Probabilistic principal component analysis to identify profiles of physical activity behaviours in the presence of non-ignorable missing data

The paper is motivated by an accelerometer-based study of physical activity (PA) behaviours in a large cohort of UK school-aged children. Advances in research on PA are accompanied by a growing number of results that are contributing to form a complex picture of PA behaviours in children.One source of such complexity is intimately related to the multiplicity of dimensions associated with PA. Currently a comprehensive individual accelerometer summary can include a large number of outcomes and this clearly poses challenges for the analysis. We explore the application of principal component analysis to accelerometer measurements that are aggregated daily over several days of the week and are affected by missingness. The probabilistic approach to principal component analysis with latent scores is extended to include non-ignorable missing data. The extended likelihood is maximized through a Monte Carlo EM algorithm via adaptive rejection Metropolis sampling. Our findings suggest that physical activity and inactivity are two dimensions over which children aggregate into distinct behavioural profiles, characterized by gender and season but not by anthropometric factors.