Dynamical Systems Modeling of Early Childhood Self-Regulation

Abstract

Self-regulation can be conceptualized in terms of dynamic tension between highly probable reactions (prepotent responses) and use of strategies that can modulate those reactions (executive processes). This study investigated the value of a dynamical systems approach to the study of early childhood self-regulation. Specifically, ordinary differential equations (ODEs) were used to model the interactive influences of 115 36-month-olds’ executive processes (strategy use) and prepotent responses to waiting to open a gift (desire for the gift and frustration about waiting to open it). Using a pair of coupled second-order ODEs in a non-linear mixed effects framework, the study tested predictions for specific within- and between-child patterns of prepotent response-executive process coupling. Dynamic modeling results articulated the limits of 36-month olds’ strategic efforts. They engaged executive processes when their prepotent responding levels were high, which delayed the resurgence of prepotent responses, but ultimately did not damp prepotent responding over the course of the wait. There was, however, preliminary evidence that the effectiveness of 36-month-olds’ self-regulation depended upon child characteristics. Externalizing behavior problems were associated with more regulatory interference. Temperamental negative affectivity was marginally associated with more regulatory inefficiency. Compared with conventional methods of studying self-regulation, dynamic modeling yielded complementary and unique findings, suggesting its potential.

Self-regulation, widely acknowledged as essential for healthy, competent functioning (NIH RFA-AG-11-010), refers to an individual’s control over prepotent responses through the engagement of executive processes (Baumeister, 1998; Carver & Scheier, 1990; Duckworth & Steinberg, 2015; Mischel, 2014). Prepotent responses are highly probable, automatic reactions to eliciting situations; they are relatively automatic due to being well-learned or biologically prepared. Executive processes involve the use of cognitive and behavioral strategies that have potential to override prepotent responses, such as refocusing attention or calmly verbalizing one’s need to another.

Self-regulation develops throughout the lifespan, with development during early childhood being of particular interest. The first indications of self-regulation appear during the third year (Kopp, 1982). By first grade most children’s capacity to regulate their emotions and actions has developed to the point that they are ready to handle the everyday frustrations and challenges of the school day (Blair & Raver, 2015). That is, by school age, most children are usually able to tolerate the difficulties of learning new material, to delay and inhibit selfish responses in order to get along with others, to comply with adult directions and prohibitions even if they conflict with their goals, and to control impulsive action even if frustrated or disappointed (Cole, 1986; Eccles & Wang, 2015; Eisenberg & Spinrad, 2004; Kochanska, Coy, & Murray, 2001; Mischel, 2014). Two longitudinal studies further suggest that differences in preschool age self-regulation predict may adolescent and adult outcomes (Moffitt et al., 2011; Shoda, Mischel, & Peake, 1990). Thus, thorough investigation of the early development of self-regulation is important for our understanding of lifelong healthy, competent functioning.

A limitation of early childhood research on self-regulation is that the evidence does not directly address regulation, which inherently involves dynamic changes. A thermostat, for example, has a mechanism that adjusts temperature in relation to moment-to-moment environmental changes. In psychological terms, self-regulation involves the engagement of executive processes to change (inhibit, delay, minimize, or amplify) prepotent responses (Baumeister, 1998; Boker & Laurenceau, 2006; Carver & Scheier, 1990; Gross, 2015). Consider the moment your computer screen flickers unexpectedly and you have not saved work recently. You likely have a prepotent response to act in a particular way (e.g., scream “No!”). Whether and how you behave depends upon how you modulate that reaction as the situation unfolds. If the flicker is followed by signs of an impending crash, your prepotent tendency may intensify but you may regulate it by calling upon executive processes—recalling solutions, explaining your problem to a computer support person. Self-regulation is a dynamic process involving the interplay of a prepotent readiness to act out of panic and the engagement of executive processes that help modulate acting on that readiness.

Typically, studies of young children’s self-regulation observe and record behaviors, usually acting on prepotent responses and sometimes engagement of executive processes, during laboratory tasks. These behaviors are later coded, discerning which if any occurred during each of many small time intervals across the task period. The conventional approach is to summarize these time series data using static variables (e.g., total number of intervals in which anger appeared) and to compare groups, e.g., the number of angry intervals for children who are high or low in behavior problems (Cole, Martin, & Dennis, 2004; Diaz & Eisenberg, 2015). Although useful, this approach does not capitalize on the rich information embedded in time series data. More robust descriptions of self-regulation may be obtained using methods specifically designed for studying how psychological processes unfold and interact over time (Boker & Laurenceau, 2006; Chow, Ram, Boker, Fujita, & Clore, 2005; Ram & Pedersen, 2008). The purpose of this study is to explore how dynamic systems modeling can be used to investigate interplay between children’s prepotent responses (PR) and executive processes (EP). Specifically, we used ordinary differential equation (ODE) models to study how the ebb and flow of 36-month-olds’ desire for and frustration (PR index) about being made to wait to open a desired gift related to the ebb and flow of their use of purported regulatory strategies (EP index). In addition, we tested whether individual differences in children’s temperamental characteristics and behavior problems were associated with differences in the dynamic relations between PR and EP.

Conventional Approaches to Studying Young Children’s Self-Regulation

As noted, young children’s self-regulation is often studied by video recording behavior during laboratory tasks designed to frustrate their goals to attain a desirable toy, snack or gift (Block & Block, 1980; Calkins & Johnson, 1998). The video record is later used to create binary or ordinal time-series that indicate which emotions and behaviors were observed in each of many brief time intervals (e.g., 1s or 15s epochs). The behavior codes usually track behaviors that indicate acting on a prepotent response (PR), such as touching a prohibited item (did/did not occur) or expressing anger (none, mild, medium, intense display), and track behaviors that indicate enactment of executive processes (EP), such as redirecting attention (distraction) or verbalizing calmly about the problem situation. The conventional approach is to sum the total number of intervals in which a specific emotion or behavior was observed.

Analysis of those sums suggest that as children age, they are less likely to express anger and more likely to engage their linguistic and cognitive capacities in ways that might help them manage their prepotent tendencies (Cole et al., 2011; Grolnick, Bridges, & Connell, 1996; Kopp, 1982; Rothbart, Ziaie, & O’Boyle, 1992). For example, infants engage in self-soothing behavior, toddlers seek adult support, and preschool age children can redirect their attention away from a desired but restricted item. Most studies focus on individual differences among same-age children, finding that, relative to less well-adjusted children, better adjusted children have fewer angry epochs and more epochs in which they use strategies. On the basis of this evidence, it is inferred that adjusted children have “better” self-regulation. Occasionally, studies use specific bivariate relations to infer self-regulation. For example, studies of within-person correlations find that young children who display less anger tend to use more strategies (Calkins & Johnson, 1998; Gilliom, Shaw, Beck, Schonberg, & Lukon, 2002). A sequential analysis showed that very young children’s anger intensity decreases in epochs after a strategy was enacted (Buss & Goldsmith, 1998).

Although these approaches use time-series data to quantify levels of PR and EP behavior, they do not articulate self-regulation as a dynamic process that unfolds over the course of a situation. Rather than modeling how one process shapes the action of another process over time, summary descriptive statistics (means, correlations) are used to describe the outcome of that process. In contrast, dynamic modeling approaches facilitate analysis of patterns of change that emerge over the full length of a time series, provide for more direct articulation and testing of the ways EP may influence PR and how self-regulatory attempts ultimately succeed or fail. In doing so, dynamic modeling approaches may provide valuable information that furthers our understanding of development and inter-individual differences in self-regulation, and may potentially inform and improve interventions intended to promote effective self-regulation.

Dynamical Systems Approach to Self-Regulation

Dynamical systems approaches can model a time-varying process, such as PR, as a function of itself and of another variable, such as EP (Gelfand & Engelhart, 2012). We operationally define effective self-regulation as the influence of EP (strategy use) on the eventual damping of PR intensity (desire, frustration). We also consider two patterns of ineffective self-regulation: one in which EP contributes to the eventual amplification of PR (regulatory inefficiency) and one in which PR contributes to the eventual damping of EP (regulatory interference). Ordinary differential equations (ODEs) allow examination of these patterns of change over time (Boker, 2001; Zill, 1993). ODE models can be used to examine intrinsic (within-process) dynamics and how one process influences another (Ram & Pedersen, 2008). They have been used to study the ebb and flow of emotions (Chow et al., 2005) and of dyadic interactions (Boker & Laurenceau, 2006; Helm, Sbarra, & Ferrer, 2012; Ram et al., 2014). Applied to young children’s self-regulation during a wait for an object they desire, we used ODEs to model (a) intrinsic changes in PR and EP and (b) how each influences the other over the course of the task. That is, we modeled the ebb and flow of children’s desire for the gift and frustration about waiting to open it (PR), the ebb and flow of their strategy use (EP), and reciprocal influences between PR and EP. We focused on effective self-regulation as EP contributing to PR damping as the task unfolded. In complement, we examined two patterns of ineffective self-regulation that have been implied in research on young children’s temperament and behavior problems. Regulatory inefficiency in which EP contributes to the amplification rather than damping of PR, and regulatory interference in which PR contributes to the damping in EP rather than EP damping PR.

