Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
Introduction to Structural Equations 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), J\ddot{o}reskog and S\ddot{o}rbom (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 may 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 may be as small as half of the reported 329 observations.

The correlation matrix is taken from J\ddot{o}reskog and S\ddot{o}rbom (1988).

   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 J\ddot{o}reskog and S\ddot{o}rbom (1988) is displayed in the following path diagram:

icaf4.gif (5830 bytes)

Figure 14.17: Path Diagram: Career Aspiration, J\ddot{o}reskog and S\ddot{o}rbom

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 and socioeconomic status, his perception of his parents' aspiration for him, and his friend's socioeconomic status and ambition. It is assumed that his friend's intelligence and socioeconomic status affect the respondent's ambition only indirectly through his friend's ambition. Ambition is indexed by the manifest variables of occupational and educational aspiration, 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 CALIS using the LINEQS statement as follows, where the names of the parameters correspond to those used by J\ddot{o}reskog and S\ddot{o}rbom (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 degrees of freedom (sample size minus one) with the EDF= option in the PROC CALIS statement.

   title2 'Joreskog-Sorbom (1988) analysis 1';
   proc calis data=aspire edf=328;
      lineqs    /* measurement model for aspiration */
             rea=lambda2 f_ramb + e_rea,
             roa=f_ramb + e_roa,
             fea=lambda3 f_famb + e_fea,
             foa=f_famb + e_foa,
                /* structural model of influences */
             f_ramb=gam1 rpa + gam2 riq + gam3 rses +
                gam4 fses + beta1 f_famb + d_ramb,
             f_famb=gam8 fpa + gam7 fiq + gam6 fses +
                gam5 rses + beta2 f_ramb + d_famb;
      std d_ramb=psi11,
          d_famb=psi22,
          e_rea e_roa e_fea e_foa=theta:;
      cov d_ramb d_famb=psi12,
          rpa riq rses fpa fiq fses=cov:;
   run;

Specify a name followed by a colon to represent a list of names formed by appending numbers to the specified name. For example, in the COV statement, the line

   rpa riq rses fpa fiq fses=cov:;

is equivalent to

   rpa riq rses fpa fiq fses=cov1-cov15;

The results from this analysis are as follows.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Joreskog-Sorbom (1988) analysis 1

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0814
Goodness of Fit Index (GFI) 0.9844
GFI Adjusted for Degrees of Freedom (AGFI) 0.9428
Root Mean Square Residual (RMR) 0.0202
Parsimonious GFI (Mulaik, 1989) 0.3281
Chi-Square 26.6972
Chi-Square DF 15
Pr > Chi-Square 0.0313
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0488
RMSEA 90% Lower Confidence Limit 0.0145
RMSEA 90% Upper Confidence Limit 0.0783
ECVI Estimate 0.2959
ECVI 90% Lower Confidence Limit 0.2823
ECVI 90% Upper Confidence Limit 0.3721
Probability of Close Fit 0.4876
Bentler's Comparative Fit Index 0.9859
Normal Theory Reweighted LS Chi-Square 26.0113
Akaike's Information Criterion -3.3028
Bozdogan's (1987) CAIC -75.2437
Schwarz's Bayesian Criterion -60.2437
McDonald's (1989) Centrality 0.9824
Bentler & Bonett's (1980) Non-normed Index 0.9576
Bentler & Bonett's (1980) NFI 0.9694
James, Mulaik, & Brett (1982) Parsimonious NFI 0.3231
Z-Test of Wilson & Hilferty (1931) 1.8625
Bollen (1986) Normed Index Rho1 0.9082
Bollen (1988) Non-normed Index Delta2 0.9864
Hoelter's (1983) Critical N 309

Figure 14.18: Career Aspiration Data: J&S Analysis 1

J\ddot{o}reskog and S\ddot{o}rbom (1988) present more detailed results from a second analysis in which two constraints are imposed:

This analysis can be performed by changing the names beta1 and beta2 to beta and omitting the line from the COV statement for psi12:

   title2 'Joreskog-Sorbom (1988) analysis 2';
   proc calis data=aspire edf=328;
      lineqs    /* measurement model for aspiration */
             rea=lambda2 f_ramb + e_rea,
             roa=f_ramb + e_roa,
             fea=lambda3 f_famb + e_fea,
             foa=f_famb + e_foa,
                /* structural model of influences */
             f_ramb=gam1 rpa + gam2 riq + gam3 rses +
                gam4 fses + beta f_famb + d_ramb,

             f_famb=gam8 fpa + gam7 fiq + gam6 fses +
                gam5 rses + beta f_ramb + d_famb;
      std d_ramb=psi11,
          d_famb=psi22,
          e_rea e_roa e_fea e_foa=theta:;
      cov rpa riq rses fpa fiq fses=cov:;
   run;

The results are displayed in Figure 14.19.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Joreskog-Sorbom (1988) analysis 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0820
Goodness of Fit Index (GFI) 0.9843
GFI Adjusted for Degrees of Freedom (AGFI) 0.9492
Root Mean Square Residual (RMR) 0.0203
Parsimonious GFI (Mulaik, 1989) 0.3718
Chi-Square 26.8987
Chi-Square DF 17
Pr > Chi-Square 0.0596
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0421
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0710
ECVI Estimate 0.2839
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.3592
Probability of Close Fit 0.6367
Bentler's Comparative Fit Index 0.9880
Normal Theory Reweighted LS Chi-Square 26.1595
Akaike's Information Criterion -7.1013
Bozdogan's (1987) CAIC -88.6343
Schwarz's Bayesian Criterion -71.6343
McDonald's (1989) Centrality 0.9851
Bentler & Bonett's (1980) Non-normed Index 0.9683
Bentler & Bonett's (1980) NFI 0.9692
James, Mulaik, & Brett (1982) Parsimonious NFI 0.3661
Z-Test of Wilson & Hilferty (1931) 1.5599
Bollen (1986) Normed Index Rho1 0.9183
Bollen (1988) Non-normed Index Delta2 0.9884
Hoelter's (1983) Critical N 338

