Burstiness and Stochasticity in the Malleability of Physical Activity

Abstract

This study examined whether patterns of self-organization in physical activity (PA) predicted long-term success in a yearlong PA intervention. Increased moderate to vigorous PA (MVPA) was targeted in insufficiently active adults (N=512) via goal setting and financial reinforcement. The degree to which inverse power law distributions, which are reflective of self-organization, summarized (a) daily MVPA and (b) time elapsed between meeting daily goals (goal attainment interresponse times) was calculated. Goal attainment interresponse times were also used to calculate burstiness, the degree to which meeting daily goals clustered in time. Inverse power laws accurately summarized interresponse times, but not daily MVPA. For participants with higher levels of MVPA early in the study, burstiness in reaching goals was associated with long-term resistance to intervention, while stochasticity in meeting goals predicted receptiveness to intervention. These results suggest that burstiness may measure self-organizing resistance to change, while PA stochasticity could be a precondition for behavioral malleability.

Insufficient physical activity (PA) is a key risk factor in the most common and costly chronic health conditions including heart disease, chronic pain, and Type 2 diabetes (Benjamin et al., 2019). The annual combined monetary costs worldwide for these chronic health conditions is in the range of trillions of dollars (Pincus & Guastello, 2012), with approximately 2% of all health care expenditures accounted for specifically by physical inactivity (Benjamin et al., 2019). Only a small percentage (<8%) of U.S. adults meet moderate-to-vigorous PA (MVPA) guidelines from the federal government (Troiano et al., 2008), and the population levels of activity over the past two decades have not increased substantially, even with significant public health investments (Brownson et. al 2005; Kruger et al., 2005; Ward et al., 2015).

Information concerning the inadequacy of PA is most often presented from an epidemiological perspective that aims to establish general thresholds where average risk is sufficiently reduced, which can then be recommended as targets for intervention. For example, the U.S. Department of Health and Human Services recommends adults engage in a minimum of 150 min of MVPA per week to reduce all-cause mortality (U.S. Department of Health and Human Services, 2018). The health assumptions underlying this approach are clear, but the behavioral assumptions are rarely discussed, perhaps because such general targets are simple, measurable, and practical for clinicians and the general public. While benchmarks like this provide useful data-driven targets, they do not address the critical questions of individuals’ resistance to change. The presence of this behavioral inertia is demonstrated by a meta-analysis showing that 33–50% of intervention participants fail to increase PA (Dishman & Buckworth, 1996) and other studies indicating similar outcomes (Marcus et al., 2000; Williams et al., 2008). Resistance on a comparable scale has been identified in other domains (Silverman et al., 1996) and for clinicians charged with promoting evidence-based guidelines (Grol, 2001). In fully examining the factors that lead to this phenomenon, it will be helpful to consider self-organization theory and human behavioral dynamics to help understand the malleability/inflexibility of certain behavioral patterns in response to a known perturbation (i.e., intervention).

Self-organization is a process within complex adaptive systems whereby interactions among system components lead to the emergence of a higher level global order, such as self-regulation and structural evolution (Kauffmann, 1995; Pincus & Metten, 2010). Pincus and Metten (2010) have proposed a general theory of biopsychosocial resilience, which is defined as a correction in behavior toward functionality. Their framework describes biological, psychological, and social systems as highly interactive network components that allow self-organizing processes to naturally arise and produce patterned outputs across these systems over time. This theory posits that biopsychosocial processes facilitate shifts toward coherence (i.e., restricted experiential range and behavioral coping) under conditions of threat and toward flexibility during times of exploration and growth. Therefore, Pincus and Metten proposed that the resilience of behavior in response to a perturbation (e.g., an intervention) may be functionally defined as meta-flexibility: the ability of a system to shift smoothly between coherence and flexibility, without getting stuck or decomposing (p. 359). Hypothetically, this functional resilience depends upon the structural resilience of one’s biopsychosocial system, defined as a network structure with optimal integrity (i.e., connectivity) and flexibility.

There are several dynamical-systems modeling strategies that measure structural integrity and flexibility within a system and can be used for testing hypotheses based on self-organization and human resilience (Pincus & Metten, 2010). Since PA is a complex behavior nested in time and within individuals’ dynamical “life space,” identifying the factors underlying resistance requires intensive longitudinal data that can reveal the potential complexity of nonlinear temporal patterns. With such data, two overlapping strategies for identifying self-organization within time series data are most appropriate: (a) the analysis of fat-tailed or inverse power law (IPL) distributions, which describe the exponential relationship between large and small bouts of activity (see Figure 1); and (b) the consideration of burstiness, the degree to which bouts of activity tend to cluster together in time (Karsai et al., 2018).