ODEs are mathematical specifications of relations among time-varying features of a dynamic process, including but not limited to: (a) levels (location; formally quantified as deviations of PR and EP from person-specific baselines), first derivatives (rates of change; velocity) and second derivatives (changes in rates of change, e.g., acceleration or deceleration; Boker, 2001). The way in which one process contributes to changes in the intensity of the other process is central to our conceptualization of self-regulation. In tasks designed to test children’s self-regulation, their desire and frustration does not increase or decrease at a constant rate and it does not simply disappear. If they have mastered self-regulation in such a task, their strategy use should contribute to a waning of the intensity of their desire and frustration. That is, they still want the object for which they must wait, and find the wait a challenge, but they use strategies that contribute to diminishing action upon their desire and frustration such that they do not keep trying to get what they want or have emotional outbursts. Process-oriented conceptions of self-regulation imply a coupled system wherein EP contributes to the damping of PR, similar to stepping on the brakes of a car. For clarity, the focus is not on the specific level of EP or PR (e.g., location of the car), but on the dynamic interplay between PR and EP. Young children, however, are just acquiring strategies that can contribute to effective damping of PR as a situation endures. Therefore, as we will explain, other patterns can be observed that reflect attempts at self-regulation that are ineffective.

Theoretical models often discuss the ways self-regulation strategies may contribute to acceleration/deceleration of emotion and behavior, but it has been lamented that it is rare that these dynamics are explicitly articulated in empirical studies (Calkins & Fox, 2003; Cole, Hall, & Hajal, 2013). Consider how a child’s behavior unfolds while waiting to open a gift. Initially, a child may wait, periodically and briefly glancing at the gift; but as the requirement to wait endures, the child’s desire and frustration may grow, becoming increasingly stronger and larger in magnitude – PR amplification. At the same time, EP may be changing from moment to moment. The child’s strategy use may decelerate; the child may use less optimal strategies that change at a progressively slower pace – EP damping. ODE models provide a framework for articulating and testing hypotheses about the acceleration/deceleration and amplifying/damping of PR and EP.

Theoretically, effective self-regulation should be represented by EP influencing PR damping. Ineffective self-regulation manifests when EP does not contribute to PR damping or when PR damps EP. ODEs allow us to investigate the extent to which the ebb and flow of a child’s behavior reflects effective or ineffective self-regulation. Moving beyond the overall, average levels of PR and EP used in conventional analysis of self-regulation (usually captured using means or sums), we model how PR and EP accelerate/decelerate over time as a function of location, velocity and dynamic relations of PR and EP.

We contend that to infer self-regulation it is necessary to demonstrate that EP (e.g., calm information seeking and distraction) contributes to the damping of PR (e.g., desire and frustration). That is, it is important to show that EP modulates changes in PR intensity and that PR does not intrinsically damp on its own. Moreover, it is useful to anticipate patterns of ineffective self-regulation in young children (Cole et al., 2013; Diaz & Eisenberg, 2015). We next discuss the two patterns of ineffective self-regulation that have been suggested in the child development literature.

The Dynamics of Self-Regulation in Early Childhood

At age 36 months, most children begin to initiate putative regulatory strategies without adult direction (Cole et al., 2011; Kopp, 1982). At such a young age, their strategy use is not necessarily developed to the point of effectiveness, but there is little research to address this issue. Young children’s strategies may have only momentary effects on their prepotent responses (Buss & Goldsmith, 1998). This may be one reason that inverse relations between behaviors reflecting PR and EP are modest and differ substantially across children (e.g., Calkins & Johnson, 1998; Gilliom et al., 2002). Analysis of PR and EP temporal dynamics can address the important question of how young children’s inchoate self-regulation efforts may succeed or falter—are the strategies they deploy ineffective, or do their emotions interfere with strategy deployment (Cole et al., 2013)? ODE models provide for both possibilities. Models can be formulated with parameters that specifically indicate the effectiveness and ineffectiveness of children’s self-regulatory efforts. A parameter can be used to indicate the extent to which EP contributes to amplification of PR, a pattern that has been suggested as a form of dysregulation associated with children’s risk for psychopathology (Calkins & Fox, 2003; Cole et al., 2013; Henderson, Pine, & Fox, 2015). Similarly, models can be formulated with a parameter that specifically indicates the extent to which PR interferes with regulatory efforts. That is, desire and frustration may overwhelm strategy use, such that children’s strategic efforts become less mature when they are highly aroused (Calkins & Dedmon, 2000; Cole, Michel, & Teti, 1994).

In sum, self-regulation implicitly involves change and there are methods that can investigate change dynamics and can yield more direct evidence of regulation and how it succeeds or falters over the course of a task. Quantitative methods for fitting nonlinear ODEs are an exciting possibility for studying self-regulation in early childhood (Chow, Bendezú, Cole, & Ram, 2016). Here, we use a specific class of ODE model, a second order coupled-oscillator, to investigate how PR and EP contribute to the dynamics of young children’s self-regulation.

Second Order Coupled-Oscillator Model

Using second-order ODEs, the ebb and flow of PR and EP are modeled as oscillations. Figure 1 depicts (a) how in theory PR and EP oscillations may be inversely related and how their damping or amplifying may depend on the other process, and (b) actual PR and EP oscillations of one child waiting to open a gift. Oscillations like those that appear in Figure 1 may be represented by models in which accelerations/decelerations are considered as a function of location and velocity (Boker, 2001). Formally, the moment-to-moment accelerations and decelerations in PR and EP, i.e. second derivatives $(\frac{d^2\mathit{\text{PR}}(t)}{{\mathit{\text{dt}}}^2},\frac{d^2\mathit{\text{EP}}(t)}{{\mathit{\text{dt}}}^2})$, are modeled as a function of current locations (PR(t),EP(t)), and velocities $(\frac{\mathit{\text{dPR}}(t)}{\mathit{\text{dt}}},\frac{\mathit{\text{dEP}}(t)}{\mathit{\text{dt}}})$ of PR and EP. Following the theoretical propositions outlined above, we formulate a model for self-regulation dynamics as:

$$\frac{d^2\mathit{\text{PR}}(t)}{{\mathit{\text{dt}}}^2}={\eta }_{\mathit{\text{PR}}}(\mathit{\text{PR}}(t)-b_{\mathit{\text{PR}}})+{\gamma }_{1,\mathit{\text{PR}}}(\mathit{\text{EP}}(t)-b_{\mathit{\text{EP}}})+{\gamma }_{2,\mathit{\text{PR}}}((\mathit{\text{PR}}(t)-b_{\mathit{\text{PR}}})*(\mathit{\text{EP}}(t)-b_{\mathit{\text{EP}}}))+{\gamma }_{3,\mathit{\text{PR}}}((\mathit{\text{EP}}(t)-b_{\mathit{\text{EP}}})*\frac{\mathit{\text{dPR}}(t)}{\mathit{\text{dt}}})\frac{d^2\mathit{\text{EP}}(t)}{{\mathit{\text{dt}}}^2}={\eta }_{\mathit{\text{EP}}}(\mathit{\text{EP}}(t)-b_{\mathit{\text{EP}}})+{\gamma }_{1,\mathit{\text{EP}}}(\mathit{\text{PR}}(t)-b_{\mathit{\text{PR}}})+{\gamma }_{2,\mathit{\text{EP}}}((\mathit{\text{PR}}(t)-b_{\mathit{\text{PR}}})*(\mathit{\text{EP}}(t)-b_{\mathit{\text{EP}}}))+{\gamma }_{3,\mathit{\text{EP}}}((\mathit{\text{PR}}(t)-b_{\mathit{\text{PR}}})*\frac{\mathrm{dEP}(t)}{\mathit{\text{dt}}})$$ (1)

Eight parameters define specific aspects of PR and EP dynamics. Figure 2 provides a visual illustration of how changes in select parameters values influence these dynamics. Specifically:

