# Exploring boundary conditions of sustained physical activity: implications for theory and practice.

Dario Baretta, Guillaume Chevance, Victoria Lawhun Costello, Shadia Mansour-Assi, David Wing, Eric Hekler, Jennifer Inauen, Job Godino, Claudio Nigg

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Dario Baretta, Guillaume Chevance, Victoria Lawhun Costello, Shadia Mansour-Assi, David Wing, et al.. Exploring boundary conditions of sustained physical activity: implications for theory and practice.. 2025. hal-05033494

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# Exploring boundary conditions of sustained physical activity: implications for theory and practice.

Baretta, D.¹, Chevance, G.², ³, Lawhun Costello, V.⁴, Mansour-Assi, S.J.⁴, Wing, D.⁴, Hekler, E.⁴, Inauen, J.¹, Godino, G.⁴, ⁵ & Nigg, C.R.⁶

¹Institute of Psychology, University of Bern, Switzerland

²Univ Rennes, Inserm, EHESP, Irset, Rennes, France

³Barcelona Institute for Global Health, Barcelona, Spain

⁴Center for Wireless and Population Health Systems, UC San Diego, La Jolla, CA, USA

⁵Laura Rodriguez Research Institute, Family Health Centers of San Diego, San Diego, CA, USA

⁶Institute of Sport Science, University of Bern, Switzerland

**Authors contributions:** DB: Conceptualization, Formal analysis, Methodology, Visualization, Writing – original draft, Writing – review & editing. GC: Conceptualization, Methodology, Writing – review & editing. VLC: Data curation, Project administration, Resources, Writing – review & editing. SJMA: Data curation, Project administration, Resources, Writing – review & editing. DW: Data curation, Project administration, Resources, Writing – review & editing. EH: Conceptualization, Writing – review & editing. JI: Conceptualization, Writing – review & editing. JG: Funding acquisition, Data curation, Project administration, Resources, Writing – review & editing. CRN: Conceptualization, Writing – review & editing.

**Study preregistration:** ClinicalTrials.gov Identifier: NCT03907462. **Data availability:** Core R-scripts and data are available online at https://osf.io/6kjwg/. **Conflict of interest disclosure:** The authors have no conflicts of interest to declare.

**Correspondence:** Correspondence should be addressed to dario.baretta@unibe.ch. Institute of Psychology, University of Bern, Fabrikstrasse 8.
Abstract

**Objective:** Sustained physical activity is crucial for health and is often referred to as 'physical activity maintenance.' However, this concept lacks consensus, possibly due limited study of the 'boundary conditions' under which this phenomenon occurs. This gap limits both theoretical development and interventional applications. This study examined the boundary conditions of two operational definitions of physical activity maintenance (*activity threshold* and *timescale*) and how their formulation impacts the detection of sustained physical activity phenomena. **Methods:** We analyzed 352 time series (length=182 days) of moderate-to-vigorous physical activity (MVPA) collected daily with Fitbit from participants in a weight loss intervention. Two activity thresholds were defined based on participants' baseline intentions and physical activity levels: a) 150 minutes of MVPA/week for participants below this threshold at baseline (subsample BL-Inactive; n=101), and b) current level of MVPA/week for those exceeding 150 minutes of MVPA/week at baseline (subsample BL-Active; n=251). Regression trees were used to examine changes in physical activity trajectories using different boundary definitions of timescales (7, 14, 28, and 56 days). **Results:** At finer-grained timescales (7 days), physical activity exhibited repeated fluctuations, with sustained activity patterns in 27% of BL-Inactive and 5% of BL-Active participants. At coarser-grained timescales (56 days), activity trajectories were more stable, with sustained activity percentages increasing to 71% and 26%. **Conclusions:** Varying boundary conditions reveal different ways of conceptualizing sustained physical activity phenomena, leading to plausibly distinct conclusions about behavioral maintenance. These results highlight the importance of carefully studying boundary conditions as critical for robust theory and practice development.

**Keywords:** Physical activity maintenance, moderate-to-vigorous physical activity, boundary conditions, phenomena-driven research, temporal trajectories.
# 1 Introduction

Regular physical activity is associated with reduced risks of mortality caused by non-communicable diseases (Rhodes et al., 2017; Warburton & Bredin, 2017). Many of the benefits and long-term health effects are achieved when physical activity is consistently performed over time (i.e., years) (Reiner et al., 2013; Shortreed et al., 2013), which is therefore a major goal for physical activity interventions (Murray et al., 2017). The broad concept of long-term and sustained physical activity is known in the literature as “physical activity maintenance” (Nigg et al., 2008).