Figure 1 —

Gaussian (normal) distribution versus Pareto inverse power law distribution. The Gaussian distribution stabilizes around a standardized mean = 0 and finite variance with increasing x. The Pareto distribution approaches infinite mean and variance with increasing x and describes fluctuations in highly interactive processes unfolding over time. Pareto distributions are scale free, or fractal, and can be defined by an exponential relationship between size and frequency.

Fat-tailed IPL distributions are scale free (i.e., fractal) in nature, which makes extreme events rare, but expected, outcomes if given sufficient time. This stands in contrast to normal distributions, where the probabilities of extreme events rapidly decay toward zero. Self-organizing systems tend to produce fat-tailed outputs, which are ubiquitous in physiological and psychosocial health processes, such as movement, balance, cardiovascular health (Goldberger, 2006; Pincus & Metten, 2010), neurological activity (Dave et al., 2018), cognition (Kello et al., 2008), self-esteem (Wong et al., 2016), personality (Pincus et al., 2019), group dynamics (Pincus, 2014), and the sequential flow of discrete individual behaviors (Pincus et al., 2014). Several studies have examined IPLs and similar “fat-tailed” distributions within the context of PA and health. Paraschiv-Ionescu et al. (2013) found that fatter tails in activity bout length (i.e., relatively more periods of long activity) were associated with worse functionality in chronic pain subjects, and Bellettiere et al. (2017) observed that fatter tails in the distribution of sedentary bouts (relatively more periods of inactivity) predicted a variety of negative health indicators. Lee et al. (2019) also found that the distribution of activity and rest times were IPL distributed, but they did not see the expected relationship between IPL characteristics and bipolar disorder. Nakamura et al. (2008) observed that depressed individuals had fatter tails in IPL distributions of their resting period durations (i.e., relatively more long rest periods) compared with healthy individuals. They replicated these results experimentally with mice that had alterations in their daily circadian rhythms (Nakamura et al., 2008), suggesting effects are present across species. Within an intervention context, Zhang et al. (2018) found that fat tails in activity patterns over time were significantly associated with improved posttreatment measures of functional movement in elderly patients with chronic pain, while more conventional PA metrics (e.g., percentage of walking and sedentary time, step counts, mean cadence) showed no such association.

The IPLs and other fat-tailed distributions can also be used to describe the distribution of the times between two events of interest. A common feature associated with this context is bursty activity patterns, where events occur in rapid succession over short time periods, followed by long periods of nonoccurrence (Karsai et al., 2018). Importantly, this clustering of activity within time is how interevent IPLs emerge, meaning that bursty patterns may be considered the building blocks of fat-tailed distributions in the context of self-organization. At the level of the individual, activity becomes a self-sustaining process for the duration of the burst, rather than something that turns on/off stochastically. Bursty activity patterns have been observed in a variety of natural phenomena, including e-mail patterns (Vazquez et al., 2007), earthquakes (Corral, 2003), and gene expression (Golding et al., 2005), and impact the behavior of many complex processes, such as network dynamics and the development of coherent structures (Delvenne et al., 2015; Eckmann et al., 2004; Iribarren & Moro, 2011; Jo et al., 2014; Karsai et al., 2011; Starnini et al., 2012).

A variety of mechanisms have been proposed to account for the presence of bursty behavior patterns observed in humans. Barabási (2005) suggested that burstiness is a result of diverse, competing stimuli and incentives that drive a prioritization among tasks, as opposed to Poisson processes that are associated with the stochastic completion of tasks, or other general heuristics like “first-come-first served.” Malmgrenet et al. (2008) argued that prioritization is not necessary and that, instead, cyclic constraints in life cause burstiness; however, Jo et al. (2012) removed known cyclic patterns from data and found that burstiness remained. Gandica et al. (2016) examined the timing of Wikipedia edits and concluded that burstiness reflected the high costs of initiating an activity versus continuing it, which they indicated could be reflective of the queuing/prioritization process suggested by Barabási. While Sorribes et al. (2011) focused on fruit flies rather than humans, they experimentally demonstrated the role of task prioritization in producing bursty behavior.

While the task prioritization explanation of bursty behavior does not explicitly refer to self-organization, it is a small logical step to suggest that self-organizing habits may exist within self-organizing task networks. Through this lens, PA habits potentially exist within a relatively complex decision-making matrix involving competition or cooperation with other potential behavioral options that could either interfere with or facilitate the uptake of exercise. For example, adopting a dog could enable increases in PA (Cutt et al., 2008), while taking on a new office job could interfere with it (Blackwell & Clarke, 2016). For individuals with many interacting task priorities, one may expect higher burstiness and fatter tails in PA patterns, to the degree that a decision to exercise versus engaging in other activities represents a task queuing process. By contrast, other people may have relatively few entanglements with PA and, thus, should produce more stochastic intervals between bouts of PA and more exponential distributions. Importantly, the tangled stimuli and incentives need not be overt, like a dog or new job, but may be covert—composed of physiological states, moods, attitude, and/or interpersonal contexts. As such, they may be highly individualized and difficult to fully assess at the start of a PA intervention.

