A Universal Primal-Dual Convex Optimization Framework

In recent years, longitudinal data have become increasingly relevant in many applications, heightening interest in selecting the best longitudinal model to analyze them. Too often traditional practices rather than substantive theory guide the specific longitudinal model selected for the analysis. This opens the possibility that alternative models might better correspond to the data. In this regard, Bollen and Curran (2004) developed the Autoregressive Latent Trajectory (ALT) model. It captures the desirable features of both latent growth curve and autoregressive models, having the ability to discriminate between these two approaches to model panel data. The purpose of this paper is to develop the Latent Variable ALT (LV-ALT) model as a generalization of the autoregressive latent trajectory model. We show how the LV-ALT under constraints can specialize to a wide variety of other longitudinal models. Hence, if theory or prior work dictate the model, then latent variable ALT is likely capable of specializing to that structure. On the other hand, if there is little guidance on the best model, then the LV-ALT provides a way to empirically compare a wide variety of models and determine the most appropriate for the data. The latent variable ALT model also provides a framework which reveals the connections between many longitudinal models that were previously considered as distinct.