# Introduction to Structural Equation Modeling with Latent Variables

## A Combined Measurement-Structural Model

Subsections:

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

The study concerns 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 data from these 329 boys paired with the data from their best friends are analyzed.

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

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

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