The present secondary analysis investigated whether indicators of PA self-organization were present and associated with resistance to behavior change within a PA intervention that measured daily MVPA using accelerometers with 512 participants over the course of 1 year. We first determined whether IPLs sufficiently summarized (a) distributions of daily MVPA bout minutes and (b) time intervals between days where assigned MVPA goals were met. Second, we investigated whether markers of self-organization differed by study incentive arms (i.e., contingent, immediate vs. noncontingent, delayed financial reinforcement), which tests the hypothesis that task incentives play a role in the self-organization process. Finally, we examined IPLs summarizing behavior from the beginning of the yearlong intervention and determined whether the features of these distributions predicted resistance to the sustained performance of PA later in the trial. Because this is the first time these dynamical questions have been asked, it is difficult to make predictions, so this work should be considered exploratory. This point notwithstanding, we hypothesized that more stochastic PA habits at the outset of the study would be indicative of malleability that is conducive to sustained behavior change in response to an intervention. We anticipate that the results of this exploratory work will help determine how the structural organization of PA either hinders or facilitates targeted health behavior change efforts and the degree to which task-incentive networks may be involved in this process.

Methods

The WalkIT Arizona Design

The WalkIT Arizona was a yearlong clinical trial with 512 insufficiently active adults (64.3% female, mean age = 45.5 ± 9.1 years, 18.8% Hispanic, 84% White) from Maricopa County, Arizona (Adams et al., 2019). The participants were asked to wear an ActiGraph GT9X wrist-worn accelerometer during awake hours and to sync (i.e., upload) data from their accelerometer each day to project servers. Each day, a new MVPA goal was provided via text message to the participants (e.g., “Goal for 7/1 is 30 min”), and after syncing with the data servers, the participants were informed of whether this goal was met, with select participants receiving financial reinforcement for achieving their goal. An automated, cloud-based system was designed for this study and online 24 hr per day, 365 days per year to receive and process the accelerometer data, determine whether goals were met, calculate financial rewards, transmit text message feedback to the participants, and send e-gift cards, when appropriate. A full description of the WalkIT Arizona methodology and sample characteristics is available in Adams et al. (2019), and a brief summary is provided below.

Study recruitment primarily occurred through Facebook. Eligible participants were initially enrolled in a baseline phase, during which MVPA was passively recorded using a blinded accelerometer. After approximately 10 days in the baseline, each participant was enrolled into one of two reinforcement conditions: Immediate Reinforcement, where participants earned monetary rewards immediately upon meeting a daily goal and syncing their device by noon the next day, or Delayed Reinforcement, where participants received a reward every 60 days if they wore their accelerometer for at least 10 hr on four out of the last 7 days. Similarly, the participants were also enrolled into one of two goal type conditions: Static Goals, where the daily MVPA goal was set to 30 min or Adaptive Goals, where the goal was set to equal the 60th rank-order percentile based on a moving window of MVPA minutes accrued over the previous 9 days. The study was carried out as a 2 × 2 factorial design with the participants block randomized into one of four groups defined by the reinforcement/goal combinations. Regardless of the study group, after successfully meeting a daily goal and syncing their accelerometer, the participants were provided a feedback text message with praise and their next goal (e.g., Cheers, James! Goal met! 63 min yesterday. Goal for 7/2 is 35 min).

Measures

Daily MVPA.

The number of MVPA bout minutes on a given day was calculated. Previous work (Freedson et al., 1998; Troiano et al., 2008) has outlined the specifics of defining bout minutes and establishing PA intensity.

Daily Goal Attainment Interresponse Time.

We also considered the time interval (in days) between consecutive instances in which a participant met a daily goal and synced their device by noon the next day. By definition, the minimum time interval is 1 day. This measure is analogous to interresponse time (IRT) from behavior analysis; consequently, we refer to it as goal attainment IRT.

Inverse Power Law Features.

Each participant’s IPL features were calculated for both daily MVPA and goal attainment IRT over their first 100 days of study enrollment, excluding the baseline. This allowed us to quantify aspects of participant behavior at the beginning of their enrollment and to determine if features of this behavior were predictive of continued success in meeting intervention goals. This calculation was performed by creating a frequency table of each measure and fitting the following equation to this data:

$$\log f=a+b\log x,$$

where (x, f) pairs represent the frequency, f, of x daily bout minutes or goal attainment IRTs recorded in the first 100 days. a and b are parameters that were estimated. Exponentiating both sides of this equation yields the IPL:

$$f=C{x}^b,$$

where C is the constant e^a. b is known as the scaling exponent, which quantifies the “fatness” of the IPL tail, while C is associated with the relative frequency of smaller values in the distribution. The adjusted coefficient of determination r² for the fit is a measure of the degree to which the measures conformed to an IPL, with a penalty for larger numbers of predictors. See Figure 2 for an illustration of the procedures for calculating IPL features.