Despite an intuitive understanding of what maintenance means, this research topic is characterized by conceptual arguments and confusion, and discordant operationalization resulting from a wide array of experiences, ideas, conventions, and notions that have formed over time within the field of health psychology (Dunton et al., 2022; Kahlert, 2015; Rhodes & Sui, 2021). This lack of established consensus hampers comparisons across studies, and most concerningly, limits our understanding of the mechanisms underlying long-term, sustained physical activity, as well as how to support it. As outlined in the following sections, we argue that this confusion stems from two interconnected factors. First, a lack of phenomena-driven research and insufficient attention to careful study and consensus development around the boundary conditions of this concept, that is, the conditions defining when and how the phenomenon occurs. Second, the historical absence of epistemic tools and methods (e.g., intensive longitudinal methods) needed to properly ground it in an observable form, which has hindered making definitive claims about its nature. This paper aims to address this gap by providing the field of health psychology with a foundation for a principled, phenomena-driven approach to examining the impact of boundary conditions on the demarcation of the phenomenon of sustained physical activity<fnref n="1" />.

<footnote n="1">¹ When it comes to the study aim, we intentionally avoided using the term 'physical activity maintenance,' as it would require an agreed-upon definition, which, as we have explained, does not currently exist. Instead, we opted for a simpler and more</footnote>
1 *The lack of phenomena-driven research to understand physical activity.*

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The concept of physical activity maintenance represents a putative phenomenon that is acknowledged and discussed by various theories; however, its boundaries are vague and its definitions seldom clearly articulated (Rhodes & Sui, 2021). Phenomena are empirical regularities – relatively stable, recurrent, general features of the world – that are detected from the data and, once firmly identified, serve as the focal point for theoretical explanations, which aim at describing how they came about (see Figure 1; Borsboom et al., 2021). Therefore, the systematic exploration of data represents the main task in the process leading to the detection of empirical phenomena which, in turn, constrain the space of possible theories. Despite the critical scientific value of this bottom-up (data-to-theory) research approach (Haig, 2013), research on physical activity maintenance has been predominantly characterized by the top-down (theory-to-data) approach, with theoretical and conceptual debates taking precedence, to the detriment of phenomena-driven research (Dunton et al., 2022; Kwasnicka et al., 2016; Rhodes & Sui, 2021). A key aspect of the bottom-up research approach is that it allows for the study of the boundary conditions of a given phenomenon. Boundary conditions are essential for establishing stable descriptions of a system's behaviors and generating descriptions of novel phenomena (Bursten, 2021), ultimately playing a critical role in the development and testing of scientific theories (Busse et al., 2017; Whetten, 1989). In the context of physical activity maintenance, boundary conditions may define the values of certain key operational definitions that are central to the maintenance debate such as the timescale of change processes, physical activity thresholds, and the optimal period of observation (Dunton et al., 2022; Kahlert, 2015; Spruijtmetz et al., 2015). In this study, we examined how different formulations of the boundary conditions of timescale of change and activity threshold, among other possible factors, impact our ability to observe,

<footnote>straightforward concept—sustained physical activity (i.e., whether a person meets a given physical activity threshold)—
which captures a major aspect of physical activity maintenance (Dunton et al., 2022).</footnote>
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describe, and understand the phenomenon of sustained physical activity. The timescale of change refers to the minimum period (e.g., number of days) required to detect changes in physical activity, while the activity threshold denotes the target level of behavior change (Dunton et al., 2022; Hekler et al., 2016).

***The role of intensive longitudinal data in exploring temporal dynamics of physical activity.***

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When adopting a bottom-up approach, access to high-quality data is essential for detecting empirical phenomena (Eronen & Bringmann, 2021). In the case of physical activity behavior, data often take the form of observations of specific physical activity signals, such as those gathered via accelerometry. These signals are collected and then processed with algorithms to derive estimates of step count or time spent in moderate-to-vigorous physical activity (MVPA) (Burchartz et al., 2020). To examine how behavior changes over time, a critical factor defining data quality is the sampling frequency (Chevance, Perski, et al., 2021). High-resolution behavioral assessments allow the collection of intensive longitudinal data (i.e., time series) which are essential to explore the temporal dynamics and trajectories of physical activity behavior and its maintenance (Dunton et al., 2021).

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In contrast, past physical activity research has predominantly relied on methods (e.g., study designs, assessment tools) that offer only low-resolution behavioral assessments, such as panel studies, leading to unclear boundaries and an under-defined understanding of physical activity maintenance. For instance, Murray and colleagues (2017) found that the existing evidence on physical activity maintenance comes from studies with only a few behavioral assessments. A typical example is represented by panel studies with 6-month follow-ups, where physical activity is assessed using two main approaches. One method involves questionnaires that ask participants to self-report an overall estimate of their activity over a given period (e.g., the past few weeks or months), resulting in an average that is prone to recall bias. The other approach uses activity trackers worn for limited time windows (e.g., 7-day assessment bouts), providing high-resolution data but capturing only a relatively brief snapshot of physical activity over time.

