Matrix of associations. The basic unit of analysis in an EFA is a matrix of associations—either a correlation or a covariance matrix. If you input a data set into your EFA, the program will estimate this as step 1. Alternatively, you can input the correlation or covariance matrix directly, reading it in as the raw data. This can be useful when trying to replicate someone’s analyses based on published results or when wanting to analyze ordinal or dichotomous variables through a corrected correlation matrix (i.e., polychoric or tetrachoric). In either case, the extraction methods above will yield slightly different results based on the matrix of association being analyzed. The default method in PROC FACTOR is the simple correlation matrix (the most commonly used type of association for EFA). Correlations are most commonly used in EFA as they are only influenced by the magnitude of the association of the two variables. By contrast, covariances are influenced by association, as well as the variance of each of the two variables in question (Thompson, 2004). The default method in PROC FACTOR can be changed using the COVARIANCE option.

Communalities. In EFA, the communalities are the estimates of the shared variance in each variable, or the variance that will be accounted for by all the factors. They are computed from the matrix of associations, and their decomposition and partitioning is the goal of all subsequent analysis. The estimation of the communalities is a defining characteristic of EFA that distinguishes it from PCA. In EFA, the communalities are always less than 1.00 for each variable because EFA seeks to decompose the shared variance; while in PCA, they are initially 1.00 because there is no distinction between shared and unique variance.

Although the different extraction methods generally yield different estimates of communalities, each method typically starts with the same initial estimates. The initial estimate aims to get a quick and simple idea of the shared variance in each variable. In PROC FACTOR, the default process for the EFA techniques[1] is to estimate the initial communalities as the squared multiple correlation of a variable with all other variables. They are called the “Prior Communality Estimates” in the output, and they should appear as one of the first tables. Starting with the initial estimates, the communalities are then iteratively re-estimated via the selected extraction method to produce the final estimates.

The communalities can be thought of as a row statistic. When looking at a table of factor loadings, with variables as the rows and factor loadings in columns, the communalities for a variable are a function of the factor loadings. Squaring and summing each factor loading for a variable should equal the extracted communality (within reasonable rounding error).[2]

Eigenvalues. Eigenvalues are a representation of the aggregated item-level variance associated with a factor. They can be viewed as a column statistic—again imagining a table of factor loadings. If you square each factor loading and sum them all within a column, you should get an approximation of the eigenvalue for that factor (again within rounding error). Thus, eigenvalues are higher when there are at least some variables with high factor loadings, and lower when there are mostly low loadings. You will notice that eigenvalues (and communalities) change from initial statistics (which are estimates and should be identical regardless of extraction method[3]) to extraction, which will vary depending on the mathematics of the extraction. The cumulative percent variance accounted for by the extracted factors will not change (to be discussed later), but the distribution of the variance will change along with changing factor loadings during rotation. Thus, if the extracted eigenvalues account for a cumulative 45% of the variance overall, the cumulative variance accounted for will still be 45% after the factors are rotated, but that 45% might have a slightly different distribution across factors after rotation. This will become clearer in a little bit, as we look at some example data.

Iterations and convergence. The majority of the methods described below rely on an iterative procedure to “converge” on a final solution. Convergence occurs when the change between one model’s communalities and the next model’s communalities is less than .001. This can essentially be interpreted to mean that the two models are yielding the same results. Although we have not yet come across a good reason to reset the convergence criteria, it is possible to do so through the CONVERGE option. If an EFA analysis fails to “converge,” that means that these coefficients failed to stabilize and continued changing dramatically. This is most commonly due to inappropriately small sample sizes. One potential solution to this problem is to increase the default number of iterations. The default number of iterations is 30 in PROC FACTOR and it can be reset with the MAXIT ER option.