In PROC CALIS, you can use many different modeling languages to specify the same model. The choice of modeling language depends on personal preferences and the purposes of the analysis. See the section Which Modeling Language? for guidance. In this example, the data and the model in Example 29.17 are used to illustrate how a particular model can be specified by various general modeling languages.
In Example 29.17, you use the PATH modeling language to specify the model because of its close resemblance to the path diagram. In this example, you consider another modeling language of PROC CALIS that is also closely related to the path diagram representation of structural equation models. The socalled RAM model language has syntax that represents the single and doubleheaded paths (or arrows) in the path diagram. However, unlike the PATH modeling language, the RAM modeling language is matrixbased. The following statements show how you can specify the same path model with the RAM model specification for the data in Example 29.17:
proc calis nobs=932 data=Wheaton; ram var = Anomie67 /* 1 */ Powerless67 /* 2 */ Anomie71 /* 3 */ Powerless71 /* 4 */ Education /* 5 */ SEI /* 6 */ Alien67 /* 7 */ Alien71 /* 8 */ SES, /* 9 */ _A_ 1 7 1.0, _A_ 2 7 0.833, _A_ 3 8 1.0, _A_ 4 8 0.833, _A_ 5 9 1.0, _A_ 6 9 lambda, _A_ 7 9 gamma1, _A_ 8 9 gamma2, _A_ 8 7 beta, _P_ 1 1 theta1, _P_ 2 2 theta2, _P_ 3 3 theta1, _P_ 4 4 theta2, _P_ 5 5 theta3, _P_ 6 6 theta4, _P_ 7 7 psi1, _P_ 8 8 psi2, _P_ 9 9 phi, _P_ 1 3 theta5, _P_ 2 4 theta5; run;
In the RAM model for covariance structure analysis, you have two important matrices to specify. The first one is the _A_ matrix, which is for the specification of the singleheaded paths (arrows) in the path diagram. The second one is the _P_ matrix, which is for the specification of the doubleheaded paths (arrows) in the path diagram. Hence, to specify the RAM model is much like mapping the path diagram arrows into the parameter of the RAM model matrices.
In the RAM statement, you can specify the variables in the model in the VAR= option. The VAR= list contains all observed and latent variables in your path diagram (without the use of error terms). Although you can specify the variables in the VAR= list in any order you like, the variable order in the list is also the order of variables in the RAM model matrices. In VAR= list of the RAM statement, you put comments to note the order of the variables.
After you specify the variable list, you can specify the model parameter locations in the RAM statement entries. In the first
nine entries, you specify the singleheaded paths by mapping them into the elements of the _A_ matrix of the RAM model. For
example, the first entry represents the singleheaded path of variable 1 (Anomie67
) from variable 7 (Alien67
). The corresponding path effect or coefficient is fixed at 1, which is also the value for _A_[1,7]. Another example is the
ninth path entry. You specify a singleheaded path of variable 8 (Alien71
) from variable 7 (Alien67
). The corresponding path effect or coefficient is a free parameter named beta
, which is also the parameter for _A_[8,7]. Hence, you can specify all singleheaded paths in the path diagram as elements
in the _A_ matrix of the RAM model.
To facilitate the comparisons between the RAN and PATH modeling languages, the PATH model specification in Example 29.17 for the same data is reproduced in the following:
proc calis nobs=932 data=Wheaton plots=residuals; path Anomie67 Powerless67 <=== Alien67 = 1.0 0.833, Anomie71 Powerless71 <=== Alien71 = 1.0 0.833, Education SEI <=== SES = 1.0 lambda, Alien67 Alien71 <=== SES = gamma1 gamma2, Alien71 <=== Alien67 = beta; pvar Anomie67 = theta1, Powerless67 = theta2, Anomie71 = theta1, Powerless71 = theta2, Education = theta3, SEI = theta4, Alien67 = psi1, Alien71 = psi2, SES = phi; pcov Anomie67 Anomie71 = theta5, Powerless67 Powerless71 = theta5; run;
It is clear that each of the path entries specified in the PATH statement corresponds to an matrix element entry of the _A_ matrix in the RAM statement. How about the specifications of the doubleheaded arrows in the path diagram? Do the RAM and PATH model specifications correspond to each other?
The answer is yes. In the PATH modeling language, you specify all doubleheaded arrows in the path diagram as entries either
in the PVAR or PCOV statement. In the RAM modeling language, you specify the corresponding entries as matrix element entries
of the _P_ matrix in the RAM statement. For example, the error variance of Anomie67
is a parameter called _Variabletheta1 in the PVAR statement of the PATH model. You specify the same parameter for the _P_[1,1]
element in an entry of the RAM statement. Another example is the error covariance between Powerless67
and Powerless71
. You specify this a parameter called theta5
in the last entry of the PCOV statement in the PATH model. You specify the same parameter for the _P_[2,4] element in the
last entry of the RAM statement. Therefore, it is not difficult to find that the specifications in the PATH and the RAM model
have some kind of onetoone correspondence.
Output 29.23.1 shows the RAM model estimates for the Wheaton
data. These RAM model estimates match the set of estimates using the PATH model specification, as shown in Output 29.17.12.
Output 29.23.1: RAM Model Estimates
RAM Pattern and Estimates  

