REMARKS ON PARALLEL ANALYSIS

We investigate parallel analysis (PA), a selection rule for the number -of-factors problem, from the point of view of permutation assessment. The idea of applying permutation test ideas to PA leads to a quasi-in ferential, non-parametric version of PA which accounts not only for fi nite-sample bias but sampling variability as well. We give evidence, h owever, that quasi-inferential PA based on normal random variates (as opposed to data permutations) is surprisingly independent of distribut ional assumptions, and enjoys therefore certain non-parametric propert ies as well. This is a justification for providing tables for quasi-in ferential PA. Based on permutation theory, we compare PA of principal components with PA of principal factor analysis and show that PA of pr incipal factors may tend to select too many factors. We also apply par allel analysis to so-called resistant correlations and give evidence t hat this yields a slightly more conservative factor selection method. Finally, we apply PA to loadings and show how this provides benchmark values for loadings which are sensitive to the number of variables, nu mber of subjects, and order of factors. These values therefore improve on conventional fixed thresholds such as 0.5 or 0.8 which are used ir respective of the size of the data or the order of a factor.