Modeling Intraindividual Dynamics Using Stochastic Differential Equations: Age Differences in Affect Regulation

Abstract

Objectives

Life-span theories of aging suggest improvements and decrements in individuals’ ability to regulate affect. Dynamic process models, with intensive longitudinal data, provide new opportunities to articulate specific theories about individual differences in intraindividual dynamics. This paper illustrates a method for operationalizing affect dynamics using a multilevel stochastic differential equation (SDE) model, and examines how those dynamics differ with age and trait-level tendencies to deploy emotion regulation strategies (reappraisal and suppression).

Method

Univariate multilevel SDE models, estimated in a Bayesian framework, were fit to 21 days of ecological momentary assessments of affect valence and arousal (average 6.93/day, SD = 1.89) obtained from 150 adults (age 18–89 years)—specifically capturing temporal dynamics of individuals’ core affect in terms of attractor point, reactivity to biopsychosocial (BPS) inputs, and attractor strength.

Results

Older age was associated with higher arousal attractor point and less BPS-related reactivity. Greater use of reappraisal was associated with lower valence attractor point. Intraindividual variability in regulation strategy use was associated with greater BPS-related reactivity and attractor strength, but in different ways for valence and arousal.

Discussion

The results highlight the utility of SDE models for studying affect dynamics and informing theoretical predictions about how intraindividual dynamics change over the life course.

Developmental researchers generally formulate process-based accounts of behavioral change from a systems perspective wherein individuals’ actions, behaviors, cognitions, and emotions are governed by a collection of homeostatic and circular processes that emerge and adapt to internal and external demands (e.g., Baltes, Lindenberger, & Staudinger, 2006; Ford & Lerner, 1992; Gottlieb, 1997; Sameroff, 1983; Thelen & Smith, 1994). Individuals are conceived as dynamic systems capable of both adaptive self-stabilization—wherein coordinated action of subsystems compensate for changing conditions in the environment to maintain homeostasis, and adaptive self-organization—wherein the system as a whole transforms to accommodate change in or challenge to the existing configuration (Cox & Paley, 1997; Thelen & Ulrich, 1991). These two basic principles provide a parsimonious framework for examination of intraindividual variability and change at multiple time scales (Nesselroade, 1991; Ram & Diehl, 2015). Observation of shorter-term change (over minutes, hours, or days) provides for study of intraindividual dynamics—the processes by which individuals self-stabilize toward homeostasis (e.g., affect regulation); observation of longer-term change (or age differences) in intraindividual dynamics chronicle how a system develops and reorganizes with age and/or in accordance with other individual/contextual differences (Ram & Gerstorf, 2009). The purpose of this paper is to advance this general framework by illustrating how long-term age differences in short-term intraindividual dynamics can be articulated and studied using a multiple time-scale study design and continuous-time process modeling. Specifically, we will fit a stochastic differential equation (SDE) model (Arnold, 1974) to experience sampling data to capture theoretically relevant aspects of individuals’ affect dynamics, and examine how these dynamics differ with age and individuals’ typical deployment of regulation strategies.

Intraindividual Affect Dynamics

A person’s affective state changes from moment to moment. The challenge of describing the intraindividual dynamics of affect has prompted both precision in theoretical conceptions and development of methods to articulate those theories (see Hamaker, Ceulemans, Grasman, & Tuerlinckx, 2015 for review). One such theoretical-methodological approach is the dynamics-of-affect model (DynAffect; Kuppens, Oravecz, & Tuerlinckx, 2010b).

A framework for studying affect dynamics: DynAffect and the Ornstein–Uhlenbeck process model

DynAffect maps theoretical propositions about short-term (regulatory) changes in core affect to a formal mathematical modeling framework. In brief, affective experience is located in a two-dimensional core affect space defined by affect valence and affect arousal (Russell, 2003; or an alternative 45° rotation defined by positive affect and negative affect dimensions). Then, short-term changes in individuals’ affect valence and arousal are described using SDEs—the Ornstein–Uhlenbeck (OU) process model (Uhlenbeck & Ornstein, 1930)—and placed within a Bayesian multilevel framework (Gelman et al., 2013) that accommodates simultaneous modeling of multiple individuals. Altogether, the multilevel OU model provides a rigorous framework for studying how individual differences in intraindividual core affect dynamics are related to age, deployment of regulation strategies, and other variables.

The DynAffect model highlights three conceptual features of intraindividual affect dynamics: attractor point, reactivity to biopsychosocial (BPS) inputs, and attractor strength. Here, we purposively engage with different labels than those used by Kuppens et al. (2010b) in order to consistently highlight the conceptual/theoretical meaning of each parameter as a substantive component of the affect system, as opposed to a statistical summary of the data. First, the attractor point (previous label = home base) is conceptualized as an individual’s equilibrium state. That is, the goal of the affect system is always to return to the attractor point—with the possibility for individual differences in the location of the attractor point. To illustrate, the Panel A of Figure 1 depicts repeated measures of affect obtained from two hypothetical persons with different attractor points (position on the y-axis indicated with dashed line). As shown, the person on top has a low (μ = 3.0) attractor point, and the person on the bottom a high (μ = 7.0) attractor point. Empirical evidence generally finds that individuals with higher positive affect (corresponding to higher affect valence and arousal equilibrium/attractor points) enjoy greater life satisfaction and well-being (Diener, Suh, Lucas, & Smith, 1999). Second, variability in affect state around the attractor point can be thought of conceptually as the reactivity of the affect system to BPS input (affect variability). The BPS input is stochastic, time-varying and external to the core affect system—including all the biological, psychological, and social factors that provoke change in core affect. For example, Russell (2003) notes how dynamic features of core affect are caused by the simultaneous influences of individuals’ internal biological factors (e.g., genes, hormone levels), psychological factors (e.g., cognitive appraisal, attention), and social environmental factors. The Panel B of Figure 1 depicts two persons whose affect systems are described with different levels BPS-related reactivity. Noting the difference in the variance of the two time series, we see that the person on top exhibits substantially less reactivity to input (γ = 0.5) than the person on the bottom (γ = 3.0). Empirical associations between affect variability and successful regulation have been mixed, although high variability (corresponding to high BPS-related reactivity) is generally associated with lower well-being and indicative of coping deficits (Röcke & Brose, 2013). Third, attractor strength (attractor strength) is conceptualized as the strength of the “pull” of the attractor point. When the system is perturbed, attractor strength governs how quickly the system brings an individual’s affect state back to the attractor point. As seen in Panel C of Figure 1, low attractor strength (β = 0.01, top) indicates that the pull back toward the attractor point is not very strong. Affect goes on long excursions before returning to or passing through the attractor point. In contrast, high attractor strength (β = 2.0, bottom) indicates quick, almost immediate (thus “spiky”) return toward the attractor point even though the stochastic BPS inputs may continue to drive the system away from the attractor point. Note that BPS-related reactivity represents dynamic change due to factors external to the affect system, whereas the attractor strength is conceived as a regulatory force internal to the affect system. Generally, low attractor strength (formally operationalized in other studies as high inertia) has been linked to ineffective emotion regulation and decreased well-being (Kuppens, Allen, & Sheeber, 2010a). Together, these three features of the DynAffect model provide a framework for modeling how individuals’ core affect change over the short term.

