Bayesian Piecewise Linear Mixed Models With a Random Change Point: An Application to BMI Rebound in Childhood

Body mass index (BMI) rebound refers to the beginning of the second rise in BMI during childhood. Accurate estimation of an individual's timing of BMI rebound is important because it is associated with health outcomes in later life.

We estimated BMI trajectories for 6545 children from the Avon Longitudinal Study of Parents and Children. We used a novel Bayesian two-phase piecewise linear mixed model where the "change point" was an individual-level random effect corresponding to the individual-specific timing of BMI rebound. The model's individual-level random effects (intercept, prechange slope, postchange slope, change point) were multivariate normally distributed with an unstructured variance-covariance matrix, thereby, allowing for correlation between all random effects.

Average age at BMI rebound (mean change point) was 6.5 (95% credible interval: 6.4 to 6.6) years. The standard deviation of the individual-specific timing of BMI rebound (random effects) was 2.0 years for females and 1.6 years for males. Correlation between the prechange slope and change point was 0.57, suggesting that faster rates of decline in BMI prior to rebound were associated with rebound occurring at an earlier age. Simulations showed that estimates from the model were less biased than those from models, assuming a common change point for all individuals or a nonlinear trajectory based on fractional polynomials.

Our model flexibly estimated the individual-specific timing of BMI rebound, while retaining parameters that are meaningful and easy to interpret. It is applicable in any situation where one wishes to estimate a change-point process which varies between individuals.