Linear Structure Model Definition

The linear structure model definition part of the model specification file begins with the key word model--see the example model specification files. As in every other section, comments are not allowed to appear between a keyword and the arguments it takes. Please see Program Syntax for details.

GENBLIS is optimized for speed. Because of this, GENBLIS implements only the ``endogenous'' parameterization of LISREL. Implementing only one of the parameterizations is more computationally efficient than implementing them all. Computation efficiency is of great concern because the LISREL likelihood has to be evaluated a large number of times by GENBLIS's evolutionary program and bootstrap routines. Implementing only the ``endogenous'' model does not in any way restrict the generality or scope of the GENBLIS program, because any model that can be defined using the exogenous or united specification (with all eight of the classical LISREL matrices) can in fact be estimated by using only the ``endogenous'' parameterization. Although the names endogenous and exogenous are conventionally used to refer to different parts of the eight-matrix LISREL parameterization, the names can be misleading. The ``endogenous'' parameterization is sufficient to express any pattern of linear structural relations among the latent variables.

To treat one set of variables as exogenous in relation to another set, it is sufficient to impose constraints on parameters such that the former (exogenous) variables do not in any way depend on the random disturbances of the latter (endogenous) variables. For instance, if one estimates a model with two latent variables, letting the variables be related through one parameter in the $\text{B}$ matrix and specifying their disturbances to be uncorrelated, then the $\text{B}$ parameter is estimating the effect of an ``exogenous'' variable on an ``endogenous'' variable. Therefore, one can estimate a model using only GENBLIS's ``endogenous'' parameterization and then write up the results as having been obtained from the exogenous or full model.

The ``endogenous'' parameterization consists of four matrices: $\Lambda_{y}$, $\Theta_{\epsilon}$, $\text{B}$, and $\Psi$. Using these matrices, the model for the covariance matrix of the observed variables is

$$\Sigma = \Lambda_{y}\text{B}^{-1}\Psi \text{B}^{-\top} \Lambda_{y}^{\top} + \Theta_{\epsilon}$$ (1)

GENBLIS attempts to minimize the discrepancy function (or ``fit statistic'')

$$F_{\text{ML}} = \log\vert\Sigma\vert + \text{tr}(\mathbf{S}\Sigma^{-1}) -\log\vert\mathbf{S}\vert - p$$

where p is the number of observed variables and S is the sample covariance matrix of the observed variables. The parameters in each of the four model-specification matrices are specified in the model section of the model specification file by numerical sequences that follow the appropriate one of the following four keywords: ly, te, b, and psi.

With multigroup processing (see general options ngroups, datafile and observedvars), S and $\Sigma$ are block diagonal, with one block for each group. That is, for m groups,

$$\mathbf{S} = \begin{bmatrix} \mathbf{S}_1 & \dots & 0 \\ ... ... \ddots & \vdots \\ 0 & \dots & \Sigma_m & \\ \end{bmatrix}$$

with S_k and $\Sigma_k$ being respectively the sample covariance matrix and the model covariance matrix for the data from group k. Different groups may have different numbers of observed variables and different numbers of observations. To enhance computational efficiency, the GENBLIS program ignores covariance matrix elements that are not in one of the group-specific submatrices. It is not valid to specify a model that implies a covariance between observed variables that are in different groups. GENBLIS does not try to detect such erroneous covariances and will not necessarily issue any warnings or error messages, but results in such cases are unpredictable and wrong. It is perfectly fine to specify equality constraints across groups between parameters of the same type. To test such constraints is often the main reason to estimate a multigroup model. Note that in the model section all the observed variables and model matrix parameters are numbered consecutively in a unified fashion across all groups. There is no separate indexing of variables or parameters by group: see Examples 3 and 4. GENBLIS also supports analysis of the sample mean crossproducts matrix (i.e., the simple crossproduct of the observed data, divided by the number of observations), which allows structures to be specified on group means. To analyze the crossproducts matrix instead of the covariance matrix, specify the usecrossp option. GENBLIS does not add a constant to the set of input variables, so if one is desired it must be included by the user.

All parameters which are not mentioned by the user in this section are fixed to zero. Therefore only the non-zero elements of each matrix need to be described by the user. Indeed, a specified value of 0 for a parameter is interpreted to mean that the parameter is a free parameter that is to be estimated. The definition of each matrix element must follow one of the following formats. It is not necessary for each set three or five arguments to appear on a separate line, though doing so may make the specification easier to proofread.

ly
v w x
v w x y z

te
v w x
v w x y z

b
v w x
v w x y z

psi
v w x
v w x y z

Combinations of argument values not covered by the following descriptions are errors. All array indexes are counted from 1 (technical programming note: GENBLIS converts them internally to count from 0 for indexing the C-language arrays). Comments cannot be placed between any of the keywords (e.g., ly, te, b, and psi) and their numerical arguments. For example, comments can appear after the ly arguments and before the te keyword.

  The definition of $\Lambda_{y}$ (ly):

ly
v w x
v w x y z


To indicate that a single parameter is to be estimated (a free parameter), or to set a fixed parameter to a nonzero value, use 3 arguments:
if v>0 w>0 then v,w indexes $\Lambda_{y}$[v,w] and x=0 indicates a free parameter while x$\ne$0 indicates a fixed parameter with value fixed at x.


To specify that one parameter is constrained to equal another parameter, use 5 arguments:
if v=0 and w>0, x>0, y>0, and z>0 then w,x and y,z index $\Lambda_{y}$ such that $\Lambda_{y}$[w,x] is constrained to equal $\Lambda_{y}$[y,z]. If $\Lambda_{y}$[y,z] is to be estimated that must be specified separately (using a 3-argument entry).