Figure 1.

Simulated time series (125 occasions at unequal intervals) from the Ornstein–Uhlenbeck model of affect dynamics. Each panel demonstrates low (top plots) and high (bottom plots) values for DynAffect parameters, attractor point (Panel A), biopsychosocial (BPS)-related reactivity (Panel B), and attractor strength (Panel C).

Moving from conceptual model to mathematical model, Kuppens and colleagues (2010b) mapped the features of the DynAffect model to the parameters of an OU process model commonly used in physics and economics for describing dynamic systems with stochastic noise. The advantage of mapping the dynamics of affect with a formal mathematical model such as the OU model is that we can assign specific mathematical forms to the conceptual intrinsic and extrinsic processes that govern the behavior of a dynamic system—thus allowing researchers to interpret the model parameters as direct representations of the theory (Ram & Grimm, 2015).

In brief, an OU process evolves continuously over time and space and is specified using a first-order SDE that pairs a control process (regulation) with a stochastic input term (random input). Formally, the univariate OU process model for person p is defined as

$$\{\begin{array}{l}{y}_p(t)={\theta }_p(t)+{\epsilon }_p(t)\hfill \\ d{\theta }_p(t)={\beta }_p({\mu }_p-{\theta }_p(t))dt+{\sigma }_{p}dW(t)\hfill \end{array}$$ (1)

Following the specifications routinely used in behavioral science for state space systems (e.g., Oud & Jansen, 2000), the model is given in two parts. The first line, the measurement equation, defines the observed score (e.g., affect valence or arousal rating) for person p at time t, $y_p(t),$ as the sum of a latent time-specific true score, ${\theta }_p(t),$ and measurement error, ${\epsilon }_p(t).$ Then, in the second line, the transition equation describes the continuously ongoing changes in individual p’s latent true score, $d{\theta }_p(t)$ , as a function of the same three features identified as core components of individuals’ affect dynamics in the DynAffect model. First, the person-specific attractor point is captured by ${\mu }_p$ . Assuming intrinsic regulation, anytime there is a discrepancy between the current affect state and attractor point (i.e., when $[{\mu }_p-{\theta }_p(t)]\ne 0$ ), the affect system restores the system to the attractor point. Second, the restorative force with which the system pulls the current affect state back to the attractor point is scaled by ${\beta }_p$ (constrained to be >0), the person-specific attractor strength. Formally, $e^{-\beta *\Delta t}$ (Δt = time difference) gives the continuous-time exponential autocorrelation function of the time series (see Oravecz, Tuerlinckx, & Vandekerckhove, 2009, 2016). Together, the attractor point and attractor strength govern the intrinsic regulatory dynamics of the system—the restorative forces that bring the individual’s affect state back to equilibrium after perturbation. Finally, the random input into the system is defined as an unbiased continuous-time random walk, a Wiener process, $dW(t)$ , scaled by ${\sigma }_p$ (the diffusion parameter). The Wiener process can be thought of as dynamic “noise” in the system—process-related perturbations that carry forward in time, as opposed to measurement errors that do not influence future behavior of the system (Schuurman, Houtveen, & Hamaker, 2015). Because the dynamic variability exhibited by the process is a function of the (“intrinsic”) ${\beta }_p$ and (“extrinsic”) ${\sigma }_p$ parameters, it is useful to derive the intraindividual variance

$${\gamma }_p=\frac{{\sigma }_p^2}{2{\beta }_p},$$ (2)

which indicates the total extent of BPS-related reactivity of the affect system—reactivity to all of the internal and environmental inputs encountered over time.

In sum, the OU process model provides a concise and formal mathematical articulation of a collection of theoretical propositions on the operation of individual core affect dynamics: (a) the affect system has an equilibrium goal state—an attractor point; (b) affect-relevant information (measured or unmeasured, internal or external to the individual) continually streams into the system—creating BPS-related reactivity; and (c) the resulting perturbations to the affect state are met with an intrinsic restorative force that attempts to move the affect state back to equilibrium—attractor strength. The mathematical operationalization provides opportunity to map between theory and data.

Intensive Longitudinal Data From Experience Sampling Studies

Modeling the intraindividual dynamics of affect, and the individual differences therein, requires intensive longitudinal data collected at relatively fast cadences (e.g., every few minutes or hours; see, e.g., Ram, Conroy, Pincus, Hyde, & Molloy, 2012). Studies wherein multiple individuals provide many reports on their emotions, behaviors, and cognitions as they go about their daily lives begin to obtain the types of longitudinal data needed for such inquiries (Mehl & Conner, 2012). The data emerging from experience sampling studies are increasingly dense, collected in situ and obtained in real time (Stone, Shiffman, Atienza, & Nebeling, 2007). With reports obtained every few minutes or hours, and over increasingly long time periods (e.g., 3 weeks), there is increasing opportunity to observe the processes governing short-term behavior as it emerges and evolves in its natural context.

However, the benefits of experience sampling data are often accompanied with an analytic challenge. When reports are obtained at unequal time intervals (e.g., using random signaling) and when participants do not follow the same schedule (unstructured), the data are not easily modeled using traditional time-series methods (e.g., ARIMA models) that assume equal/regular spacing. Rather, the unequally spaced and unstructured data require use of continuous-time methods (see, e.g., Oud & Delsing, 2010; Voelkle & Oud, 2013). The continuous-time nature of the OU process model provides the opportunity to make use of unequally spaced, unstructured experience sampling data to estimate parameters that can be mapped directly to theoretical propositions.

Individual Differences in Intraindividual Affect Dynamics

When collected from multiple persons, intensive longitudinal data from experience sampling studies facilitates study of interindividual differences in intraindividual dynamics (see Ram & Gerstorf, 2009). The DynAffect model suggests that there are interindividual differences in all three aspects of individuals’ affective dynamics: attractor point, BPS-related reactivity, and attractor strength. Developmental and social psychological theories suggest how these features might differ with age, trait-level patterns of deployment of specific regulation strategies, and a variety of other individual difference characteristics.