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Of concern, both these approaches misses entire assessment periods and important behavioral patterns
(e.g., sudden changes) that characterize physical activity time series (Chevance, Baretta, et al., 2021).

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***Timescales and thresholds of physical activity behavior.***

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Recent research on physical activity has revealed its highly volatile and fluctuating nature (Shang et al., 2018). This behavior is characterized by intra-individual variability not only over days (Baretta et al., 2023; Chevance, Baretta, et al., 2021) but also within the same day (Nigg et al., 2022; ten Broeke et al., 2020). An effective approach to explore such non-linear patterns and describe how physical activity changes over time is to analyze the trajectory – or temporal trend – underlying the time series data during a relevant period of time (e.g., Baretta et al., 2023; Chevance, Baretta, Heino, et al., 2021).

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When running this type of analysis, determining the model parameters that define the temporal
processes (e.g., changes in mean or trend) and their timescale holds substantial conceptual implications
for identifying empirical phenomena. By varying these model parameters, it is possible to examine the
processes of change at various timescales (e.g., weekly, monthly, etc.), each potentially yielding different
temporal patterns. For example, it is possible to fit models with a non-linear temporal trend that closely
mimics the observed data or, conversely, models that filter out most of the variance, providing a static
summary of the time series (e.g., mean, median, or linear trend). Between these opposite modeling cases,
there exist an array of plausible patterns of change to be examined, each unfolding at different timescales
(Spruijt-Metz et al., 2015). Dunton and colleagues (2022) noted that prior research on physical activity
maintenance has defined the timescale of change for identifying maintenance failures in various ways,
from a single session to one or a few consecutive weeks, or even up to three months. These operational
definitions of the timescale of change represent key boundary conditions when investigating temporal
phenomena, like maintenance. However, despite its theoretical importance, a nuanced understanding of
physical activity change processes across different timescales remains limited (Dunton et al., 2022).

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A significant advantage of analyzing the physical activity trajectories is to compare them with specific thresholds which represent the amount of physical activity that is the target of behavior change. They might correspond to a 'clinical' standard for reducing the risk of non-communicable diseases (e.g., 150 minutes of MVPA/week) (Ding et al., 2020), an adaptive goal that changes alongside individuals' progress during the course of an intervention (Adams et al., 2013), or a specific physical activity level to be restored after the occurrence of a disruptive event (Baretta et al., 2023). Comparing physical activity trajectories with specific thresholds is crucial for understanding change processes and exploring the existence and timescale of sustained physical activity patterns, whereby 'sustained' refers to being active according to the threshold itself. Defining such thresholds thus implies setting a further key boundary condition to study the phenomenon of sustained physical activity.

## Aim of the study

In this paper, we adopted a phenomena-driven research approach to examine how different formulations of two boundary conditions—timescale of change and activity threshold—impact the observability, description, and understandability of patterns of sustained physical activity in time series data. Specifically, we systematically varied the modeling parameters to define the timescale of change in physical activity (i.e., 7, 14, 28, and 56 days) and considered different thresholds based on participants' intentions (i.e., 150 minutes or baseline level of MVPA per week). The aim of this study is intended as 'descriptive', which is particularly suited for phenomena-driven research. Unlike tasks such as prediction or explanation, a descriptive task minimally relies on underlying theory and does not involve formal hypothesis testing (Shmueli, 2010). Instead, it aims at summarizing and representing the structure of the data and revealing underlying patterns, thereby facilitating the detection of phenomena. This empirical exploration aims to provide a more precise and clearly defined understanding of the phenomenon of sustained physical activity and its boundary conditions.
# 1 Methods

# 2 Study design

We used data from the SMART 2.0 study, a 24-month (96 weeks), three-arm, parallel-group randomized controlled trial targeting weight loss among young adults with overweight or obesity (Mansour-Assi et al., 2022; ClinicalTrials.gov Identifier: NCT03907462). The trial was conducted between April 2019 and February 2024, in San Diego, USA. All relevant ethical regulations have been followed and the ethical committee of the University of California, San Diego approved the study protocol.

After providing informed consent, the participants were stratified by sex and university/college affiliation and were randomized within each stratum at an 1:1:1 ratio to one of three groups: 1) treatment one, 2) treatment two, or 3) a control group. Group assignment remained unknown for the present analyses. Participants in each group received a consumer-level wearable activity tracker and connected scale from Fitbit, which includes access to the Fitbit smartphone and web-based application ecosystem. Participants in the two treatment conditions were given the goal to achieve 225 min/week of MVPA. Further details on the intervention content are thoroughly described in the trial protocol (Mansour-Assi et al., 2022). For the current study, we examined physical activity trajectories and their relationship with target thresholds considering the first 182 days (i.e., 6 months) of data. We chose this length because it reflects participants' intentions on being active according to a specific goal as assessed at baseline (see next section).