Matrix  Row  Column  Parameter  Estimate  Standard Error 
t Value  Pr > t  
_A_ (1)  Anomie67  1  Alien67  7  1.00000  
Powerless67  2  Alien67  7  0.83300  
Anomie71  3  Alien71  8  1.00000  
Powerless71  4  Alien71  8  0.83300  
Education  5  SES  9  1.00000  
SEI  6  SES  9  lambda  5.36883  0.43371  12.3788  <.0001  
Alien67  7  SES  9  gamma1  0.62994  0.05634  11.1809  <.0001  
Alien71  8  SES  9  gamma2  0.24086  0.05489  4.3884  <.0001  
Alien71  8  Alien67  7  beta  0.59312  0.04678  12.6788  <.0001  
_P_ (2)  Anomie67  1  Anomie67  1  theta1  3.60796  0.20092  17.9572  <.0001 
Powerless67  2  Powerless67  2  theta2  3.59488  0.16448  21.8556  <.0001  
Anomie71  3  Anomie71  3  theta1  3.60796  0.20092  17.9572  <.0001  
Powerless71  4  Powerless71  4  theta2  3.59488  0.16448  21.8556  <.0001  
Education  5  Education  5  theta3  2.99366  0.49861  6.0040  <.0001  
SEI  6  SEI  6  theta4  259.57639  18.31151  14.1756  <.0001  
Alien67  7  Alien67  7  psi1  5.67046  0.42301  13.4050  <.0001  
Alien71  8  Alien71  8  psi2  4.51479  0.33532  13.4639  <.0001  
SES  9  SES  9  phi  6.61634  0.63914  10.3519  <.0001  
Anomie67  1  Anomie71  3  theta5  0.90580  0.12167  7.4447  <.0001  
Powerless67  2  Powerless71  4  theta5  0.90580  0.12167  7.4447  <.0001 
Another way to specify the model in Example 29.17 is to use the LINEQS modeling language, which is shown in the following:
proc calis nobs=932 data=Wheaton; lineqs Anomie67 = 1.0 * f_Alien67 + e1, Powerless67 = 0.833 * f_Alien67 + e2, Anomie71 = 1.0 * f_Alien71 + e3, Powerless71 = 0.833 * f_Alien71 + e4, Education = 1.0 * f_SES + e5, SEI = lambda * f_SES + e6, f_Alien67 = gamma1 * f_SES + d1, f_Alien71 = gamma2 * f_SES + beta * f_Alien67 + d2; variance E1 = theta1, E2 = theta2, E3 = theta1, E4 = theta2, E5 = theta3, E6 = theta4, D1 = psi1, D2 = psi2, f_SES = phi; cov E1 E3 = theta5, E2 E4 = theta5; run;
As compared with the PATH and RAM modeling languages, the most distinct feature of the LINEQS modeling language is the explicit use of error terms in equation specifications. In the LINEQS statement, you specify exactly one equation for each endogenous variable. In each equation, you list an endogenous variable on the lefthandside of the equation and all its predictors on the righthandside of the equation. You must also include an error term in each equation. Because each endogenous variable in the LINEQS statement can only be specified in exactly one equation, the number of equations in the LINEQS model and the number of paths in the corresponding path diagram do not match necessarily. In this example, there are eight equations in the LINEQS statement, but there are nine paths in the corresponding path diagram.
In addition, in the LINEQS model, you need to follow a convention of naming latent variables. For latent variables that are
neither errors nor disturbances, you must use either the 'F
' or 'f
' prefix. For error terms, you must use either the 'E
' or 'e
' prefix. For disturbances, you must use either the 'D
' or 'd
' prefix. However, in the PATH or RAM model specification, no such convention is imposed. For example, f_Alien67
, f_Alien71
, and f_SES
are latent factors in the LINEQS model. They are not error terms, and so they must start with the 'f
' prefix. However, this prefix is not needed in the PATH or RAM model. Furthermore, there are no explicit error terms that
need to be specified in the PATH or RAM model, let alone specific prefixes for the error terms.
The PVAR statement in the PATH model is replaced with the VARIANCE statement in the LINEQS model, and the PCOV statement with the COV statement. The PVAR and PCOV statements in the PATH model are for the partial variance and partial covariance specifications. The partial variance or covariance concepts are used in the PATH or RAM model specification because error terms are not named explicitly. Specification of error variances in the PATH and RAM model is conceptualized as the specification of the partial variances of the corresponding variables. But in the LINEQS model, because errors or disturbances are named explicitly as exogenous variables, the partial variance or covariance concepts are no longer necessary. Instead, you specify the variances of the error terms directly, which reflects the conceptualization behind the VARIANCE statement of the LINEQS modeling language. Similarly, you use the COV, but not PCOV, statement in the LINEQS modeling language because you can specify the covariances among variables or error terms without using the partial covariance conceptualization.
In this example, the variances of the errors ("E
"variables) and disturbances ("D
"variables) specified in the VARIANCE
statement of the LINEQS model correspond to the partial variances of the endogenous variables specified in the PVAR
statement of the PATH model. Similarly, covariances of errors specified in the COV
statement of the LINEQS model correspond to the partial covariances of endogenous variables specified in the PCOV
statement of the PATH model. The estimation results of the LINEQS model are shown in Output 29.23.2. Again, they are essentially the same estimates obtained from the PATH model specified in Example 29.17, as shown in Output 29.17.12.
Output 29.23.2: LINEQS Model Estimates
Linear Equations  