Age differences in affect dynamics

Theories of successful aging suggest that individuals generally orchestrate their changing resources (e.g., declining cognitive and physical abilities) through selective compensation and optimization of goal pursuits (Baltes & Baltes, 1990). Formulated with respect to affective well-being, both Socioemotional Selectivity Theory (SST; Carstensen, Isaacowitz, & Charles, 1999) and the Strength and Vulnerability Integration (SAVI) model (Charles, 2010) suggest that affect dynamics should change with age. The key developmental changes identified in these theories include (a) age-related prioritization of emotion regulation and maintenance of well-being, (b) age-related increases in ability to regulate (emotional strengths developed over many years of experience), and (c) age-related decreases in effectiveness of regulatory efforts (vulnerabilities imposed by age-related declines in physiological adaptability to stressors; e.g., cardiovascular stiffness). Both theories propose that healthy older adults are more efficient at regulating their affect than younger adults, in part because older adults limit exposure to potentially perturbing stimuli. Extrapolating to the dynamic features articulated in the OU model, the location of the attractor point likely does not change with age—equilibrium (average) levels of affect are similar between younger and older age groups (for a review, see Rӧcke & Brose, 2013). In contrast, the theories imply that the extent of BPS-related reactivity is actively titrated so that older adults’ affect fluctuates less than younger adults’ affect, particularly for affect arousal (Charles, 2010). Indeed, intraindividual variability of both positive and negative affect tends to be lower for (healthy) older adults than younger adults (e.g., Carstensen et al., 2011; Rӧcke, Li, & Smith, 2009). Although theories of aging are somewhat ambiguous about whether the experiential increases in regulatory ability outweigh the vulnerabilities imposed by declining ability (e.g., physiological inflexibility), it seems that among select populations of generally healthy adults we might see greater attractor strength at older ages. To date, there have not been any studies that have specifically examined age-related differences in attractor strength (or affect inertia, see recommendations in Rӧcke & Brose, 2013).

Interindividual differences in deployment of regulation strategies

Adults of all ages invoke specific strategies to regulate their affect state. Gross (1998) outlined a framework that describes how individuals attempt to influence their emotions through engagement in cognitive reappraisal, changing the way a situation is construed to decrease its emotional impact, and suppression, inhibiting the outward signs of inner feelings. Both strategies are meant to move affect state toward a specific goal state (i.e., the attractor point). Extrapolating to the dynamic features articulated in the OU model, reappraisal and suppression are considered here as “extrinsic” influences that might be engaged when the intrinsic dynamics of the affect system are ineffective or unable to keep things moving back toward homeostasis. From this perspective, trait-like individual differences in the extent and variability of strategy deployment may be related to all three features of individuals’ core affect dynamics. Although exploratory, our guiding hypotheses are that individuals who tend to engage more strategy deployment will tend to be those with lower attractor points for affect valence (i.e., less positive affect) as these are the cases where regulation is most needed, and that individuals who demonstrate greater variability in deployment of regulation strategies will be those who have greater reactivity to BPS input (perhaps from receiving more input) and/or who are more adept at intrinsic regulation, as indicated by greater attractor strength.

A framework for studying individual differences in affect dynamics: multilevel OU process model

With repeated observations nested within persons from intensive longitudinal data collected in experience sampling studies, we can place Equation 1 within a Bayesian multilevel framework (Gelman & Hill, 2006) that allows for simultaneous estimation of each person’s intraindividual dynamic parameters (attractor point, BPS-related reactivity, attractor strength) and how the interindividual differences in those parameters are systematically related to other between-person variables. Formally, the joint population distributions of the OU model parameters are conditioned on a vector of k person-specific, time-invariant covariates, $x_p={(x_{p0},x_{p1},x_{p2},\dots ,x_{pk})}^T$ , where $x_{p0}=1$ (a unit constant to define the intercept parameter for the pth person), $x_{p1 to k}$ are the predictors of interest (e.g., age, typical strategy deployment) and ^T denotes the transpose operator. The model for person-specific attractor points (separately for affect valence and arousal), ${\mu }_p$ , is defined as

$${\mu }_p={\alpha }_{\mu }{x}_p+e_{p\mu },\text{ with }{e}_{p\mu }~N\left(0,{\sigma }_{\mu }^2\right)$$ (3)

where vector ${\alpha }_{\mu }$ , of dimension 1 × (k + 1), contains the regression weights for the covariates (e.g., association between age and attractor point) and e_p are residual unexplained interindividual differences (random effects). The time-invariant covariate effects on extent of BPS-related reactivity, ${\gamma }_p$ , are modeled similarly, but with a lognormal structure that accommodates that variance parameters must be positive (see details in Oravecz, Tuerlinckx, & Vandekerckhove, 2011; Oravecz et al., 2016),

$$\text{log}({\gamma }_p)={\alpha }_{\gamma }{x}_p+e_{p,\text{log}(\gamma )},\text{with }{e}_{p,\text{log}(\gamma )}~N\left(0,{\sigma }_{\text{log}(\gamma )}^2\right)$$ (4)

where ${\alpha }_{\gamma }$ contains the regression weights indicating how the extent of between-person differences in BPS-related reactivity are related to the time-invariant covariates. Similarly, because attractor strength, ${\beta }_p$ , is by definition also constrained to be positive (in order that the attractor does not become a “repeller”), these parameters are also log-transformed to get values on the real line when regressed on the covariates,

$$\text{log}({\beta }_p)={\alpha }_{\beta }{x}_p+e_{p,\text{log}(\beta )},\text{with\hspace{0.17em}\hspace{0.17em}}{e}_{p,\text{log}(\beta )}~N\left(0,{\sigma }_{\text{log}(\beta )}^2\right)$$ (5)

where ${\alpha }_{\beta }$ contains the regression weights indicating how individual differences in attractor strength are related to the covariates. Finally, for complete model formulation in the Bayesian framework, all model parameters must be assigned prior distributions. In the current analysis, we use diffuse priors that distribute the probability weight equally over a wide range of possible parameter values (details in Oravecz et al., 2016). Together, Equations 1 to 5 and the priors provide a rigorous statistical framework for studying interindividual differences in intraindividual dynamics that unfold continuously in time.

The Present Study

To summarize, the purpose of the paper is to model intraindividual dynamics of core affect with a class of SDE models, following (with some concept-oriented renaming) the DynAffect model (Kuppens et al., 2010b). Specifically, we (a) articulate three specific aspects of individuals’ core affect dynamics (attractor point, BPS-related reactivity, attractor strength) with an OU process model, (b) fit that model to multi-person experience sampling data obtained from an age-heterogeneous community sample, and (c) examine how interindividual differences in intraindividual affect dynamics are related to age and deployment of emotion regulation strategies. Working from general principles laid out in SST and SAVI (Carstensen et al., 1999; Charles, 2010), we hypothesize that older age will be associated with less BPS-related reactivity in arousal, but not to differences in location of the valence or arousal attractor points or attractor strengths. Working from Gross’ (1998) model of emotion regulation, we hypothesize that overall deployment of regulation strategies indicates a general matching of emotion regulation strategy with the need to regulate (undesirable levels and perturbations pull for regulatory action). Thus, without assumption as to whether the deployment of regulation strategies is successful or not, we expect that individuals with greater deployment and variability in deployment of emotion regulation strategies (suppression and reappraisal) will have lower valence attractor points, greater BPS-related reactivity, and greater attractor strength.