• a)Two parameters, η_PR and η_EP, represent intrinsic, non-interactive dynamics of PR and EP. These parameters describe the frequency at which (i.e. how rapidly) each process oscillates around its baseline levels (static within-person average PR or EP level, represented as b_PR and b_EP), and thus capture the extent of intrinsic pressure for PR or EP to “turn back” toward its baseline. Oscillations in the data (Figure 1) are in part driven by a negative relation between location and the second derivatives. The further away a process is from its baseline the more pressure there is for it to return to baseline. Generally, η_PR and η_EP will be negative with more negative values indicating more rapid oscillation. Because our conceptual framework focuses on the interactive coupling of PR and EP, these parameters account for intrinsic processes but are not used as indices of self-regulation (see Boker, 2001 for extensive discussion and display of how differences in frequency alter individual trajectories).
• b)Two parameters, γ_1,PR and γ_1,EP,, capture direct influences that EP exerts on PR to “turn back” toward baseline, and that PR exerts on EP to “turn back” towards baseline. Complementing the intrinsic turns represented by η_PR and η_EP, these parameters indicate how the location (i.e. level) of PR or EP directly influences the other process’ turn around (Figure 2).
• c)Two parameters, γ_2,PR and γ_2,EP, capture how the intrinsic dynamics of PR and EP are moderated by the location of the other process. For example, PR’s return towards baseline may be delayed when EP is above baseline, i.e. high. In this case, when EP is high, children’s PR bursts become less frequent (Figure 2).

These first two sets of parameters are used to describe the intrinsic dynamics of EP and PR. The third set of parameters are used to describe how one process moderates, or contributes to changes in, the frequency of the other process - in other words, how fast it turns around. They represent ways in which EP and PR may exert influences on each other but they are not specific to our conceptual framework. In our conceptualization, self-regulation is not defined by overall level of PR and EP, but is defined by how the levels of the opposing process moderate changes (i.e. damping or amplification) in the intensity of the other process, effects that are captured by the last two coupling parameters.

• d)The final two parameters, γ_3,PR and γ_3,EP, capture the specific aspects of self-regulation targeted in our conceptual framework. These parameters describe, respectively, the extent to which amplification (values > 0) or damping (< 0) of PR is moderated by EP level—regulatory efficiency/inefficiency, and the extent to which amplification/damping of EP is moderated by PR level—regulatory interference.

Figure 1.

Observed data and predicted trajectories for one randomly selected participant generated using parameter estimates from the final model.

Figure 2.

Illustration of different hypothetical (A) linear and (B) nonlinear coupling parameter value effects on PR and EP dynamics. In panels A and B, regulatory dynamics occur when the level of the opposing process (PR and EP, respectively) is high.

Recall that effective self-regulation is defined by EP level moderating PR damping in that higher levels of strategic effort slow the ebb and flow of desire and frustration (significant negative γ_3,PR parameter). However, self-regulation is not a well-established skill at age 36 months. Therefore, it is possible that the γ_3,PR parameter for 36-month-olds could be closer to zero, i.e. low effectiveness. Furthermore, as discussed, for some children there may be a pattern of inefficient self-regulation even when EP levels are above baseline. Specifically, for some children arguably “effective” strategies may nonetheless amplify PR. Thus when the parameter that is intended to capture effective self-regulation has a positive rather than negative value, it reflects this pattern of regulatory inefficiency. Consider Panel A in Figure 3 as a visual depiction of PR dynamics in the face of high EP levels. When regulatory inefficiency is high (i.e. the parameter, γ_3,PR is positive), high levels of EP contribute to amplification of PR (left side, Panel 3A). In contrast, when regulatory inefficiency is low (i.e. γ_3,PR is negative), high EP levels facilitate damping of PR, with greater facilitative effects at more negative values of γ_3,PR (right side, Panel 3A). The low regulatory inefficiency pattern is conceptually and mathematically similar to effective self-regulation dynamics.

Figure 3.

Illustration of high and low A) regulatory inefficiency and B) regulatory interference at various ?₃ parameter values. In panels A and B, regulatory dynamics occur when the level of the opposing process (EP and PR, respectively) is high.

Similarly, regulatory interference describes how PR level moderates the amplification or damping of EP. Panel 3B is a visual depiction of EP dynamics in the face of high PR levels. When regulatory interference is high (i.e., γ_3,EP is negative), high levels of PR contribute to damping of EP (left side, Panel 3B), with more negative values of γ_3,EP leading to greater damping; in contrast, when regulatory interference is low (i.e., γ_3,EP is positive); high levels of PR contribute to EP amplification (right side, Panel 3B), with more positive values of γ_3,EP leading to greater amplification in EP. In this case, at least for some children, high PR levels may trigger effort to engage in more mature, sustained strategic efforts. These two aspects of self-regulation capture distinct patterns that may be related to other individual differences in children.

Individual Differences in Early Childhood Self-Regulation Dynamics

Children’s temperament and externalizing behavior problems are reliably correlated with aspects of behavior that are attributed to skill at self-regulation (Diaz & Eisenberg, 2015). Although dynamic patterns of PR-EP relations that reflect ineffective attempts at self-regulation have been implied in the literature, they have not been studied until the present investigation.

Negative affectivity (NA) refers to a predisposition to react to environmental change with strong negative emotion (Rothbart & Bates, 2006). In young children, high NA is linked to more anger expression, more attention to restricted objects, and less strategy use (Calkins, Dedmon, Gill, Lomax, & Johnson, 2002; Santucci et al., 2008). Thus, high NA should be related to higher baseline PR and lower baseline EP. Fox and Calkins (2003) posited that NA can compromise self-regulation in ways that imply dynamic relations between PR and EP. They posited that NA leads to slower development of age-appropriate, effective strategies. This can be seen in moment-to-moments terms by failure to sustain effective strategies and resort to less mature strategies. In ODE terms, for children with higher NA, strategy use (EP) may amplify desire and frustration (PR), i.e. regulatory inefficiency, and/or strong emotion (PR) may damp strategy use (EP), i.e. regulatory interference.

Effortful control (EC) refers to a predisposition to be easily soothed and readily engage in attention and behavior control (Rothbart & Bates, 2006). In young children, EC has been associated with less intense anger and more mature strategy use (Calkins & Dedmon, 2000; Kochanska et al., 2001). Thus, children with higher EC should be associated with lower levels of PR and higher levels of EP. An open question though is whether it is associated with higher EP contributing to PR damping as the task unfolds, which we define as evidence of effective self-regulation.

Externalizing behavior problems refers to disobedience, impulsivity, and defiance, which have been linked to higher levels of anger expression and rule violations that reflect poor self-regulation (Calkins & Dedmon, 2000; Cole et al., 2013; Cole, Teti, & Zahn-Waxler, 2003; Gilliom et al., 2002). Higher NA is regarded as a risk factor, associated with greater demands on children’s self-regulation, for externalizing problems (Bates, Maslin, & Frankel, 1985; Campbell, Shaw, & Gilliom, 2000). Thus, like NA, externalizing problems may also be associated with higher baseline PR levels and with patterns of self-regulation failures, such as regulatory inefficiency and interference. ODE modeling can be used to assess multiple dynamic patterns within and between individuals.

The Present Study

In sum, this study evaluated the value of a dynamical systems approach to studying young children’s self-regulation. We studied the interplay of 36-month-olds’ interest in and frustration about waiting to open a gift (PR) and use of strategies purported to help them tolerate waiting (EP). Using archived data with standard second-by-second coding of observed child behavior, we calculated PR and EP composites. To shed light on ways that a dynamic approach provided unique information relative to conventional methods, we first examined bivariate correlations between PR and EP within-person average levels, and regressions to assess the degree to which child characteristics accounted for variance in those levels. Then, to assess self-regulation dynamics, we fit a set of coupled, second-order ODE models that provided a set of baseline parameters (static information) and a set of coupling parameters (dynamic information) that captured specific aspects of effective self-regulation, its inverse (regulatory inefficiency), and regulatory interference. Parent ratings of child negative affectivity (NA), effortful control (EC), and externalizing behavior problems were then introduced as potential predictors of between-person differences in baseline and coupling parameters. Because higher quality parenting is related to better self-control in young children (Fox & Calkins, 2003), a measure of parenting quality was included in all analyses.

