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 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), Jreskog and Srbom (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 Jreskog and Srbom (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 Jreskog and Srbom (1988) is displayed in the following path diagram:

Figure 14.17: Path Diagram: Career Aspiration, Jreskog and Srbom

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 Jreskog and Srbom (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 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

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

• The coefficents connecting the latent ambition variables are equal.
• The covariance of the disturbances of the ambition variables is zero.

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 f_ramb + 1.0000 e_roa rea = 1.061 * f_ramb + 1.0000 e_rea Std Err 0.0892 lambda2 t Value 11.8923 foa = 1 f_famb + 1.0000 e_foa fea = 1.0736 * f_famb + 1.0000 e_fea Std Err 0.0806 lambda3 t Value 13.315

 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 + 0.254 * 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 + 0.152 * 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 5.532

 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 StandardError 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 StandardError 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.

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 = 0.894 f_riq + 1.0000 e_riq rpa = 0.837 f_rpa + 1.0000 e_rpa rses = 0.949 f_rses + 1.0000 e_rses roa = 1 f_ramb + 1.0000 e_roa rea = 1.084 * f_ramb + 1.0000 e_rea Std Err 0.0942 lambda2 t Value 11.5105 fiq = 0.894 f_fiq + 1.0000 e_fiq fpa = 0.837 f_fpa + 1.0000 e_fpa fses = 0.949 f_fses + 1.0000 e_fses foa = 1 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 = 0.119 * f_famb + 0.1837 * f_rpa + 0.28 * f_riq + 0.2262 * f_rses + 0.087 * f_fses + 1.0000 d_ramb Std Err 0.114 bet1 0.0504 gam1 0.0614 gam2 0.0522 gam3 0.0548 gam4 t Value 1.044 3.642 4.5618 4.33 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 4.206

 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 StandardError 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 StandardError 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 Jreskog and Srbom'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 = 0.894 f_riq + 1.0000 e_riq rpa = 0.837 f_rpa + 1.0000 e_rpa rses = 0.949 f_rses + 1.0000 e_rses roa = 1 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 = 0.894 f_fiq + 1.0000 e_fiq fpa = 0.837 f_fpa + 1.0000 e_fpa fses = 0.949 f_fses + 1.0000 e_fses foa = 1 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 0.047 gam2 0.0363 gam3 0.0375 gam4 t Value 1.3801 5.013 6.8557 6.1373 2.017 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 0.047 gam2 0.0363 gam3 t Value 1.3801 2.017 5.013 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 StandardError 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 StandardError 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 = 0.894 f_riq + 1.0000 e_riq rpa = 0.837 f_rpa + 1.0000 e_rpa rses = 0.949 f_rses + 1.0000 e_rses roa = 1 f_ramb + 1.0000 e_roa rea = 1.1051 * f_ramb + 1.0000 e_rea Std Err 0.068 lambda t Value 16.2416 fiq = 0.894 f_fiq + 1.0000 e_fiq fpa = 0.837 f_fpa + 1.0000 e_fpa fses = 0.949 f_fses + 1.0000 e_fses foa = 1 f_famb + 1.0000 e_foa fea = 1.1051 * f_famb + 1.0000 e_fea Std Err 0.068 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 StandardError 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 StandardError 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 = 0.894 f_riq + 1.0000 e_riq rpa = 0.837 f_rpa + 1.0000 e_rpa rses = 0.949 f_rses + 1.0000 e_rses roa = 1 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 = 0.894 f_fiq + 1.0000 e_fiq fpa = 0.837 f_fpa + 1.0000 e_fpa fses = 0.949 f_fses + 1.0000 e_fses foa = 1 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 + 0.077 * f_fses + 1.0000 d_ramb Std Err 0.0428 beta 0.035 gam1 0.0435 gam2 0.0353 gam3 0.0323 gam4 t Value 2.5854 5.0308 7.4435 6.3215 2.387 f_famb = 0.1107 * f_ramb + 0.077 * 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 0.035 gam1 0.0435 gam2 0.0353 gam3 t Value 2.5854 2.387 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 StandardError 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 StandardError 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 = 0.894 f_riq + 1.0000 e_riq rpa = 0.837 f_rpa + 1.0000 e_rpa rses = 0.949 f_rses + 1.0000 e_rses roa = 1 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 = 0.894 f_fiq + 1.0000 e_fiq fpa = 0.837 f_fpa + 1.0000 e_fpa fses = 0.949 f_fses + 1.0000 e_fses foa = 1 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 0.036 gam1 0.0461 gam2 0.0363 gam3 0.0306 gam4 t Value 4.8981 7.752 6.6671 3.628 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 0.036 gam1 0.0461 gam2 0.0363 gam3 t Value 3.628 4.8981 7.752 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 StandardError 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 StandardError 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 Jreskog and Srbom'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 df p-value SBC 1. Full model 12.0132 13 0.5266 -63.3356 2. Equality constraints 19.0697 28 0.8960 -143.2200 3. No SES path 23.0365 29 0.7749 -145.0492 4. No reciprocal influence 20.9981 29 0.8592 -147.0876 5. No disturbance correlation 19.0745 29 0.9194 -149.0111 6. No error correlation 33.4475 30 0.3035 -140.4342 7. Constraints from both 4 & 5 25.3466 30 0.7080 -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);
(13 28 29 29 29 30 30);
retain indent 16;
file print;
input ho ha @@;
chisq = achisq[ho] - achisq[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.

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