Exploratory Factor Analysis

For the exploratory factor model with orthogonal factors, PROC TCALIS assumes the following model structures for the population covariance or correlation matrix :

where is the factor loading matrix and is a diagonal matrix of error variances. In this section, denotes the number of manifest variables corresponding to the rows and columns of matrix , and denotes the number of factors (or components, if the COMPONENT option is specified in the FACTOR statement) corresponding to the columns of the factor loading matrix . While the number of manifest variables is set automatically by the number of variables in the VAR statement or in the input data set, the number of factors can be set by the N= option in the FACTOR statement.

The unrestricted exploratory factor model is not identified because any orthogonal rotated factor loading matrix satisfies the same model structures as does, where is any orthogonal matrix so that . Mathematically, the covariance or correlation structures can be expressed as:

To obtain an identified orthogonal factor solution as a starting point, the elements in the upper triangle of are constrained to zeros in PROC TCALIS. Initial estimates for factor loadings and unique variances are computed by an algebraic method of approximate factor analysis. Given the initial estimates, final estimates are obtained through the iterative optimization of an objective function, which depends on the estimation method specified in the METHOD= option (default with ML—maximum likelihood) of the PROC TCALIS statement.

To make the factor solution more interpretable, you can use the ROTATE= option in the FACTOR statement to obtain a rotated factor loading matrix with a "simple" pattern. Rotation can be orthogonal or oblique. The rotated factors remain uncorrelated after an orthogonal rotation but would be correlated after an oblique rotation. The model structures of an oblique solution are expressed in the following equation:

where is the rotated factor loading matrix and is a symmetric matrix for factor correlations. See the sections The FACTOR Model and Exploratory Factor Analysis Models for more details about exploratory factor models.

You can also do exploratory factor analysis by the more dedicated FACTOR procedure. Even though extensive comparisons of the factor analysis capabilities between the FACTOR and TCALIS procedures are not attempted here, some general points can be made here. In general, the FACTOR procedure provides more factor analysis options than the TCALIS procedure does, although both procedures have some unique factor analysis features that are not shared by the other. PROC TCALIS requires more computing time and memory than PROC FACTOR because it is designed for more general structural estimation problems and is not able to exploit all the special properties of the unconstrained factor analysis model. For maximum likelihood analysis, you can use either PROC FACTOR (with METHOD=ML, which is not the default method in PROC FACTOR) or PROC TCALIS. Because the initial unrotated factor solution obtained by PROC FACTOR uses a different set of identification constraints than that of PROC TCALIS, you would observe different initial ML factor solutions for the procedures. Nonetheless, the initial solutions by both procedures are statistically equivalent.

The following exploratory factor analysis options are available in the FACTOR statement.

Confirmatory Factor Analysis

FACTOR factor-variable-relations ;

PVAR partial variance parameters ;

COV covariance parameters ;

MEAN mean parameters ;

If you specify at least one factor-variable-relation, a confirmatory factor analysis is assumed. Whenever applicable, you can use the PVAR, COV, and MEAN statements to specify the variance, partial variance, covariance, and mean parameters. The model structures for the covariance matrix of the confirmatory factor model are described in the following equation:

where is the factor loading matrix, is a symmetric matrix for factor correlations, and is a diagonal matrix of error variances.

If the mean structures are also analyzed, the model structures for the mean vector of the confirmatory factor model are described in the following equation:

where is the intercept vector for the observed variables and is the vector for factor means. See the sections The FACTOR Model and Confirmatory Factor-Analysis Models for more details about confirmatory factor models.

You can use the PVAR statement to specify factor variance parameters in the diagonal of matrix , or the error variances in the diagonal of matrix . These variances, by default, are automatically added as free parameters in the factor model even if you do not specify them. However, if you want to set constraints on these parameters, it is useful to define them explicitly by using the PVAR statement. For example, you might like to set some variances to fixed constants, or you might want to set equality constraints by using the same parameter name at different parameter locations in your model.

You can use the COV statement to specify factor covariance parameters. These factor covariances are the off-diagonal elements of matrix . Unlike the factor variances, factor covariances are assumed to be fixed zeros by default. Note that you cannot use the COV statement to specify the error covariances—they are always fixed zeros in the confirmatory factor-analysis model.

You can use the MEAN statement to specify the means of the factors in vector and the intercepts of the manifest variables in vector . All the factor means and intercepts are fixed zeros if you do not specify them explicitly.

