Estimating latent trends in multivariate longitudinal data via Parafac2 with functional and structural constraints

Longitudinal data are inherently multimode in the sense that such data are often collected across multiple modes of variation, for example, time × variables × subjects. In many longitudinal studies, multiple variables are collected to measure some latent construct(s) of interest. In such cases, the goal is to understand temporal trends in the latent variables, as well as individual differences in the trends. Multimode component analysis models provide a powerful framework for discovering latent trends in longitudinal data. However, classic implementations of multimode models do not take into consideration functional information (i.e., the temporal sequence of the collected data) or structural information (i.e., which variables load onto which latent factors) about the study design. In this paper, we reveal how functional and structural constraints can be imposed in multimode models (Parafac and Parafac2) in order to elucidate trends in longitudinal data. As a motivating example, we consider a longitudinal study on per capita alcohol consumption trends conducted from 1970 to 2013 by the U.S. National Institute on Alcohol Abuse and Alcoholism. We demonstrate how functional and structural information about the study design can be incorporated into the Parafac and Parafac2 alternating least squares algorithms to understand temporal and regional trends in three latent constructs: beer consumption, spirits consumption, and wine consumption. Our results reveal that Americans consume more than the recommended amount of alcohol, and total alcohol consumption trends show no signs of decreasing in the last decade.