Deriving models of change with interpretable parameters: linear estimation with nonlinear inference

In the modeling of change over time, there is often a disconnect between developmental theories advanced in substantive research and statistical models specified in longitudinal analysis. That is, theory is understood and advanced in terms of meaningful developmental quantities (e.g., peaks, inflections, timing, and tempo) while common polynomial models estimate the effect of powered terms of time in a linear, additive form. This linear parameterization approach has many advantages, especially its computation efficiency in obtaining stable results, but the quantities estimated in these models are often difficult to directly connect to theoretical ideas of change over time. To bridge the gap between estimation and theory development, I propose a series of approaches for linear estimation with nonlinear inference (LENI), where the results of the stable, easily-estimated linear model are converted through a set of principled transformation functions into nonlinear estimates which align more closely with theoretical quantities of interest.I first lay out how to derive these interpretable nonlinear parameters, then show how to transform the results of the linear model – including fixed and random effects and the conditional effects of covariates – into the effects we would have obtained by fitting a nonlinear version of the model. I conclude by summarizing a linearized structural equation model approach which can be flexibly applied to model any known nonlinear target function into a linearly-estimable model. I conclude with recommendations for applied researchers and directions for fruitful future work in this area.