Integrating Developmental Theory and Methodology: Using Derivatives to Articulate Change Theories, Models, and Inferences

AN0110318279;7lf01oct.15;2019Feb12.16:50;v2.2.500

Integrating Developmental Theory and Methodology: Using Derivatives to Articulate Change Theories, Models, and Inferences. 

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 the change 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 zeroth, first, and second 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.

Of great interest to developmental researchers is the association between two or more changing constructs, and whether and how they are related over time. To examine these relations, methodological factors must be considered, such as the type of statistical model needed to investigate questions about growth, development, and change, and the time scale over which change is expected (Ferrer & McArdle, [14]; Little, [28]; Nesselroade, [30]; Sliwinski, [39]). Models exploring the longitudinal relations between constructs can take many forms (Little, [28]). For example, studies examining relations between mothers' and children's depressive symptoms have used correlations of observed scores, hierarchical linear models of change (e.g., Pilowsky et al., [32]), models of latent growth curves with correlated growth parameters (e.g., Garber & Cole, [15]; Gross, Shaw, & Moilanen, [20]), latent change score models (e.g., Kouros & Garber, [24]), and cross-lagged panel designs (e.g., Ge, Conger, Lorenz, Shanahan, & Elder, 1995; Jaffee & Poulton, [22]). Each of these models is useful for describing developmental processes, but rarely are the commonalities across the models explicitly noted. This disconnect burdens researchers with the task of figuring out how each analytic approach is or is not appropriate for modeling different types of developmental processes and theories. This confusion also influences the inferences that can or cannot be drawn from specific models.

A bridge between theoretical rationale for development and the appropriate model may be achieved through a common language for statisticians and substantive researchers. In this article we present derivative terminology, which can aid in the articulation of change and thereby better link developmental theories, models, and inferences. Derivative terminology overlaps with both developmental and statistical concepts, and could serve as a common vocabulary in modeling change. In doing so, developmental psychologists may be able to better understand a wide range of models as part of a common framework, instead of as distinct options whose overlap in commonalities and discrepancies are muddled, at best. We begin by presenting a typical developmental research question on the relation of change between mothers' and children's behavior, after which we introduce derivative vocabulary and apply this language to substantive examples to discuss the inferences drawn from differing statistical models.

DYNAMIC RELATION BETWEEN MOTHERS AND CHILDREN

Developmentalists often are interested in the mother-child dyad, and how change in one member of the dyad could affect the other. The interest in this bidirectional relation between members of a dyad is reflected in many prominent theories and developmental perspectives (dynamic systems; Thelen & Smith, [40]; family systems theory; Cox & Paley, [9]; see also Sameroff, [36]). One dynamic, bidirectional relation of this sort, the transactional association between mothers' and children's psychiatric symptoms, also has theoretical and empirical justification (see Elgar, McGrath, Waschbusch, Stewart, & Curtis, 2004; Gunlicks & Weissman, [21]). Theoretically, different mechanisms have been proposed to account for the effect of maternal depression on children's psychological symptoms (e.g., impairments in parenting and quality of family environment, see Goodman & Gotlib, [19]), as well as the effect of children's psychological symptoms on maternal depression (e.g., stress of caring for a child with psychological problems, see Elgar et al.). Given strong theoretical support for bidirectional relations, research has focused on providing empirical support for this link; this has been examined with several different models. For example, Gross et al. ([20]) and Garber and Cole ([15]) tested parallel process models, in which the latent growth parameters representing linear change in mothers' and children's symptoms were correlated. Jaffee and Poulton ([22]) tested an autoregressive, cross-lag model to test for time-dependent bidirectional relations. Kouros and Garber ([24]) used a latent change score model to test for transactional relations. Finally, Nicholson, Deboeck, Farris, Boker, and Borkowski (2011) used differential equation modeling techniques to fit a coupled damped linear oscillator model. These studies have attempted to assess the bidirectional relation using a variety of statistical methods. Notably, some of these methods ask very similar questions about the change processes, despite using different models, whereas others make unique assumptions about how the constructs are related. These different change models may (and frequently do) yield different results and conclusions, and the similarities between models may not be readily apparent (e.g., Kouros, Quasem, & Garber, [25]).

