AP 41- Retaining principal components for discrete variables

The present study discusses retention criteria for principal components analysis (PCA) applied to Likert scale items
typical in psychological questionnaires. The main aim is to recommend applied researchers to restrain from relying only on
the eigenvalue-than-one criterion; alternative procedures are suggested for adjusting for sampling error. An additional objective
is to add evidence on the consequences of applying this rule when PCA is used with discrete variables. The experimental
conditions were studied by means of Monte Carlo sampling including several sample sizes, different number of variables
and answer alternatives, and four non-normal distributions. The results suggest that even when all the items and thus
the underlying dimensions are independent, eigenvalues greater than one are frequent and they can explain up to 80% of the
variance in data, meeting the empirical criterion. The consequences of using Kaiser’s rule are illustrated with a clinical
psychology example. The size of the eigenvalues resulted to be a function of the sample size and the number of variables,
which is also the case for parallel analysis as previous research shows. To enhance the application of alternative criteria,
an R package was developed for deciding the number of principal components to retain by means of confidence intervals
constructed about the eigenvalues corresponding to lack of relationship between discrete variables.

Keywords: principal components analysis, eigenvalues, parallel analysis, discrete items.