Other Issues and Statistics - Matrix Ill-conditioning

Computationally, in order to perform a factor analysis, STATISTICA needs to invert the correlation matrix. If, in this correlation matrix there are variables that are 100% redundant, then the inverse of the matrix cannot be computed. For example, if a variable is the sum of two other variables selected for the analysis, then the correlation matrix of those variables cannot be inverted, and the factor analysis can basically not be performed. In practice this happens when you are attempting to factor analyze a set of highly intercorrelated variables, as it, for example, sometimes occurs in correlational research with questionnaires. The Factor Analysis module will detect matrix ill-conditioning and issue a respective warning. For all extraction methods other than principal components, STATISTICA will artificially lower all correlations in the correlation matrix by adding a small constant to the diagonal of the matrix, and then restandardizing it. This procedure will usually yield a matrix that now can be inverted and thus factor-analyzed; moreover, the factor patterns should not be affected by this procedure. However, note that the resulting estimates are not exact.

In the case of principal components analysis the number of components that can be extracted is equal to the number of positive eigenvalues for the respective correlation matrix; no adjustment (restandardization) of the correlation matrix will be made in this case.