Figure 2 —

Illustration of the IPL fitting procedure for a representative individual. Above the figure is a frequency table for goal attainment IRT that will serve as the data for the IPL fit. (a) A linear function is fit to the log of the data and (b) these fitted parameters specify an IPL relationship when the raw data are plotted. IRT = interresponse time; IPL = inverse power law.

Goal Attainment.

The primary outcome for regression analyses was long-term intervention performance, operationalized as the number of times after the 100th day in the intervention on which the daily goal was met and the accelerometer was synced by noon by the next day. This was denoted as g_>100. By focusing on intervention performance toward the end of the yearlong intervention, we aimed to assess behavior most consistent with MVPA maintenance. Statistical analyses involving g_>100 were adjusted for the number of goals met over the first 100 days in the study, which was denoted as g_≥100.

Burstiness.

The burstiness of goal attainment IRT was calculated (for clarity, we note that burstiness cannot be calculated for daily MVPA since there is no “event” threshold). Typically, burstiness is measured by the simple metric $\frac{\sigma -\mu }{\sigma +\mu }=\frac{r-1}{r+1}$, where μ and σ are the mean and SD of interevent times and $r=\frac{\sigma }{\mu }$. However, this measure was developed for summaries of an infinite or very large number of events and, as such, it fails to consider the effects of a finite number of events. Kim and Jo (2016) addressed this shortcoming and proposed the following alternative measure of burstiness:

$$B=\frac{(n-2)(r\sqrt{n+1}-[1-n\overline{y}]\sqrt{n-1})}{r(n\sqrt{n+1}-2[n-1])+(n-2\sqrt{n+1})\sqrt{n-1}(1-n\overline{y})},$$

where n is the number of unique goal attainment IRTs and $\overline{y}=\frac{y_m}{y_d}$, with y_m equal to the minimum goal attainment IRT and y_d equal to the duration of the time series. Burstiness values are expressed on a continuum, ranging from −1 (regularly spaced events) to 0 (stochastic events) to 1 (bursty events). Burstiness was calculated separately for goal attainment IRTs in the first 100 days of the study and after the 100th day of enrollment, denoted as B_≤100 and B_>100, respectively. Examples of bursty and nonbursty patterns are shown in Figure 3.

Figure 3 —

Representative cases of low burstiness (B = −0.11) and high burstiness (B = 0.82) in participants’ goal attainment interresponse time. Each vertical line represents a day on which participants met their daily moderate to vigorous physical activity goal and synced their device by noon the following day.

Covariates.

The following covariates were included in the regression analyses outlined below. The design variables were census block walkability (higher vs. lower) and census block socioeconomic status (higher vs. lower). The demographic characteristics were ethnicity (White vs. any non-White ethnicity), marital status (married vs. not married [e.g., single, separated, divorced]), educational attainment (college graduate and higher vs. other), gender, number of children living in the household, and body mass index (in kilograms per meters squared).

Sample Description

The analytical sample criteria were based on valid wear days, defined as days on which at least 6 hr of accelerometer wear time was recorded or the daily PA goal was met. Days not meeting these criteria were eliminated from all analyses. Participants who did not have at least 50 valid days both before and after their 100th day in the intervention were eliminated from the analyses, leading to a sample size of 415 individuals. By definition, a participant who met g goals had g – 1 goal attainment IRTs; for small values of g, the IPLs fit to the small number of goal attainment IRTs may not have been robust summaries of the data. To account for this scenario, an additional inclusion criterion was used for goal attainment IRTs, such that participants were required to have a minimum of five unique goal attainment IRT measures and to have met their goal on at least 10% (i.e., 10) of days before their 100th day of enrollment. This eliminated an additional 94 participants from the study, which led to a total sample size of 321. The demographics for this sample group are provided in Table 1.

Table 1.

Descriptive Statistics for Demographics of the Sample Defined by the Inclusion Criteria for Goal Attainment Interresponse Times (N = 321)

Quantitative variables	M	SD
Age (years)	45.9	9.1
Body mass index	33.0	7.0
Number of children under 18 years	0.97	1.2
Categorical variables	Frequency	%
Ethnicity—White	301	72.5
Marital status—married	284	68.4
Education—college graduate+	282	68.0
Census block walkability—high	223	53.7
Census block socioeconomic status—high	222	53.5

Statistical Analyses

Summary Statistics.

