Multi-set factor analysis by means of Parafac2

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