Method

Our empirical illustration makes use of data from the Intraindividual Study of Affect, Health, and Interpersonal Behavior (iSAHIB), a multiple time-scale experience sampling study designed for articulation of process-oriented theory and methods (Ram et al., 2014).

Participants

The iSAHIB sample consists of 150 adults (50% women), recruited from the Pennsylvania State University and surrounding community, who were stratified by gender and age to cover the full adult life span. Participants ranged in age from 18 to 89 years (M_Age = 47.10, SD_Age = 18.76), had obtained between 2 and 24 years of formal education (M_Educ = 16.36, SD_Educ = 3.90), with 91% self-identifying as Caucasian (4% African American, 1% Asian American, and 4% Mixed or Other ethnicity). Most individuals identified as heterosexual (93%) with a few identifying as bisexual/gay/lesbian (6%). After participants were recruited, informed of the intensive nature of the assessments, and self-selected into the study, they began the assessment protocol, providing extensive reports about their lives during the next year through a combination of web-based (completed during visits to the laboratory) and smartphone-based questionnaires (completed multiple times per day during regular daily life).

Procedure

Participants completed three 21-day “measurement bursts” spaced approximately evenly over 1 year. During each 21-day burst, individuals used a study-provided smartphone (Verizon XV6900) with a customized “iSAHIB Surveys” application to report on their social interactions. Specifically, participants provided event-contingent reports of face-to-face social interactions that lasted longer than 5 min by repeatedly completing 27 items on the smartphone survey application throughout the day. Each report included descriptions of when and where the interaction occurred, whom the interaction was with, their and their partner’s behaviors during the interaction, and how they felt after the interaction. To facilitate compliance, the smartphones were programmed to chime a prompt if the participant had not provided a report for any 2-hr span between 8 a.m. and 8 p.m. Data flow was monitored in real time, a process that enabled the research staff to make periodic “check-in” calls that supported, motivated, and helped participants (e.g., solving technical problems) provide high quality data. At the beginning and end of each 21-day burst, individuals visited the laboratory, received training or debriefing, picked-up or dropped-off smartphones, and completed web-based batteries of demographic, health, personality, and other questionnaires. Participants were compensated $500 for completing the entire three-burst protocol.

Measures

Our illustration of an SDE approach makes use of the intensive repeated measures of core affect (valence and arousal) and emotion regulation strategies (cognitive reappraisal and expressive suppression) obtained during the first measurement burst: data from 150 adults who provided, on average, 145.46 reports (SD = 39.59, range = 35–265) over 21 days, for a total of 21,819 reports.

Core affect

The two dimensions of core affect, affect valence and affect arousal (Russell, 2003), were measured after each social interaction by asking “How do you feel right now?,” followed by two “touch-point-continuums” (slider-type interface) with end-point anchors labeled “Unpleasant … Pleasant” (valence) and “Sleepy … Activated/aroused” (arousal). Responses were digitally coded on a 0 to 100 scale (numbers not visible to participants) and rescaled 0 to 10 here for ease of parameter estimation. Repeated measures of affect valence and affect arousal responses exhibited substantial variation, of which 33.36% and 36.19%, respectively, could be considered between-person variance, with the remaining 66.64% and 63.81%, considered within-person variance (and measurement error). The average within-person correlation of affect valence and arousal was r = 0.30 (range = −0.38 to 0.80). Descriptive statistics are included in Table 1.

Table 1.

Descriptives of Main Study Variables

	Min	Max	Mean	SD
iMean valence	3.81	9.97	7.85	1.06
iMean arousal	1.79	9.37	6.06	1.30
Age	18.98	88.71	46.27	17.30
Extent reappraisal deployment (iMean)	0.23	6.02	2.33	1.55
Variabil. reappraisal deployment (iSD)	0.19	4.44	1.78	0.81
Extent suppression deployment (iMean)	0.24	8.79	3.02	1.77
Variabil. suppression deployment (iSD)	0.14	4.29	2.13	0.81
Sex (% male)	0 = female	1 = male	49%	—

Note: N = 150. iMean = intraindividual mean; iSD = intraindividual standard deviation calculated as person-specific summaries of the 21-day burst time series; Variabil. = variability.

Age

Chronological age was calculated as the difference between an individual’s birthdate (from demographic questionnaire) and the day on which the study commenced (May 12, 2010; M = 47.10, SD = 18.76).

Sex

Participants’ sex was also drawn from the demographic questionnaire, and coded as a binary variable (0 = female, 1 = male).

Emotion regulation

The extent of individuals’ engagement of emotion regulation strategies was measured after each social interaction using two items adapted from the Emotion Regulation Questionnaire (ERQ; Gross & John, 2003). Specifically, individuals’ use of cognitive reappraisal was assessed by prompting participants to rate the extent to which they agreed with the statement, “I controlled my emotions by changing how I thought about the interaction,” on a slider-type 0 (not at all) to 100 (very much) scale. Similarly, individuals’ use of expressive suppression was assessed by prompting participants to rate “During this interaction, I kept my emotions to myself,” on the 0 (not at all) to 100 (very much) scale. Although time-varying, we summarized individuals’ use of emotion regulation strategies using intraindividual metrics, also rescaling both to a 0 to 10 scale. Specifically, individual differences in trait-level extent of strategy deployment (calculated as the intraindividual mean, iMean, of the repeated measures) and trait-level variability of strategy deployment (calculated as the intraindividual standard deviation, iSD, of the repeated measures; see, e.g., Ram et al., 2012). Between-person differences in extent of deployment of expressive suppression ranged from 0.24 to 8.79 (M_iMean = 3.02, SD_iMean = 1.77), and were correlated r = 0.79 with between-person differences in extent of deployment of cognitive reappraisal that ranged from 0.23 to 6.02 (M_iMean = 2.33, SD_iMean = 1.55). Between-person differences in variability of deployment of expressive suppression ranged from 0.14 to 4.29 (M_iSD = 2.13, SD_iSD = 0.81), and were correlated r = 0.88 with between-person differences in variability of deployment of cognitive reappraisal that ranged from 0.19 to 4.44 (M_iSD = 1.78, SD_iSD = 0.81).

Data Analysis

Using these data, we illustrate how the multilevel OU model can be used to articulate three features of individuals’ affect dynamics (attractor point, BPS-related reactivity, attractor strength) and how these features are related to individual differences in age and use of emotion regulation strategies, controlling for sex differences. The analyses were carried out using the Bayesian Hierarchical Ornstein–Uhlenbeck Modeling (BHOUM) toolbox (Oravecz et al., 2016), a stand-alone software program with a user-friendly graphical user interface (freely available for download at www.zitaoravecz.net).

