parallelanalys

Parallelanalys, also known as parallel analysis, is a statistical procedure used to determine how many components or factors to retain in dimensionality reduction techniques such as principal component analysis or exploratory factor analysis. The method compares observed data eigenvalues to those obtained from random data of the same size, helping to distinguish meaningful structure from random variation. It was first proposed by Horn in 1965 and has since become a standard tool in multivariate analysis.

The typical procedure is as follows: compute the eigenvalues of the observed correlation (or covariance) matrix; generate many random data sets with the same number of observations and variables (often from a standard normal distribution); for each simulated data set, calculate its eigenvalues to form a distribution for each component; retain components for which the observed eigenvalues exceed a chosen threshold, commonly the 95th percentile of the random distribution. The number of retained components is usually the largest index for which this condition holds.

Parallel analysis can be applied to PCA or to factor analysis. It can use correlation or covariance matrices and requires standardization of variables if they are on different scales. It is implemented in many statistical software packages and has several variants, including permutation-based or percentile-based approaches, to accommodate different data conditions and assumptions.

Limitations include sensitivity to sample size and variable count, assumptions of continuous and roughly multivariate normal data, and potential effects of outliers or nonnormal distributions. While it provides an objective criterion, parallelanalys should be used alongside substantive theory and other fit indicators when deciding on the number of factors or components to retain.