Which model should developmentalists use to test such transactional relations? The difficulty, in part, is that "transactional relation" is vaguely defined from a mathematical perspective, and could imply many different models. Figure 1 depicts hypothetical trajectories for maternal depressive symptoms and child externalizing behavior problems for two dyads. In both Figures 1A and 1B, the mother's trajectory of depressive symptoms increases over time. Prior to the onset of maternal depression, the child could have many possible initial trajectories, corresponding to increasing or decreasing rate of externalizing behaviors. Had there not been an increase in mother's depression, we might expect children to continue on these initial trajectories, corresponding to the dashed, gray dotted lines in Figure 1. As mother's depressive symptoms increase, however, in both cases the children may respond by showing an increase in the number of behavioral problems relative to their expected trajectory (solid gray line).

Graph: FIGURE 1 Plots of hypothetical trajectories for two mother-child dyads where the mother's depression affects the child's externalizing behaviors. Note. 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.

We expect that a child's behaviors would change in response to mother's depression, but the word change is mathematically vague and does not imply a specific model that can be tested. In these examples, the children in both dyads demonstrate changes in their trajectories in response to the increase in their mother's depression, but they show very different changes over time despite their starting at baseline with similar scores on externalizing behaviors. The first child (Figure 1A) follows a pattern that would be expected, as there is a positive correlation between maternal depression and children's externalizing scores due to both variables having positive slopes. In contrast, there would be a correlation of approximately zero between the mother and the second child (Figure 1B), as depression shows a positive slope whereas the child's corresponding behavioral trajectory appears neutral or slightly negative. A model focused on correlating scores or correlating linear slopes would suggest that the second child's behavior problems were not related to the mother's depression, as the change is masked by an initially negative trajectory.

Herein are some of a few questions that can be raised related to how change is defined. In saying that the children have changed in relation to their mothers, are we expecting an increase in the observed levels of their externalizing scores or an increasing slope as with the child in the first dyad (Figure 1A), or a change in the trajectory of the child (i.e., curvature) as seen in both children? Moreover, it must be considered if the child is responding to the level of the mother's depression, the steepness of her slope, or changes in the steepness of her slope. The imprecise vernacular of using the word change fails to articulate the forms of the relations depicted in Figure 1. Similarly, the word trajectory does not convey a specific form of change, and can be interpreted mathematically in multiple ways. Derivatives offer the potential of specifically defining what is meant by change and the ability to specify components of a trajectory.

Derivatives, being mathematically defined, also allow for translation to and from mathematical models; that is, they allow selection of models that more closely match theory and provide more precise interpretations of the inferences that can or cannot be made with a specific model. Using derivatives as a precise common denominator between theory and method would reduce misunderstandings among developmental researchers and mitigate translating theory into inappropriate statistical models. Using derivative terms to represent change easily identifies models that fit the theory and provides a framework for understanding whether different models address similar questions about change. Additionally, by identifying how current models fit in this framework, new modeling possibilities may be identified.

DERIVATIVES: THE VOCABULARY OF CHANGE

In mathematics, the change in one construct with respect to change in another construct is expressed using derivatives. In this article we will specifically consider the changes in constructs with respect to time. The value of a construct at a specified point in time is the zeroth derivative; the zeroth derivative is usually described by the name of the substantive construct to which it is referring, but despite differing names in developmental science the zeroth derivative can be generally considered the level of a construct at a specific time. In a plot of a construct (y-axis) versus time (x-axis), the zeroth derivative would be represented with a single point. Changes in the level of a construct, with respect to time, are called the first derivative; the first derivative is the velocity (speed in a specific direction) at which a construct is changing. In a plot of a construct versus time, the first derivative would be the slope of a straight line, and can also be referred to as the rate of change, the rise over the run, and the change in a construct with respect to a change in time. The rate of change in the velocity with respect to time, that is how quickly the slope of a line changes in a single direction, is the second derivative or the acceleration. Acceleration in a construct plotted versus time would appear as a curved line. While some variation in terms occurs, the present article utilizes level, velocity, and acceleration to refer to the zeroth, first and second derivatives.

To gain familiarity with this vocabulary, consider driving a car as a metaphor for a changing construct. The zeroth derivative expresses the level of a construct at a specific time, which in a car would correspond to the position of the car at a specific time. How this position changes with respect to time, or how much of a change in position occurs divided by the elapsed amount of time, is the velocity (first derivative) of the car if it is assumed the car is traveling in a single direction along a straight road. The velocity of the car, however, may not be constant as there may be increases or decreases in velocity over some period of time. These changes in velocity are represented in the acceleration of the car (second derivative); the term acceleration is used to represent both positive (acceleration) and negative (deceleration) changes to velocity. Using this metaphor to describe derivatives, we can articulate more clearly the property or properties of any construct of interest to inform our decision of which analysis to choose to investigate how constructs change over time.