The appropriateness of using an IPL to summarize participants’ (a) daily minutes of MVPA bouts and (b) goal attainment IRT was assessed by examining the distributions of adjusted r² values for each of these variables over all participants meeting the inclusion criteria. Descriptive statistics for all independent/dependent variables were calculated. Correlations between burstiness and goal attainment in the first 100 days of enrollment (B_≤100 and g_≤100) and after the 100th day (g_>100 and B_>100) were also calculated, as was the correlation in burstiness during these two time periods (B_≤100 and B_>100).

Burstiness and Reward Timing.

To test the hypothesis that burstiness in the goal attainment IRT was associated with immediate rewards, which can be viewed as a process that induces task prioritization, the following regression model was fit to the data:

$$B_{\text{ij}}=\alpha +{\beta }_1{t}_j+{\beta }_2{r}_i+\sum _{m=1}^M{\gamma }_m{z}_{\text{mi}}+u_i,$$

where b_ij is the burstiness for individual i recorded at measurement occasion j = 1 or 2, corresponding to B_≤100 and B_>100. The term t_j represents the effect of the measurement occasion (≤100 vs. >100), r_i is the reward timing (immediate vs. delayed) for individual i, z_mi are the remaining covariates for individual i, and u_i is an individual-level intercept. r_i could have been incorporated into the z_mi summation term, but we chose to show it on its own, since this is our main variable of interest in assessing task prioritization. The individual level intercept (u_i) is often accounted for by a random intercept, mixed-effects linear model approach, but because nearly all covariates are time invariant, this approach makes the strong assumption that covariates are uncorrelated with all unobserved variables. To avoid this assumption, we followed the example of Plümper and Troeger (2007) and decomposed the random effects into explained and unexplained sources before running a fixed-effects model. To determine the effect size, Cohen’s d was calculated by dividing the difference in the estimated marginal means for the immediate and delayed groups by the pooled SD.

Burstiness and Success Meeting MVPA Goals.

To determine the factors associated with sustained success in meeting goals throughout enrollment in the study, g_>100 was regressed on B_≤100, B_>100, and g_≤100 in both simple linear regression and multiple regression models. An additional model included a B_≤100 × g_≤100 interaction term to assess if burstiness in goal attainment early in the trial had a differential effect based on participants’ early success in meeting daily MVPA goals. To further explore the B_≤100 × g_≤100 interaction, the data were stratified into four groups defined by the quartiles of g_≤100, and a distinct multiple regression model was calculated for each quartile group. Demographic covariates were included in all regression models. In addition, since previous research with IPLs has noted the effects of the fit index r² and shape parameter b on outcomes, all nonstratified regression analyses were rerun with these values included. Table 2 illustrates the components of the regression models fit to the data.

Table 2.

Components of Regression Approaches for (a) Burstiness and Reward Timing and (b) Burstiness and Success Meeting MVPA Goals Analysis

Burstiness and reward timing	Burstiness and success meeting MVPA goals
Dependent variables: Burstiness (B≤100 and B>100)	Dependent variable: g>100
Independent variables: Reward timing (immediate vs. delayed) and measurement interval (>100 days vs. ≤100 days)	Independent variables: g≥100, B≤100, and B>100
Covariates: Goal type, census block walkability, census block SES, ethnicity, marital status, education, gender, number of children living in the household, and BMI	Covariates: Reward timing, goal type, census block walkability, census block SES, ethnicity, marital status, education, gender, number of children living in the household, and BMI
Methodology: B≤100, B>100 clustered by individual in a modified fixed-effects model (see “Burstiness and Reward Timing” section of “Statistical Analyses” section for details)	Methodology: (a) Bivariate regression of dependent variable on each independent variable; (b) multiple regression; and (c) multiple regression with g≤100 × B≤100 interaction

Note. BMI = body mass index; SES = socioeconomic status; MVPA = moderate to vigorous physical activity; B_≤100=burstiness during first 100 days of enrollment; B>100 = burstiness during remainder of enrollment; g_≤100 = number of goals met during the first 100 days of enrollment.

Nonlinear Effects and Sensitivity Analysis.

The regression models described above both assume that all effects are linear and additive, which is likely to be violated. A supplementary analysis with quantile regression (results not shown) yielded qualitatively similar results to those outlined below. Furthermore, while a preliminary analysis indicated that 100 days was a reasonable selection as a stratification point between predictor and outcome measures, this choice was somewhat arbitrary. To test the sensitivity of the findings to this parameter, we performed a sensitivity analysis that is fully detailed in the Appendix.

