The full principal component extraction model assumes that all the variance is common, and so the communalities are all equal to 1 (i.e. there is no specific variance). It is only when we reduce the number of factors that specific variance is introduced into the model.
In the principal axis factoring method, we make an initial estimate of the common variance in which the communalities are less than 1. This initial estimate assumes that the communality of each variable is equal to the square multiple regression coefficient of that variable with respect to the other variables. The principal axis factoring method is implemented by replacing the main diagonal of the correlation matrix (which consists of all ones) by these initial estimates of the communalities. The principal component is now applied to this revised version of the correlation matrix, as described above
In the principal axis method the following iterative approach is used:
R0 = R, the original correlation matrix
Rp+1 = Rp with the main diagonal of Rp replaced by the communalities Cp of Rp
This algorithm is repeated until a predefined maximum number of iterations are performed or the communalities converge (and so there is too little difference between Cp and Cp+1 (and therefore between Rp and Rp+1. The final version of Rp is then used as in the principal component method of extraction.
We now discuss how to calculate the communalities Cp. To calculate the initial communalities C0 for principal axis factoring we use the value of R2 between each factor and all the other factors. For Example 1 of Factor Extraction, the initial communalities are given in range V33:V41 of Figure 1.
Figure 1 – Initial Communalities
Referring to the sample data in Figure 1 of Factor Analysis Example, the communality for the first factor (cell V33) can be computed by the formula =RSquare(B4:J123,U33), which is has the same value as RSquare(C4:J123,B4:B123), and similarly for the other eight factors.
It turns out that the vector of initial communalities V33:V41 can also be computed by the array formula
where M4:U12 is the correlation matrix (see Figure 3 of Factor Analysis Example).
For each p we show how to compute the communalities Cp+1 in the next example.
Example 1: Repeat the factor analysis on the data in Example 1 of Factor Extraction using the principal axis factoring method.
As calculate the correlation matrix and then the initial communalities as described above. We next substitute the initial communalities in the main diagonal of the correlation matrix and calculate the factor matrix as we did in the principal component method of extraction. This is shown in Figure 2.
Figure 2 – Iteration #1
The revised correlation matrix R1 in range Y6:AG14 is equal to the original correlation matrix with the entries in the main diagonal replaced by the communalities calculated in the previous step (i.e C0 in this case). We can calculate this correlation via the array formula
where M4:U12 is the original correlation matrix R0 (Figure 3 of Factor Analysis Example) and V33:V41 are the communities C0 (from Figure 1).
The eigenvalues and eigenvectors in range Y18:AG28 is calculated by =eVECTORS(Y6;AG114). The Factor Matrix in range Y33;AG41 is calculated as in Principal Component extraction, except where the corresponding eigenvalues are not positive. While this is not possible for Principal Component extraction, it is possible for Principal Axis extraction. When an eigenvalue is non-positive (as is the case with the final 5 eigenvalues in Figure 2) the corresponding loading factors are set to zero. For example the formula for calculating the first entry in the Factor Matrix (cell Y33) is
The new communalities C1 (range AH33:AH41) is now computed as in Principal Component extraction. E.g. AH33 is computed by the formula =SUMSQ(Y33:AG33).
As explained above we next calculate C2 and R2 in the same manner, and continue in this manner until a fixed number of predetermined iterations is reached (e.g. we will use p = 25 as the default maximum number of iterations) or until Cp and Cp+1 are sufficiently close. For this we test whether the sum of the squares of the differences in the communalities are less than some predetermined precision amount (we will use .00001 as the default).
For iteration #1 this metric is found in cell AH43 and is calculated by the formula
(referring to Figure 1 and 2).
It turns out that after 19 iterations convergence goal of .00001 is reached with the difference between the communalities C18 and C19 of 8.81E-06. The values of the communalities after the 19th iteration are given in range IP33:IP41 of Figure 3.
Figure 3 – Iteration #19
The Real Statistics Resource Pack provides a supplemental array function which automates the process of finding the converged values of the communalities, thus avoiding the tedious calculations described above.
Real Statistics Function: If R1 is a k × k correlation matrix then
ExtractCommunalities(R1, iter, prec, eigen) = the 1 × k row vector with the communalities after convergence based on a precision value of prec but with a maximum number of iter iteration. As described above we use .00001 as the default value of prec and 25 as the default value of iter.
Since the eigenvalues and eigenvectors of the correlation matrix is calculated (using the eVECTORS supplemental function) in each iteration, a fourth argument eigen can be used to specify the number of iterations used to calculate these eigenvalues/vectors (with a default of 100).
Once these values for the communalities is found the Principal Axis extraction method proceeds exactly as for the Principal Component extraction method, except that these communalities are used instead of 1’s in the main diagonal of the correlation matrix. This is illustrated in Real Statistics Support for Factor Analysis.