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 directly observed. Each latent variable in the FACTOR model can be either 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 described in the following subsections.

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 CALIS, 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, where * denotes free parameters in the model matrices:

and

Most of these patterns are put only on the initial factor solutions. 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 eliminates the fixed zero pattern in the upper triangle of the factor loading matrix . If you apply an orthogonal rotation, the factor covariance matrix does 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 is not an identity matrix and the off-diagonal elements are not 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, where * denotes free parameters in the model matrices:

and

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 CALIS, 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, and N=.

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

For example, the following statement requests that three factors be 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

Default parameters in the FACTOR model are different for exploratory and confirmatory factor models.

For the exploratory FACTOR model, all fixed and free parameters of the model are prescribed. These prescribed 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 . This means that factors are uncorrelated in the estimation. However, if you specify an oblique rotation after the estimation of the factor solution, the factors could become correlated. See the section Exploratory Factor-Analysis Models for more details about the patterns of the exploratory FACTOR model. Because all these patterns are prescribed, you cannot override any of these parameters for the exploratory FACTOR model.

For the confirmatory FACTOR model, the set of default free parameters of the confirmatory FACTOR model includes the following:

• the error variances of the observed variables; these correspond to the diagonal elements of the uniqueness matrix

• the variances and covariances among the factors; these correspond to all elements of the factor covariance matrix

• the intercepts of the observed variables if the mean structures are modeled; these correspond to all elements of the intercept vector

PROC CALIS names the default free parameters with the _Add prefix, followed by a unique integers for each parameter. You can override the default free parameters by explicitly specifying them as free, constrained, or fixed parameters in the COV, MEAN, or PVAR statement.

In addition to default free parameters, another type of default parameter is the fixed zeros applied to the unspecified parameters in the loading matrix and the factor means in the vector. Certainly, you use the FACTOR and MEAN specifications to override those default zero loadings or factor means and set them to free, constrained, or fixed parameters. Notice that the uniqueness matrix in the confirmatory factor model is a diagonal element. You cannot specify any of its off-diagonal elements—they are always fixed zeros by the model restriction.