Deriving Models of Change with Interpretable Parameters: Linear Estimation with Nonlinear Inference

Psychometrika, vol. 90, pp. 47–73

Abstract

Abstract In modeling change over time, developmental theories often emphasize meaningful quantities like peaks, inflections, timing, and tempo. However, longitudinal analyses typically rely on simple polynomial models that estimate powered terms of time in a linear, additive form which are disconnected from these meaningful quantities. While these linear parameterizations are computationally efficient and produce stable results, the quantities estimated in these models are difficult to directly connect to theoretical hypotheses. To address this disconnect between estimation and theory development, I propose several approaches for linear estimation with nonlinear inference (LENI), a framework that transforms results from stable, easily-estimated linear models into nonlinear estimates which align with theoretical quantities of interest through a set of principled transformation functions. I first outline derivations for the interpretable nonlinear parameters, and transform the results of the corresponding linear model—including fixed and random effects as well as conditional covariates effects —into the results we would have obtained by fitting a nonlinear version of the model. I conclude by summarizing a linearized structural equation model approach which can flexibly accommodate any known nonlinear target function within a linearly-estimable framework. I conclude with recommendations for applied researchers and directions for fruitful future work in this area.

Authors 1

1. Ethan Michael McCormick corresponding Aachen Methodology and Statistics Department Institute of Psychology, Leiden University

   Leiden University · University of Delaware

   Affiliation as printed

   Educational Statistics and Research Methods , School of Education , University of Delaware , Newark , DE , United States

   Methodology & Statistics Department , Institute of Psychology , Leiden University , Leiden , Netherlands, and