Data Preparation

Data are organized as per usual for modeling repeated measures data within a multilevel framework, with time-varying outcomes, time-invariant covariates, and time and person identifiers organized in a long (multiple records per person) format file (see BHOUM user’s manual). Although implicit in the model equations, time is used to organize and appropriately space the observed data in continuous time. Specifically, the timing of each observation is marked cumulatively from the beginning of the study, here coded as hours from midnight on the first day of each individuals’ measurement burst.

Model Specification

The nested nature of the data (repeated affect reports nested within persons) was accommodated using a Bayesian multilevel modeling framework (Gelman & Hill, 2006). As an explicit example, Equation 6 presents the Level 1 model we fit for individual affect dynamic parameters for affect arousal (the equations for the affect valence outcome are structured identically).

$$\{\begin{array}{l}{y}_{Arousal,p}(t)={\theta }_{Arousal,p}(t)+{\epsilon }_{Arousal,p}(t)\hfill \\ d{\theta }_{Arousal,p}(t)={\beta }_{Arousal,p}({\mu }_{Arousal,p}-{\theta }_{Arousal,p}(t))dt+{\sigma }_{Arousal,p}dW(t)\hfill \end{array}$$ (6)

To reiterate, $y_{Arousal,p}(t)$ denotes the actual observed measurements (self-reports) on arousal from person p, whereas ${\theta }_{\text{Arousal},p}$ denotes the latent, unobserved, continuously evolving arousal quality of the affect system. Individual differences in the person-specific parameters were modeled at Level 2 as

$$\begin{array}{l}{\mu }_{\text{Arousal},p}={\alpha }_{{\mu }_{Arousal}0}+{\alpha }_{{\mu }_{Arousal}1}Ag{e}_p+{\alpha }_{{\mu }_{Arousal}2}Se{x}_p\\ +{\alpha }_{{\mu }_{Arousal}3}iMeanReappraisa{l}_p\\ +{\alpha }_{{\mu }_{Arousal}4}iMeanSuppressio{n}_p\\ +{\alpha }_{{\mu }_{Arousal}5}iSDReappraisa{l}_p\\ +{\alpha }_{{\mu }_{Arousal}6}iSDSuppressio{n}_p+e_{p,{\mu }_{Arousal}}\end{array}$$ (7)

with the random effects, $e_{p,{\mu }_{Arousal}}$ , having variance ${\sigma }_{\mu Arousal}^2$ , and with parallel structure for $\log ({\gamma }_{Arousal,p})$ and $\log ({\beta }_{Arousal,p})$ as in Equations 3 to 5. In this model specification, αs are sample-level parameters that describe how covariates (e.g., Age_p) are related to the attractor point, BPS-related reactivity, and attractor strength.

Parameter Estimation

The OU process model, an SDE model, has more complexity than conventional linear multilevel models typically used in analysis of experience sampling data. Model fitting for SDEs is greatly facilitated by estimation in a Bayesian framework. In brief, estimation within a frequentist (i.e., maximum likelihood) framework requires high-dimensional integration of the differential equation over many distributions, most of which do not have closed-form solutions. Bayesian estimation avoids the need for explicit integration by constructing the posterior distribution (which is proportional to the prior and the likelihood) of the parameters through iterative sampling. Specifically, the BHOUM toolbox employed here uses tailored Metropolis-within-Gibbs Markov chain Monte Carlo (MCMC) algorithms to sample from the conditional posterior distribution of each parameter (or vector of parameters). For a detailed discussion of the mathematical model and statistical inference for the Bayesian hierarchical OU model, see Oravecz et al. (2011).

Following this approach, univariate models for affect arousal and affect valence were fit to the complete data (<<1% missing) using the BHOUM toolbox. Although core affect dimensions are commonly considered in bivariate space, we chose to model core affect dimensions separately (univariately) to facilitate convergence of the estimation algorithm. All process model parameters are regressed on numerous covariates. Estimating the regression coefficients and all person- and population-level process model parameters in one step becomes a complex task. The univariate simplification is consistent with the main emphasis here, which is to investigate how process model parameters relate to person characteristics (e.g., age), as opposed to zooming in on coupling between the two affect dimensions.

Person-level, time-invariant predictors (age, trait-level extent of deployment of emotion reappraisal and suppression strategies, and trait-level variability of deployment of expressive suppression and cognitive reappraisal strategies) were standardized to facilitate interpretation of model parameters as representing effects for the prototypical female (sex kept as a binary variable). Univariate models of affect valence and affect arousal were fitted using two chains, taking 50,000 samples per chain after 4,000 samples of burn-in per chain. Convergence criterion was met for every parameter (Gelman–Rubin $\stackrel{⌢}{R}$ statistic smaller than 1.1, see Gelman et al., 2013). Posterior distributions (the probability distribution of the likely values of the parameter), each consisting of 100,000 samples, were summarized with the posterior means and 95% credible intervals (CIs) to support inference (see, e.g., Morey, Hoekstra, Rouder, Lee, & Wagenmakers, 2015). Absolute model fit was evaluated using two approaches (see Oravecz et al., 2016). First, model parameters were used to simulate 1,000 data sets that were then evaluated for overlap with the observed data. High correlations (>0.9) indicate that the estimated-model-generated data occupy the same “spaces” with similar frequency to the observed data (i.e., good fit). Second, as another representation of how much the model-generated data physically resemble the observed data, the turning angles (the clockwise angles that connect three consecutive observations) in the generated data were evaluated for overlap with the turning angles of the observed data (percent of model-generated turning angles that fell in a 95% CI of observed turning angles).

Results

Results from the multilevel OU models are shown in Table 2, with select results plotted in Figure 2. After assessing overall fit of the model, we examine how age and deployment of emotion regulation strategies were related to interindividual differences in the three aspects of affect dynamics captured by the OU model (attractor point, BPS-related reactivity, attractor strength).

Table 2.

