Modeling Longitudinal Data Using Matrix Completion

In clinical practice and biomedical research, measurements are often collected sparsely and irregularly in time, while the data acquisition is expensive and inconvenient. Examples include measurements of spine bone mineral density, cancer growth through mammography or biopsy, a progression of defective vision, or assessment of gait in patients with neurological disorders. Practitioners often need to infer the progression of diseases from such sparse observations. A classical tool for analyzing such data is a mixed-effect model where time is treated as both a fixed effect (population progression curve) and a random effect (individual variability). Alternatively, researchers use Gaussian processes or functional data analysis, assuming that observations are drawn from a certain distribution of processes. While these models are flexible, they rely on probabilistic assumptions, require very careful implementation, and tend to be slow in practice. In this study, we propose an alternative elementary framework for analyzing longitudinal data motivated by matrix completion. Our method yields estimates of progression curves by iterative application of the Singular Value Decomposition. Our framework covers multivariate longitudinal data, and regression and can be easily extended to other settings. As it relies on existing tools for matrix algebra, it is efficient and easy to implement. We apply our methods to understand trends of progression of motor impairment in children with Cerebral Palsy. Our model approximates individual progression curves and explains 30% of the variability. Low-rank representation of progression trends enables identification of different progression trends in subtypes of Cerebral Palsy.