This example uses some wellknown data from Haller and Butterworth (1960). The section A Combined MeasurementStructural Model in Chapter 17: Introduction to Structural Equation Modeling with Latent Variables, analyzes some models for these data. Inspired by the examples given in Loehlin (1987), this example shows additional applications to the same data set, but with a focus on testing nested models. By manipulating the OUTMODEL= data set, this example shows how you can specify new models in an efficient way. Various models and analyses of these data are also given by Duncan, Haller, and Portes (1968), Jöreskog and Sörbom (1988), and Loehlin (1987).
The study is concerned with the career aspirations of high school students and how these aspirations are affected by close friends. The data are collected from 442 seventeenyearold 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.
Because of the dependency of the data, the effective sample size assumed in the example is 329, which you can specify in the NOBS= option in the PROC CALIS statements. See the section A Combined MeasurementStructural Model in Chapter 17: Introduction to Structural Equation Modeling with Latent Variables, for the justification of the use of this effective sample size.
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. ;
For illustration purposes, this correlation matrix is treated here as if it were a covariance matrix for PROC CALIS to analyze. The reason is that the chisquare tests shown in this example are valid only with covariance structure analysis. See Example 29.27 for an illustration of covariance structure analysis on correlations.
Loehlin (1987) analyzes the following path model for the data:
In Output 29.24.1, the observed variables rpa
, riq
, rses
, fses
, fiq
, and fpa
are measured with 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
. The error terms for these two latent factors are correlated, as indicated by a doubleheaded 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 Output 29.24.1. These correlated errors are also represented by two doubleheaded paths (arrows) in the path diagram.
Notice that the covariances among the six exogenous latent variables (f_rpa
, f_riq
, f_rses
, f_fses
, f_fiq
, and f_fpa
) are not represented in the path diagram for two reasons. First, there are 15 of these covariances and hence you need 15
doubleheaded arrows to represent them in the path diagram. Apparently, because of the space limitations, it would be difficult
to put all these doubleheaded arrows in the path diagram without cluttering it. Second, covariances among exogenous latent
variables are free parameters by default in PROC CALIS, and therefore omitting these doubleheaded arrows in the path diagram
is compatible with the default model specification in PROC CALIS. Similarly, doubleheaded arrows for the error variances
of the endogenous variables (rpa
, riq
, rses
, fses
, fiq
, fpa
, R_Amb
, and F_Amb
) in the path diagram are omitted because they are unconstrained free parameters and are set automatically by default in PROC
CALIS .
The model represented by the path diagram in Output 29.24.1 is considered to be the full model for the data, in the sense that it has the largest number of parameters among the competing models considered this example. The same model is analyzed in the section A Combined MeasurementStructural Model in Chapter 17: Introduction to Structural Equation Modeling with Latent Variables, with the following specification:
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: 5 equality constraints */ 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: 1 equality constraint */ R_Amb ===> rea , R_Amb ===> roa = 1., F_Amb ===> foa = 1., F_Amb ===> fea ; pvar f_rpa f_riq f_rses f_fpa f_fiq f_fses = 6 * 1.0; pcov R_Amb F_Amb , rea fea , roa foa ; run;
The PATH model specification represents each arrow (singleheaded and doubleheaded) in the path diagram. You transcribe each arrow in Output 29.24.1 into an entry in the PATH model. The PATH statement specifies all the singleheaded arrows in the path diagram. The PVAR statement specifies all the doubleheaded arrows that point to individual variables (that is, the fixed error variances of the exogenous latent variables) in the path diagram. The PCOV statement specifies all the doubleheaded arrows that connect paired variables (that is, the error covariances) in the path diagram.
Output 29.24.2 shows the fit summary of Model 1.
Since the pvalue for the chisquare test is 0.5266, this model clearly cannot be rejected. Both standardized RMR and RMSEA are very small. All these point to an excellent model fit. Three informationtheoretic fit indices are also shown: Akaike’s information criterion (AIC), Bozdogan’s CAIC, and Schwarz’s Bayesian Criterion (SBC). These indices are useful when you need to compare competing models for the data.
You now consider a much more restrictive model with equality constraints in the model. The path diagram for this constrained model is shown in Output 29.24.3.
The main idea about setting the equality constraints in this model is that there is some symmetry in the model components
that correspond to the respondent and his friend. In particular, the corresponding coefficients or parameters should be equal.
For example, the path f_rpa===>R_Amb
for the respondent has the same effect as that of f_fpa===>F_Amb
. In the path diagram, they are both labeled by the same parameter gam1
. Generalizing the same idea to other pairs of paths, Output 29.24.3 shows nine pairs of these equality constraints, which are all represented by the same parameter names for distinct (singleheaded
or doubleheaded) paths.
However, because of the space limitation, there are six more equality constraints that are not shown in the path diagram.