Results From Multilevel Stochastic Differential Equation (Ornstein–Uhlenbeck) Model of Adults’ Intraindividual Affect Dynamics

	Affect valence		Affect arousal
	Posterior mean	95% credible interval	Posterior mean	95% credible interval
Sample attractor point ( ${\alpha }_{\mu 0}$ )	7.84	(7.68, 8.01)	6.01	(5.80, 6.22)
Age ( ${\alpha }_{\mu 1}$ )	0.11	(−0.06, 0.29)	0.28	(0.06, 0.50)
Sex ( ${\alpha }_{\mu 2}$ )	0.02	(−0.15, 0.19)	−0.03	(−0.24, 0.18)
Extent reappraisal ( ${\alpha }_{\mu 3}$ )	−0.35	(−0.67, −0.02)	−0.23	(−0.64, 0.18)
Extent suppression ( ${\alpha }_{\mu 4}$ )	−0.09	(−0.38, 0.20)	0.19	(−0.18, 0.56)
Variabil. reappraisal ( ${\alpha }_{\mu 5}$ )	0.27	(−0.15, 0.68)	0.23	(−0.29, 0.74)
Variabil. suppression ( ${\alpha }_{\mu 6}$ )	−0.03	(−0.39, 0.33)	−0.10	(−0.55, 0.36)
Sample BPS input ${\rm E}({\alpha }_{\gamma 0})$	2.00	(1.67, 2.42)	2.59	(2.16, 3.15)
Age ( ${\alpha }_{log(\gamma 1)}$ )	−0.05	(−0.21, 0.12)	−0.19	(−0.35, −0.03)
Sex ( ${\alpha }_{log(\gamma 2)}$ )	−0.10	(−0.26, 0.06)	−0.04	(−0.20, 0.12)
Extent reappraisal ( ${\alpha }_{log(\gamma 3)}$ )	−0.23	(−0.64, 0.18)	−0.05	(−0.35, −0.03)
Extent suppression ( ${\alpha }_{log(\gamma 4)}$ )	0.19	(−0.18, 0.56)	−0.23	(−0.51, 0.07)
Variabil. reappraisal ( ${\alpha }_{log(\gamma 5)}$ )	0.23	(−0.29, 0.74)	0.43	(0.04, 0.81)
Variabil. suppression ( ${\alpha }_{log(\gamma 6)}$ )	0.73	(0.40, 1.07)	0.22	(−0.13, 0.81)
Sample attractor strength ( ${\alpha }_{log(\beta 0)}$ )	1.08	(0.73, 1.43)	0.93	(0.59, 1.27)
Age ( ${\alpha }_{log(\beta 1)}$ )	0.16	(−0.21, 0.53)	0.00	(−0.35, 0.36)
Sex ( ${\alpha }_{log(\beta 2)}$ )	−0.36	(−0.70, −0.01)	0.10	(−0.228, 0.43)
Extent reappraisal ( ${\alpha }_{log(\beta 3)}$ )	−0.38	(−1.05, 0.30)	0.08	(−0.553, 0.714)
Extent suppression ( ${\alpha }_{log(\beta 4)}$ )	0.10	(−0.50, 0.71)	−0.41	(−0.979, 0.163)
Varabil. reappraisal ( ${\alpha }_{log(\beta 5)}$ )	0.89	(0.03, 1.75)	−0.26	(−1.076, 0.556)
Varaibil. suppression ( ${\alpha }_{log(\beta 6)}$ )	0.41	(−0.35, 1.18)	1.11	(0.393, 1.832)
Var. attractor point ( ${\sigma }_{\mu }^2$ )	1.02	(0.80, 1.30)	1.62	(1.274, 2.063)
Var. BPS input ( $E({\sigma }_{\gamma }^2)$ )	5.56	(2.80, 10.81)	9.74	(4.87, 19.05)
Var. attractor strength ( ${\sigma }_{log(\beta )}^2$ )	4.08	(3.13, 5.28)	3.50	(2.48, 4.66)
Measurement error ( ${\sigma }_{\epsilon }^2$ )	0.32	(0.29, 0.34)	0.38	(0.34, 0.41)

Note: N = 150. BPS = biopsychosocial; iMean = intraindividual mean; iSD = intraindividual standard deviation; Variabil. = variability; Var. = variance. Covariate estimates for γs and βs on a log-scale. Bold print denotes estimates where credible interval does not include zero. Age, iMeans, and iSDs standardized to z-scores for analysis. Sex coded 0 = female, 1 = male. Affect valence and arousal coded on a 0 to 10 scale.

Figure 2.

Plots of selected covariate effects of the multilevel Ornstein–Uhlenbeck model. Plots depict person-specific parameter estimates (dots), with linear regression line. Biopsychosocial (BPS)-related reactivity and attractor strength parameters are on a log-scale. All regression estimates are based on standardized covariates; raw scales are presented in the figure for interpretability. Panel A: Older age is associated with higher arousal attractor point ( ${\alpha }_{{\mu }_{Arousal}1}$ = 0.28). Panel B: Older age is associated with less arousal BPS-related reactivity ( ${\alpha }_{log({\gamma }_{Arousal}1)}$ = −0.19). Panel C: Greater variability in reappraisal use is associated with more arousal BPS-related reactivity ( ${\alpha }_{log({\gamma }_{2}5)}$ = 0.43). Panel D: Greater variability in suppression use is associated with greater arousal attractor strength ( ${\alpha }_{log({\beta }_{Arousal}6)}$ = 1.11). One outlying observation has been removed from Panel D. iSD = intraindividual standard deviation.

Model fit statistics indicated that the estimated parameters accurately represented the form of the observed data. The correlation of areas visited between the observed data and 1,000 simulated data sets based on the estimated parameters was r = 0.95, indicating that the parameter estimates yield simulated time series that cover similar spaces to the observed data. As well, 94% of the turning angles in the 1,000 simulated data sets fell in a 95% CI of the observed turning angles. In sum, the global and dynamic characteristics of the data are well matched, suggesting that the model provides a good representation of the data.

Attractor Point

Conceptually, the attractor point indicates individuals’ equilibrium level of core affect. For the prototypical individual in this sample, the attractor point for affect valence was ${\alpha }_{{\mu }_{Valence}0}$ = 7.84, with 95% CI limits of 7.68 and 8.01 on the 0 to 10 scale, with substantial between-person differences in the location of the attractor point, ${\sigma }_{\mu Valence}^2$ = 1.02 (CI = 0.80, 1.30). These between-person differences in location of the valence attractor point were not systematically related to age ( ${\alpha }_{{\mu }_{Valence}1}$ = 0.11; CI = −0.06, 0.29), sex ( ${\alpha }_{{\mu }_{Valence}2}$ = 0.02; CI = −0.15, 0.19), or extent of deployment of suppression strategies (iMean: ${\alpha }_{{\mu }_{Valence}4}$ = −0.09; CI = −0.38, 0.20), but were related to differences in extent to which individuals deployed reappraisal strategies, ${\alpha }_{{\mu }_{Valence}3}$ = −0.35 (CI = −0.67, −0.02). Individuals who deployed more reappraisal strategies had less positively valenced attractor points. Variability in deployment of reappraisal or suppression strategies was not related to location of the affect valence attractor point (iSD reappraisal: ${\alpha }_{{\mu }_{Valence}5}$ = 0.27; CI = −0.15, 0.68; iSD suppression: ${\alpha }_{{\mu }_{Valence}6}$ = −0.03; CI = −0.39, 0.33).