Figure 14.19: Career Aspiration Data: J&S Analysis 2

Peer Influences on Aspiration: Haller & Butterworth (1960)
Joreskog-Sorbom (1988) analysis 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

roa = 1.0000   f_ramb + 1.0000   e_roa
rea = 1.0610 * f_ramb + 1.0000   e_rea
Std Err   0.0892   lambda2        
t Value   11.8923            
foa = 1.0000   f_famb + 1.0000   e_foa
fea = 1.0736 * f_famb + 1.0000   e_fea
Std Err   0.0806   lambda3        
t Value   13.3150            

 


Peer Influences on Aspiration: Haller & Butterworth (1960)
Joreskog-Sorbom (1988) analysis 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

f_ramb = 0.1801 * f_famb + 0.2540 * riq + 0.1637 * rpa + 0.2211 * rses + 0.0773 * fses + 1.0000   d_ramb
Std Err   0.0391   beta   0.0419   gam2   0.0387   gam1   0.0419   gam3   0.0415   gam4        
t Value   4.6031       6.0673       4.2274       5.2822       1.8626            
f_famb = 0.1801 * f_ramb + 0.0684 * rses + 0.3306 * fiq + 0.1520 * fpa + 0.2184 * fses + 1.0000   d_famb
Std Err   0.0391   beta   0.0387   gam5   0.0412   gam7   0.0364   gam8   0.0395   gam6        
t Value   4.6031       1.7681       8.0331       4.1817       5.5320            


Peer Influences on Aspiration: Haller & Butterworth (1960)
Joreskog-Sorbom (1988) analysis 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable Parameter Estimate Standard
Error
t Value
riq   1.00000    
rpa   1.00000    
rses   1.00000    
fiq   1.00000    
fpa   1.00000    
fses   1.00000    
e_rea theta1 0.33764 0.05178 6.52
e_roa theta2 0.41205 0.05103 8.07
e_fea theta3 0.31337 0.04574 6.85
e_foa theta4 0.40381 0.04608 8.76
d_ramb psi11 0.28113 0.04640 6.06
d_famb psi22 0.22924 0.03889 5.89

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
riq rpa cov1 0.18390 0.05246 3.51
riq rses cov3 0.22200 0.05110 4.34
rpa rses cov2 0.04890 0.05493 0.89
riq fiq cov8 0.33550 0.04641 7.23
rpa fiq cov7 0.07820 0.05455 1.43
rses fiq cov9 0.23020 0.05074 4.54
riq fpa cov5 0.10210 0.05415 1.89
rpa fpa cov4 0.11470 0.05412 2.12
rses fpa cov6 0.09310 0.05438 1.71
fiq fpa cov10 0.20870 0.05163 4.04
riq fses cov12 0.18610 0.05209 3.57
rpa fses cov11 0.01860 0.05510 0.34
rses fses cov13 0.27070 0.04930 5.49
fiq fses cov15 0.29500 0.04824 6.12
fpa fses cov14 -0.04380 0.05476 -0.80


The difference between the chi-square values for the two preceding models is 26.8987 - 26.6972= 0.2015 with 2 degrees of freedom, which is far from significant. However, the chi-square test of the restricted model (analysis 2) against the alternative of a completely unrestricted covariance matrix yields a p-value of 0.0596, which indicates that the model may 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 aspects. 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 may 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 following path diagram for Loehlin's model 1.

icaf5.gif (8703 bytes)

Figure 14.20: Path Diagram: Carrer Aspiration -- Loehlin

The statements for fitting this model are as follows:

   title2 'Loehlin (1987) analysis: Model 1';
   proc calis data=aspire edf=328;
      lineqs    /* measurement model for aspiration */
             rea=lambda2 f_ramb + e_rea,
             roa=f_ramb + e_roa,
             fea=lambda3 f_famb + e_fea,
             foa=f_famb + e_foa,
             /* measurement model for intelligence and environment */
             rpa=.837 f_rpa + e_rpa,
             riq=.894 f_riq + e_riq,
             rses=.949 f_rses + e_rses,
             fpa=.837 f_fpa + e_fpa,
             fiq=.894 f_fiq + e_fiq,
             fses=.949 f_fses + e_fses,
                /* structural model of influences */
             f_ramb=gam1 f_rpa + gam2 f_riq + gam3 f_rses +
                gam4 f_fses + bet1 f_famb + d_ramb,
             f_famb=gam8 f_fpa + gam7 f_fiq + gam6 f_fses +
                gam5 f_rses + bet2 f_ramb + d_famb;
      std d_ramb=psi11,
          d_famb=psi22,
          f_rpa f_riq f_rses f_fpa f_fiq f_fses=1,
          e_rea e_roa e_fea e_foa=theta:,
          e_rpa e_riq e_rses e_fpa e_fiq e_fses=err:;
      cov d_ramb d_famb=psi12,
          e_rea e_fea=covea,
          e_roa e_foa=covoa,
          f_rpa f_riq f_rses f_fpa f_fiq f_fses=cov:;
   run;

The results are displayed in Figure 14.21.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 1