Results

Summary Statistics

Table 3 provides descriptive statistics, and Figure 4 illustrates the distribution of adjusted r² values for the IPL fit over all participants. The adjusted r² values for daily bouts of MVPA were relatively small, indicating that IPLs do not appear to be an appropriate summarization of this feature. Goal attainment IRTs were much better summarized by IPLs, so all subsequent analyses solely focused on this measure. The burstiness of participants’ goal attainment IRTs in both the first 100 days (B_>100) and beyond the first 100 days (B_>100) is also summarized in Table 3 and Figure 4. The values ranged from approximately 0 to 1, meaning that the participants’ occurrences of meeting goals roughly varied from stochastic through bursty, but did not extend to regular patterns (which would have been indicated by B approaching −1).

Table 3.

Descriptive Statistics for Regression Variables

Variable	M (SD)	Minimum/maximum	Median	IQR
IPL adjusted r2 (daily MVPA bout minutes)	0.30 (.25)	−0.04/.91	0.27	0.46
IPL adjusted r2 (goal attainment IRTs)	0.69 (.24)	−0.20/.99	0.75	0.27
IPL scaling exponent b (goal attainment IRTs)	−1.06 (0.51)	−2.63/0.05	−1.02	0.68
	−1.40 (0.42)	−2.63/−0.45	−1.36	0.54
g ≤100	35.7 (15.4)	10/77	35	21
g >100	66.1 (41.2)	0/254	63	57
B ≤100	0.32 (0.17)	−0.11/0.82	0.32	0.21
B >100	0.28 (0.19)	−0.41/1.00	0.26	0.23

Note. The inclusion criteria for goal attainment interresponse times were used to define the sample for the calculation of all statistics except adjusted r² for daily bout minutes. Bold values are scaling exponent summary statistics for only those individuals with an adjusted r² >.8 (n = 125), which were shown since the scaling exponent is valid/interesting only when distributions are reasonably approximated by IPLs. IPL = inverse power law; MVPA = moderate to vigorous physical activity; IQR = interquartile range; g_≤100 = number of goals met during the first 100 days of enrollment; B_≤100 = burstiness during first 100 days of enrollment; g_>100 = number of goals during remainder of enrollment; B_>100 = burstiness during remainder of enrollment.

Figure 4 —

The distribution of (a) adjusted r² values for inverse power law fits of daily minutes of moderate to vigorous physical activity bouts and (b) goal attainment interresponse times. The distribution of (c) B_≤100 and (d) B_>100 values for goals. B_≤100 = burstiness during first 100 days of enrollment; B_>100 = burstiness during remainder of enrollment.

Burstiness and Reward Timing

Table 4 illustrates the results of the hierarchical linear model built to test the hypothesis that burstiness is associated with task prioritization and therefore would be higher in the immediate (vs. delayed) reinforcement arm of the trial. The results confirmed this hypothesis and indicated that burstiness was significantly higher for immediate versus delayed reinforcement. The estimated marginal means were then calculated for these two arms, which were 0.339 for the immediate arm and 0.262 for the delayed arm. Cohen’s d for this difference was 0.43, which reflects a small to moderate effect. The model also indicated that burstiness was significantly lower for individuals from low socioeconomic status block groups, males, and when calculated after the 100th day of enrollment as opposed to before.

Table 4.

Results of Modified Fixed-Effects Model to Assess the Effect of Reward Timing and Covariates on the Burstiness of Goal Attainment Interresponse Times

Variable	β	SE	p
Reward timing (immediate vs. delayed)	0.077	0.009	<.001
Goal type (static vs. adaptive)	0.014	0.009	.151
Measurement interval (>100 days vs. ≤100 days)	−0.048	0.009	<.001
Neighborhood walkability (low vs. high)	0.0003	0.009	.973
Neighborhood socioeconomic status (low vs. high)	−0.020	0.010	.042
Agea	−0.0002	0.001	.729
Sex (male vs. female)	−0.048	0.010	<.001
Marital status (not married vs. married)	−0.004	0.011	.730
Ethnicity (White vs. other)	−0.015	0.011	.146
Education (college graduate + vs. < college graduate)	−0.018	0.010	.070
Number of children in homea	−0.007	0.004	.079
BMIa	0.0004	0.001	.591

Note. Bold values indicate p < .05. BMI = body mass index.

^aContinuous variable. The remaining variables are noncontinuous (i.e., qualitative) variables in which the levels follow each variable with the referent level italicized.

Burstiness and Success Meeting MVPA Goals

The correlation between B_≤100 and B_>100 was .25, indicating a considerable variation in burstiness between these two intervals. The relationships between burstiness and goals met during these two intervals were not consistent, with a correlation between B_≤100 and g_≤100 of .43 and a correlation between B_>100 and g_>100 of .09, suggesting that burstiness and concurrent success in meeting MVPA goals are more closely linked earlier in the trial.

