Multi‐set factor analysis by means of Parafac2. (3rd November 2015) - Stegeman, Alwin; Lam, Tam T. T.
Abstract : We consider multi‐set data consisting of N k observations, k  = 1, …, K (e.g., subject scores), on J variables in K different samples. We introduce a factor model for the J  ×  J covariance matrices Σ k, k  = 1, …, K, where the common part is modelled by Parafac2 and the unique variances U k, k  = 1, …, K, are diagonal. The Parafac2 model implies a common loadings matrix that is rescaled for each k, and a common factor correlation matrix. We estimate the unique variances U k by minimum rank factor analysis on Σ k for each k . The factors can be chosen orthogonal or oblique. We present a novel algorithm to estimate the Parafac2 part and demonstrate its performance in a simulation study. Also, we fit our model to a data set in the literature. Our model is easy to estimate and interpret. The unique variances, the factor correlation matrix and the communalities are guaranteed to be proper, and a percentage of explained common variance can be computed for each k . Also, the Parafac2 part is rotationally unique under mild conditions.