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0366
Goodness of Fit Index (GFI) 0.9927
GFI Adjusted for Degrees of Freedom (AGFI) 0.9692
Root Mean Square Residual (RMR) 0.0149
Parsimonious GFI (Mulaik, 1989) 0.2868
Chi-Square 12.0132
Chi-Square DF 13
Pr > Chi-Square 0.5266
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0512
ECVI Estimate 0.3016
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.3392
Probability of Close Fit 0.9435
Bentler's Comparative Fit Index 1.0000
Normal Theory Reweighted LS Chi-Square 12.0168
Akaike's Information Criterion -13.9868
Bozdogan's (1987) CAIC -76.3356
Schwarz's Bayesian Criterion -63.3356
McDonald's (1989) Centrality 1.0015
Bentler & Bonett's (1980) Non-normed Index 1.0041
Bentler & Bonett's (1980) NFI 0.9862
James, Mulaik, & Brett (1982) Parsimonious NFI 0.2849
Z-Test of Wilson & Hilferty (1931) -0.0679
Bollen (1986) Normed Index Rho1 0.9523
Bollen (1988) Non-normed Index Delta2 1.0011
Hoelter's (1983) Critical N 612

Figure 14.21: Career Aspiration Data: Loehlin Model 1

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 1

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

riq = 0.8940   f_riq + 1.0000   e_riq
rpa = 0.8370   f_rpa + 1.0000   e_rpa
rses = 0.9490   f_rses + 1.0000   e_rses
roa = 1.0000   f_ramb + 1.0000   e_roa
rea = 1.0840 * f_ramb + 1.0000   e_rea
Std Err   0.0942   lambda2        
t Value   11.5105            
fiq = 0.8940   f_fiq + 1.0000   e_fiq
fpa = 0.8370   f_fpa + 1.0000   e_fpa
fses = 0.9490   f_fses + 1.0000   e_fses
foa = 1.0000   f_famb + 1.0000   e_foa
fea = 1.1163 * f_famb + 1.0000   e_fea
Std Err   0.0863   lambda3        
t Value   12.9394            

 


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 1

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

f_ramb = 0.1190 * f_famb + 0.1837 * f_rpa + 0.2800 * f_riq + 0.2262 * f_rses + 0.0870 * f_fses + 1.0000   d_ramb
Std Err   0.1140   bet1   0.0504   gam1   0.0614   gam2   0.0522   gam3   0.0548   gam4        
t Value   1.0440       3.6420       4.5618       4.3300       1.5884            
f_famb = 0.1302 * f_ramb + 0.0633 * f_rses + 0.1688 * f_fpa + 0.3539 * f_fiq + 0.2154 * f_fses + 1.0000   d_famb
Std Err   0.1207   bet2   0.0522   gam5   0.0493   gam8   0.0674   gam7   0.0512   gam6        
t Value   1.0792       1.2124       3.4205       5.2497       4.2060            


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 1

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable Parameter Estimate Standard
Error
t Value
f_rpa   1.00000    
f_riq   1.00000    
f_rses   1.00000    
f_fpa   1.00000    
f_fiq   1.00000    
f_fses   1.00000    
e_rea theta1 0.32707 0.05452 6.00
e_roa theta2 0.42307 0.05243 8.07
e_fea theta3 0.28715 0.04804 5.98
e_foa theta4 0.42240 0.04730 8.93
e_rpa err1 0.29584 0.07774 3.81
e_riq err2 0.20874 0.07832 2.67
e_rses err3 0.09887 0.07803 1.27
e_fpa err4 0.29987 0.07807 3.84
e_fiq err5 0.19988 0.07674 2.60
e_fses err6 0.10324 0.07824 1.32
d_ramb psi11 0.25418 0.04469 5.69
d_famb psi22 0.19698 0.03814 5.17

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.24677 0.07519 3.28
f_rpa f_rses cov2 0.06184 0.06945 0.89
f_riq f_rses cov3 0.26351 0.06687 3.94
f_rpa f_fpa cov4 0.15789 0.07873 2.01
f_riq f_fpa cov5 0.13085 0.07418 1.76
f_rses f_fpa cov6 0.11517 0.06978 1.65
f_rpa f_fiq cov7 0.10853 0.07362 1.47
f_riq f_fiq cov8 0.42476 0.07219 5.88
f_rses f_fiq cov9 0.27250 0.06660 4.09
f_fpa f_fiq cov10 0.27867 0.07530 3.70
f_rpa f_fses cov11 0.02383 0.06952 0.34
f_riq f_fses cov12 0.22135 0.06648 3.33
f_rses f_fses cov13 0.30156 0.06359 4.74
f_fpa f_fses cov14 -0.05623 0.06971 -0.81
f_fiq f_fses cov15 0.34922 0.06771 5.16
e_rea e_fea covea 0.02308 0.03139 0.74
e_roa e_foa covoa 0.11206 0.03258 3.44
d_ramb d_famb psi12 -0.00935 0.05010 -0.19


