Multilevel Mixture with Known Mixing Proportions: Applications to School and Individual Level Overweight and Obesity Data from Birmingham, England

Multilevel (ML) models allow for total variation in the outcome to be decomposed as level one and level two or ‘individual and group’ variance components. Multilevel Mixture (MLM) models can be used to explore unobserved heterogeneity that represents different qualitative relationships in the outcome. In this paper, we extend the standard MLM by introducing constraints to guide the MLM algorithm towards a more appropriate data partitioning. Our constraint-based methods combine the mixing proportions estimated by parametric Expectation Maximization (EM) of the outcome and the random component from the ML model. This forms a new Multilevel Mixture known (MLMk) mix method. This framework allows for smaller residual variances and permits meaningful parameter estimates for distinct classes in the coefficient space. We also provide an illustrative example demonstrating the advantage of the MLMk compared with the MLM approach. We show the benefit of our method using overweight and obesity from Body Mass Index (BMI) measurements for students in year 6. We apply these methods on multi-level BMI data to estimate student multiple deprivation and school sport effects.