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Table 1 Overview of linear spline LME models, natural cubic spline LME models, SITAR, and latent trajectory models for analysing nonlinear growth trajectories of a single repeatedly measured continuous outcome

From: Using linear and natural cubic splines, SITAR, and latent trajectory models to characterise nonlinear longitudinal growth trajectories in cohort studies

 

Linear spline LME model

Natural cubic spline LME model

SITAR

Latent trajectory model

Description

linear mixed-effects model with a linear spline function of the independent time variable

linear mixed-effects model with a restricted cubic spline function of the independent time variable

nonlinear mixed-effects model based on the shape invariant growth model

heterogenous growth curves fit to unknown subgroups of individuals

Advantages

easy to interpret the spline slope coefficients; can describe growth rate during different periods of the growth process

continuous 1st & 2nd derivatives give smoother trajectory and can identify points of peaks/troughs; linearity constraint gives a more reliable trajectory shape as less erratic at the tails of distribution

has useful features of the natural cubic spline, easy to estimate individual growth features – most notably individual ages at peak growth velocity

can identify unobserved sub-groups of individuals sharing distinct growth trajectories if any exist

Limitations

biologically implausible sudden changes in velocity (i.e., at the knots); erratic at the tails; cannot identify points of velocity maxima/minima; position (and location) of knots important

coefficients difficult to interpret (so plotting is more useful); can be challenging to estimate the individual growth curves due to complex spline basis functions used by the statistical software

may not work well for complex growth patterns e.g., with multiple peaks and troughs or where the growth curve does not plateau in adulthood

difficult to identify the optimal number of sub-groups; may identify implausible subgroups; trajectories tend not to replicate in other cohorts

R package(s)

lme4; lspline

lme4; splines

sitar

lcmm, splines

  1. All models can include all individuals with at least one observed outcome measure, with valid estimates obtained under the assumption of outcome data missing at random (MAR) depending on observed values of the outcome and/or covariates. Note all models assume no autocorrelation (in our example, there are wide enough gaps between measures to assume that here)