These six constraints concern the covariance structures of the exogenous latent factors f_rpa
, f_riq
, f_rses
, f_fses
, f_fiq
, and f_fpa
. The first three factors are for the respondent, and the last three are for his friend. Using the same symmetry argument,
the covariance structures imposed on these exogenous latent factors are shown in the following:
f_rpa f_riq f_rses f_fpa f_fiq f_fses f_rpa 1. f_riq c1 1. f_rses c2 c3 1. f_fpa c4 c5 c6 1. f_fiq c5 c7 c8 c1 1. f_fses c6 c8 c9 c2 c3 1.
In this pattern of covariance structures, the covariance matrix (upper left portion) for the latent factors of the respondent is the same as that (lower right portion) for the latent factors of his friend. The crosscovariances among the factors between the friends (lower left portion) also display a symmetry pattern. There are six pairs of equality constraints in the covariance structures. Imposing these six pairs of equality constraints and the nine pairs of equality constraints in the path diagram lead to Model 2 of Loehlin (1987).
You can specify the current constrained model by the following PATH modeling language of PROC CALIS:
proc calis data=aspire nobs=329 outmodel=model2; 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: 5 equality constraints */ f_rpa ===> R_Amb = gam1, f_riq ===> R_Amb = gam2, f_rses ===> R_Amb = gam3, f_fses ===> R_Amb = gam4, f_rses ===> F_Amb = gam4, f_fses ===> F_Amb = gam3, f_fiq ===> F_Amb = gam2, f_fpa ===> F_Amb = gam1, F_Amb ===> R_Amb = beta, R_Amb ===> F_Amb = beta, /* measurement model for aspiration: 1 equality constraint */ R_Amb ===> rea = lambda, R_Amb ===> roa = 1., F_Amb ===> foa = 1., F_Amb ===> fea = lambda; pvar f_rpa f_riq f_rses f_fpa f_fiq f_fses = 6 * 1.0, R_Amb F_Amb = 2 * psi, /* 1 ec */ rea fea = 2 * theta1, /* 1 ec */ roa foa = 2 * theta2; /* 1 ec */ pcov R_Amb F_Amb = psi12, rea fea = covea, roa foa = covoa, f_rpa f_riq f_rses = cov1cov3, /* 3 ec */ f_fpa f_fiq f_fses = cov1cov3, f_rpa f_riq f_rses * f_fpa f_fiq f_fses = /* 3 ec */ cov4 cov5 cov6 cov5 cov7 cov8 cov6 cov8 cov9; run;
In the current PATH model specification, you specify the same set of paths as in Model 1. In addition, to set the required
constraints in this path model, you use parameter names to label the related paths, variances, or covariances. Same parameter
names mean equality constraints. The 15 equality constraints are labeled with comments in the specification. In the PROC CALIS
statement, you use the OUTMODEL= option to output the model estimation results into the output data set model2
, which is used for subsequent hypotheses tests.
Output 29.24.4 shows the fit summary of Model 2.
The test of Model 2 against Model 1 (Loehlin, 1987) yields a chisquare of 19.0697 – 12.0132 = 7.0565 with 15 degrees of freedom, which is clearly not significant. This indicates that the restricted Model 2 fits at least as well as Model 1. Schwarz’s Bayesian criterion (SBC) is also much lower for Model 2 (175.5623) than for Model 1 (255.4476). 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 R_Amb
and from f_rses
to F_Amb
designated by the parameter name gam4
, yielding Model 3 of Loehlin (1987). The corresponding path diagram is shown in Output 29.24.5.
In Output 29.24.5, you drop the paths f_rses===>F_Amb
and f_fses===>R_Amb
from the previous model. Using the path diagram in Output 29.24.5, you can specify the current model the same way you do for Model 2. However, because you have the estimation results from
Model 2 in the SAS data set model2
, you can modify this SAS data set to reflect the current model specification and then input the modified SAS data set as
an INMODEL= file for PROC CALIS to analyze.
First, you create a new SAS data set model3
by the following DATA step:
data model3(type=calismdl); set model2; if _name_='gam4' then do; _name_=' '; _estim_=0; end; run;
Essentially, by blanking out the parameter name for the target paths, you are stating that these paths are no longer associated
with the free parameter gam4
in the new model. Instead, you put a fixed zero to these paths. This way you eliminate the paths f_rses===>F_Amb
and f_fses===>R_Amb
for Model 3, of which the model specification is now saved in the model3
data set.
Next, you input model3
as the INMODEL= data set for PROC CALIS to analyze, as shown in the following statements:
proc calis data=aspire nobs=329 inmodel=model3; run;
PROC CALIS can now use the previous estimation results for fitting the required model. Output 29.24.6 shows the fit summary of Model 3.