Table 5 illustrates the results of the three regression models that examined the effect of IPL features on g_>100. In bivariate models, g_≤100 and B_≤100 were both significantly associated with higher values of g_>100, but B_>100was not. After controlling for each other, the relationship between g_≤100 and g_>100 remained similar, but the positive association between B_≤100 and g_>100 reversed, with greater burstiness values now associated with smaller g_>100, at a level approaching significance.

Table 5.

Regression Results With Goals Met After 100th Day of Enrollment (g_>100) as the Dependent Variable

	Bivariate models			Multiple-regression model			Interaction model
Regression term	β (SE)	$\widehat{\beta }$	p	β (SE)	$\widehat{\beta }$	p	β (SE)	$\widehat{\beta }$	p
g ≤100	1.65 (0.12)	0.62	<.001	1.71 (0.13)	0.65	<.001	2.16 (0.29)	0.82	<.001
B ≤100	40.02 (13.81)	0.16	.004	−21.11 (12.11)	−0.09	.08	12.88 (23.17)	0.05	.58
B >100	4.97 (12.61)	0.02	.88	1.60 (10.19)	0.007	.87	2.44 (10.17)	0.01	.81
g≤100 × B≤100	—	—	—	—	—	—	−1.10 (0.64)	−0.41	.09

Note. Bold values indicate p < .05. g_≤100 = number of goals met during the first 100 days of enrollment; g_≤100 = number of goals during remainder of enrollment; B_≤100 = burstiness during first 100 days of enrollment; B>100 = burstiness during the remainder of enrollment; β (SE) = regression coefficient (SE); $\widehat{\beta }$ = standardized regression coefficient.

When an interaction term between g_≤100 and B_≤100 was included, g_≤100 had a larger effect on g_>100, but B_≤100 lost its trend toward significance. The interaction term neared significance such that high burstiness before Day 100 was more strongly associated with fewer goals met later in the intervention for individuals meeting a relatively large number of goals before Day 100. To further illustrate this point, multiple regression models, stratified by quartiles of g_≤100, were fit to the data. Figure 5 illustrates the B_≤100 versus g_>100 relationship for the different quartiles of g_≤100, with all other covariates set equal to their means. A statistically significant association between burstiness and goals attained after Day 100 was present only for individuals in the fourth quartile of g_≤100, that is, those who were most successful in meeting goals over the first 100 days of the intervention. The B_≤100 coefficient for this group was −101.7.

Figure 5 —

Results of quartile regression: B_≤100 versus g_>100 stratified by g_≤100 quartiles. g_≤100 = number of goals met during the first 100 days of enrollment; B_≤100 = burstiness during first 100 days of enrollment; g_>100 = number of goals during remainder of enrollment.

When introducing r² and b into these analyses, there was a statistically significant relationship between each of these terms and g_>100 (positive association for r², negative for b). But this was not the case for the multiple regression and interaction models, where their inclusion minimally changed the results. Therefore, the results for the most parsimonious IPL were presented in this section.

Discussion

Our central finding was that stochasticity in meeting goals at the outset of a PA intervention was associated with greater malleability of behavior, while burstiness was associated with resistance to change during the trial. Quartile regression indicated that this phenomenon occurred specifically for participants who were successful in meeting goals early in the intervention. In other words, greater goal attainment early in the intervention was more strongly associated with later success when the patterns of that early goal attainment were relatively stochastic (i.e., less bursty). When early goal attainment was highly clustered in time, goal attainment after 100 days was relatively poor. For individuals meeting fewer goals early in the intervention, burstiness did not predict later goal attainment, which may suggest that, for individuals who are less responsive to the intervention, meeting goals in any capacity at the beginning of treatment, regardless of the temporal pattern, is predictive of later success. That is, low baseline MVPA was the primary obstacle to sustained intervention success, with established habits (as revealed by greater burstiness) providing headwinds for only those subjects who best responded to treatment.

For practitioners, the results may be understood most intuitively within the context of “breaking down” prior habits. As an extreme example, one may consider the strategies of a military drill sergeant who prescribes PA at random intervals during basic training. While this level of “breaking down” is certainly overkill for standard studies, mHealth interventions may be more effective if PA prompts are more stochastic early in a behavioral intervention, particularly for those who show high levels of engagement. More broadly, stochasticity may be a sign of openness or readiness to change PA patterns and has been identified in other work as a critical process for PA adoption (Iso-Ahola, 2011). Inducing greater stochasticity may provide a “softening up” in preparation for a lifestyle transformation in one’s established PA habits.