Suppose that you want to fit a factor model with nine manifest variables V1-V9. In this factor model, you assume a general factor and three group-factors, with three variables defining each group-factor. The following FACTOR statement shows the specification of such a model:

 
   factor
      g_factor -> V1-V9 (9 * g_load__),
      factor_a -> V1-V3 (3 * load_a__),  
      factor_b -> V4-V6 (3 * load_b__),  
      factor_c -> V7-V9 (3 * load_c__);

In this example, you are fitting a factor matrix of the following form:

 
            g_factor   factor_a   factor_b   factor_c

     V1      g_load1    load_a1
     V2      g_load2    load_a2
     V3      g_load3    load_a3
     V4      g_load4               load_b1
     V5      g_load5               load_b2
     V6      g_load6               load_b3
     V7      g_load7                          load_c1
     V8      g_load8                          load_c2
     V9      g_load9                          load_c3

The blanks in the factor matrix denotes fixed zeros. Because you do not specify the covariances by using the COV statement, g_factor, factor_a, factor_b, and factor_c are all uncorrelated in the model.

Modifying a FACTOR Model from a Reference Model

In this section, it is assumed that you use a REFMODEL statement within the scope of a MODEL statement and the reference model (or base model) is a factor model, either exploratory or confirmatory. The reference model is referred to as the old model, while the model makes reference to this old model is referred to as the new model. If the new model is not intended to be an exact copy of the old FACTOR model, you can use the following extended FACTOR modeling language to make modifications from the old model before transferring the specifications to the new model.

Before discussing the syntax for modifying factor models, there are two cases where using the REFMODEL statement for defining new factor models are not recommended. First, if your old model is an exploratory factor-analysis model, then specification by using the FACTOR modeling language in the new model will replace the old model completely. In this case, the use of the REFMODEL statement is superfluous and should be avoided. Second, if your old model is a confirmatory factor-analysis model, then specification of an exploratory factor model by using the FACTOR statement in the new model will also replace the old model completely. Again, in this case the use of the REFMODEL statement is superfluous and should be avoided.

The nontrivial case where you might find the REFMODEL statement useful is when you modify an old confirmatory factor model to form a new confirmatory factor model. This nontrivial case is the focus of discussion in the remaining of the section.

The extended FACTOR modeling language for modifying model specification bears the same syntax as that of the ordinary FACTOR modeling language (see the section Confirmatory Factor Analysis). The syntax is:

FACTOR factor-variable-relation ;

PVAR partial variance parameters ;

COV covariance parameters ;

MEAN mean parameters ;

The new model is formed by integrating with the old model in the following ways:

Duplication:

If you do not specify in the new model a parameter location that exists in the old model, the old parameter specification is duplicated in the new model.

Addition:

If you specify in the new model a parameter location that does not exist in the old model, the new parameter specification is added in the new model.

Deletion:

If you specify in the new model a parameter location that also exists in the old model and the new parameter is denoted by the missing value '.', the old parameter specification is not copied into the new model.

Replacement:

If you specify in the new model a parameter location that also exists in the old model and the new parameter is not denoted by the missing value '.', the new parameter specification replaces the old one in the new model.

For example, consider the following two-group analysis:

   proc tcalis;
      group 1 / data=d1;
      group 2 / data=d2;
      model 1 / group=1;
         factor 
            F1 -> V1-V3 = 1. load1 load2,
            F2 -> V4-V6 = 1. load3 load4,
            F3 -> V7-V9 = 1. load5 load6;
         cov
            F1 F2 = c12,
            F2 F3 = c23;
         pvar
            F1-F3 = c1-c3,
            V1-V9 = ev1-ev9;
      model 2 / group=2;
         refmodel 1;
         factor
            F1 -> V1 = loada,
            F2 -> V4 = loadb,
            F3 -> V7 = loadc;
         cov
            F1 F2 = .,
            F1 F3 = c13;
      run;

In this specification, you specify model 2 by referring to model 1 in the REFMODEL statement; model 2 is the new model which refers to the old model, model 1. Because the PVAR statement is not used in new model, all variance and partial variance parameter specifications in the PVAR statement of the old model are duplicated in the new model. The covariance parameter c23 for covariance between F2 and F3 in the COV statement of the old model is also duplicated in the new model. Similarly, loading parameters load1–load6 for some specific factor matrix locations are duplicated from the old model to the new model.

In the new model, there is an additional parameter specification that does not exist in the old model. In the COV statement of the new model, covariance parameter c13 for the covariance between F1 and F3 is added.

In the same statement, the covariance between F1 and F2 is denoted by the missing value '.'. This means that this parameter location in the old model should not be included in the new model. The consequence of this deletion from the old model is that the covariance between F1 and F2 is a fixed zero in the new model.

Finally, the three new loading specifications in the FACTOR statement of the new model replaces the fixed ones in the old model. They are now free parameters loada, loadb, and loadc in the new model.