  A recommender system learns to predict the user-specific preference or
intention over many items simultaneously for all users, making personalized
recommendations based on a relatively small number of observations. One central
issue is how to leverage three-way interactions, referred to as user-item-stage
dependencies on a monotonic chain of events, to enhance the prediction
accuracy. A monotonic chain of events occurs, for instance, in an article
sharing dataset, where a ``follow'' action implies a ``like'' action, which in
turn implies a ``view'' action. In this article, we develop a multistage
recommender system utilizing a two-level monotonic property characterizing a
monotonic chain of events for personalized prediction. Particularly, we derive
a large-margin classifier based on a nonnegative additive latent factor model
in the presence of a high percentage of missing observations, particularly
between stages, reducing the number of model parameters for personalized
prediction while guaranteeing prediction consistency. On this ground, we derive
a regularized cost function to learn user-specific behaviors at different
stages, linking decision functions to numerical and categorical covariates to
model user-item-stage interactions. Computationally, we derive an algorithm
based on blockwise coordinate descent. Theoretically, we show that the
two-level monotonic property enhances the accuracy of learning as compared to a
standard method treating each stage individually and an ordinal method
utilizing only one-level monotonicity. Finally, the proposed method compares
favorably with existing methods in simulations and an article sharing dataset.