We hypothesized that 36-month-olds, as a group, would demonstrate a capacity for self-regulation (H1). If so, child within-person average PR and EP levels would be inversely correlated (H1a), children would self-initiate EP when needed (H1b; negative γ_1,EP), above baseline EP would delay resurgence of below baseline PR (H1c; positive γ_2,PR), and higher EP would moderate damping in PR (H1d; negative γ_3,PR). However, as age 36 months is an early period in the development of self-regulation and children mature at different paces (Kopp, 1989), we hypothesized individual differences in patterns of self-regulation, explained in part by child temperament and externalizing behavior problems (H2). First, EC would be associated with higher EP and lower PR within-person average levels and NA and externalizing problems would be associated with lower EP and higher PR within-person average levels (H2a). Second, EC would be associated with higher EP (b_EPi) and lower PR (b_PRi) baseline levels, and NA and externalizing would predict higher PR (b_PRi) and lower EP (b_EPi) baseline levels (H2b). Third, in terms of self-regulation dynamics, higher EC would contribute to effective self-regulation (H2c; e.g., more negative γ_3,PRi), and higher NA and externalizing would contribute to ineffective self-regulation (H2d), either regulatory inefficiency (closer to zero or positive γ_3,PRi) and/or regulatory interference (more negative γ_3,EPi).

Method

Data were taken from a longitudinal study of the development of emotion regulation in early childhood in which 120 children from rural and semi-rural families were seen eight times between ages 18 months and 48 months. In the present study, we used parent ratings of child behavior at ages 30 and 36 months and coded observations of children’s emotions/behavior during a wait task at age 36 months.

Participants

Of the 120 children, 115 (62 boys) had complete data for analyses. The other five cases either missed the 36-month lab visit (n = 3) or had less than 50% of the behavioral data because the task ended early due to child distress (n = 2); these cases did not differ from the analysis sample on child gender or family income (Fs < 1, ps > .250). Most children (93%) were identified by their mothers as White (7.0% as biracial). Average household annual income at child age 30 months was $44,481.83 (SD= $17,197.48). The lab visit took place within two weeks of a child’s third birthday (M_age = 35.67 months, SD = 0.87).

Procedure

Mothers completed mailed questionnaires in which they rated child temperament (age 30 months) and behavior problems (age 36 months). Mother and child participated in lab visits, during which trained staff administered a series of activities that alternated between enjoyable activities (e.g., free play) and tasks commonly used to tax child self-regulation. The present study used data from an 8 minute waiting task completed at the age 36 months.

Waiting task

The waiting task elicits child desire and frustration and self-regulatory attempts (Cole et al., 2011). A research assistant (RA) cleared the room of toys, handed the mother a clipboard with task instructions and questionnaires, placed a wrapped gift on a child size table, and handed the child a toy car with missing wheels. When the RA left, the mother told her child to wait to open the gift until she finished her work (completing questionnaires). Mothers were told to interact as they usually would if their child had to wait for her to finish a task before giving the child something the child wanted. After 8 minutes, the RA returned and let the mother allow the child to open the gift. The gift contents were magnetic marbles.

Measures

Wait task behavior was video-recorded and later coded by three trained, independent teams. One team coded the occurrence and intensity of children’s nonverbal emotional expressions in each 1s epoch of the 480s task (Cole, Teti, & Zahn-Waxler, 2003). A second team coded the occurrence of all other behaviors, including strategy use, in the same 1s epochs (Grolnick, Bridges, & Connell, 1996). A third team coded the occurrence of maternal efforts to encourage child self-regulation (Structuring) in each 15s epoch. All coders were trained to 85% accuracy with master coded records. Inter-rater reliability was assessed periodically via double-coding of 15% of cases.

Prepotent responses (PR)

PR was indexed by behaviors indicating children’s desire for the gift and their frustration about waiting to open it. PR in each 1s epoch was calculated as the sum of child anger intensity (Cohen’s κ = .86) and behaviors directed at the gift (Cohen’s κ = .83): nonverbal anger (facial and vocal cues such as brow furrowing, pursed or pressed lips, clenched jaw, square open mouth, harsh loud vocal quality) was weighted by intensity rating (0 = no anger, 1 = mild anger, 2 = moderate anger, 3 = strong anger), angry bids to the mother about the demands of the wait (e.g., “I want it!”), attention focused on the gift (staring at, touching, and trying to open the gift), and disruptive behavior (throwing the boring toy, trying to hurt the mother). Higher scores indicated higher PR.

Executive processes (EP)

EP was indexed by child-initiated, non-disruptive strategy use and by strategy use encouraged by the mother if the child autonomously engaged in it 15s after the mother’s prompt. Because it is rare to engage in more than one strategy in a 1s epoch variability is limited. Following published approaches applying dynamic models to observational data (e.g., Gottman et al., 2002; Thomas & Martin, 1976), we computed an EP index by assigning each strategy a rank score based on developmental research on strategy development (Calkins & Johnson, 1998; Cole et al., 2011; Grolnick et al., 1996; Rothbart, Ziaie, & O’Boyle, 1992): self-soothing = 1, calmly seeking mother’s attention = 2, calmly seeking task-relevant information (e.g., I can open it when you’re done, right?) = 3, brief, unfocused distraction (shifts of attention away from the gift) = 4, and focused distraction = 5. Child-initiated distraction is widely held as the best strategy for delaying acting on impulse in a task in which a reward is delayed. Thus, a rise in EP level reflected use of more mature strategies. Higher scores indicated higher EP.

Child temperament

At child age 30 months, mothers completed the 96-item Toddler Behavior Assessment Questionnaire (TBAQ-R; Goldsmith, 1996). Mothers rated how well each item described their child’s typical behavior in the past month (1 = extremely untrue to 7 = extremely true). Two factors reliably emerge for this measure (Rothbart, Ahadi, Hershey, & Fisher, 2001). Negative Affectivity (NA; Cronbach’s α = .88), or the tendency to react strongly, is the average of items from Anger, Sadness, Social Fear, and Soothability (reversed) subscales. Effortful Control (EC; Cronbach’s α = .86), or readiness to engage in self-control, scores were calculated as the average of items from Attention Focusing, Attention Shifting, and Inhibitory Control subscales.

Child externalizing behavior problems

At child age 36 months, mothers completed the 100-item Child Behavior Checklist (CBCL/1½-5; Achenbach & Rescorla, 2001), rating how often their child engaged in problem behavior (0 = never, 1 = sometimes, 2 = often). Externalizing Behavior scores were calculated as the sum of responses to 24 items asking about impulsivity, defiance, disobedience, and aggression (Cronbach’s α = .91).

Parenting quality

Because parenting is correlated with children’s self-control (Fox & Calkins, 2003), scores for maternal efforts to encourage child self-regulation were included in analyses. Maternal Structuring was calculated as the total number of 15s epochs in which mothers tried to help children (e.g., remind them to wait, suggest ways to play; Cohen’s κ = .89).

Data Preparation

Horizontal aggregation

ODE modeling requires (within-person) variability. In standard behavioral coding of children’s self-regulation, variability is constrained by the binary (a strategy did or did not occur) or ordinal (3 point rating of emotion intensity) coding schema. To generate variability within small time intervals, PR and EP composites were aggregated in 3s epochs, reducing the time series from 480 seconds to 160 3s epochs. This yielded possible ranges for PR and EP of 0 to 18 and 0 to 15, respectively, while still providing sufficient repeated measures for modeling PR and EP change over time. Composite scores were standardized within person (z-scores). Then, following procedures used in other dynamic modeling applications (e.g., Chow & Zhang, 2013; Chow et al., 2010), between-person differences in PR and EP baselines were retained by adding the original person-specific means back into the standardized scores. This within-person standardization facilitates efforts to find a “common” dynamic model that captures critical dynamics at the group level, while retaining some (albeit attenuated) extent of inter-individual differences to be captured via mixed effects modeling.

Derivative estimation

Micro-coded observational data have inherent noise that may obscure patterns of change in a time series. We obtained smoothed estimates for PR and EP levels and first and second derivatives using functional data analytic (FDA) techniques (Ramsay, Hooker, & Graves, 2009). Each child’s PR and EP time series was smoothed and approximated using 6^th order penalized B-splines with knot-points at all observed time points. Smoothed first and second derivative estimates were computed using standard FDA-procedures, e.g., eval.fd( ) function in the ‘fda’ R library (Chow et al., in press; Liang & Wu, 2008; Ramsay et al., 2009; Trail et al., 2014). The Generalized Cross-Validation Index was used to guide the extent of smoothing. For each child, we obtained 160-epoch time series of level (PR(t), EP(t)), velocity $(\frac{\mathit{\text{dPR}}(t)}{\mathit{\text{dt}}},\frac{\mathit{\text{dEP}}(t)}{\mathit{\text{dt}}})$, and acceleration/deceleration $(\frac{d^2\mathit{\text{PR}}(t)}{{\mathit{\text{dt}}}^2},\frac{d^2\mathit{\text{EP}}(t)}{{\mathit{\text{dt}}}^2})$.