Anomie67  =  1.0000  f_Alien67  +  1.0000  e1  
Powerless67  =  0.8330  f_Alien67  +  1.0000  e2  
Anomie71  =  1.0000  f_Alien71  +  1.0000  e3  
Powerless71  =  0.8330  f_Alien71  +  1.0000  e4  
Education  =  1.0000  f_SES  +  1.0000  e5  
SEI  =  5.3688  (**)  f_SES  +  1.0000  e6  
f_Alien67  =  0.6299  (**)  f_SES  +  1.0000  d1  
f_Alien71  =  0.2409  (**)  f_SES  +  0.5931  (**)  f_Alien67  +  1.0000  d2 
Effects in Linear Equations  

Variable  Predictor  Parameter  Estimate  Standard Error 
t Value  Pr > t 
Anomie67  f_Alien67  1.00000  
Powerless67  f_Alien67  0.83300  
Anomie71  f_Alien71  1.00000  
Powerless71  f_Alien71  0.83300  
Education  f_SES  1.00000  
SEI  f_SES  lambda  5.36883  0.43371  12.3788  <.0001 
f_Alien67  f_SES  gamma1  0.62994  0.05634  11.1809  <.0001 
f_Alien71  f_SES  gamma2  0.24086  0.05489  4.3884  <.0001 
f_Alien71  f_Alien67  beta  0.59312  0.04678  12.6788  <.0001 
Estimates for Variances of Exogenous Variables  