Since the p-value for the chi-square test is 0.5266, this model clearly cannot be rejected. However, Schwarz's Bayesian Criterion for this model (SBC = -63.3356) is somewhat larger than for J\ddot{o}reskog and S\ddot{o}rbom's (1988) analysis 2 (SBC =-71.6343), 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 friend. In particular, the corresponding coefficients should be equal. By imposing equality constraints on the 15 pairs of corresponding coefficents, this example obtains Loehlin's model 2. The LINEQS model is as follows, where an OUTRAM= data set is created to facilitate subsequent hypothesis tests:

   title2 'Loehlin (1987) analysis: Model 2';
   proc calis data=aspire edf=328 outram=ram2;
      lineqs    /* measurement model for aspiration */
             rea=lambda f_ramb + e_rea,             /* 1 ec! */
             roa=f_ramb + e_roa,
             fea=lambda f_famb + e_fea,
             foa=f_famb + e_foa,
             /* measurement model for intelligence and environment */
             rpa=.837 f_rpa + e_rpa,
             riq=.894 f_riq + e_riq,
             rses=.949 f_rses + e_rses,
             fpa=.837 f_fpa + e_fpa,
             fiq=.894 f_fiq + e_fiq,
             fses=.949 f_fses + e_fses,
                /* structural model of influences */     /* 5 ec! */
             f_ramb=gam1 f_rpa + gam2 f_riq + gam3 f_rses +
                gam4 f_fses + beta f_famb + d_ramb,
             f_famb=gam1 f_fpa + gam2 f_fiq + gam3 f_fses +
                gam4 f_rses + beta f_ramb + d_famb;
      std d_ramb=psi,                            /* 1 ec! */
          d_famb=psi,
          f_rpa f_riq f_rses f_fpa f_fiq f_fses=1,
          e_rea e_fea=thetaea thetaea,           /* 2 ec! */
          e_roa e_foa=thetaoa thetaoa,
          e_rpa e_fpa=errpa1 errpa2,
          e_riq e_fiq=erriq1 erriq2,
          e_rses e_fses=errses1 errses2;
      cov d_ramb d_famb=psi12,
          e_rea e_fea=covea,
          e_roa e_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 results are displayed in Figure 14.22.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0581
Goodness of Fit Index (GFI) 0.9884
GFI Adjusted for Degrees of Freedom (AGFI) 0.9772
Root Mean Square Residual (RMR) 0.0276
Parsimonious GFI (Mulaik, 1989) 0.6150
Chi-Square 19.0697
Chi-Square DF 28
Pr > Chi-Square 0.8960
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0194
ECVI Estimate 0.2285
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.2664
Probability of Close Fit 0.9996
Bentler's Comparative Fit Index 1.0000
Normal Theory Reweighted LS Chi-Square 19.2372
Akaike's Information Criterion -36.9303
Bozdogan's (1987) CAIC -171.2200
Schwarz's Bayesian Criterion -143.2200
McDonald's (1989) Centrality 1.0137
Bentler & Bonett's (1980) Non-normed Index 1.0174
Bentler & Bonett's (1980) NFI 0.9781
James, Mulaik, & Brett (1982) Parsimonious NFI 0.6086
Z-Test of Wilson & Hilferty (1931) -1.2599
Bollen (1986) Normed Index Rho1 0.9649
Bollen (1988) Non-normed Index Delta2 1.0106
Hoelter's (1983) Critical N 713

Figure 14.22: Career Aspiration Data: Loehlin Model 2

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

riq = 0.8940   f_riq + 1.0000   e_riq
rpa = 0.8370   f_rpa + 1.0000   e_rpa
rses = 0.9490   f_rses + 1.0000   e_rses
roa = 1.0000   f_ramb + 1.0000   e_roa
rea = 1.1007 * f_ramb + 1.0000   e_rea
Std Err   0.0684   lambda        
t Value   16.0879            
fiq = 0.8940   f_fiq + 1.0000   e_fiq
fpa = 0.8370   f_fpa + 1.0000   e_fpa
fses = 0.9490   f_fses + 1.0000   e_fses
foa = 1.0000   f_famb + 1.0000   e_foa
fea = 1.1007 * f_famb + 1.0000   e_fea
Std Err   0.0684   lambda        
t Value   16.0879            

 


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

f_ramb = 0.1158 * f_famb + 0.1758 * f_rpa + 0.3223 * f_riq + 0.2227 * f_rses + 0.0756 * f_fses + 1.0000   d_ramb
Std Err   0.0839   beta   0.0351   gam1   0.0470   gam2   0.0363   gam3   0.0375   gam4        
t Value   1.3801       5.0130       6.8557       6.1373       2.0170            
f_famb = 0.1158 * f_ramb + 0.0756 * f_rses + 0.1758 * f_fpa + 0.3223 * f_fiq + 0.2227 * f_fses + 1.0000   d_famb
Std Err   0.0839   beta   0.0375   gam4   0.0351   gam1   0.0470   gam2   0.0363   gam3        
t Value   1.3801       2.0170       5.0130       6.8557       6.1373            


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 2

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable Parameter Estimate Standard
Error
t Value
f_rpa   1.00000    
f_riq   1.00000    
f_rses   1.00000    
f_fpa   1.00000    
f_fiq   1.00000    
f_fses   1.00000    
e_rea thetaea 0.30662 0.03726 8.23
e_roa thetaoa 0.42295 0.03651 11.58
e_fea thetaea 0.30662 0.03726 8.23
e_foa thetaoa 0.42295 0.03651 11.58
e_rpa errpa1 0.30758 0.07511 4.09
e_riq erriq1 0.26656 0.07389 3.61
e_rses errses1 0.11467 0.07267 1.58
e_fpa errpa2 0.28834 0.07369 3.91
e_fiq erriq2 0.15573 0.06700 2.32
e_fses errses2 0.08814 0.07089 1.24
d_ramb psi 0.22456 0.02971 7.56
d_famb psi 0.22456 0.02971 7.56