Our results also revealed that immediate, PA-contingent rewards versus delayed, noncontingent rewards were associated with higher levels of burstiness in PA goal attainment, which is consistent with previous assertions of burstiness as a consequence of task prioritization. By definition, the burstiness engendered by immediate rewards comes at the expense of stochasticity. If stochasticity is indeed associated with breaking down old habits, then immediate rewards may inhibit this effect by freezing counterproductive habits in place, unless rewards are specifically aimed at generating stochastic PA. Behavioral literature suggests that discontinuing reinforcement increases variability in behavior (Skinner, 1953); therefore, an effective intervention strategy, particularly for people successfully meeting PA goals, may be to initially provide incentives, but then to eliminate them to allow goal achievement to become more stochastic. If these hypotheses prove correct, burstiness may be used as a relatively simple dynamical index that can be used to tailor feedback for individuals attempting to change their behavior. For example, participants who enter an intervention with relatively bursty PA patterns may be informed that rewards will be temporarily delayed while they are encouraged to explore a greater variety of opportunities for PA within a broader range of life contexts—for example, walking or stair climbing during the work day rather than exclusively after work or on weekends.

There are several implications for measurement of self-organization in daily PA that arise from the current results. IPLs were observed for some participants with respect to daily MVPA bouts, with the largest fits having adjusted r² values near .75. But the mode was zero, indicating that IPLs may not be the best model to use in general for PA magnitude. On the other hand, IPLs were far more common for goal attainment IRTs, clustering at the higher range of adjusted r² with a mode near .9. Fit index, r², and shape parameter, b, were significantly related to g_>100 in bivariate regression models, but not within the multiple regression and interaction models. As such, the most parsimonious model included burstiness of goals met within the first 100 days of the intervention. A full spectrum of burstiness was observed within this sample, ranging from stochastic (at the low end) to highly clustered at the high end, within a Gaussian-type distribution with a mode near 0.3. This variance in burstiness may have played some role in its power to predict malleability above and beyond the strong effect of g_>100. Yet many other factors could also have lent power to burstiness, including its embeddedness in time compared with IPLs, or factors unique to this sample or to the selection of daily PA as an outcome measure. These results stand in minor contrast to other studies involving different types of PA in which direct IPL features were used as predictors and may serve to integrate the PA literature with the literature on non-PA behaviors (e.g., flows of work tasks). Future studies may wish to continue to examine both IPLs and burstiness as indicators of the self-organization of habits.

This study had several limitations. It was designed to be exploratory due to the novelty of the design within the context of a PA intervention, so replication and extension are needed before applying the current results with confidence to future interventions. Over the course of a 1-year trial, most participants did not wear the accelerometer every day, so the data used in these analyses likely exclude MVPA that was performed, but not measured. The regression effect sizes were small, and notable inferential statistical tests tended toward, but did not meet, significance at α = .05. Without additional analyses that suggest robust IPLs in both temporal and bout-size dynamics, there is also the potential for the overgeneralization of results as self-organization. This is especially true since our sample size and measurement periods were relatively small and concerns over the accuracy of the log–log fitting routine we used have been raised (Clauset et al., 2009). Similarly, less constrained regression approaches that model nonlinear associations among covariates may affect findings and should be explored in future work. In addition, the notion of incentives and task prioritization were not tested directly and, so, should be interpreted with caution.

Most broadly, the current results support the importance of ongoing research that moves beyond the identification of static mean targets. For decades, PA research has shown us where to aim intervention efforts (e.g., 150 min per week). But only recently have we gained access to mobile data that can show us how best to get there. Ideally, this data will allow us to better tailor treatments to individuals and their life situations as they unfold in time.

Appendix: Sensitivity Analysis

A sensitivity analysis was performed to test the robustness of findings to changes in the stratification point between the predictor and outcome measures, which was set to 100 days in the analyses within this manuscript. This was performed by recalculating the predictor and outcome variables for stratification points ranging from 50 to 150 days in 10-day intervals (i.e., 50, 60, … , 140, 150 days) and refitting the multiple regression models described in the “Burstiness and Success Meeting MVPA Goals” section of the “Statistical Analyses” section for each scenario. Figure A1 shows the results of these analyses. Regression coefficients are qualitatively similar for delineation points ranging approximately from 80 to 140 days. With the exception of a small uptick at 90 days, the p values were also qualitatively similar for delineation points ranging from 80 to 140 days. These results indicate that the negative association between burstiness and g_>100 found when using a delineation point of 100 days (summarized in Table 5) is a robust finding and qualitatively similar results would be found for a large range of delineation points. They also indicate that the interaction results approaching significance in Table 5 can be assumed to reflect actual trends.

Figure A1 —

Results of regression analyses when considering alternate delineation points as the boundary between predictor and outcome variables. (a) The regression coefficient for B_≤100 in the multiple-regression model (M–B), B_≤100 in the interaction model (I–B), and g_≤100 × B_≤100 interaction coefficient (I–I) for various stratification points. (b) The p value for these terms has dashed lines at α = .05 and .1. g_≤100 = number of goals met during the first 100 days of enrollment; B_≤100 = burstiness during first 100 days of enrollment