Reducing Boundary Estimates and Increasing Power in Detecting Individual Differences in Change: The Effect of Adding a Quadratic Term

Accurately estimating and powerfully testing individual differences in change—via the variance of the average rate of change (ARC)—is important in longitudinal studies, yet boundary (i.e., zero) variance estimates with non-estimable standard errors and insufficient power for variance testing remain a common challenge. This study examines how frequently boundary estimates and biased ARC variance estimates occur when a straight-line growth (SLG) model is applied to quadratic growth data or a quadratic model is applied to SLG data. Analytical derivations using fully observed data and a simulation study with varying missing data conditions were conducted to evaluate three models: SLG, a quadratic growth model with a fixed quadratic effect (QGF), and one with a random quadratic effect (QGR). Results show that adding a quadratic term—fixed for three-wave data and random for four or more waves—reduced boundary estimates and improved power to detect individual differences in ARC, when growth data follow quadratic patterns. While underfitting (e.g., fitting SLG to quadratic data) yielded more boundary estimates and reduced power, overfitting (applying QGF/QGR to SLG data) did not harm ARC variance estimation and testing when sample size is adequate (N > 50). An empirical example illustrates model applications.