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.26470 0.05442 4.86
f_rpa f_rses cov2 0.00176 0.04996 0.04
f_riq f_rses cov3 0.31129 0.05057 6.16
f_rpa f_fpa cov4 0.15784 0.07872 2.01
f_riq f_fpa cov5 0.11837 0.05447 2.17
f_rses f_fpa cov6 0.06910 0.04996 1.38
f_rpa f_fiq cov5 0.11837 0.05447 2.17
f_riq f_fiq cov7 0.43061 0.07258 5.93
f_rses f_fiq cov8 0.24967 0.05060 4.93
f_fpa f_fiq cov1 0.26470 0.05442 4.86
f_rpa f_fses cov6 0.06910 0.04996 1.38
f_riq f_fses cov8 0.24967 0.05060 4.93
f_rses f_fses cov9 0.30190 0.06362 4.75
f_fpa f_fses cov2 0.00176 0.04996 0.04
f_fiq f_fses cov3 0.31129 0.05057 6.16
e_rea e_fea covea 0.02160 0.03144 0.69
e_roa e_foa covoa 0.11208 0.03257 3.44
d_ramb d_famb psi12 -0.00344 0.04931 -0.07


The test of Loehlin's model 2 against model 1 yields a chi-square of 19.0697 - 12.0132 = 7.0565 with 15 degrees of freedom, which is clearly not significant. Schwarz's Bayesizn Criterion (SBC) is also much lower for model 2 (-143.2200) than model 1 (-63.3356). 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 model 3:

   title2 'Loehlin (1987) analysis: Model 3';
   data ram3(type=ram);
      set ram2;
      if _name_='gam4' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;
   proc calis data=aspire edf=328 inram=ram3;
   run;

The output is displayed in Figure 14.23.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 3

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0702
Goodness of Fit Index (GFI) 0.9858
GFI Adjusted for Degrees of Freedom (AGFI) 0.9731
Root Mean Square Residual (RMR) 0.0304
Parsimonious GFI (Mulaik, 1989) 0.6353
Chi-Square 23.0365
Chi-Square DF 29
Pr > Chi-Square 0.7749
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0295
ECVI Estimate 0.2343
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.2780
Probability of Close Fit 0.9984
Bentler's Comparative Fit Index 1.0000
Normal Theory Reweighted LS Chi-Square 23.5027
Akaike's Information Criterion -34.9635
Bozdogan's (1987) CAIC -174.0492
Schwarz's Bayesian Criterion -145.0492
McDonald's (1989) Centrality 1.0091
Bentler & Bonett's (1980) Non-normed Index 1.0112
Bentler & Bonett's (1980) NFI 0.9736
James, Mulaik, & Brett (1982) Parsimonious NFI 0.6274
Z-Test of Wilson & Hilferty (1931) -0.7563
Bollen (1986) Normed Index Rho1 0.9590
Bollen (1988) Non-normed Index Delta2 1.0071
Hoelter's (1983) Critical N 607

Figure 14.23: Career Aspiration Data: Loehlin Model 3

The chi-square value for testing model 3 versus model 2 is 23.0365 - 19.0697 = 3.9668 with 1 degree of freedom and a p-value of 0.0464. Although the parameter is of marginal significance, the estimate in model 2 (0.0756) 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 model 4:

   title2 'Loehlin (1987) analysis: Model 4';
   data ram4(type=ram);
      set ram2;
      if _name_='beta' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;

   proc calis data=aspire edf=328 inram=ram4;
   run;

The output is displayed in Figure 14.24.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 4

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0640
Goodness of Fit Index (GFI) 0.9873
GFI Adjusted for Degrees of Freedom (AGFI) 0.9760
Root Mean Square Residual (RMR) 0.0304
Parsimonious GFI (Mulaik, 1989) 0.6363
Chi-Square 20.9981
Chi-Square DF 29
Pr > Chi-Square 0.8592
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0234
ECVI Estimate 0.2281
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.2685
Probability of Close Fit 0.9994
Bentler's Comparative Fit Index 1.0000
Normal Theory Reweighted LS Chi-Square 20.8040
Akaike's Information Criterion -37.0019
Bozdogan's (1987) CAIC -176.0876
Schwarz's Bayesian Criterion -147.0876
McDonald's (1989) Centrality 1.0122
Bentler & Bonett's (1980) Non-normed Index 1.0150
Bentler & Bonett's (1980) NFI 0.9759
James, Mulaik, & Brett (1982) Parsimonious NFI 0.6289
Z-Test of Wilson & Hilferty (1931) -1.0780
Bollen (1986) Normed Index Rho1 0.9626
Bollen (1988) Non-normed Index Delta2 1.0095
Hoelter's (1983) Critical N 666

Figure 14.24: Career Aspiration Data: Loehlin Model 4

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 4

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

riq = 0.8940   f_riq + 1.0000   e_riq
rpa = 0.8370   f_rpa + 1.0000   e_rpa
rses = 0.9490   f_rses + 1.0000   e_rses
roa = 1.0000   f_ramb + 1.0000   e_roa
rea = 1.1051 * f_ramb + 1.0000   e_rea
Std Err   0.0680   lambda        
t Value   16.2416            
fiq = 0.8940   f_fiq + 1.0000   e_fiq
fpa = 0.8370   f_fpa + 1.0000   e_fpa
fses = 0.9490   f_fses + 1.0000   e_fses
foa = 1.0000   f_famb + 1.0000   e_foa
fea = 1.1051 * f_famb + 1.0000   e_fea
Std Err   0.0680   lambda        
t Value   16.2416            

 