Overview of Analyses

Conventional approach

In preliminary analyses we used conventional methods to examine individual differences in children’s PR and EP behavior. Specifically, we summarized the 160 epoch time series data for PR and EP as within-person averages, and examined the correlation between the two summary scores (H1), and how each was correlated with child Effortful Control, Negative Affectivity, and Externalizing Behavior Problem scores (H2a). Multiple regressions were used to examine the extent to which child factors accounted for variance in the PR and EP summaries, controlling for Maternal Structuring. These analyses serve as a foundation from which to assess the unique and complementary information provided by the dynamic models.

Dynamic approach

Two ODE models were fit using the nlme library in R (Pinheiro, Bates, DebRoy, Sarkar, & RCore Team, 2014) with all individual differences predictors sample-mean centered prior to analyses. Model 1 was used to examine 36-month-olds’ self-regulatory dynamics at the group level (H1b, H1c, H1d). Following Equation 1 the coupled ODEs were fit with no additional between-person predictors. Specifically, Model 1 included parameters for baseline PR and EP levels (b_PRi, b_EPi), intrinsic PR and EP dynamics (η_PR, η_EP), the dynamic interplay or coupling between PR and EP (γ_1,PR, γ_1,EP, γ_2,PR, γ_2,EP γ_3,PRi, γ_3,EPi) and four random effects (φ_bPRi, φ_bEPi, φ_γ3,PRi φ_γ3,EPi).

Model 2 was then used to examine how individual differences in self-regulatory dynamics were related to other child factors (H2b, H2c, H2d). Specifically, mean-centered Maternal Structuring, Effortful Control, Negative Affectivity, and Externalizing Behavior Problems were added as predictors of four parameters, b_PRi, b_EPi, γ_3,PRi, and γ_3,EPi¹. The general form of these models was:

$$b_{\mathit{\text{PRi}}}=b_{\mathit{\text{PR}}0}+b_{\mathit{\text{PR}}1}({\mathit{\text{MS}}}_i)+b_{\mathit{\text{PR}}2}({\mathit{\text{EC}}}_i)+b_{\mathit{\text{PR}}3}({\mathit{\text{NA}}}_i)+b_{\mathit{\text{PR}}4}({\mathit{\text{EXT}}}_i)+{\phi }_{b_{\mathit{\text{PRi}}}}{b}_{\mathit{\text{EPi}}}=b_{\mathit{\text{EP}}0}+b_{\mathit{\text{EP}}1}({\mathit{\text{MS}}}_i)+b_{\mathit{\text{EP}}2}({\mathit{\text{EC}}}_i)+b_{\mathit{\text{EP}}3}({\mathit{\text{NA}}}_i)+b_{\mathit{\text{EP}}4}({\mathit{\text{EXT}}}_i)+{\phi }_{b_{\mathit{\text{EPi}}}}{\gamma }_{3,\mathit{\text{PRi}}}={\gamma }_{3,\mathit{\text{PR}}0}+{\gamma }_{3,\mathit{\text{PR}}1}({\mathit{\text{MS}}}_i)+{\gamma }_{3,\mathit{\text{PR}}2}({\mathit{\text{EC}}}_i)+{\gamma }_{3,\mathit{\text{PR}}3}({\mathit{\text{NA}}}_i)+{\gamma }_{3,\mathit{\text{PR}}4}({\mathit{\text{EXT}}}_i)+{\phi }_{{\gamma }_{3,\mathit{\text{PRi}}}}{\gamma }_{3,\mathit{\text{EPi}}}={\gamma }_{3,\mathit{\text{EP}}0}+{\gamma }_{3,\mathit{\text{EP}}1}({\mathit{\text{MS}}}_i)+{\gamma }_{3,\mathit{\text{EP}}2}({\mathit{\text{EC}}}_i)+{\gamma }_{3,\mathit{\text{EP}}3}({\mathit{\text{NA}}}_i)+{\gamma }_{3,\mathit{\text{EP}}4}({\mathit{\text{EXT}}}_i)+{\phi }_{{\gamma }_{3,\mathit{\text{EPi}}}}$$ (2)

where individual differences in PR and EP baseline levels, b_PRi and b_EPi, were modeled as a function of sample-level intercepts, b_PR0 and b_EP0, Maternal Structuring (MS_i), Effortful Control (EC_i), Negative Affectivity (NA_i), Externalizing Behavior Problems (EXT_i), and residual unexplained individual-specific deviations (i.e. random effects), φ_bpRi, φ_bEPi, that had zero means, variances ${\sigma }_{{\phi }_{b_{\mathit{\text{PR}}}}}^2,{\sigma }_{{\phi }_{b_{\mathit{\text{EP}}}}}^2$, and correlations r(φ_bPRi, φ_bEPi). Similarly, individual differences in the self-regulation coupling parameters, γ_3,PRi and γ_3,EPi, were modeled as a function of sample-level intercepts, γ_3,PR0. and γ_3,EP0, the individual difference variables, and unexplained person-specific deviations, φ_γ3,PRi, and φ_γ3,PRi, that had zero means and variances ${\sigma }_{{\phi }_{{\gamma }_{3,\mathit{\text{PR}}}}}^2\text{ and }{\sigma }_{{\phi }_{{\gamma }_{3,\mathit{\text{EP}}}}}^2$, and correlations r(φ_γ3,PRi, φ_γ3,EPi). A series of models that sequentially freed up remaining (four) random effects correlations revealed non-significant (three) or small (one) effects. Thus, those correlations were constrained to 0 to aid convergence². All other parameters were modeled as fixed effects. Sample code and simulated data are available online as supplementary materials. Data were simulated using a simplified version of our final model, modified to emphasize key features of our hypothesized self-regulation model.

Multiple comparisons

As our modeling contained novel extensions of available ODE models, some of the results reported may be exploratory in nature. Thus, we applied the Benjamini-Hochberg (B–H) multiple comparison correction procedure (Benjamini & Hochberg, 1995, see also Thissen, Steinberg, & Kuang, 2002) to the entire dynamic parameter set. We report final dynamic modeling results both with and without the correction and include cautionary notes on particular effects of interest where appropriate.

Results

The results are organized in the following manner. First, group level results are presented for the static and dynamic approaches. Second, individual differences results are presented for static and dynamic approaches. A summary of study hypotheses and test results is presented in Table 1. Based on model fit statistics, Model 2 provided the best fit to the data (Table 3). Thus, we report group and individual difference level results from Model 2 only.

Table 1.

Summary of Study Hypotheses, Tests, and Results

Hypotheses by Approach	Results
Conventional approach
Group-level (correlation)
36m PR & EP within-person average levels are inversely correlated	r = −.66, p < .001
Between-person (multiple regression)
30m EC & 36m EP within-person average levels are positively associated	B = 0.842 (SE = 0.446), p = .062
30m EC & 36m PR within-person average levels are negatively associated	B = −0.110 (SE = 0.187), p > .250
30m NA & 36m EP within-person average levels are negatively associated	B = 0.724 (SE = 0.479), p = .133
30m NA & 36m PR within-person average levels are positively associated	B = −0.154 (SE = 0.200), p > .250
36m EXT & 36m EP within-person average levels are negatively associated	B = −0.011 (SE = 0.032), p > .250
36m EXT & 36m PR within-person average levels are positively associated	B = 0.003 (SE = 0.013), p > .250
Dynamic approach
Group-level (ODE model)
36m olds would self-initiate EP when needed most	γ1,EP= −0.077 (SE = 0.004), p < .001
36m olds’ EP engagement would delay PR resurgence	γ2,PR = 0.063 (SE = 0.003), p < .001
36m olds would display effective self-regulation	γ3,PR0= 0.026 (SE = 0.026), p > .250
Between-person (ODE model)
30m EC & 36m EP baseline levels are positively associated	bEP2 = 0.925 (SE = 0.430), p = .032a
30m EC & 36m PR baseline levels are negatively associated	bPR2 = −0.031 (SE = 0.184), p > .250
30m NA & 36m EP baseline levels are negatively associated	bEP3 = 0.761 (SE = 0.461), p = .099
30m NA & 36m PR baseline levels are positively associated	bPR3 = −0.027 (SE = 0.198), p > .250
36m EXT & 36m EP baseline levels are negatively associated	bEP4 = −0.009 (SE = 0.031), p > .250
36m EXT & 36m PR baseline levels are positively associated	bPR4 = 0.002 (SE = 0.013), p > .250
30m EC & 36m effective self-regulation are positively associated	γ3,PR2 = 0.045 (SE = 0.052), p > .250
30m NA & 36m regulatory inefficiency are positively associated	γ3,PR3 = 0.113 (SE = 0.057), p = .046a
30m NA & 36m regulatory interference are positively associated	γ3,EP3= 0.013 (SE = 0.041), p > .250
36m EXT & 36m regulatory inefficiency are positively associated	γ3,PR4= −0.002 (SE = 0.004), p > .250
36m EXT & 36m regulatory interference are positively associated	γ3,EP4= −0.007 (SE = 0.003), p = .011

Note. PR = Prepotent Response; EP = Executive Process; EC = Effortful Control; NA = Negative Affectivity; EXT = Externalizing Behavior Problems.

