Integrating developmental theory and methodology: Using derivatives to articulate change theories, models, and inferences

Abstract

Matching theories about growth, development, and change to appropriate statistical models can present a challenge, which can result in misuse, misinterpretation, and underutilization of different analytical approaches. We discuss the use of derivatives --- the change of a construct with respect to changes in another construct. Derivatives provide a common language linking developmental theory and statistical methods. Conceptualizing change in terms of derivatives allows precise translation of theory into method and highlights commonly overlooked models of change. A wide variety of models can be understood in terms of the level, velocity and acceleration of constructs: the 0th, 1st, and 2nd derivatives, respectively. We introduce the language of derivatives, and highlight the conceptually differing questions that can be addressed in developmental studies. A substantive example is presented to demonstrate how common and unfamiliar statistical methodology can be understood as addressing relations between differing pairs of derivatives.

Keywords: derivatives; developmental theory; growth curves; hierarchical linear modeling; models of change; theory-method interface.

Figures

Figure 1

Plots of hypothetical trajectories for two mother-child dyads where mother’s depression affects the child’s externalizing behaviors. In both figures the change in child’s behavior (solid gray line) is coupled to the change in mother’s depression (solid black line). The figures differ in the initial trajectories of the children (dashed gray lines). Panel A shows a relation where the scores of the mother and child would be highly correlated. In Panel B, whereas the mother’s depression is leading to changes in the child’s trajectory from his/her initial trajectory, the scores of the mother and child would have a correlation near zero.

Figure 2

Plot of a developmental trajectory (light gray line) with the level (black circles), instantaneous velocity (black lines), and instantaneous accelerations (dark grey lines) at three points in time. In this figure the straight black lines indicate a positive first derivative (velocity) for the first and second points at which derivatives are estimated, and a negative first derivative at the third point. The upwards curved dark gray lines indicate a positive second derivative (acceleration) at the first and last points at which derivatives are estimated; the downwards curve at the second point indicates a negative second derivative.

Figure 3

Figures demonstrating that the presence of a relation between a pair of derivatives does not necessarily imply that other derivatives will be related. The top row of each panel shows trajectories of 4 hypothetical dyads; the bottom row of each panel plots the levels, velocities, and accelerations of the dyads against each other. The symbols for the dyads in the top row correspond to the symbols plotted in the second row. Panel A demonstrates trajectories with a significant velocity-velocity relation when level-level and acceleration-acceleration relations are equal to zero. Panel B demonstrates trajectories with a significant acceleration-acceleration relation when level-level and velocity-velocity relations are equal to zero. The presence of correlated velocities (Panel A) or correlated accelerations (Panel B) does not necessarily imply the levels of constructs will be correlated. The same can be shown for correlated levels of constructs; the presence of correlated levels of constructs does not necessarily imply that correlated velocities or accelerations occur.

Figure 4

SEM tested in substantive example. Four models were examined: (1) no relations between levels and slopes (no path A or B), (2) depression level predicting the child behavior problems level (inclusion of path A), (3) depression velocity predicting the child behavior problems velocity (inclusion of path B), and (4) a model that included both relations from models 2 and 3 (inclusion of both A and B).