The prototypical attractor point for affect arousal was ${\alpha }_{{\mu }_{Arousal}0}$ = 6.01 (CI = 5.80, 6.21) on the 0 to 10 scale, again with substantial interindividual differences in where the arousal attractor point was located, ${\sigma }_{\mu Arousal}^2$ = 1.62 (CI = 1.27, 2.06). Here, the arousal attractor point was not related to any aspect of regulation strategy deployment, but was related positively to age ( ${\alpha }_{{\mu }_{Arousal}1}$ = 0.28; CI = 0.06, 0.50), against expectation. As shown in Figure 2A, older age was associated with a higher attractor point for arousal, such that a young adult’s (−1 SD = age 27 years) attractor point would be at arousal 5.73, and an older adult’s (+1 SD = age 67 years) attractor point would be at arousal 6.29.

BPS-Related Reactivity

Conceptually, the extent of stochastic variability in affect indicates the reactivity of the system to BPS input injected into an individual’s system. Sample-level means and variances for BPS-related reactivity (γ) were estimated on the log-scale. However, for better interpretability, we transformed these γ parameters to the original scale. For the prototypical individual in this sample, expected BPS-related reactivity in affect valence ratings was $E({\alpha }_{\gamma {}_{Valence}}0)$ = 2.00 (95% CI = 1.67, 2.42), with considerable between-person differences, $E({\sigma }_{\gamma Valence}^2)$ = 5.56 (CI = 2.80, 10.81). These between-person differences were related to trait-level variability in deployment of suppression strategies, ${\alpha }_{\text{log}({\gamma }_{Valence}6)}$ = 0.73 (CI = 0.40, 1.07). Greater variability in BPS-related reactivity in affect valence was accompanied by greater variability in how suppression strategies were deployed, but was not systematically related to age, sex, or deployment of reappraisal strategies.

For the prototypical person, extent of BPS-related reactivity in affect arousal was $E({\alpha }_{{\gamma }_{Arousal}0})$ = 2.59 (CI = 2.16, 3.15), with substantial between-person differences, $E({\sigma }_{\gamma Arousal}^2)$ = 9.74 (CI = 4.87, 19.05). As hypothesized, these differences were related to age, ${\alpha }_{log({\gamma }_{Arousal}1)}$ = −0.19 (CI = −0.35, −0.02), such that older individuals tended to exhibit less BPS-related reactivity in arousal than younger individuals. As shown in Figure 2B, a young adult’s (−1 SD = age 27 years) expected BPS-related reactivity in arousal was projected at = 3.32, whereas an older adult’s (+1 SD = age 67 years) BPS-related reactivity was projected at = 2.27. Interestingly, while BPS-related reactivity in valence was related to extent of trait-level variability in deployment of suppression strategies, BPS-related reactivity in arousal was, as shown in Figure 2C, related to extent of trait-level variability in deployment of reappraisal strategies, ${\alpha }_{log({\gamma }_{2}5)}$ = 0.43 (CI = 0.04, 0.81).

Attractor Strength

Conceptually, the attractor strength indicates the extent to which, or how quickly, individuals intrinsically regulate their affective states toward their attractor point. For the prototypical person (female), individual attractor strength for affect valence was ${\alpha }_{\text{log}({\beta }_{Valence}0)}$ = 1.08 (CI = 0.73, 1.43), with substantial between-person differences in the strength of the pull back toward the valence attractor point, ${\sigma }_{\text{log}(\beta Valence)}^2$ = 4.08 (CI = 3.13, 5.28). On average, males had less pull back toward the valence attractor point ( ${\alpha }_{\text{log}({\beta }_{Valence}2)}$ = −0.36; CI = −0.70, −0.01) than females. Prototypical attractor strength for affect arousal was ${\alpha }_{\text{log}({\beta }_{Arousal})}$ = 0.93 (CI = 0.59, 1.27), also with substantial between-person differences in the strength of the pull back toward the attractor point, ${\sigma }_{\text{log}(\beta Arousal)}^2$ = 3.50 (CI = 2.48, 4.66). Although not related to age, sex, or deployment of reappraisal strategies, arousal attractor strength was related to trait-level intraindividual variability in deployment of suppression strategies. As shown in Figure 2D, greater trait-level variability in deployment of suppression strategies was associated with a faster return to equilibrium level of arousal, ${\alpha }_{\text{log}({\beta }_{Arousal}6)}$ = 1.11 (CI = 0.39, 1.83), an indication that individuals may generally be mapping deployment to need.

Discussion

Purposively using mathematical models to map theory to intensive longitudinal data, this study illustrates how multilevel SDE models can be used to articulate and investigate intraindividual affect dynamics, and interindividual differences in those intraindividual dynamics. Specifically, we used an OU process model to study how individuals’ core affect state evolves in continuous time and how individual differences in three specific aspects of that evolution (attractor point, BPS-related reactivity, and attractor strength) differ with age and trait-level deployment of emotion regulation strategies (reappraisal and suppression). This application of an SDE model to 21-day experience sampling data obtained from 150 adults (age 18–89 years) empirically demonstrates the benefits of multiple time-scale (age-heterogeneous) designs and multilevel SDE process models for studying interindividual differences in intraindividual dynamics (Ram et al., 2014).

Multilevel SDEs

Following the DynAffect model (Kuppens et al., 2010b), we used an SDE model to operationalize features of intraindividual affect dynamics. In particular, the OU process model provided for precise articulation of a collection of theoretical propositions via three parameters: (a) the attractor point, representing the affect system’s equilibrium; (b) BPS-related reactivity, created by affect-relevant information (internal or external to the individual) that is continually streaming into the system; and (c) attractor strength that quantifies the restorative force that moves the affect state back toward equilibrium. Importantly, the OU process model is explicitly situated in continuous time (as opposed to discrete time) and in a stochastic world (as opposed to a deterministic world). Thus, this SDE framework provides for more precise articulation of core affect as an uninterrupted stream of experiences in real time (a live show) that is flexible (free will), rather than as a stop-motion animation film that jumps from moment to moment in a predetermined way. Formulated as a Bayesian multilevel model, this SDE parameterization articulates the additional propositions that (d) individuals differ in each aspect of the intraindividual affect dynamics, and (e) that these individual differences may be related to other variables (e.g., age). In sum, the multilevel SDE provides a framework where theoretical propositions can be mapped to empirical data precisely and with statistical rigor.

Age and Strategy-Deployment Differences

Our empirical example was constructed to demonstrate how the multilevel SDE model could be applied to experience sampling data to examine interindividual differences in intraindividual dynamics. Framed with respect to developmental meta-theory (self-stabilization and reorganization; Thelen & Ulrich, 1991) and successful aging (Baltes & Baltes, 1990), we examined how attractor point, BPS-related reactivity, and attractor strength differ with age and trait-level deployment of emotion regulation strategies (reappraisal and suppression). Two main findings emerged.

