In order to talk about principal axis factors (PAF) the term communality must first be introduced. In the common factor model, each original variable x is thought of to be a combination of common factors and a unique factor. The variance of x can then also be thought of as being composed of a common portion and a unique portion, that is . It is the common portion of the variance of x that is called the communality of x, that is the variance that the variable has in common through the common factors with all the other variables.
With the idea of communality thus in place it is straightforward to describe the principal axis factors algorithm. Begin by estimating the communality of each variable and replacing this value in the appropriate position in the diagonal of the correlation or covariance matrix being factored. Then a principal components solution is found in the usual manner, as described earlier. As before, the user has the option of specifying either a fixed number of desired factors or a minimum eigenvalue by which to reduce the number of factors in the solution. Finally, the new communalities are calculated as the sum of the squared factor loadings, and these values are substituted into the correlation or covariance matrix. This process is repeated until the communalities change by only a small amount.
Through its use of communality estimates, the principal axis factor method attempts to find independent common factors that account for the covariance or correlation between the original variables in the model, while ignoring the effect of unique factors. It is then possible to use the factor loadings matrix to reproduce the correlation or covariance matrix and compare this to the original as a way of assessing the effectiveness of the model. The reproduced correlation or covariance matrix is simply the factor loadings matrix times its transpose (i.e., CCT). The user may optionally request that the signs of the factor loadings be inverted if there are more minus signs than positive ones. This is purely cosmetic and does not affect the solution in a substantive way. However, if signs are reversed, this must be kept in mind when attempting to interpret or assign meaning to the factors.