The Role of a Quadratic Term in Estimating the Average Treatment Effect From Longitudinal Randomized Controlled Trials With Missing Data

Longitudinal randomized controlled trials (RCTs) have been commonly used in psychological studies to evaluate the effectiveness of treatment or intervention strategies. Outcomes in longitudinal RCTs may follow either straight-line or curvilinear change trajectories over time, and missing data are almost inevitable in such trials. The current study aims to investigate (a) whether the estimate of average treatment effect (ATE) would be biased if a straight-line growth (SLG) model is fit to longitudinal RCT data with quadratic growth and missing completely at random (MCAR) or missing at random (MAR) data, and (b) whether adding a quadratic term to an SLG model would improve the ATE estimation and inference. Four models were compared via a simulation study, including the SLG model, the quadratic growth model with arm-invariant and fixed quadratic effect (QG-AIF), the quadratic growth model with arm-specific and fixed quadratic effects (QG-ASF), and the quadratic growth model with arm-specific and random quadratic effects (QG-ASR). Results suggest that fitting an SLG model to quadratic growth data often yielded severe biases in ATE estimates, even if data were MCAR or MAR. Given four or more waves of longitudinal data, the QG-ASR model outperformed the other methods; for three-wave data, the QG-ASR model was not applicable and the QG-ASF model performed well. Applications of different models are also illustrated using an empirical data example.