Trilinear structure and factor interaction of three-way data structures obtained using two-factor designs

Experiments performed according to two-way factor designs monitored using multivariate methods produce three-way data structures. The relationship between the presence of factor interaction and the trilinear model structure of the three-way data is investigated. In this work, this problem is investigated for a lipidomic study of the effects of time and tributyltin TBT dose on multiple samples of Daphnia Magna species during their molting and reproduction. Data simulations were performed using previously obtained experimental data [1]. In this simulations, the presence or not of interaction between the two factors, time and TBT dose, were considered, as well as the number of components (sources of data variance), bilinear vs trilinear behavior. Results show that for one single component systems (one single pattern of variation of the data), no factor interaction implies trilinearity, whereas factor interaction implies the lack of trilinear structure and the data are only bilinear. In the case of multicomponent systems, this relationship is more complex and the presence of interaction was shown in all cases for the global data response. The trilinear model gives individual responses for every separated component without interaction, although the global (all components) response appears to show the interaction between the two factors. In the case of only fulfillment of the bilinear model, the interaction between factors was always present.