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 4

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

f_ramb = 0   f_famb + 0.1776 * f_rpa + 0.3486 * f_riq + 0.2383 * f_rses + 0.1081 * f_fses + 1.0000   d_ramb
Std Err           0.0361   gam1   0.0463   gam2   0.0355   gam3   0.0299   gam4        
t Value           4.9195       7.5362       6.7158       3.6134            
f_famb = 0   f_ramb + 0.1081 * f_rses + 0.1776 * f_fpa + 0.3486 * f_fiq + 0.2383 * f_fses + 1.0000   d_famb
Std Err           0.0299   gam4   0.0361   gam1   0.0463   gam2   0.0355   gam3        
t Value           3.6134       4.9195       7.5362       6.7158            


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 4

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable Parameter Estimate Standard
Error
t Value
f_rpa   1.00000    
f_riq   1.00000    
f_rses   1.00000    
f_fpa   1.00000    
f_fiq   1.00000    
f_fses   1.00000    
e_rea thetaea 0.30502 0.03728 8.18
e_roa thetaoa 0.42429 0.03645 11.64
e_fea thetaea 0.30502 0.03728 8.18
e_foa thetaoa 0.42429 0.03645 11.64
e_rpa errpa1 0.31354 0.07543 4.16
e_riq erriq1 0.29611 0.07299 4.06
e_rses errses1 0.12320 0.07273 1.69
e_fpa errpa2 0.29051 0.07374 3.94
e_fiq erriq2 0.18181 0.06611 2.75
e_fses errses2 0.09873 0.07109 1.39
d_ramb psi 0.22738 0.03140 7.24
d_famb psi 0.22738 0.03140 7.24


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 4

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.27241 0.05520 4.94
f_rpa f_rses cov2 0.00476 0.05032 0.09
f_riq f_rses cov3 0.32463 0.05089 6.38
f_rpa f_fpa cov4 0.16949 0.07863 2.16
f_riq f_fpa cov5 0.13539 0.05407 2.50
f_rses f_fpa cov6 0.07362 0.05027 1.46
f_rpa f_fiq cov5 0.13539 0.05407 2.50
f_riq f_fiq cov7 0.46893 0.06980 6.72
f_rses f_fiq cov8 0.26289 0.05093 5.16
f_fpa f_fiq cov1 0.27241 0.05520 4.94
f_rpa f_fses cov6 0.07362 0.05027 1.46
f_riq f_fses cov8 0.26289 0.05093 5.16
f_rses f_fses cov9 0.30880 0.06409 4.82
f_fpa f_fses cov2 0.00476 0.05032 0.09
f_fiq f_fses cov3 0.32463 0.05089 6.38
e_rea e_fea covea 0.02127 0.03150 0.68
e_roa e_foa covoa 0.11245 0.03258 3.45
d_ramb d_famb psi12 0.05479 0.02699 2.03

The chi-square value for testing model 4 versus model 2 is 20.9981 - 19.0697 = 1.9284 with 1 degree of freedom and a p-value of 0.1649. Hence, there is little evidence of reciprocal influence.

Loehlin's 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 model 5:

   title2 'Loehlin (1987) analysis: Model 5';
   data ram5(type=ram);
      set ram2;
      if _name_='psi12' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;

   proc calis data=aspire edf=328 inram=ram5;
   run;

The output is displayed in Figure 14.25.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 5

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0582
Goodness of Fit Index (GFI) 0.9884
GFI Adjusted for Degrees of Freedom (AGFI) 0.9780
Root Mean Square Residual (RMR) 0.0276
Parsimonious GFI (Mulaik, 1989) 0.6370
Chi-Square 19.0745
Chi-Square DF 29
Pr > Chi-Square 0.9194
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0152
ECVI Estimate 0.2222
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.2592
Probability of Close Fit 0.9998
Bentler's Comparative Fit Index 1.0000
Normal Theory Reweighted LS Chi-Square 19.2269
Akaike's Information Criterion -38.9255
Bozdogan's (1987) CAIC -178.0111
Schwarz's Bayesian Criterion -149.0111
McDonald's (1989) Centrality 1.0152
Bentler & Bonett's (1980) Non-normed Index 1.0186
Bentler & Bonett's (1980) NFI 0.9781
James, Mulaik, & Brett (1982) Parsimonious NFI 0.6303
Z-Test of Wilson & Hilferty (1931) -1.4014
Bollen (1986) Normed Index Rho1 0.9661
Bollen (1988) Non-normed Index Delta2 1.0118
Hoelter's (1983) Critical N 733

Figure 14.25: Career Aspiration Data: Loehlin Model 5

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 5

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

riq = 0.8940   f_riq + 1.0000   e_riq
rpa = 0.8370   f_rpa + 1.0000   e_rpa
rses = 0.9490   f_rses + 1.0000   e_rses
roa = 1.0000   f_ramb + 1.0000   e_roa
rea = 1.1009 * f_ramb + 1.0000   e_rea
Std Err   0.0684   lambda        
t Value   16.1041            
fiq = 0.8940   f_fiq + 1.0000   e_fiq
fpa = 0.8370   f_fpa + 1.0000   e_fpa
fses = 0.9490   f_fses + 1.0000   e_fses
foa = 1.0000   f_famb + 1.0000   e_foa
fea = 1.1009 * f_famb + 1.0000   e_fea
Std Err   0.0684   lambda        
t Value   16.1041            

 


