Nonlinearity in the analysis of longitudinal data

In recent years, the use of longitudinal designs has increased appreciably and the study of change has become an essential component of research in the behavioural sciences. The availability of "micropanels", that consists of large cross-sections of individuals observed for short time periods, provides informations for answering questions about (i) how each individual perform over time, and (b) what are the predict differences among individuals in their change. These questions form the core of every study about achievement growth, and we need suitable models to investigate the dependence structure of these longitudinal data. Two important features have to be taken into account: (i) the clustering of responses within units, and (ii) the chronological ordering of the responses. Both imply dependence among responses on the same unit. Including random effects into statistical models is a common way of distinguishing between-subject and within-subject source of variability in view of reflecting unobserved heterogeneity in the individual behaviour. In particular, the random effects can be incorporated into Structural Equation Models (SEM) by considering them as latent variables. We refer to these models as Latent Curve Models (LCMs). The basic idea is that individuals differ in their growth over time, and they are likely to have different temporal behaviours as a function of differences in particular characteristics, such as gender, scholar background etc. etc. The model allows both the level of the response and the effects of covariates to vary randomly across units. In this context, an issue that researchers have to address is the nonlinearity of the functional form and particularly in the parameters. The individual growth is generally assumed to be linear, but many behavioural processes exhibit differential rates of change. Since LCM is a confirmatory factor model, it cannot estimate complex nonlinear functions directly. Hence, several methods have been developed in view of treating nonlinear dynamics via models which are linear in the parameters. This chapter considers nonlinear latent curve models for the study of longitudinal developmental data. Section 2 describes linear LCM, in terms of model specification and estimation. In Section 3, we review parametric and nonparametric methods for the estimation of nonlinear LCMs. A simulation study is performed in order to evaluate the goodness of these methods in the approximation of nonlinear trajectories. Finally, Section 4 illustrates an application on nonlinear academic performance data based on a cohort of students enrolled at the University of Bologna.