Journal of Education and Psychology

Abstract

Longitudinal data consist of measurements on the same subject repeatedly over time. Such data typically posses a hierarchical structure that repeated measurements are nested within individuals. Longitudinal data with large numbers of time points typically have shifts in the shapes of relationship between performance over time at certain time points, differences between individuals, and dependence and heteroscedasticity in the residuals. These characteristics pose particular challenges to the development of methodologies for analyzing longitudinal data.

Polynomial regressions are used for analyzing longitudinal data. However, there exist some limitations in utilizing polynomial regressions in analyzing longitudinal data. The residuals in longitudinal data often exhibit heteroscedasticity and dependence characteristics, which violate the assumptions of homogeneity and independence for multiple regressions. Moreover, the residuals need specific covariance models to describe the residual structure. If the number of occasions is large, the use of polynomial functions is inadequate to describe the whole model shifts for the entire time range because polynomial functions are globally determined in a small interval of time. As an alternative functional form, spline regressions can be fit to the sub-ranges of time with the adjacent functions joined together smoothly to adapt the whole model shift.

The main objective of this study was to investigate a methodology that incorporate spline regressions with extended linear mixed-effects models (spline extended LMEs) in modeling multilevel longitudinal data with large number of time points. First the literature of spline regressions and extended linear mixed-effects models are first reviewed. Then a systematic approach which is generally applicable to modeling various multilevel longitudinal data with large number of time points is proposed. A detailed illustration of the proposed methodology is further demonstrated through reanalyzing the visual-search dataset of Peterson and Kramer (2001). Results indicate that spline extended LMEs are flexible in specifying the covariance models, can indicate the between-subjects variability that occurred at certain knots, as well as can incorporate them into variance-covariance structures for random effects. Several recommendations on the application of spline extended LMEs in longitudinal analysis, including the importance of visualization, knots placement, variability at knots, and the possibility of over parameterization, are discussed.

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