The chisquare value for testing Model 3 versus Model 2 is 23.0365 – 19.0697 = 3.9668 with one degree of freedom and a pvalue of 0.0464. The chisquare test shows a marginal significance, which means that the paths might be needed in the model. However, the SBC (173.7340) indicates that Model 3 is slightly preferable to Model 2, which has an SBC value of 175.5632.
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, you can set the parameter beta
for the paths linking R_Amb
and F_Amb
to zero to obtain Model 4 of Loehlin (1987).
Similar to Model 3, you can modify the model2
data set to form the new model data set model4
for PROC CALIS to analyze, as shown in the following statements:
data model4(type=calismdl); set model2; if _name_='beta' then do; _name_=' '; _estim_=0; end; run;
proc calis data=aspire nobs=329 inmodel=model4; run;
Output 29.24.7 shows the fit summary of Model 4.
The chisquare value for testing Model 4 versus Model 2 is 20.9981 – 19.0697 = 1.9284 with one degree of freedom and a pvalue of 0.1649. Hence, there is little evidence of reciprocal influence.
Model 2 of Loehlin (1987) has the direct paths connecting the latent ambition factors R_Amb
and F_Amb
and a covariance between the disturbance or error terms (that is, a doubleheaded arrow connecting the two factors in the
path diagram shown in Output 29.24.3). The presence of this disturbance correlation serves as a "wastebasket" that enables other omitted variables to have joint
influences on the respondent’s and his friend’s ambition factors. To test the hypothesis that this disturbance correlation
is zero, you use the following statements to set the parameter psi12
to zero in the model5
data set and fit the new model by PROC CALIS:
data model5(type=calismdl); set model2; if _name_='psi12' then do; _name_=' '; _estim_=0; end; run;
proc calis data=aspire nobs=329 inmodel=model5; run;
Output 29.24.8 displays the fit summary of Model 5.
The chisquare value for testing Model 5 versus Model 2 is 19.0745 – 19.0697 = 0.0048 with one degree of freedom. This test statistic is insignificant. Therefore, omitting the covariance between the disturbance terms causes hardly any deterioration in the fit of the model.
The test in Model 4 fails to provide evidence of a direct reciprocal influence between the respondent’s and friend’s ambitions,
and the test in Model 5 fails to provide evidence of a covariance or correlation between the disturbance terms for the ambition
factors. Because you consider these two tests separately, you cannot establish evidence to eliminate the reciprocal influence
and the disturbance correlation jointly. Instead, to make such a joint inference, it is important to test both hypotheses
together by setting both beta
and psi12
to zero as in Model 7 of Loehlin (1987). The following statements show how you can do that by modifying the model2
data set to form a new INMODEL= data set model7
for PROC CALIS to analyze:
data model7(type=calismdl); set model2; if _name_='psi12'_name_='beta' then do; _name_=' '; _estim_=0; end; run;
proc calis data=aspire nobs=329 inmodel=model7; run;
Output 29.24.9 shows the fit summary of Model 7.
When Model 7 is tested against Models 2, 4, and 5, the pvalues 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 (covea
and covoa
) between the error terms for educational aspiration (that is, between rea
and fea
) and occupational aspiration (that is, between roa
and foa
), because these terms are omitted from Jöreskog and Sörbom (1988) models. Constraining covea
and covoa
to zero produces Model 6 of Loehlin (1987). You can use the following statements to fit this model:
data model6(type=calismdl); set model2; if _name_='covea'_name_='covoa' then do; _name_=' '; _estim_=0; end; run;
proc calis data=aspire nobs=329 inmodel=model6; run;
Output 29.24.10 shows the fit summary of Model 6.
The chisquare value for testing Model 6 versus Model 2 is 33.4476 – 19.0697 = 14.3779 with two degrees of freedom and a pvalue of 0.0008, indicating that there is considerable evidence of correlation between the error terms.
The following table summarizes the results from the seven models described in Loehlin (1987).
Model 

df 
pvalue 
SBC 

1. Full model 
12.0132 
13 
0.5266 
255.4476 
2. Equality constraints 
19.0697 
28 
0.8960 
175.5632 
3. No SES path 
23.0365 
29 
0.7749 
173.7340 
4. No reciprocal influence 
20.9981 
29 
0.8592 
171.6956 
5. No disturbance correlation 
19.0745 
29 
0.9194 
169.7721 
6. No error correlation 
33.4475 
30 
0.3035 
178.3489 
7. Constraints from both 4 and 5 
25.3466 
30 
0.7080 
170.2480 
For comparing models, you can use a DATA step to compute the differences of the chisquare statistics and pvalues, as shown in the following statements:
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 pvalue' / +indent ''; put +indent +3 ho ' versus ' ha @18 +indent chisq 8.4 df 5. p 9.4; datalines; 2 1 3 2 4 2 5 2 7 2 7 4 7 5 6 2 ;
The DATA step displays the table in Output 29.24.11.
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.