Fitting the Correct Correlation Structures

This main idea of this solution is to embed the intended correlation structures (with correct constraints on the diagonal elements of the correlation matrix) into a covariance structure model so that the estimation methods of PROC TCALIS can be applied legitimately to the specially constructed covariance structures.

First, the issue of the fixed ones on the diagonal of the correlation structure model is addressed. That is, the diagonal elements of the correlation structures represented by must be fitted by ones. This can be accomplished by constraining the error variances as dependent parameters of the loadings, as shown in the following:

Other constraints might also serve the purpose, but the ones just presented are the most convenient and intuitive.

Now, due to the fact that discrepancy functions used in PROC TCALIS are derived for covariance matrices rather than correlation matrices, PROC TCALIS is essentially set up for analyzing covariance structures (with or without mean structures), but not correlation structures. Hence, the statistical theory behind PROC TCALIS applies to covariance structure analysis, but it might not generalize to correlation structure analysis in all situations. Despite that, with some manipulations PROC TCALIS can fit the correct correlation structures to the current data without compromising the statistical theory. These manipulations are now discussed. Recall that the correlation structures are represented by:

As before, in the matrix, there are six linear constraints on the factor loadings. In addition, the diagonal elements of are constrained to ones, as done by defining the error variances as dependent parameters of the loadings in the preceding equation. To analyze the correlation structures by using PROC TCALIS, a covariance structure model with such correlation structures embedded is now specified. That is, the covariance structure to be fitted by PROC TCALIS is as follows:

where is a 6 x 6 diagonal matrix containing the population standard deviations for the manifest variables. Theoretically, it is legitimate that you analyze this covariance structure model for studying the embedded correlation structures. In addition, it does not matter whether your input matrix is a correlation or covariance matrix, or any rescaled covariance matrix (by multiplying any variables by any positive constants). You would get correct results if you could somehow specify these covariance structures correctly in PROC TCALIS. However, there seems to be nowhere in PROC TCALIS that you can specify the diagonal matrix for the population standard deviations. So what can one do with this formulation? The answer is to rewrite the covariance structure model in a form similar to the usual confirmatory factor model, as presented in the following.

Let and . The covariance structure model of interest can now be rewritten as:

This form of covariance structures implies a confirmatory factor model with factor loading matrix and error covariance matrix . This confirmatory factor model can certainly be specified using the FACTOR modeling language, in much the same way you specify a confirmatory factor model in the preceding section. However, because you are actually more interested in estimating the basic set of parameters in matrices and of the embedded correlation structures, you would define the model parameters as functions of this basic set of parameters of interest. This can be accomplished by using the PARAMETERS and the SAS programming statements.

All in all, you can use the following statements to set up such a confirmatory factor model with the desired correlation structures embedded.

   proc tcalis data=Kinzer nobs=326 nose;
      factor
         factor1 -> var1-var6 = t11 t21 t31 t41 t51 t61,
         factor2 -> var1-var6 = t12 t22 t32 t42 t52 t62;
      pvar 
         factor1-factor2 = 2 * 1.,
         var1-var6       = k1-k6;
      parameters alpha (1.) d1-d6 (6 * 1.)
                 b11 b21 b31 b41 b51 b61 (6 *.6),
                 b12 b22 b32 b42 b52 b62
                 psi1-psi6;
      /* SAS Programming Statements */
      /* 12 Constraints on Correlation structures */
      b12  = alpha - b11;
      b22  = alpha - b21;
      b32  = alpha - b31;
      b42  = alpha - b41;
      b52  = alpha - b51;
      b62  = alpha - b61;
      psi1 = 1. - b11 * b11 - b12 * b12;
      psi2 = 1. - b21 * b21 - b22 * b22;
      psi3 = 1. - b31 * b31 - b32 * b32;
      psi4 = 1. - b41 * b41 - b42 * b42;
      psi5 = 1. - b51 * b51 - b52 * b52;
      psi6 = 1. - b61 * b61 - b62 * b62;
      /* Defining Covariance Structure Parameters */
      t11  = d1 * b11;
      t21  = d2 * b21;
      t31  = d3 * b31;
      t41  = d4 * b41;
      t51  = d5 * b51;
      t61  = d6 * b61;
      t12  = d1 * b12;
      t22  = d2 * b22;
      t32  = d3 * b32;
      t42  = d4 * b42;
      t52  = d5 * b52;
      t62  = d6 * b62;
      k1   = d1 * d1 * psi1;
      k2   = d2 * d2 * psi2;
      k3   = d3 * d3 * psi3;
      k4   = d4 * d4 * psi4;
      k5   = d5 * d5 * psi5;
      k6   = d6 * d6 * psi6;
      fitindex on(only)=[chisq df probchi];
   run;

First, you notice that specifications in the FACTOR and the PVAR statements are essentially unchanged from the previous specification, except that the parameters are named differently here to reflect different model matrices. In the current specification, the factor loading parameters in matrix are named with prefix 't,' and the error variance parameters in matrix are named with prefix 'k.' Specification of these parameters reflects the covariance structures. As you see in the last block of the SAS programming statements statements, all these parameters are functions of the correlation structure parameters in , , and .

Next, in the PARAMETERS statement, all correlation structure parameters are defined with initial values provided. These are the parameters of interest: alpha is used to define dependencies among loadings, d’s are the population standard deviations, b’s are the loading parameters, and psi’s are the error variance parameters. There are 25 parameters specified in this statement, but not all of them are free or independent.

In the first block of SAS programming statements, parameter dependencies or constraints on the correlation structures are specified. The first six statements realize the required linear relations among the factor loadings:

The next six statements constrain the error variances so as to ensure that an embedded correlation structure model is being fitted. That is, each error variance is dependent on the corresponding loadings, as prescribed by the following equation:

These twelve constraints reduce the number of independent parameters to 13, as expected.

The next block of SAS programming statements are essentially for relating the correlation structure parameters to the covariance structures that are specified in the FACTOR and the PVAR statements. These SAS programming statements realize the required relations: and , but in non-matrix forms:

where denotes the j-th diagonal element of .

The fit summary is presented in Output 88.9.4. The chi-square test statistic is 14.63 with =8 (=0.067). This shows that the previous chi-square test based on fitting a wrong covariance structure model is indeed questionable.

Estimates of the loadings and error variances are presented in Output 88.9.5. These estimates are for the covariance structure model with loading matrix and error covariance matrix . They are rescaled versions of the correlation structure parameters and are not of primary interest themselves.

The parameter estimates of the embedded correlation structures are shown in Output 88.9.6 as "additional" parameters.

Except for the population standard deviation parameter d’s, all other parameters estimated in the current model can be compared with those from the previous fitting of an incorrect covariance structure model. Although estimates in the current model do not differ very much from those in the previous specification, it is at least reassuring that they are obtained from fitting a correctly specified covariance structure model with the intended correlation structures embedded.