Variable Type 
Variable  Parameter  Estimate  Standard Error 
t Value  Pr > t 
Error  e1  theta1  3.60796  0.20092  17.9572  <.0001 
e2  theta2  3.59488  0.16448  21.8556  <.0001  
e3  theta1  3.60796  0.20092  17.9572  <.0001  
e4  theta2  3.59488  0.16448  21.8556  <.0001  
e5  theta3  2.99366  0.49861  6.0040  <.0001  
e6  theta4  259.57639  18.31151  14.1756  <.0001  
Disturbance  d1  psi1  5.67046  0.42301  13.4050  <.0001 
d2  psi2  4.51479  0.33532  13.4639  <.0001  
Latent  f_SES  phi  6.61634  0.63914  10.3519  <.0001 
You can also specify general structural models by using the LISMOD modeling language. See the section The LISMOD Model and Submodels for details.
To use the LISMOD modeling language, you must recognize four types of variables in the model. The variables (etavariables) are latent factors that are endogenous, or predicted by other latent factors. The variables (xivariables) are exogenous latent variables that are not predicted by any other variables. The variables are manifest variables that are indicators of the variables, and the variables are manifest variables that are indicators of the variables. In this example, Alien67
and Alien71
are the variables, and SES
is the variable in the model. Manifest indicators for Alien67
and Alien71
include Anomie67
, Powerless67
, Anomie71
, and Powerless71
, which are the variables. Manifest indicators for SES
include Education
and SEI
, which are the variables.
After defining these four types of variables, the parameters of the model are defined as entries in the model matrices. The _LAMBDAY_, _LAMBDAX_, _GAMMA_, and _BETA_ are matrices for the path coefficients or effects. The _THETAY, _THETAX_, _PSI_, and _PHI_ are matrices for the variances and covariances.
The following is the LISMOD specification for the model in Example 29.17:
proc calis nobs=932 data=Wheaton; lismod yvar = Anomie67 Powerless67 Anomie71 Powerless71, xvar = Education SEI, etavar = Alien67 Alien71, xivar = SES; matrix _LAMBDAY_ [1,1] = 1, [2,1] = 0.833, [3,2] = 1, [4,2] = 0.833; matrix _LAMBDAX_ [1,1] = 1, [2,1] = lambda; matrix _GAMMA_ [1,1] = gamma1, [2,1] = gamma2; matrix _BETA_ [2,1] = beta; matrix _THETAY_ [1,1] = theta1theta2 theta1theta2, [3,1] = theta5, [4,2] = theta5; matrix _THETAX_ [1,1] = theta3theta4; matrix _PSI_ [1,1] = psi1psi2; matrix _PHI_ [1,1] = phi; run;
In the LISMOD statement, you specify the four lists of variables in the model. The orders of the variables in these lists define the order of the row and column variables in the model matrices, of which the parameter locations are specified in the MATRIX statements.
The estimated model is divided into three conceptual parts. The first part is the measurement model that relates the variables with the variables, as shown in Output 29.23.3:
Output 29.23.3: LISMOD Model Measurement Model for the Variables
_THETAY_ Matrix: Estimate/StdErr/tvalue/pvalue  

Anomie67  Powerless67  Anomie71  Powerless71  
Anomie67 





Powerless67 





Anomie71 





Powerless71 




The _LAMBDAY_ matrix contains the coefficients or effects of the variables on the variables. All these estimates are fixed constants as specified. The _THETAY_ matrix contains the error variances and covariances
for the variables. Three free parameters are located in this matrix: theta1
, theta2
, and theta5
.
The second part of the estimated model is the measurement model that relates the variable with the variables, as shown in Output 29.23.4:
The _LAMBDAX_ matrix contains the coefficients or effects of the variable SES
on the variables. The effect of SES
on Education
is fixed at one. The effect of SES
on SEI
is represented by the free parameter lambda
, which is estimated at 5.3688. The _THETAX_ matrix contains the error variances and covariances for the variables. Two free parameters are located in this matrix: theta3
and theta4
.
The last part of the estimated model is the structural model that relates the latent variables and , as shown in Output 29.23.5:
The _BETA_ matrix contains effects of variables on themselves. In the current example, there is only one such effect. The effect of Alien67
on Alien71
is represented by the free parameter beta
. The _GAMMA_ matrix contains effects of the variable, which is SES
in this example, on the variables Alien67
on Alien71
. These effects are represented by the free parameters gamma1
and gamma2
. The _PSI_ matrix contains the error variances and covariances in the structural model. In this example, psi1
and psi2
are two free parameters for the error variances. Finally, the _PHI_ matrix is the covariance matrix for the variables. In this example, there is only one variable so that this matrix contains only the estimated variance of SES
. This variance is represented by the parameter phi
.
The estimates obtained from fitting the LISMOD model are the same as those from fitting the equivalent PATH, RAM, or LINEQS model. To some researchers the LISMOD modeling language might be more familiar, while for others modeling languages such as PATH, RAM, or LINEQS are more convenient to use.