Figure 2 presents another way to visualize level (zeroth derivative), velocity (first derivative), and acceleration (second derivative). The light gray line represents a hypothetical developmental trajectory. The black circles represent the zeroth derivative, or the level of a construct at any given point in time along this trajectory. The straight black lines tangential to the trajectory represent the velocity of the construct at some particular time. If we examine the slopes just before and after a given point in time, we would see a change in velocity (i.e., acceleration, curved dark gray lines) that could be described using the second derivative. Table 1 presents a summary of several ways of expressing derivatives. Although using derivative notation may be unfamiliar, researchers who are comfortable thinking in terms of the level of a construct at a specific time (point), straight line change, and curved lines can begin to identify derivatives that correspond to these same concepts.

Graph: 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. Note. 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 downward curve at the second point indicates a negative second derivative.

TABLE 1 Vocabulary for Describing Change With Respect to Time

RELATING CONSTRUCTS

Considering levels, velocities, and accelerations on individual constructs expands the possible ways in which two or more constructs may be related. We return to our example of maternal depression and child externalizing behavior problems. In the context of related constructs, the questions about correlated levels, velocities, and accelerations would respectively address: (1) Are high levels of maternal depression scores observed with high levels of child behavioral problem scores? (2) Does the rate at which maternal depression change (i.e., velocity) correlate with the rate at which child behavioral problems are changing? (3) When the rate of change in maternal depression is increasing or decreasing (i.e., accelerating), do child behavioral problems also show increasing or decreasing rates of change? These three questions each capture a uniquely different component of change and the presence of any one relation does not imply any other relation. For example, if high maternal depression scores are correlated with high child behavior problem scores, it does not necessarily mean that the dyads' slopes are also related. Figure 3 explores two examples (Panels A and B) of how the presence of correlated velocities or accelerations individually do not necessarily imply other related derivatives: each relation can occur independently of the other two. The two examples each have trajectories for four participants (upper row of each panel) that are used to produce the plots of the relations between derivatives (lower row of each panel). In Panel A, there is a clear velocity–velocity relation, but level–level and acceleration–acceleration relations are equal to zero. In Panel B, there is a significant acceleration–acceleration relation, but level–level and velocity–velocity relations are equal to zero. As the significance of each relation can be independently manipulated it becomes even more necessary to explicitly express what is meant when saying that two constructs show related change.

Graph: 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. Note. The top row of each panel shows trajectories of four 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.

Thinking about level, velocity, and acceleration also offers opportunities to examine relations that may be underutilized. Table 2 presents possible relations between two constructs and examples of possible research questions regarding the relation between maternal depression and child behavioral problems. Those along the diagonal already have been discussed: level–level, velocity–velocity, and acceleration–acceleration. Relations between constructs, however, also can occur between differing derivatives; three opportunities presented in this table are level–velocity, level–acceleration, and velocity–acceleration relations. For example, a mother's level of depression, regardless of whether her symptoms are changing or not, might be related to the rate at which a child's behavior problems are changing (level–velocity). A mother's level of depression may also lead to an increase or decrease in the rate at which the child's behavioral problems are changing, such that her child's behavior shows a curvilinear relation (level–acceleration). Finally, the absolute level of a mother's depression could be inconsequential, whereas the rate at which a mother's depression is changing could precipitate an increase or decrease in the rate at which the child's behavior problems are changing (velocity–acceleration). Such different relations between derivatives can lead to important research questions that are more in line with the need for appropriate models to better explain the true nature of human behavior (Lewis, [27]; Richters, [34]; Sameroff, [37]; Sameroff & MacKenzie, [38]). Theoretical approaches, like presentations of dynamical systems by Thelen and Smith ([40]), encourage a way to conceptualize, operationalize, and formalize complex patterns for how variables may differ in their change relations.

TABLE 2 Possible Relations Among Level, Velocity, and Acceleration of Two Constructs With Sample Questions Using the Example of Mother's Depression and Child Behavioral Problems

SUBSTANTIVE EXAMPLE

Many models are available for the description of change; models that are often presented as separate and different entities. Derivatives can provide a framework for understanding the inferences that can be drawn from particular models and understanding how they relate to concepts of change. We present here analyses to demonstrate how both familiar and unfamiliar analyses can be conceptualized as testing inferences about differing derivative relations. The first examines a Hierarchical Linear Model (HLM), a common analytic approach for developmental researchers. The second examines the same data, from the perspective of Latent Growth Curve Modeling (LGCM). We re-express both of these familiar models in terms of derivatives, and consequently discuss the inferences of parameters in terms of level, velocity, and acceleration. Finally, we introduce a more novel methodology for estimating derivatives from repeated intraindividual observations; while this method will be less familiar to most readers, the inferences that are being drawn can still be understood in terms of relations between pairs of derivatives. The present examples are provided only for pedagogical value and any specific results should be considered exploratory, if for no other reason than we have selected to focus on unidirectional relations using data where a bidirectional transactional relation is certainly plausible. Additionally, readers should use these examples to try to appreciate the differences in the derivative relations examined by differing models, and not draw conclusions about the value of any one particular method based on the significance or nonsignificance of the relations examined.

