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

Several mixed-effects models for Gaussian longitudinal outcomes have been proposed to accommodate the non-linearity of late-life cognitive trajectories and assess the putative influence of covariates on it. No prior research provides a side-by-side examination of these models to offer guidance on their proper application and interpretation. In this work, we examined five statistical approaches commonly used to answer research questions related to non-linear changes in cognitive aging: the linear mixed model with a quadratic term, the linear mixed model with a function of time approximated with splines, the functional mixed model, the piecewise linear mixed model, and the sigmoidal mixed model. We first theoretically describe the models. Next, we conducted a simulation study to evaluate the models empirically and provide practical recommendations. Lastly, using real data from two prospective cohorts with annual cognitive testing, we compared the interpretation of the models by examining the associations between education and cognitive change before death. Except for the linear mixed model with a quadratic term for time, the fit of all models was generally adequate in capturing the non-linearity of cognitive change, and models were relatively robust. Although spline-based models have time-related parameters that are not directly interpretable, their convergence was easier to achieve, and they allow for graphical interpretation. In contrast, piecewise and sigmoidal models provide directly interpretable parameters but may require more data to achieve convergence. This study provided insights into how each strategy can help better characterize cognitive change over time. By performing simulations, we also emphasized that a quadratic function of time is not recommended in the context of these data. Lastly, we highlighted that more sophisticated models for longitudinal data, such as the piecewise and sigmoidal mixed models, should be considered more frequently in applications than they currently are.