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Using fractional polynomials to model non-linear trends in longitudinal data.

Jeffrey Long1, Jihoon Ryoo

  • 1University of Minnesota, Minneapolis, Minnesota, USA. longj@umn.edu

The British Journal of Mathematical and Statistical Psychology
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Fractional polynomials (FPs) offer a flexible way to model non-linear growth curves within linear mixed models. FPs provide a better fit and more favorable prediction characteristics than conventional polynomials for psychological data.

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Area of Science:

  • Statistics
  • Psychometrics
  • Biostatistics

Background:

  • Non-linear growth trends are common in longitudinal data.
  • Conventional polynomials (CPs) can be limited in modeling complex non-linear patterns.
  • Linear mixed models provide a robust framework for analyzing such data.

Purpose of the Study:

  • To introduce and evaluate fractional polynomials (FPs) as a method for modeling non-linear growth curves.
  • To compare the performance of FPs against conventional polynomials (CPs) within linear mixed models.
  • To explore the interpretability and predictive advantages of FPs.

Main Methods:

  • Modeling non-linearity using time transformations with fractional polynomials (FPs).
  • Utilizing linear mixed models to incorporate FPs and static correlates.
  • Employing penalized and non-penalized indices for model selection.
  • Fitting FPs and CPs to psychological data.

Main Results:

  • FPs demonstrated equal or superior model fit compared to higher-order CPs.
  • FP prediction curves exhibited more favorable characteristics, including less extreme behavior at interval edges.
  • FPs offered advantages in parsimony, flexibility of curve shape, and asymptote approximation.

Conclusions:

  • Fractional polynomials are a valuable tool for modeling smooth, non-linear growth trends.
  • FPs offer a flexible and parsimonious alternative to CPs for analyzing complex growth patterns.
  • FPs show potential for improving predictions and understanding of non-linear trajectories in psychological research.