _null_;
      array achisq[7] _temporary_
         (12.0132 19.0697 23.0365 20.9981 19.0745 33.4475 25.3466);
      array adf[7] _temporary_
         (13 28 29 29 29 30 30);
      retain indent 16;
      file print;
      input ho ha @@;
      chisq = achisq[ho] - achisq[ha];
      df = adf[ho] - adf[ha];
      p = 1 - probchi( chisq, df);
      if _n_ = 1 then put
         / +indent 'model comparison   chi**2   df  p-value'
         / +indent '---------------------------------------';
      put +indent +3 ho ' versus ' ha @18 +indent chisq 8.4 df 5. p 9.4;
   datalines;
   2 1    3 2    4 2    5 2    7 2    7 4    7 5    6 2
   ;

The DATA step displays the table in Figure 17.34.

Figure 17.34 Career Aspiration Data: Model Comparisons
                model comparison   chi**2   df  p-value                         
                ---------------------------------------                         
                   2  versus 1     7.0565   15   0.9561                         
                   3  versus 2     3.9668    1   0.0464                         
                   4  versus 2     1.9284    1   0.1649                         
                   5  versus 2     0.0048    1   0.9448                         
                   7  versus 2     6.2769    2   0.0433                         
                   7  versus 4     4.3485    1   0.0370                         
                   7  versus 5     6.2721    1   0.0123                         
                   6  versus 2    14.3778    2   0.0008                         

Although none of the seven models can be rejected when tested against the alternative of an unrestricted covariance matrix, the model comparisons make it clear that there are important differences among the models. Schwarz’s Bayesian criterion indicates model 5 as the model of choice. The constraints added to model 5 in model 7 can be rejected (p=0.0123), while model 5 cannot be rejected when tested against the less constrained model 2 (p=0.9448). Hence, among the small number of models considered, model 5 has strong statistical support. However, as Loehlin (1987, p. 106) points out, many other models for these data could be constructed. Further analysis should consider, in addition to simple modifications of the models, the possibility that more than one friend could influence a boy’s aspirations, and that a boy’s ambition might have some effect on his choice of friends. Pursuing such theories would be statistically challenging.

Previous Page | Next Page | Top of Page