Introduction to Structural Equation Modeling with Latent Variables

The Measurement Model for x

The second component of the LISMOD specification is the measurement model for x, as shown in the following equation:

$x = \bLambda _\mb {x} \xi + \delta$

The measurement model for x is similar to that for y. Assuming that x and $\xi$ are centered, this equation states that x is a function of the true scores vector $\xi$ plus the error term $\delta$ , which is independent of $\xi$ . The model matrices involved in this measurement model are $\bLambda _\mb {x}$ (effects of $\xi$ on x) and $\bTheta _\mb {x}$ , which is the covariance matrix of $\delta$ .

For the career aspiration data, you specify the following MATRIX statement for this measurement model:

   matrix _lambdax_ [1,1] = 0.837 0.894 0.949 0.949 0.894 0.837;

Figure 17.51 shows the output related to the specification of the measurement model for x.

Figure 17.51: Career Aspiration Analysis 3: Initial Measurement Model for x

Initial _LAMBDAX_ Matrix
	f_rpa	f_riq	f_rses	f_fses	f_fiq	f_fpa
rpa	0.8370	0	0	0	0	0
riq	0	0.8940	0	0	0	0
rses	0	0	0.9490	0	0	0
fses	0	0	0	0.9490	0	0
fiq	0	0	0	0	0.8940	0
fpa	0	0	0	0	0	0.8370

rpa

[_Add01]

riq

[_Add02]

rses

[_Add03]

fses

[_Add04]

fiq

[_Add05]

fpa

[_Add06]

In Figure 17.51, the initial _LAMBDAX_ matrix is a 6 $\times$ 6 matrix. The _LAMBDAX_ matrix contains information about the relationships between the row indicator variables x (XVAR= variables) and the column factors $\xi$ (XI= variables). As specified in the MATRIX statement for _LAMBDAX_, the diagonal elements are filled with the fixed values provided. The [1,1] specification in the MATRIX statement for _LAMBDAX_ provides the starting element for the subsequent parameter list to fill in. In this case, the list contains six fixed values, and PROC CALIS proceeds from [1,1] to [2,2], [3,3] and so on until the entire list of parameters is consumed. This kind of notation is a shortcut of the following equivalent specification:

   matrix _lambdax_ [1,1]=0.837, [2,2]=0.894, [3,3]=0.949,
                    [4,4]=0.949, [5,5]=0.894, [6,6]=0.837;

PROC CALIS provides many different kinds of shortcuts in specifying matrix elements. See the MATRIX statement of Chapter 29: The CALIS Procedure, for details.

At the bottom of Figure 17.51, the initial _THETAX_ matrix is shown. Even though you did not specify any elements of this matrix in any MATRIX statements, the diagonal elements of this matrix are set as default parameters by PROC CALIS. Default parameters added by PROC CALIS are all denoted by names with the prefix '_Add'.