# Introduction to Structural Equation Modeling with Latent Variables

### Career Aspiration: Analysis 3

Loehlin (1987) points out that the models considered are unrealistic in at least two respects. First, the variables of parental aspiration, intelligence, and socioeconomic status are assumed to be measured without error. Loehlin adds uncorrelated measurement errors to the model and assumes, for illustrative purposes, that the reliabilities of these variables are known to be 0.7, 0.8, and 0.9, respectively. In practice, these reliabilities would need to be obtained from a separate study of the same or a very similar population. If these constraints are omitted, the model is not identified. However, constraining parameters to a constant in an analysis of a correlation matrix might make the chi-square goodness-of-fit test inaccurate, so there is more reason to be skeptical of the p-values. Second, the error terms for the respondent’s aspiration are assumed to be uncorrelated with the corresponding terms for his friend. Loehlin introduces a correlation between the two educational aspiration error terms and between the two occupational aspiration error terms. These additions produce the path diagram for Loehlin’s model shown in Figure 17.47.

Figure 17.47: Path Diagram for Career Aspiration: Analysis 3

In Figure 17.47, the observed variables `rpa`, `riq`, `rses`, `fses`, `fiq`, and `fpa` are all measured with measurement errors. Their true scores counterparts `f_rpa`, `f_riq`, `f_rses`, `f_fses`, `f_fiq`, and `f_fpa` are latent variables in the model. Path coefficients from these latent variables to the observed variables are fixed coefficients, indicating the square roots of the theoretical reliabilities in the model. These latent variables, rather than the observed counterparts, serve as predictors of the ambition factors `R_Amb` and `F_Amb` in the current model (Analysis 3). The error terms for these two latent factors are correlated, as indicated by a double-headed path (arrow) that connects the two factors. Correlated errors for the occupational aspiration variables (`roa` and `foa`) and the educational aspiration variables (`rea` and `fea`) are also shown in Figure 17.47. Again, these correlated errors are represented by two double-headed paths (arrows) in the path diagram.

You use the following statements to specify the path model for Analysis 3:

```proc calis data=aspire nobs=329;
path
/* measurement model for intelligence and environment */
rpa     <===  f_rpa    = 0.837,
riq     <===  f_riq    = 0.894,
rses    <===  f_rses   = 0.949,
fses    <===  f_fses   = 0.949,
fiq     <===  f_fiq    = 0.894,
fpa     <===  f_fpa    = 0.837,

/* structural model of influences */
f_rpa   ===>  R_Amb,
f_riq   ===>  R_Amb,
f_rses  ===>  R_Amb,
f_fses  ===>  R_Amb,
f_rses  ===>  F_Amb,
f_fses  ===>  F_Amb,
f_fiq   ===>  F_Amb,
f_fpa   ===>  F_Amb,
F_Amb   ===>  R_Amb,
R_Amb   ===>  F_Amb,

/* measurement model for aspiration */
R_Amb   ===>  rea        ,
R_Amb   ===>  roa    = 1.,
F_Amb   ===>  foa    = 1.,
F_Amb   ===>  fea    ;
pvar
f_rpa f_riq f_rses f_fses f_fiq f_fpa = 6 * 1.0;
pcov
R_Amb F_Amb  ,
rea  fea     ,
roa  foa     ;
run;
```

In this specification, the measurement model for the six intelligence and environment variables are added. They are the first six paths in the PATH statement. Fixed constants are set for these path coefficients so as to make the measurement model identified and to set the required reliabilities of these measurement indicators. The structural model of influences and the measurement model for aspiration are the same as specified in Analysis 1. (See the section Career Aspiration: Analysis 1.) All the correlated errors are specified in the PCOV statement.

The fit summary of the current model is displayed in Figure 17.48.

Figure 17.48: Career Aspiration Data: Fit Summary for Analysis 3

Fit Summary
Chi-Square 12.0132
Chi-Square DF 13
Pr > Chi-Square 0.5266
Standardized RMR (SRMR) 0.0149
Adjusted GFI (AGFI) 0.9692
RMSEA Estimate 0.0000
Akaike Information Criterion 96.0132
Schwarz Bayesian Criterion 255.4476
Bentler Comparative Fit Index 1.0000

Since the p-value for the chi-square test is 0.5266, this model clearly cannot be rejected. Both the standardized RMSR and the RMSEA are very small, and both the adjusted GFI and the comparative fit index are high. All these point to an excellent model fit. However, Schwarz’s Bayesian criterion for this model (SBC = 255.4476) is somewhat larger than for Jöreskog and Sörbom (1988) Analysis 2 in Figure 17.45 (SBC = 247.1489), suggesting that a more parsimonious model would be desirable.

The estimation results are displayed in Figure 17.49.