The examples are based on a sample of 240 youth (M age = 11.86 years, SD = .56; 54% female; 81.5% European American) and their mothers followed annually from grade 6 through grade 12. A subset of these children had a mother with a history of a mood disorder (n = 185), whereas the remaining mothers were life-time free of psychopathology (n = 55). At each annual assessment, mothers reported on their depressive symptoms on the Beck Depression Inventory (BDI; Beck, Ward, Mendelson, Mock, & Erbaugh, 1961; the term depression is used here for the sake of brevity) and about their child's externalizing behavioral problems on the Child Behavior Checklist (CBCL; Achenbach, [1]). Externalizing behavioral problem t-scores were used in the subsequent analyses.

Example 1: Hierarchical Linear Modeling

A hierarchical linear model (HLM) was fit to predict changes in child behavior problems in relation to mother's depression. We begin with a level 1 model where child behavioral problems (Yit) are predicted by time:

(1)

Graph

where T represents time (Grade 6 = 0), and the subscripts i and t indicate observations for each individual child over several occasions of time. Although researchers are familiar with the growth parameter terms of intercept, linear slope, and quadratic slope modeled in Equation 1, the relation between variables modeled in Equation 1 also can be understood as an expression of how the change in one variable is related to another variable. Equation 1 can be re-written as:

(2)

Graph

Notice, that the β-parameters from Equation 1 have now been replaced with their corresponding derivatives (i.e., the changes in Y with respect to time). Re-written this way, Equation 2 shows that the observed score of child behavior problems: Yit for child i at some time t is equal to the value of the level of the child's behavior problems when T = 0 (Y0i), plus the velocity of the child's behavioral problems multiplied by the elapsed time (T), plus one-half times[1] the rate at which the velocity is changing (acceleration, ) multiplied by time squared (T2). Whereas replacing the β-parameter notation to which we are accustomed may seem unnecessarily complicated, doing so allows us to see that the β0, β1, and β2 parameters respectively represent information about the level when T = 0, the velocity when T = 0, and the acceleration of a child's behavior problems. Clearly mapping the derivative associated with these commonly tested growth parameters will ensure that the interpretation of change is accurately articulated.

For the models that follow, all models were tested using R (version 3.0.2; R Core Team, [33]), and the package lme4 (version 1.0-6; Bates, Maechler, Bolker, & Walker, [2]). In any cases where more than one random effect was estimated, the models allowed for all covariances between random effects (i.e., unstructured covariance matrix). The lme4 package does not allow for specification of the variance-covariance structures for the residuals. When available, the REML = FALSE command was used, so that ML estimates were produced for the likelihood ratio tests presented.

The effect of mothers' depression (BDI score) was considered in two ways. The first way (Model 1) tested the initial BDI score as a level 2 predictor of the β's (intercept, linear slope, and quadratic slope). Incorporating mother's initial depression into the model as a level 2 predictor of the β's (Model 1) provides the following level 2 equations:

Graph

Graph

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where β0i, β1i, and β2i are equivalently seen as Y0i, , and , or read as level, velocity, and one-half times acceleration. Full results are provided in Table 3. A series of six models was tested, each which added an additional component to the prior model; changes in model fit due to additions to the model were evaluated through the likelihood ratio test and AIC. The top-half of the table provides model comparison results testing whether including the random effect of time and the random effect of time2 improved model fit (models 1a–1c), and whether adding initial BDI scores as a Level 2 predictor of the three random effects improved model fit (models 1d–1f).

TABLE 3 Parameter Estimates and Model Fit Information for Hierarchical Linear Models Testing Mothers' Initial Depression as a Predictor (Level 2) of Children's Behavioral Problems

Note. Parameter estimates presented from Model 1f, which had the lowest deviance. BDI = Beck Depressive Inventory (mother self-report).

Results of the model comparisons suggested a need for incorporating both the random effects of time and time2. Adding initial BDI as a predictor of differing intercepts was significant (1c vs. 1d; χ2(1) = 117.2, p