Novel Approach Growth Mixture Modeling Using Natural Cubic Smoothing Splines

• In Event: Machine Learning Algorithms, New Modeling Approaches, and Simulation Methods to Understand Model Performance

Abstract

The accurate depiction of longitudinal data to reveal individual differences has immense consequences for the understanding and classification of developmental change in educational research (Ruscio, 2007). A strategy that is frequently used is to model growth trajectories as linear functions. While linear patterns are regularly encountered in educational research, nonlinear patterns are much more prevalent with extended measurements over time (Grimm, Ram, & Estabrook, 2016). A number of different approaches have been proposed in the literature to capture growth patterns. For example, a direct extension of the linear model frequently used to capture nonlinear components of change is the polynomial function. Other nonlinear extensions include B-Splines, Bezier Curves, Hermite Splines, Gompertz Curves, and Piecewise Splines. However, the necessity to a priori determine the functional form or the location of the change point indicating the occurrence of shifts in the studied process (also referred to as knots) has limited their overall utility (Bollen & Curran, 2006).

Though it is frequently assumed that sampled individuals in a given longitudinal study exhibit similar overall growth trends, there can be situations involving typological differences in change that require individuals be treated as stemming from heterogeneous populations (Muthén & Shedden, 1999). Heterogeneity may either be a function of observed variables, or heterogeneity is due to unobserved features, such that the composition of individuals is not known ahead of time and must be inferred from the data using growth mixture modeling approaches. A noted limitation with commonly applied growth mixture modeling methods is that researchers often assume the growth trajectories to be the same for all individuals within a latent class (Diallo, Morin, & Lu, 2016).

The purpose of this presentation is to introduce a novel mixture modeling approach that can help researchers better understand patterns of growth trajectories and effectively be used to determine homogeneous and heterogeneous individual trajectories without imposing potentially problematic constraints on the model. The approach uses estimated derivatives of individual natural cubic smoothing spline functions to algorithmically group or cluster individuals who follow the same growth trajectory patterns. In order to determine groupings of individuals (should they exist), it is assumed that subjects in an observed sample belong to K different clusters or classes, such that individuals in each group have similar growth function derivatives. All natural cubic smooth spline functions and their derivatives are obtained separately for each individual with no assumptions regarding their distributions. The algorithm then utilizes a closeness measure to evaluate the distance between observations and estimates the unknown number of classes with a hierarchical clustering algorithm via a threshold rule applied to the generated dendrogram.

We will present an illustrative example that demonstrates the use of the proposed approach to determine groupings of individual observations in longitudinal data settings based on their computed growth function derivatives. We will then discuss the results of a simulation study that examines the performance of this new approach under a variety of longitudinal data design conditions.

Authors

• Raj Karl Wahlquist, University of Minnesota

• Katerina M. Marcoulides, University of Minnesota

• Laura Trinchera, NEOMA Business School