^ap-values rose above conventional thresholds (α = .05) following Benjamini-Hochberg correction.

Table 3.

Parameter Estimates for Nested Taxonomy of Second-Order Coupled Oscillator Models

	Group-Level Dynamics (Mod el 1)			Between-Person Predictors (Model 2)			Adjusted p-value
Parameters	Estimates (SE)		95%CI	Estimates (SE)		95%CI
ηPR	−0.146*	(0.003)	[−0.153, −0.140]	−0.142*	(0.003)	[−0.148, −0.135]	.001
ηEP	−0.256*	(0.005)	[−0.265, −0.246]	−0.246*	(0.005)	[−0.256, −0.237]	.001
γ1,PR	0.005	(0.004)	[−0.002, 0.012]	0.005	(0.004)	[−0.002, 0.012]	.332
γ1,EP	−0.074*	(0.004)	[−0.083, −0.066]	−0.077*	(0.004)	[−0.086, −0.069]	.001
γ2,PR	0.066*	(0.003)	[−0.092, −0.079]	0.063*	(0.003)	[0.057, 0.068]	.001
γ2,EP	−0.112*	(0.004)	[0.121, 0.104]	−0.098*	(0.004)	[−0.106, −0.090]	.001
Intercept (γ3,PR0)	0.033	(0.026)	[−0.017, 0.084]	0.026	(0.026)	[−0.026, 0.078]	.537
36m MS (γ3,PR1)				−0.005	(0.004)	[−0.014, 0.003]	.434
30m EC (γ3,PR2)				0.045	(0.052)	[−0.058, 0.147]	.605
30m NA (γ3,PR3)				0.113*	(0.057)	[0.002, 0.224]	.108
36mEXT(γ3,PR4)				−0.002	(0.004)	[−0.009, 0.007]	.895
Intercept (γ3,EP0)	−0.003	(0.019)	[−0.040, 0.034]	0.003	(0.019)	[−0.040, 0.034]	.905
36m MS (γ3,EP1)				0.001	(0.004)	[−0.005, 0.007]	.895
30m EC (γ3,EP2)				−0.026	(0.037)	[−0.098, 0.047]	.700
30m NA (γ3,EP3)				0.013	(0.041)	[−0.067, 0.092]	.895
36mEXT(γ3,EP4)				−0.007*	(0.003)	[−0.012, −0.002]	.033
Baseline PR(bPR0)	1.373*	(0.091)	[1.194, 1.552]	1.409*	(0.094)	[1.201, 1.574]	.001
36m MS (bPR1)				0.016	(0.015)	[−0.013, 0.046]	.495
30m EC (bPR2)				−0.031	(0.184)	[−0.393, 0.330]	.905
30m NA (bPR3)				−0.027	(0.198)	[−0.415, 0.360]	.905
36m EXT (bPR4)				0.002	(0.013)	[−0.024, 0.027]	.905
Baseline EP(bEP0)	6.260*	(0.239)	[5.791, 6.729]	6.189*	(0.217)	[5.718, 6.573]	.001
36m MS (bEP1)				−0.194*	(0.036)	[−0.264, −0.124]	.001
30m EC (bEP2)				0.925*	(0.430)	[0.082, 1.767]	.081
30m NA (bEP3)				0.761	(0.461)	[−0.143, 1.664]	.215
36m EXT (bEP4)				−0.009	(0.031)	[−0.069, 0.050]	.905
σφbPR	0.958*	—	[0.852, 1.078]	0.956*	—	[0.922, .989]	—
σφbEP	2.561*	—	[2.269, 2.891]	2.248*	—	[1.987, 2.554]	—
r(φbPRi, φbEPi)	−.53*	—	[−.64, −.40]	−.57*	—	[−.68, −.44]	—
σφγ3,PR	0.255*	—	[0.212, 0.308]	0.256*	—	[0.219, 0.299]	—
σφγ3,EP	0.172*	—	[0.144, 0.205]	0.168*	—	[0.141, 0.200]	—
r(φγ3,PRi, φγ3,EPi)	−.01	—	[−.12, .10]	.03*	—	[.01 .06]	—
σεPR	0.746*	—	[0.735, 0.758]	0.771*	—	[0.758, 0.783]	—
σεEP	0.404*	—	[0.400, 0.409]	0.375*	—	[0.371, 0.379]	—
AIC	27639.97			22054.43
BIC	27792.57			22340.47
Log Likelihood	−13801.98			−10993.21

Note. N= 115. PR = Prepotent Response; EP = Executive Process; MS = Maternal Structuring Frequency; EC = Effortful Control; NA = Negative Affectivity; EXT = Externalizing Behavior Problems. Models 1 and 2 allowed for b_PRi, b_EPi, γ_3,PRi, and γ_3,EPi random effects.

^*p < .05.

Group level static approach

As expected (H1a), within-person average PR and EP levels were inversely related, (r = −.66, p < .001). Descriptive statistics are shown in Table 2.

Table 2.

Sample-Level Descriptive Statistics and Simple Correlations among Prepotent Response (PR), Executive Process (EP), and Covariates

	1	2	3	4	5	6
1. 36m PR: Within-person average level
2. 36m EP: Within-person average level	−.66*	-
3. 36m Maternal Structuring Frequency	.19*	−.46*	-
4. 30m Effortful Control (TBAQ-R)	−.01	.09	.07	-
5. 30m Negative Affectivity (TBAQ-R)	−.05	.05	.04	−.42*	-
6. 36m Externalizing Problems (CBCL/1½-S)	.01	−.06	.05	−.39*	.41*	-
M	1.15	6.65	15.01	3.57	4.76	12.90
SD	0.96	2.62	6.29	0.54	0.58	7.97
Min	0.10	0.86	3.00	2.27	3.56	0.00
Max	5.91	11.52	30.00	4.96	6.15	34.00

Note. N= 115. m = month; TBAQ-R = Toddler Behavior Assessment Questionnaire - Revised; CBCL/1½-5 = Child Behavior Checklist 1½-5.

^*p < .05.

Group level dynamic approach

As well, PR and EP baseline levels derived in the dynamic model were inversely related, r (φ_bPRi, φ_bEPi) = −57, p < .001. For PR, baseline levels, b_PR0 = 1.409 (SE = 0.094), indicated low average PR relative to the 0 to 18 scale, and significant between-person variation, σ_φbPR = 0.956. For EP, baseline levels, b_EP0 = 6.189, SE = 0.217, indicated moderate average EP relative to the 0 to 15 scale, and substantial between-person variation, σ_φbEP = 2.248. PR and EP fluctuated significantly around their baselines, η_PR = −0.142 (SE = 0.003), p < .001, adjusted p < .001 and η_EP = −0.246 (SE = 0.005), p < .001, adjusted p < .001.

At the group level, PR and EP dynamics unfolded in both expected and unexpected ways. There was no significant extrinsic effect of EP on PR at the group level, γ_l,PR = 0.005 (SE = 0.004), p = .166, adjusted p > .250. Rather, as anticipated for such young children (H1b), PR levels contributed to EP deceleration, γ_1,EP = −0.077 (SE = 0.004), p < .001, adjusted p < .001. On average, when PR levels were above baseline, EP slowed and changed direction back to baseline.

Also as expected (H1c), EP level moderated intrinsic PR dynamics, γ_2,PR = 0.063 (SE = 0.003), p < .001, adjusted p < .001; higher than baseline EP levels, when interacting with lower than baseline PR levels, decelerated and delayed the resurgence of PR toward baseline. That is, high EP levels led to longer periods of low PR. Also, PR level moderated intrinsic EP dynamics, γ_2,EP= −0.098 (SE = 0.004), p < .001, adjusted p < .001; higher than baseline PR levels led to more frequent, if briefer, EP bouts.

