Main effects and interactions in mixed and incomplete data frames

TL;DR

This paper introduces a novel estimation method called mimi for simultaneously recovering main effects and low-rank interactions in mixed data frames, with statistical guarantees and efficient optimization, outperforming existing methods especially with large main effects and missing data.

Contribution

The work presents a new method that incorporates main effects and interactions in mixed data frames with theoretical guarantees and an efficient convergent algorithm.

Findings

mimi performs well with sparse main effects and low-rank interactions

Outperforms existing methods when main effects are large

Effective with high proportions of missing data

Abstract

A mixed data frame (MDF) is a table collecting categorical, numerical and count observations. The use of MDF is widespread in statistics and the applications are numerous from abundance data in ecology to recommender systems. In many cases, an MDF exhibits simultaneously main effects, such as row, column or group effects and interactions, for which a low-rank model has often been suggested. Although the literature on low-rank approximations is very substantial, with few exceptions, existing methods do not allow to incorporate main effects and interactions while providing statistical guarantees. The present work fills this gap. We propose an estimation method which allows to recover simultaneously the main effects and the interactions. We show that our method is near optimal under conditions which are met in our targeted applications. We also propose an optimization algorithm which provably converges to an optimal solution. Numerical experiments reveal that our method, mimi, performs well when the main effects are sparse and the interaction matrix has low-rank. We also show that mimi compares favorably to existing methods, in particular when the main effects are significantly large compared to the interactions, and when the proportion of missing entries is large. The method is available as an R package on the Comprehensive R Archive Network.