First, age was associated with higher affect arousal attractor points and lower BPS-related reactivity in arousal, but not with the dynamics of affect valence. Although we did not hypothesize that age would be associated with differences in attractor points, both SST (Carstensen et al., 1999) and the SAVI model (Charles, 2010) provide interpretations of the age-arousal association. From a strengths perspective, higher equilibrium and less stochastic variability may reflect age-related increases in ability to direct or filter BPS input toward valued experiences. From a vulnerability perspective, age-related decreases in cardiovascular and neuroendocrine flexibility may be reorganizing the system toward a new equilibrium/homeostasis. Although the age-related decrease in BPS-related reactivity of affect generally aligns with findings that affect variability is lower in older adults compared with younger adults, the dimension-specific nature of our finding underscores calls for more research on affect arousal (see Röcke & Brose, 2013).

Second, we found an association between overall extent and variability of individuals’ use of emotion regulation strategies and intraindividual affect dynamics. Specifically, extent of deployment of reappraisal was associated with lower valence attractor points, and variability in deployment of suppression and reappraisal strategies was associated with greater BPS-related reactivity in valence and arousal, respectively. Together, these associations suggest that individuals generally matched their use of regulation strategies to their regulatory need. Turbulent (stochastic) lives pull for more varied deployment, whereas relatively comfortable and stable environments require less varied extrinsic regulation. Concordant with this interpretation, intraindividual variability in use of reappraisal and suppression was also associated with greater attractor strength for valence and arousal, respectively—perhaps indicating that individuals with more “flexible” (i.e., iSD as indicator of flexibility; Hollenstein, 2015) patterns of strategy use are returning to their equilibrium points faster than those with more “rigid” deployment patterns. However, some interpretative caution is warranted without additional information or tracking of the success of each strategy deployment. Although there are some hints of specificity in how reappraisal and suppression map to valence and arousal, inconsistency across relations with iMeans and iSDs calls for more research on intraindividual variability of strategy deployment (currently being missed by cross-sectional assessment of emotion regulation). All together, the findings highlight how much may be learned through principled articulation and examination of interindividual differences in intraindividual affect dynamics.

Limitations

There are some limitations that need to be addressed in future application of SDE models to experience sampling data. Although precisely mapped to a specific OU process model, the theoretical propositions of the DynAffect model (Kuppens et al., 2010b) may be questioned. For example, the affect system may not have one, and only one, attractor point (see Hollenstein, 2015). Individuals may simultaneously have multiple goals—multiple attractor points, all of which are pulling with differential strength. Additional theoretical and mathematical work is needed to accommodate the possibility (or fact) that humans can have multiple goals that may change over time. As well, we modeled the valence and arousal dimensions separately, even though they are often considered as a unified bivariate space. Next steps will include moving to a coupled bivariate model where the hierarchical OU model is expanded to include coupling terms that model additional covariance between affect valence dynamics and affect arousal dynamics, and that allow for all three dynamic parameters to be time-varying (BHOUM toolbox currently only allows time-varying attractor points). Looking at the models used for physical and chemical systems suggests that at least some of the complexity of multidimensional human systems can be mapped using expanded SDE models.

At the practical level, fitting the OU model to data is facilitated by recent advances in computational speed and specialized software. However, we caution that the MCMC-based model fitting of SDEs is still computationally demanding. Estimation of the final models here, fit to ~22,000 observations nested within N = 150 persons, took about 20 hours of computing time. Although already quite short in relation to the time (i.e., years) the participants and research staff spent providing and preparing the data, computational time may be further shortened through parallelization, memory expansion, and algorithm optimization. Improved and more flexible, cross-platform versions of BHOUM are forthcoming.

Models may be evaluated in a number of ways. Working in a confirmatory framework (to articulate specific theoretical propositions), we evaluated absolute model fit. However, researchers with more exploratory aims should consider comparing several models. For example, relative fit of an OU model with all parameters estimated as random effects (i.e., the one presented here) could be compared with an OU model where some parameters that are fixed to a certain value or are homogenous across persons. Multiple alternative models can be compared with the Deviance Information Criterion (DIC; Plummer, 2008; available in the BHOUM toolbox)—a commonly used quantification of model fit in terms of how well a model reduces uncertainty in future predictions (lower = better; see alternative OU models tested in Kuppens et al., 2010b).

While our application of the multilevel OU model to study age-related individual differences in affect regulation and dynamics is novel, the cross-sectional nature of the age variable used here means that our findings with respect to aging should be interpreted cautiously. In this illustration, we only examined individuals’ affect dynamics for one 21-day measurement burst and did not attempt to additionally examine within-person change across the three measurement bursts. Although possible analytically, the time span of the study (three bursts across one calendar year) does not truly provide for longitudinal study of developmental change, a process that likely requires study designs wherein the multiple bursts span across multiple decades. As those data get collected, we hope that these preliminary results serve as a promising illustration of the utility of SDE models for studying developmental and contextual change in affect dynamics, and may promote consideration of longer-term longitudinal intensive measurement burst designs to capture high-resolution, nuanced measurement of the aging process (Ram & Gerstorf, 2009).

Finally, some of the usual sampling and generalizability caveats apply. The experience sampling data used in our empirical illustration were obtained from a select sample that, while age-heterogeneous (age 18–89 years) only provides a first look at how individual differences may be structured in the larger adult population. While the unequally spaced individual affect measurements obtained during the 21-day measurement burst (with an average of 145 observations per person) were suited for (and/or required) continuous-time models, we are cautious that the measurements obtained in the event-contingent design, wherein individuals always provided reports after social interactions, may not capture the full range and time sequence of individuals’ affective experiences. In short, the design has restricted the range of BPS factors being considered here. Full representation of intraindividual affect dynamics requires more varied sampling protocols (i.e., random prompts). Notably, change in affect was, in accordance with the concept of differentiation embedded in differential equation models, facilitated by use of a truly continuous response scale (0 to 100 slider-type interface). However, a few measurement artifacts exist. Specifically, three individuals who never deviated too far from the valence = 10 ceiling appear to have extremely constrained affect dynamics. These cases may actually be better modeled using a qualitatively different SDE model, such as the reservoir model (DeBoeck & Bergeman, 2013), which is designed to model dynamics in time series exhibiting strong floor/ceiling effects.

Outlook

This study provides entry into the use of SDE models for articulating and examining individual differences in intraindividual dynamics. Following the DynAffect model (Kuppens et al., 2010b), we mapped theoretical propositions about individuals’ affect system to an OU process model and examined how individual differences in attractor point, BPS-related reactivity, and attractor strength were related to age and deployment of emotion regulation strategies. The results provide promising evidence that the multilevel SDE framework can be used to study developmental change by capturing both short-term homeostatic processes and long-term reorganization of the systems governing short-term change. Our approximation of long-term changes through examination of age-related differences may now lead inquiry with multiple time-scale experience sampling designs that allow for modeling of intraindividual change in intraindividual dynamics. We look forward to more discovery.

Funding

This work was supported by the National Institute on Health (RC1 AG035645, R01 HD076994, R24 HD041025, UL TR000127), and the Penn State Social Science Research Institute.