Nonlinear Growth Curve Modeling Using Penalized Spline Models: A Gentle Introduction

This didactic article aims to provide a gentle introduction to penalized splines as a way of estimating nonlinear growth curves in which many observations are collected over time on a single or multiple individuals. We begin by presenting piecewise linear models in which the time domain of the data is divided into consecutive phases and a separate linear regression line is fitted in each phase. Linear splines add the feature that the regression lines fitted in adjacent phases are always joined at the boundary so there is no discontinuity in level between phases. Splines are highly flexible raising the fundamental tradeoff between model fit and smoothness of the curve. Penalized spline models address this tradeoff by introducing a penalty term to achieve balance between fit and smoothness. The linear mixed-effects model, familiar from multilevel analysis, is introduced as a method for estimating penalized spline models. Higher order spline models using quadratic or cubic functions which further enhance a smooth fit are introduced. Technical issues in estimation, hypothesis testing, and constructing confidence intervals for higher order penalized spline models are considered. We then use data from the Early Childhood Longitudinal Study to illustrate each step in fitting a higher order penalized spline model, and to illustrate hypothesis testing, the construction of confidence intervals, and the comparison of the functions in 2 groups (boys and girls). Extensive graphical illustrations are provided throughout. Annotated computer scripts using the R package nlme are provided in online supplemental materials.