Contrary to prediction (H1d), group EP levels did not moderate changes in PR oscillation magnitude, γ_3,PR0= 0.026 (SE = 0.026), p > .250, adjusted p > .250, and PR levels did not moderate changes in EP oscillation magnitude, γ_3,EP0= 0.003 (SE = 0.019), p > .250, adjusted p > .250. Thus, at the group level, there was no evidence of either effective self-regulation or regulatory interference. However, we did find significant between-person differences in how EP level contributed to PR damping, σ_φγ3,PR = 0.256 (95% confidence intervals = [0.219, 0.299]) and how PR level contributed to EP damping, σ_φγ3,EP = 0.168 (95% confidence intervals = [0.141, 0.200]).

Individual differences static approach

Bivariate correlations yielded two significant relations involving maternal efforts to guide child self-regulation. More frequent Maternal Structuring was associated with higher child average PR (r = .19, p = .039) and lower child average EP (r = −.46, p < .001). The same associations were captured in the regression analyses. Mothers appeared to try to help children regulate, B = 0.034 (SE = 0.016), p = .030, adjusted p = .075, particularly when children initiated less mature strategic attempts, B = −0.203 (SE = 0.037), p < .001, adjusted p < .001.

Contrary to prediction, child Negative Affectivity and Externalizing Behavior Problems did not account for significant variance in within-person average PR or EP levels (H2a; Table 4), and Effortful Control only accounted for a marginal amount of variance in average EP levels, B = 0.842 (SE = 0.446), p = .062, adjusted p = .124.

Table 4.

Results from Multiple Regression models for Children’s Average Levels of Prepotent Response (PR) and Executive Process (EP)

		Prepotent Response Within-Person Average Level	Adjusted p-value	Executive Process Within-Person Average Level	Adjusted p-value
Intercept		1.176* (0.094)	.001	6.592* (0.225)	.001
36m Maternal structuring frequency		0.034* (0.016)	.075	−0.203* (0.037)	.001
30m Effortful control (TBAQ-R)		−0.110 (0.187)	.698	0.842† (0.446)	.124
30m Negative affectivity (TBAQ-R)		−0.154 (0.200)	.634	0.724 (0.479)	.222
36m Externalizing problems		0.003 (0.013)	.833	−0.011 (0.032)	.801
(CBCL/1½-5)
R2		.049		.243
F		1.338		8.269*

Note. N = 115. m = month; TBAQ-R = Toddler Behavior Assessment Questionnaire - Revised; CBCL/1½-5 = Child Behavior Checklist 1½-5. Estimates are unstandardized parameters from a model where predictors were centered at sample means.

^†p = .062.

^*p < .05.

Individual differences dynamic approach

In regard to baseline levels of PR and EP, most predictions were not supported (H2b). Higher Effortful Control was not related to lower baseline PR, b_PR2= −0.031 (SE = 0.184), p > .250, adjusted p > .250; higher Effortful Control was related to higher baseline EP, b_EP2= 0.925 (SE = 0.430), p = .032, but statistical significance was not retained after correction for multiple comparisons, adjusted p = .081. In addition, neither Negative Affectivity nor Externalizing Problems was related to baseline PR or EP. Maternal Structuring was related to baseline EP, b_EP1= −0.194 (SE = 0.036), p < .001, adjusted p < .001, but not baseline PR, b_PR1= 0.016 (SE = 0.015), p > .250, adjusted p > .250.

A focal question for this study was the degree to which child individual differences predicted coupling parameters that indexed aspects of self-regulation. Contrary to prediction (H2c), Effortful Control was not related to the coupling parameters for effective self-regulation, γ_3,PR2= 0.045 (SE = 0.052), p > .250, adjusted p > .250, and Negative Affectivity was not related to regulatory inefficiency after adjusting for multiple comparisons, γ_3,PR3= 0.113 (SE = 0.057), p = .046, adjusted p = .108. The main finding was that, as predicted (H2d), Externalizing Behavior Problems was related to regulatory interference, γ_3,EP4= −0.007 (SE = 0.003), p = .011, adjusted p = .033. As the task unfolded, in the presence of high PR, EP was damped for children with high Externalizing Problems while EP was amplified for children with low Externalizing Problems (Figure 4B). That is, for children whose mothers described them as higher in behaviors that are disruptive, aggressive, and/or defiant, their strategy use declined from use of more optimal strategies, such as distracting oneself from the restricted gift, to use of strategies that are more common in younger children (seeking adult attention, self-soothing).

Figure 4.

Predicted trajectories for covariate effects 1 SD above and below sample averages, respectively. Trajectory initial conditions based on observed smoothed level and derivative values at the first time point.

Multiple comparisons

Table 2 includes the B-H adjusted p-values for parameter estimates in the final between-person model. Following correction, statistical significance was retained for all parameters estimates with two exceptions: Effortful Control predicting baseline EP (b_EP2, p = .081) and Negative Affectivity predicting regulatory inefficiency (γ_3,PR3, p = .108). Table 4 includes the B-H adjusted p-values for multiple regression parameter estimates. Following correction, statistical significance was retained for all parameters estimates with one exception: Maternal Structuring predicting EP within-person average level (p = .075).

Model misfit

Even though there are no well-validated indices for assessing the “absolute fit” of an ODE model and its adequacy in describing a data set, some general conjectures on potential sources of misfit may be made from evaluation of the individual residuals from model fitting. To this end, we extracted and plotted (see Figure 5) the residuals of four participants whose magnitudes of average absolute residuals were in the 1^st and 99^th percentiles compared to the rest of the sample. These participants may be regarded as prototypical cases of “good” and “bad” fit, respectively. Inspection of the residual plots suggests that the model generally performed well in capturing EP and PR dynamics when the fluctuations in one process led to the damping of the opposing process toward its estimated baseline (see top panel of Figure 5). In contrast, large residuals were observed in portions of the data where EP and PR were inversely related to each other (e.g., in ID 114, between t = 50 and 100 and between t = 130 and 150). That is, as one process prevailed (i.e., high above baseline), the opposing process was suppressed to a level that was distinctly lower than its baseline level - a feature not capture by our proposed ODE model. Other notable sources of misfit stemmed from within-person over-time shifts in the dynamics between EP and PR (e.g., in ID 28 around t = 130 when the observed PR showed sudden amplifications despite the rises in observed EP even though the participant’s EP showed high regulatory efficiency in damping PR prior to t = 130).

Figure 5.

Plots of the residuals and observed data from four participants whose magnitudes of average absolute residuals were in the 1^st (top row) and 99^th (bottom row) percentiles of the sample. We removed the estimated PR and EP baseline levels from the observed data to more efficiently overlay of these two observed processes on each other. In addition, we added and subtracted 5, respectively, to the residuals of PR and EP to distinguish them visually from the observed PR and EP scores.

Discussion

The findings of this study reveal the potential of a dynamic approach to studying self-regulation. Patterns of self-regulation, both effective and ineffective, were conceptualized in terms of the dynamic interplay between executive processes (EP), indexed by children’s strategy use, and prepotent responses (PR), indexed by children’s desire for a gift and frustration about waiting to open it. A set of second-order ODE models (placed within a multilevel framework that facilitated examination of individual differences) were used to describe both the intrinsic dynamics of executive processes and prepotent responses and a theoretically-driven set of coupling parameters. The results revealed a more dynamic, finely grained portrait of child self-regulation that complemented but was also unique from results obtained using conventional methods.

As a group, this sample of 36-month-olds initiated behaviors that are purported strategies for modulating desire and frustration. This finding is consistent with the assertion that the capacity to make autonomous efforts to regulate emotion appears in the third year of life (Kopp, 1989). Despite initiating their efforts to use putative strategies, 36-month-olds as a group did not use those strategies to effectively damp their desire and frustration over the course of the wait. There was some evidence of temporary effectiveness, but not of overall effectiveness such that the children ultimately deployed strategies that diminished their desire and frustration. Thus, a dynamic modeling approach indicates the limits of 36-month-olds’ self-control attempts. These limits are informative, given that the ability to initiate strategies has been regarded as a developmental milestone achieved by age 30 months (Vaughn et al., 1984). The onset of the ability to initiate strategies, and even to use them in temporarily effective ways, is not the equivalent of the kind of effective self-regulation that teachers and parents ultimately expect. There was, however, substantial heterogeneity in the dynamics of 36-month-olds’ self-regulation. As we will discuss, one child characteristic was associated with this heterogeneity— externalizing behavior problems—and two other characteristics warrant further study— temperamental effortful control and negative affectivity.

