A comparison of six linear and non-linear mixed models for longitudinal data: application to late-life cognitive trajectories

Longitudinal characterization of cognitive change in late-life has received increasing attention to better understand age-related cognitive aging and cognitive changes reflecting pathology-related and mortality-related processes. Several mixed-effects models have been proposed to accommodate the non-linearity of cognitive decline and assess the putative influence of covariates on it. In this work, we examine the standard linear mixed model (LMM) with a linear function of time and five alternative models capturing non-linearity of change over time, including the LMM with a quadratic term, LMM with splines, the functional mixed model, the piecewise linear mixed model and the sigmoidal mixed model. We first theoretically describe the models. Next, using data from deceased participants from two prospective cohorts with annual cognitive testing, we compared the interpretation of the models by investigating the association of education on cognitive change before death. Finally, we performed a simulation study to empirically evaluate the models and provide practical recommendations. In particular, models were challenged by increasing follow-up spacing, increasing missing data, and decreasing sample size. With the exception of the LMM with a quadratic term, the fit of all models was generally adequate to capture non-linearity of cognitive change and models were relatively robust. Although spline-based models do not have interpretable nonlinearity parameters, their convergence was easier to achieve and they allow for graphical interpretation. In contrast the piecewise and the sigmoidal models, with interpretable non-linear parameters may require more data to achieve convergence.