Figure 17.49: Career Aspiration Data: Estimation Results for Analysis 3

PATH List
Path Parameter Estimate Standard
Error
t Value Pr > |t|
rpa <=== f_rpa   0.83700
riq <=== f_riq   0.89400
rses <=== f_rses   0.94900
fses <=== f_fses   0.94900
fiq <=== f_fiq   0.89400
fpa <=== f_fpa   0.83700
f_rpa ===> R_Amb _Parm01 0.18370 0.05044 3.6420 0.0003
f_riq ===> R_Amb _Parm02 0.28004 0.06139 4.5618 <.0001
f_rses ===> R_Amb _Parm03 0.22616 0.05223 4.3300 <.0001
f_fses ===> R_Amb _Parm04 0.08698 0.05476 1.5883 0.1122
f_rses ===> F_Amb _Parm05 0.06327 0.05219 1.2124 0.2254
f_fses ===> F_Amb _Parm06 0.21539 0.05121 4.2060 <.0001
f_fiq ===> F_Amb _Parm07 0.35387 0.06741 5.2497 <.0001
f_fpa ===> F_Amb _Parm08 0.16876 0.04934 3.4205 0.0006
F_Amb ===> R_Amb _Parm09 0.11898 0.11396 1.0441 0.2964
R_Amb ===> F_Amb _Parm10 0.13022 0.12067 1.0791 0.2805
R_Amb ===> rea _Parm11 1.08399 0.09417 11.5105 <.0001
R_Amb ===> roa   1.00000
F_Amb ===> foa   1.00000
F_Amb ===> fea _Parm12 1.11630 0.08627 12.9394 <.0001

Variance Parameters
Variance
Type
Variable Parameter Estimate Standard
Error
t Value Pr > |t|
Exogenous f_rpa   1.00000
f_riq   1.00000
f_rses   1.00000
f_fses   1.00000
f_fiq   1.00000
f_fpa   1.00000
Error riq _Add01 0.20874 0.07832 2.6652 0.0077
rpa _Add02 0.29584 0.07774 3.8057 0.0001
rses _Add03 0.09887 0.07803 1.2671 0.2051
roa _Add04 0.42307 0.05243 8.0695 <.0001
rea _Add05 0.32707 0.05452 5.9988 <.0001
fiq _Add06 0.19989 0.07674 2.6048 0.0092
fpa _Add07 0.29988 0.07807 3.8409 0.0001
fses _Add08 0.10324 0.07824 1.3195 0.1870
foa _Add09 0.42240 0.04730 8.9310 <.0001
fea _Add10 0.28716 0.04804 5.9776 <.0001
R_Amb _Add11 0.25418 0.04469 5.6874 <.0001
F_Amb _Add12 0.19698 0.03814 5.1653 <.0001

Covariances Among Exogenous Variables
Var1 Var2 Parameter Estimate Standard
Error
t Value Pr > |t|
f_riq f_rpa _Add13 0.24677 0.07519 3.2820 0.0010
f_rses f_rpa _Add14 0.06183 0.06945 0.8903 0.3733
f_rses f_riq _Add15 0.26351 0.06687 3.9408 <.0001
f_fses f_rpa _Add16 0.02382 0.06952 0.3427 0.7318
f_fses f_riq _Add17 0.22136 0.06648 3.3298 0.0009
f_fses f_rses _Add18 0.30156 0.06359 4.7421 <.0001
f_fiq f_rpa _Add19 0.10853 0.07362 1.4742 0.1404
f_fiq f_riq _Add20 0.42476 0.07219 5.8837 <.0001
f_fiq f_rses _Add21 0.27250 0.06660 4.0914 <.0001
f_fiq f_fses _Add22 0.34922 0.06771 5.1576 <.0001
f_fpa f_rpa _Add23 0.15789 0.07873 2.0056 0.0449
f_fpa f_riq _Add24 0.13084 0.07418 1.7639 0.0778
f_fpa f_rses _Add25 0.11516 0.06978 1.6505 0.0988
f_fpa f_fses _Add26 -0.05622 0.06971 -0.8065 0.4200
f_fpa f_fiq _Add27 0.27867 0.07530 3.7008 0.0002

Covariances Among Errors
Error of Error of Parameter Estimate Standard
Error
t Value Pr > |t|
R_Amb F_Amb _Parm13 -0.00936 0.05010 -0.1867 0.8519
rea fea _Parm14 0.02308 0.03139 0.7355 0.4621
roa foa _Parm15 0.11206 0.03258 3.4399 0.0006

Like Analyses 1 and 2, two paths that concern the validity of the indicators in the current analysis do not show significance. That is, `f_fses` does not seem to be a good indicator of a respondent’s ambition `R_Amb`, and `f_rses` does not seem to be a good indicator of a friend’s ambition `F_Amb`. The t values are 1.588 and 1.212, respectively. In addition, in the current model (Analysis 3), the structural relationships between the ambition factors do not show significance. The t value for the path from the friend’s ambition factor `F_Amb` on the respondent’s ambition factor `R_Amb` is only 1.044, while the t value for the path from the respondent’s ambition factor `R_Amb` on the friend’s ambition factor `F_Amb` is only 1.079. These cast doubts on the validity of the structural model and perhaps even the entire model.