Testing Group Differences in Nonlinear Growth Curve Modeling with Penalized Spline

Longitudinal data analysis in social and behavioral sciences often assumes a linear change over time, but human behavior is complex and frequently exhibits nonlinear patterns. Regression spline is a popular approach in machine learning to model nonlinear trajectories. While traditional machine learning prioritizes prediction, researchers in social and behavioral sciences focus on testing and interpreting the significance of group differences. This study compares six spline models for evaluating group differences in group-mean trajectories across multiple time points: the frequentist truncated power basis model, the frequentist piecewise basis model, the frequentist B-spline basis model, the Bayesian truncated power basis model, the Bayesian piecewise basis model, and the Bayesian B-spline basis model. Through simulation, we found that for non-censored data, the frequentist models demonstrated stability and robust performance. However, these models showed potential issues with statistical inference such as coverage rates and Type I error rates when the data are censored (e.g., floor and ceiling effects). For the latter cases, we recommend either using the frequentist models with an unstructured level-2 covariance matrix if computation time permits, or using the Bayesian B-spline model with an unstructured level-2 covariance matrix.