Conventional static analyses and dynamic analyses yielded three sets of complementary findings. First, both approaches buttressed prior evidence of inverse relations between anger expressions and strategy use (Calkins & Johnson, 1998; Gilliom et al., 2002). Static analyses showed that higher levels of executive process were inversely related to lower levels of prepotent responding. ODE modeling complemented this finding, indicating that executive process engagement had potential to accelerate or decelerate prepotent responding. Thus, the dynamic approach more directly captured the process of change. Furthermore, group-level dynamic analyses clarified that 36-month-olds varied in the extent to which the engagement of executive processes ultimately helped them tolerate waiting.

Second, both static and dynamic methods found that mothers more often tried to help children when children were less able to self-regulate, i.e. children displayed higher levels of desire and frustration and lower levels of strategy use. The dynamic analyses complemented this correlational evidence, indicating that mothers tried to foster self-regulation in their children when children used less mature strategies (e.g., bidding for mother’s attention versus distracting oneself through play) rather than when children acted on prepotent response tendencies. In other words, mothers may have allowed their 36-month-olds to express emotion and find solutions on their own, and intervened mainly if children’s strategies were those more commonly used by infants and toddlers and with limited effectiveness (Rothbart et al., 1992). This finding provides a dynamic image of when and how mothers try to scaffold children’s behavior when they need help to engage a more mature strategy such as shifting attention from blocked goal to self-directed play (Calkins & Johnson, 1998).

A third set of results involved children’s effortful control, the predisposition to engage attention and behavioral control (Rothbart & Bates, 2006). These addressed the prediction that children rated higher in effortful control would use more mature strategies (e.g., Eisenberg et al., 2001), which should in turn effectively damp desire and frustration during the wait. Both the static and dynamic approaches indicated nonsignificant trends for these children using more mature strategies and no evidence of effective strategy use. This is perhaps unsurprising given the very young age of the children. As suggested by sequential analyses of the effects of very young children’s strategy use on their anger intensity (Buss & Goldsmith, 1998), 36-month-olds’ newly acquired ability to initiate strategies may have temporary but not enduring effects on reducing emotional intensity and thus they may not yet succeed at helping them tolerate the wait by damping their desire and frustration.

The dynamic approach also yielded unique findings about the interplay of young children’s EP and PR, both at the group level and in terms of individual differences. Supporting prior evidence that children as young as 36 months of age initiate putative regulatory strategies when their goals are blocked (Cole et al., 2011; Grolnick et al., 1996; Kochanska et al., 2001), dynamic modeling yielded the first evidence that they did so when needed most. That is, when prepotent desire and frustration were higher, 36-month-olds deployed strategies more quickly and frequently. However, although they persevered at these brief strategic efforts, their attempts were limited in effectively damping their prepotent responses over time. Rather, only when children’s prepotent response levels were relatively low did their strategies decelerate and delay the resurgence of desire and frustration. Thus, their self-regulation efforts appeared limited. Longitudinal analyses of this sample showed that between 24 and 36 months of age children’s latency to engage strategies decreased and latency to express anger increased (Cole et al., 2011). Temporal variables, such as latencies, are thought to imply the engagement of self-control processes (Thompson, 1994). The present findings qualify prior evidence, demonstrating that 36-month-olds’ engagement of executive processes helps keep their frustrations and desire at bay as long as those prepotent responses are not at high levels. This is likely why the overall group level dynamic analyses did not reveal effective self-regulation. At age 36 months, children show that they have the ability to initiate putative strategies without adult direction but the effectiveness of their efforts is not established. This evidence adds new information about the extent of self-regulation that suggests new ways to investigate its developmental trajectory in early childhood.

Moreover, the dynamic modeling revealed information about substantial individual differences in the dynamic relations between 36-month-olds’ executive processes and prepotent responses. Specifically, a pattern of ineffective self-regulation was associated with externalizing behavior problems. Among children with higher externalizing problems, prepotent responses damped executive process engagement, such that children’s desire and frustration diminished the maturity of their strategy use. It is noteworthy that externalizing did not predict lower levels of executive process engagement in either static or dynamic models. That is, the 36-month-olds in this non-risk sample were able to use age-appropriate strategies; rather their desire and frustration overwhelmed their strategy use. It has been posited that higher emotional arousal can overwhelm children’s strategic efforts, leading to use of less mature strategies such as thumb-sucking or seeking adult support at an age at which many children are capable of the more optimal strategy of distracting themselves (Calkins, 1994; Cole et al., 1994; Gilliom et al., 2002; Kopp, 1989). The present study is the first to link this pattern to externalizing behavior problems. By contrast, the prepotent responding of 36-month-olds with fewer externalizing problems contributed to amplifying their executive process engagement. Perhaps these children are beginning to draw upon internalized rules about self-control, even if their efforts do not yet have the longer lasting effect of damping desire and frustration (Block & Block, 1980; Kochanska et al., 2001).

The results also yielded a non-significant trend associated with child negative affectivity that warrants further study. Specifically, a significant effect of temperamental negative affectivity did not survive correction for multiple tests, but is noteworthy given its importance to the study of the development of self-regulation. Do children described as generally reactive to environmental changes engage executive processes but do so ineffectively? Negative affectivity has been linked to less mature and less frequent strategy use (Blair, Denham, Kochanoff, & Whipple, 2004; Calkins & Dedmon, 2000; Santucci et al., 2008). Our results suggest that their strategic attempts may not be effective; that greater negative affectivity is associated with self-regulatory inefficiency has been suggested (Calkins & Fox, 2002). We mention this nonsignificant trend because as methods of data collection and coding become better suited to dynamic modeling (see Future Directions), it will be important to ascertain whether child negative affectivity is associated with regulatory inefficiency.

In sum, this is the first study to distinguish between the self-regulation dynamics of children with negative affectivity and externalizing behavior problems, aspects of child behavior that are typically correlated (Eisenberg et al., 2000; Nigg, 2006). Negative affectivity leads to problem behavior for children with less effortful control (Eisenberg et al., 2009). Longitudinal research using a dynamic approach can investigate the extent to which a tendency toward negative affectivity very early in life compromises the development of strategies that damp PR or consumes the executive resources needed to enact more mature, effective strategies.

Limitations and Future Directions

The present study indicated the promise of dynamic approaches to self-regulation. As a first of its kind study, it had several limitations that suggested future directions. First, the behavioral codes used to generate PR and EP indices were based on standard practices in observational research with children. These codes yield binary or ordinal scores in each epoch. Differential equation modeling should perform better with time series data that has more variability within time intervals. Methods such as continuous scoring within each small time interval may be promising (Messinger, Mahoor, Chow, & Cohn, 2009). Second, the modeling techniques used in this study represent a first step in applying them to the development of self-regulation. We adopted the theoretical view that EP damps the intensity of PR over time. Some features of the data were not captured by our proposed model and may be incorporated in future studies as modeling extensions. For instance, our hypothesized regulatory parameters, which serve to damp the opposing process toward its baseline, have been found to be inadequate at capturing more extreme damping actions that “annihilate” the opposing process to below-baseline levels. Within-person shifts in the dynamics of EP and PR were also not allowed in our proposed model and may be pursued in future extensions. In addition, we did not model within-person changes in baseline levels of EP and PR beyond the interactive effects hypothesized in our model, but such changes may arise due to fatigue or other within-person changes independent of the EP-PR interactive dynamics. Future studies will need to investigate the methods themselves, different coupling parameters that capture alternative views of self-regulation, and the degree to which different approaches add value above and beyond static approaches (Chow et al., 2016). It remains to be seen if our approach stands up to replication. Also, it would be useful to know if temporal variables such as latencies and durations (Thompson, 1994), which do not address how processes change over the course of a task, yield different or complementary information relative to dynamic models. We also note that the findings are limited to a single sample of 36-month-olds. The influence of executive processes on prepotent responding at later ages should be different as executive processes develop and the skill at recruiting them to regulate prepotent responses strengthens.

Conclusion

This study indicated the promise of dynamic systems modeling approaches to the study of self-regulation. All leading conceptualizations of self-regulation, even the term regulation itself, imply change over time. The ODE models provided for direct articulation and test of this dynamic process, and for investigating the interactive influences of prepotent responses that are highly probable in the specific situation and engagement of executive processes purported to change those responses. The findings included both results that complemented standard static approaches as well as unique results that have not been previously investigated. Together the results offer a fuller portrait of very young children’s self-regulatory efforts and new ideas about their limitations and the individual differences that relate to whether efforts are effective or ineffective. The findings are preliminary and warrant replication but the study provides promise that self-regulation can be studied as a dynamic process.