Using Higher-Order Derivatives to Estimate Damped Linear Oscillator Models with an Over-Arching Temporal Trend

Abstract

Studies that look to examine intra-individual data have become more popular in the social sciences in recent years, and thusly methods that accurately model this type of data are needed. One particular differential equation model that has been used to fit this type of data is the damped linear oscillator (DLO), which models constructs that vary about some equilibrium value over time. Currently, methods for fitting the DLO model require that no over-arching, temporal trend be present in the data, or that this trend be removed prior to fitting the model in some two-step procedure. One such two-step approach that has been used in psychology is the method of Latent Differential Equations (LDE). Using two-step procedures can cause standard errors of parameter estimates to be inflated, which makes single-step methods with simultaneous estimation of all parameters preferred. This study proposes a method using higher-order derivatives and structural equation modeling (SEM) to estimate the DLO model and trend simultaneously. A simulation was conducted to examine (a) the bias of estimates obtained using the proposed method and (b) whether the proposed method provides any improvement over the existing, two-step LDE approach. The results suggest that the proposed method does provide accurate estimates, but for a much smaller range of conditions than the LDE approach. The simultaneous estimation of the higher-order derivative method did provide more precise parameter estimates compared to the two-step approach for the range of conditions that it was found to be accurate.