Modelling series of studies with a common structure

Consider the situation where the Structuration des Tableaux à Trois Indices de la Statistique (STATIS) methodology is applied to a series of studies, each study being represented by data and weight matrices. Relations between studies may be captured by the Hilbert–Schmidt product of these matrices. Specifically, the eigenvalues and eigenvectors of the Hilbert–Schmidt matrix SS may be used to obtain a geometrical representation of the studies. The studies in a series may further be considered to have a common structure whenever their corresponding points lie along the first axis. The matrix SS can be expressed as the sum of a rank 1 matrix λuuTλuuT with an error matrix EE. Therefore, the components of the vector λu are sufficient to locate the points associated to the studies. Former models for SS where vec(E)vec(E) are mathematically tractable and yet do not take into account the symmetry of the matrix SS. Thus a new symmetric model is proposed as well as the corresponding tests for a common structure. It is further shown how to assess the goodness of fit of such models. An application to the human immunodeficiency virus (HIV) infection is used for assessing the proposed model.