  The definition of $\Theta_{\epsilon}$ (te):

te
v w x
v w x y z


To indicate that a single parameter is to be estimated (a free parameter), or to set a fixed parameter to a nonzero value, use 3 arguments:
if v>0 w>0 then v,w indexes $\Theta_{\epsilon}$[v,w]= $\Theta_{\epsilon}$[w,v] and x=0 indicates a (single) free parameter while x$\ne$0 indicates a fixed parameter with value fixed at x. GENBLIS automatically imposes symmetry on $\Theta_{\epsilon}$, so only one of any symmetric pair of off-diagonal parameters (if any) should be specified.


To indicate that all the diagonal elements of $\Theta_{\epsilon}$ are free parameters, use special values for v and w:
if v = -1 and w = -1 then all the diagonal elements of $\Theta_{\epsilon}$ are free parameters. x must be specified as a number but its value is irrelevant.


To indicate that a square symmetric block of elements of $\Theta_{\epsilon}$ are free parameters, use a special value for v:
if v = -2 and w>0 x>0 then all elements in the square, symmetric block of $\Theta_{\epsilon}$ from $\Theta_{\epsilon}$[w,w] to $\Theta_{\epsilon}$[x,x] inclusive are free--both diagonal and off-diagonal.


To specify that one parameter is constrained to equal another parameter, use 5 arguments:
if v=0 and w>0, x>0, y>0, and z>0 then w,x and y,z index $\Theta_{\epsilon}$ such that $\Theta_{\epsilon}$[w,x] is constrained to equal $\Theta_{\epsilon}$[y,z]. If $\Theta_{\epsilon}$[y,z] is to be estimated that must be specified separately (using a 3-argument entry). GENBLIS automatically imposes symmetry on $\Theta_{\epsilon}$, so equality constraints should not be used for that purpose. To do so is an error that will cause unpredictable and incorrect results.

  The definition of $\text{B}$ (b):

b
v w x
v w x y z

To indicate that a single parameter is to be estimated (a free parameter), or to set a fixed parameter to a nonzero value, use 3 arguments:
if v>0 and w>0 then v,w indexes $\text{B}$[v,w] and x=0 indicates a free parameter while x$\ne$0 indicates a fixed parameter with value fixed at x.


To indicate that all the parameters in a rectangular block of $\text{B}$ (except parameters in $\text{B}$'s diagonal, which are always fixed equal to 1) are to be estimated, use 5 arguments:
if v= -2 and w>0, x>0, y>0, and z>0, then w,x and y,z index $\text{B}$ such that all elements are free in the rectangle bounded by upper-left corner $\text{B}$[w,x] and bottom-right corner $\text{B}$[y,z], except for elements in the rectangle that are on $\text{B}$'s diagonal, which are ignored.


To specify that one parameter is constrained to equal another parameter, use 5 arguments:
if v=0 and w>0, x>0, y>0, and z>0 then w,x and y,z index $\text{B}$ such that $\text{B}$[w,x] is constrained to equal $\text{B}$[y,z]. If $\text{B}$[y,z] is to be estimated that must be specified separately (using a 3-argument entry).

  The definition of $\Psi$ (psi):

psi
v w x
v w x y z


To indicate that a single parameter is to be estimated (a free parameter), or to set a fixed parameter to a nonzero value, use 3 arguments:
if v>0 and w>0, then v,w indexes $\Psi$[v,w]=$\Psi$[w,v]. x=0 indicates a (single) free parameter while x$\ne$0 indicates a fixed parameter with value fixed at x. GENBLIS automatically imposes symmetry on $\Psi$, so only one of any symmetric pair of off-diagonal parameters (if any) should be specified.


To indicate that all the diagonal elements of $\Psi$ are free parameters, use special values for v and w:
if v = -1 and w = -1 then all the diagonal elements of $\Psi$ are free parameters. x must be specified as a number but its value is irrelevant.

To indicate that a square symmetric block of elements of $\Psi$ are free parameters, use a special value for v:
if v = -1 and w = -2 then all elements of $\Psi$ are free parameters. x must be specified as a number but its value is irrelevant.


if v = -2 and w>0, x>0, then all elements in the square, symmetric block of $\Psi$ from $\Psi$[w,w] to $\Psi$[x,x] inclusive are free (both diagonal and off-diagonal).


To specify that a square symmetric block of elements of $\Psi$ is fixed to be exactly equal to a square symmetric block of the sample covariance matrix of the observed variables, use 5 arguments:
if v = -3 and w>0, x>0, y>0, z>0, then the elements of the square, symmetric block of $\Psi$ from $\Psi$[w,w] thru $\Psi$[x,x] are set equal to the corresponding elements in the block of the sample covariance matrix, S, from S[y,y] thru S[z,z]. The sample covariance matrix is the appropriate one respectively for the original sample, jackknife and bootstrap computations. If the $\Lambda_{y}$ loading pattern is set correctly, the equivalence here will reproduce the relevant observed covariances exactly. This is useful if those variables are considered observed without error and exogenous.


To specify that one parameter is constrained to equal another parameter, use 5 arguments:
if v=0 and w>0, x>0, y>0, and z>0 then w,x and y,z index $\Psi$ such that $\Psi$[w,x] is constrained to equal $\Psi$[y,z]. If $\Psi$[y,z] is to be estimated that must be specified separately (using a 3-argument entry). GENBLIS automatically imposes symmetry on $\Psi$, so equality constraints should not be used for that purpose. To do so is an error that will cause unpredictable and incorrect results.