A one‐step method for modelling longitudinal data with differential equations

Differential equation models are frequently used to describe non‐linear trajectories of longitudinal data. This study proposes a new approach to estimate the parameters in differential equation models. Instead of estimating derivatives from the observed data first and then fitting a differential equation to the derivatives, our new approach directly fits the analytic solution of a differential equation to the observed data, and therefore simplifies the procedure and avoids bias from derivative estimations. A simulation study indicates that the analytic solutions of differential equations ( ASDE ) approach obtains unbiased estimates of parameters and their standard errors. Compared with other approaches that estimate derivatives first, ASDE has smaller standard error, larger statistical power and accurate Type I error. Although ASDE obtains biased estimation when the system has sudden phase change, the bias is not serious and a solution is also provided to solve the phase problem. The ASDE method is illustrated and applied to a two‐week study on consumers’ shopping behaviour after a sale promotion, and to a set of public data tracking participants’ grammatical facial expression in sign language. R codes for ASDE , recommendations for sample size and starting values are provided. Limitations and several possible expansions of ASDE are also discussed.