Multi-set factor analysis by means of Parafac2

Abstract

We consider multi-set data consisting of Nk 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 Σk, k = 1,..., K, where the common part is modelled by Parafac2 and the unique variances Uk, 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 Uk by minimum rank factor analysis on Σk 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.