Types of Variables in the FACTOR Model

Each variable in the FACTOR model is either manifest or latent. Manifest variables are those variables that are measured in the research. They must be present in the input data set. Latent variables are not measured. Each latent variable in the FACTOR model can either be a factor or an error term.

Factors are unmeasured hypothetical constructs for explaining the covariances among manifest variables, while errors are the unique parts of the manifest variables that are not explained by the (common) factors.

In the FACTOR model, all manifest variables are endogenous, which means that they are predicted from the latent variables. In contrast, all latent variables in the FACTOR model are exogenous, which means that they serve as predictors only.

Naming Variables in the FACTOR Model

Manifest variables in the FACTOR model are referenced in the input data set. In the FACTOR model specification, you use their names as they appear in the input data set. Manifest variable names must not be longer than 32 characters. There are no further restrictions on these names beyond those required by the SAS system.

Error variables in the FACTOR model are not named explicitly, although they are assumed in the model. You can name latent factors only in confirmatory FACTOR models. Factor names must not be longer than 32 characters and must be distinguishable from the manifest variable names in the same analysis. You do not need to name factors in exploratory FACTOR models, however. Latent factors named Factor1, Factor2, and so on are generated automatically in exploratory FACTOR models.

Model Matrices in the FACTOR Model

Suppose in the FACTOR model that there are manifest variables and factors. The FACTOR model matrices are summarized as follows:

Matrix Representation of the FACTOR Model

Let be a vector of manifest variables, be an vector of latent factors, and be a vector of errors. The factor model is written as:

With the model matrix definitions in the previous section, the covariance matrix () of manifest variables is structured as:

The mean vector () of manifest variables is structured as:

Exploratory Factor Analysis Models

So far confirmatory and exploratory models are not distinguished in deriving the covariance and mean structures. These two types of models are now distinguished in terms of the required structures or restrictions in model matrices.

Traditionally, exploratory factor analysis is applied when the relationships of manifest variables with factors have not been well-established in research. All manifest variables are allowed to have nonzero loadings on the factors in the model. First, factors are extracted and an initial solution is obtained. Then, for ease of interpretation a final factor solution is usually derived by rotating the factor space. Factor-variable relationships are determined by interpreting the final factor solution. This is different from the confirmatory factor analysis in which the factor-variable relationships are prescribed and to be confirmed.

Nonetheless, in terms of initial model fitting, exploratory factor analysis is not different from confirmatory factor analysis. In PROC TCALIS, the initial exploratory factor solution is obtained from a specific confirmatory factor model with restricted model matrices, which are described as follows:

• The factor loading matrix has fixed zeros at the upper triangle portion of the matrix.

• The factor covariance matrix is an identity matrix, which means that factors are not correlated.

• The error covariance matrix is a diagonal matrix.

• The mean structures are not modeled. That is, no specification in or vector is allowed.

For example, for an analysis with nine variables and three factors, the relevant model matrices of an exploratory FACTOR model have the following patterns:

and

where * denotes free parameters in the model matrices.

It is emphasized that most of these patterns are put on the initial factor solutions only. If an initial factor solution is rotated afterward, only the pattern of the error covariance matrix remains unchanged. That is, it is still a diagonal matrix after rotation. In general, rotating a factor solution will eliminate the fixed zero pattern in the upper triangle of the factor loading matrix . If you apply an orthogonal rotation, the factor covariance matrix will not change. It is an identity matrix before and after rotation. However, if you apply an oblique rotation, in general the rotated factor covariance matrix will not be an identity matrix and the off-diagonal elements will not be zeros.

Confirmatory Factor-Analysis Models

In confirmatory FACTOR models, there are no imposed patterns on the , , , and model matrices. All elements in these model matrices can be specified. However, for model identification, you might need to specify some factor loadings or factor variances as constants.

The only model restriction in confirmatory FACTOR models is placed on , which must be a diagonal matrix, as in exploratory FACTOR models too.

For example, for a confirmatory factor analysis with nine variables and three factors, you might specify the following patterns for the model matrices:

and

where * denotes free parameters in the model matrices.

In this confirmatory factor model, mean structures are not modeled. In addition, there are some distinctive features that underscore the differences between confirmatory and exploratory models:

• Factor loading matrix contains mostly zero elements and few nonzero free parameters, a pattern which is seen in most confirmatory factor models. In contrast, in exploratory factor models most elements in the matrix are nonzero parameters.

• Factor loading matrix contains fixed values of ones. These fixed values are used for model identification purposes (that is, identifying the scales of the latent variables). In general, you always have to make sure that your confirmatory factor models are identified by putting fixed values in appropriate parameter locations in the model matrices. However, this is not a concern in exploratory FACTOR models because identification has been ensured by imposing certain patterns on the model matrices.

• The nonzero off-diagonal parameters in the factor covariance matrix indicate that correlated factors are hypothesized in the confirmatory factor model. This cannot be the case with the initial model of exploratory FACTOR models, where the matrix must be an identity matrix before rotation.

Summary of Matrices in the FACTOR Model

Let be the number of manifest variables and be the number of factors in the FACTOR model. The names, roles, and dimensions of the FACTOR model matrices are shown in the following table:

Specification of the Exploratory Factor Model

Because all initial model matrices of exploratory FACTOR models are predefined in PROC TCALIS, you do not need to specify any other parameters in the model matrices. To obtain desired factor solutions, you can use various options for exploratory factor analysis in the FACTOR statement. These options are the EFA_options in the FACTOR statement. Two main types of EFA_options are shown as follows:

• options for factor extraction: COMPONENT, HEYWOOD, N=.

• options for factor rotation: GAMMA=, NORM=, RCONVERGE=, RITER=, ROTATE=, TAU=.

For example, the following statement requests that three factors are extracted, followed by a varimax rotation of the initial factor solution:

   factor n=3 rotate=varimax;

See the FACTOR statement for details about the EFA_options.

Specification of the Confirmatory Factor Model

To specify a confirmatory FACTOR model, you specify the factor-variable relationships in the FACTOR statement, the factor variances and error variances in the PVAR statement, the factor covariances in the COV statement, and the means and intercepts in the MEAN statement.

   factor 
      F1  -> V1-V3  = 1. parm1 (.4) parm2 (.4),
      F2  -> V4-V6  = 1. parm3 parm4,
      F3  -> V7-V9  = 1. parm5 parm6 (.3);

   pvar F1-F3  = fvar1-fvar3,
        V1-V9  = evar1-evar9 (9*10.);

   cov F1 F2  = cov12,
       F1 F3  = cov13,
       F2 F3  = cov23;

   mean F1-F3 = fmean1-fmean3,
        V1-V9 = 9*12.;

Default Parameters in the FACTOR Model

In the initial exploratory FACTOR model, all fixed and free parameters of the model are prescribed or set by default. These prescribed or default parameters include a fixed pattern for the factor loading matrix , a diagonal pattern for the error variance matrix , and an identity matrix for factor covariance matrix . See the section Exploratory Factor Analysis Models for more details about the patterns of the exploratory FACTOR model.

In the confirmatory FACTOR model, the error covariance matrix is restricted to be a diagonal matrix. In other words, covariances of errors are restricted to zeros. Other than that, you can specify any other parameter locations in the all model matrices. These includes all the parameter locations in the factor loading matrix , the factor covariance matrix , the diagonal of the error covariance matrix , the factor mean vector , and the intercept vector . If any of these parameter locations are not specified, default parameters are applied. There are two types of default parameters: one is automatic free parameter, and the other is fixed zero.