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 5

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

f_ramb = 0.1107 * f_famb + 0.1762 * f_rpa + 0.3235 * f_riq + 0.2233 * f_rses + 0.0770 * f_fses + 1.0000   d_ramb
Std Err   0.0428   beta   0.0350   gam1   0.0435   gam2   0.0353   gam3   0.0323   gam4        
t Value   2.5854       5.0308       7.4435       6.3215       2.3870            
f_famb = 0.1107 * f_ramb + 0.0770 * f_rses + 0.1762 * f_fpa + 0.3235 * f_fiq + 0.2233 * f_fses + 1.0000   d_famb
Std Err   0.0428   beta   0.0323   gam4   0.0350   gam1   0.0435   gam2   0.0353   gam3        
t Value   2.5854       2.3870       5.0308       7.4435       6.3215            


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 5

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable Parameter Estimate Standard
Error
t Value
f_rpa   1.00000    
f_riq   1.00000    
f_rses   1.00000    
f_fpa   1.00000    
f_fiq   1.00000    
f_fses   1.00000    
e_rea thetaea 0.30645 0.03721 8.24
e_roa thetaoa 0.42304 0.03650 11.59
e_fea thetaea 0.30645 0.03721 8.24
e_foa thetaoa 0.42304 0.03650 11.59
e_rpa errpa1 0.30781 0.07510 4.10
e_riq erriq1 0.26748 0.07295 3.67
e_rses errses1 0.11477 0.07265 1.58
e_fpa errpa2 0.28837 0.07366 3.91
e_fiq erriq2 0.15653 0.06614 2.37
e_fses errses2 0.08832 0.07088 1.25
d_ramb psi 0.22453 0.02973 7.55
d_famb psi 0.22453 0.02973 7.55

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.26494 0.05436 4.87
f_rpa f_rses cov2 0.00185 0.04995 0.04
f_riq f_rses cov3 0.31164 0.05039 6.18
f_rpa f_fpa cov4 0.15828 0.07846 2.02
f_riq f_fpa cov5 0.11895 0.05383 2.21
f_rses f_fpa cov6 0.06924 0.04993 1.39
f_rpa f_fiq cov5 0.11895 0.05383 2.21
f_riq f_fiq cov7 0.43180 0.07084 6.10
f_rses f_fiq cov8 0.25004 0.05039 4.96
f_fpa f_fiq cov1 0.26494 0.05436 4.87
f_rpa f_fses cov6 0.06924 0.04993 1.39
f_riq f_fses cov8 0.25004 0.05039 4.96
f_rses f_fses cov9 0.30203 0.06360 4.75
f_fpa f_fses cov2 0.00185 0.04995 0.04
f_fiq f_fses cov3 0.31164 0.05039 6.18
e_rea e_fea covea 0.02120 0.03094 0.69
e_roa e_foa covoa 0.11197 0.03254 3.44
d_ramb d_famb   0    


The chi-square value for testing model 5 versus model 2 is 19.0745 - 19.0697 = 0.0048 with 1 degree of freedom. 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 -0.003344 for model 2 to 0.05479 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 model 7:

   title2 'Loehlin (1987) analysis: Model 7';
   data ram7(type=ram);
      set ram2;
      if _name_='psi12'|_name_='beta' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;

   proc calis data=aspire edf=328 inram=ram7;
   run;
The relevant output is displayed in Figure 14.26.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 7

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.0773
Goodness of Fit Index (GFI) 0.9846
GFI Adjusted for Degrees of Freedom (AGFI) 0.9718
Root Mean Square Residual (RMR) 0.0363
Parsimonious GFI (Mulaik, 1989) 0.6564
Chi-Square 25.3466
Chi-Square DF 30
Pr > Chi-Square 0.7080
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0000
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0326
ECVI Estimate 0.2350
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.2815
Probability of Close Fit 0.9975
Bentler's Comparative Fit Index 1.0000
Normal Theory Reweighted LS Chi-Square 25.1291
Akaike's Information Criterion -34.6534
Bozdogan's (1987) CAIC -178.5351
Schwarz's Bayesian Criterion -148.5351
McDonald's (1989) Centrality 1.0071
Bentler & Bonett's (1980) Non-normed Index 1.0084
Bentler & Bonett's (1980) NFI 0.9709
James, Mulaik, & Brett (1982) Parsimonious NFI 0.6473
Z-Test of Wilson & Hilferty (1931) -0.5487
Bollen (1986) Normed Index Rho1 0.9564
Bollen (1988) Non-normed Index Delta2 1.0055
Hoelter's (1983) Critical N 568

Figure 14.26: Career Aspiration Data: Loehlin Model 7

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 7

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

riq = 0.8940   f_riq + 1.0000   e_riq
rpa = 0.8370   f_rpa + 1.0000   e_rpa
rses = 0.9490   f_rses + 1.0000   e_rses
roa = 1.0000   f_ramb + 1.0000   e_roa
rea = 1.1037 * f_ramb + 1.0000   e_rea
Std Err   0.0678   lambda        
t Value   16.2701            
fiq = 0.8940   f_fiq + 1.0000   e_fiq
fpa = 0.8370   f_fpa + 1.0000   e_fpa
fses = 0.9490   f_fses + 1.0000   e_fses
foa = 1.0000   f_famb + 1.0000   e_foa
fea = 1.1037 * f_famb + 1.0000   e_fea
Std Err   0.0678   lambda        
t Value   16.2701            

 


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 7

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

f_ramb = 0   f_famb + 0.1765 * f_rpa + 0.3573 * f_riq + 0.2419 * f_rses + 0.1109 * f_fses + 1.0000   d_ramb
Std Err           0.0360   gam1   0.0461   gam2   0.0363   gam3   0.0306   gam4        
t Value           4.8981       7.7520       6.6671       3.6280            
f_famb = 0   f_ramb + 0.1109 * f_rses + 0.1765 * f_fpa + 0.3573 * f_fiq + 0.2419 * f_fses + 1.0000   d_famb
Std Err           0.0306   gam4   0.0360   gam1   0.0461   gam2   0.0363   gam3        
t Value           3.6280       4.8981       7.7520       6.6671            


Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 7

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Variances of Exogenous Variables
Variable Parameter Estimate Standard
Error
t Value
f_rpa   1.00000    
f_riq   1.00000    
f_rses   1.00000    
f_fpa   1.00000    
f_fiq   1.00000    
f_fses   1.00000    
e_rea thetaea 0.31633 0.03648 8.67
e_roa thetaoa 0.42656 0.03610 11.82
e_fea thetaea 0.31633 0.03648 8.67
e_foa thetaoa 0.42656 0.03610 11.82
e_rpa errpa1 0.31329 0.07538 4.16
e_riq erriq1 0.30776 0.07307 4.21
e_rses errses1 0.14303 0.07313 1.96
e_fpa errpa2 0.29286 0.07389 3.96
e_fiq erriq2 0.19193 0.06613 2.90
e_fses errses2 0.11804 0.07147 1.65
d_ramb psi 0.21011 0.02940 7.15
d_famb psi 0.21011 0.02940 7.15

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value
f_rpa f_riq cov1 0.27533 0.05552 4.96
f_rpa f_rses cov2 0.00611 0.05085 0.12
f_riq f_rses cov3 0.33510 0.05150 6.51
f_rpa f_fpa cov4 0.17099 0.07872 2.17
f_riq f_fpa cov5 0.13859 0.05431 2.55
f_rses f_fpa cov6 0.07563 0.05077 1.49
f_rpa f_fiq cov5 0.13859 0.05431 2.55
f_riq f_fiq cov7 0.48105 0.06993 6.88
f_rses f_fiq cov8 0.27235 0.05157 5.28
f_fpa f_fiq cov1 0.27533 0.05552 4.96
f_rpa f_fses cov6 0.07563 0.05077 1.49
f_riq f_fses cov8 0.27235 0.05157 5.28
f_rses f_fses cov9 0.32046 0.06517 4.92
f_fpa f_fses cov2 0.00611 0.05085 0.12
f_fiq f_fses cov3 0.33510 0.05150 6.51
e_rea e_fea covea 0.04535 0.02918 1.55
e_roa e_foa covoa 0.12085 0.03214 3.76
d_ramb d_famb   0    


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 J\ddot{o}reskog and S\ddot{o}rbom's models. Constraining COVEA and COVOA to zero produces Loehlin's model 6:

   title2 'Loehlin (1987) analysis: Model 6';
   data ram6(type=ram);
      set ram2;
      if _name_='covea'|_name_='covoa' then
         do;
            _name_=' ';
            _estim_=0;
         end;
   run;

   proc calis data=aspire edf=328 inram=ram6;
   run;
The relevant output is displayed in Figure 14.27.

Peer Influences on Aspiration: Haller & Butterworth (1960)
Loehlin (1987) analysis: Model 6

The CALIS Procedure
Covariance Structure Analysis: Maximum Likelihood Estimation

Fit Function 0.1020
Goodness of Fit Index (GFI) 0.9802
GFI Adjusted for Degrees of Freedom (AGFI) 0.9638
Root Mean Square Residual (RMR) 0.0306
Parsimonious GFI (Mulaik, 1989) 0.6535
Chi-Square 33.4475
Chi-Square DF 30
Pr > Chi-Square 0.3035
Independence Model Chi-Square 872.00
Independence Model Chi-Square DF 45
RMSEA Estimate 0.0187
RMSEA 90% Lower Confidence Limit .
RMSEA 90% Upper Confidence Limit 0.0471
ECVI Estimate 0.2597
ECVI 90% Lower Confidence Limit .
ECVI 90% Upper Confidence Limit 0.3164
Probability of Close Fit 0.9686
Bentler's Comparative Fit Index 0.9958
Normal Theory Reweighted LS Chi-Square 32.9974
Akaike's Information Criterion -26.5525
Bozdogan's (1987) CAIC -170.4342
Schwarz's Bayesian Criterion -140.4342
McDonald's (1989) Centrality 0.9948
Bentler & Bonett's (1980) Non-normed Index 0.9937
Bentler & Bonett's (1980) NFI 0.9616
James, Mulaik, & Brett (1982) Parsimonious NFI 0.6411
Z-Test of Wilson & Hilferty (1931) 0.5151
Bollen (1986) Normed Index Rho1 0.9425
Bollen (1988) Non-normed Index Delta2 0.9959
Hoelter's (1983) Critical N 431

Figure 14.27: Career Aspiration Data: Loehlin Model 6

The chi-square value for testing model 6 versus model 2 is 33.4476 - 19.0697 = 14.3779 with 2 degrees of freedom and a p-value of 0.0008, indicating that there is considerable evidence of correlation between the error terms.

The following table summarizes the results from Loehlin's seven models.

Model \chi^2 df p-value SBC
1. Full model12.0132130.5266-63.3356
2. Equality constraints19.0697280.8960-143.2200
3. No SES path23.0365290.7749-145.0492
4. No reciprocal influence20.9981290.8592-147.0876
5. No disturbance correlation19.0745290.9194-149.0111
6. No error correlation33.4475300.3035-140.4342
7. Constraints from both 4 & 525.3466300.7080-148.5351

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

   title 'Comparisons among Loehlin''s models';
   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 following table.

      Comparisons among Loehlin's models

   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.

Chapter Contents
Chapter Contents
Previous
Previous
Next
Next
Top
Top

Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.