# 18 Participants

Participants (N = 638) were young adults (aged 18–35 years) recruited from three universities and five community colleges in San Diego, USA. The inclusion criteria include, 1) overweight or obese (25 ≤ BMI < 40 kg/m2); 2) available for a 24-month intervention, 3) affiliated with one of the target universities as a student, staff, or alumni, 4) willing and able to use social media, a smartphone, text messaging, and Fitbit devices and app, and 5) willing and able to engage in MVPA. The exclusion criteria include, 1)
comorbidities of obesity or conditions that prohibit compliance with the study protocol, 2) a recent cardiovascular event, 3) currently being treated for malignancy and/or an eating disorder, 4) planning to have weight loss surgery or enroll in a weight loss program, 5) loss of more than 15 pounds within the past 3 months, and 6) pregnancy or planning pregnancy within 24 months.

Additionally, for the purpose of the current study we first excluded from the analysis those participants missing more than 20% of the expected daily observations according to statistical recommendations and previous research on physical activity time series (Baretta et al., 2023; Bolger & Laurenceau, 2013; Chevance, Baretta, et al., 2021). Then, we applied further exclusion and inclusion criteria to obtain two subsamples (see Figure 2). The first analytical group (n = 101) encompassed participants who, at baseline, met two criteria: 1) engaged in less than 150 minutes of MVPA per week, and 2) expressed an intention to engage in at least 150 minutes of MVPA per week over the subsequent 6 months. For this group, the target physical activity threshold, against which the trajectory was compared, was set at 150 minutes of MVPA per week in alignment with the participants' intentions. We labelled this group as "BL-Inactive". The second analytical group (n = 251), labelled "BL-Active", included participants who, at baseline, satisfied two conditions: 1) engaged in more than 150 minutes of MVPA per week, and 2) expressed an intention to increase their physical activity level over the next 6 months (without specifying a particular physical activity target). For this group, the target physical activity threshold, corresponded to the participants' baseline MVPA value (see the next section about measures). In this way, we allowed the boundary conditions related to the threshold to vary: 150 MVPA/week for subsample "BL-Inactive" and the baseline weekly MVPA value for subsample "BL-Active". To note, the threshold for subsample "BL-Inactive" is lower than the threshold for subsample "BL-Active" as the baseline MVPA value of participants in the second subsample already exceeded 150 MVPA/week.

We opted for these formulations of the boundary condition of the threshold for the following
reasons. First, 150 MVPA/week is the recommended level of physical activity to limit the risk of
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developing non-communicable diseases (Ding et al., 2020), and it represents an intended threshold for
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those participants who do not meet this threshold yet. Second, we consider the baseline level as the
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maintenance target par excellence, in the sense that it does not require any further increase in physical
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activity. Thus, we addressed two aspects of physical activity promotion: i) the increase of physical activity
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among inactive individuals up to a clinically relevant level, and ii) the maintenance of a certain level of
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physical activity among those who already meet a given threshold.
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8 *Measures*

9 **Baseline measures**

Baseline physical activity was assessed using the Global Physical Activity Questionnaire (Armstrong & Bull, 2006), with consequent assessment of whether physical activity met current recommendations of 150 minutes of MVPA per week. Two items were used to assess the i) intention to be more active in the next 6 months, and ii) intention to be active at a moderate intensity for at least 150 minutes a week over the next 6 months. These items were measured on a 5-point Likert scale ranging from 1 (Strongly disagree) to 5 (Strongly agree). To be included in the first or second analytic sample, participants had to report a score of 4 or 5 on the corresponding item. Weight and height were measured objectively in the lab and used to compute participant's BMI in kg/m². Age and sex were self-reported.

18 **Physical activity**

MVPA was continuously collected through a wrist-worn activity tracker from Fitbit (i.e., Fitbit Charge; Fitbit Inc, San Francisco, CA, USA). Participants were asked to wear the device continuously according to manufacturer recommendations (i.e., to wear the device on one's non-dominant arm and up to three finger widths above the wrist bone). The MVPA metric was not directly available from Fitbit. Therefore, we calculated it as the sum of minutes spent in both fairly active, and very active physical
1 activity, regardless of the duration of physical activity bouts (Jakicic et al., 2019). MVPA was preferred
2 over other physical activity indicators (e.g., step-count) as the intention to engage in MVPA was an
3 inclusion criterion and reflects the participants' goal at baseline. We identified invalid days in case 1) no
4 data was collected by the Fitbit on a specific day, or 2) the number of sedentary minutes was equal to 1440
5 per day (i.e., 24 hours), which represents an unrealistic value. Invalid days were subsequently recoded as
6 missing. Previous validation studies of Fitbit Charge devices showed a tendency to overestimate MVPA
7 (Matlary et al., 2022). Even though this aspect is problematic when the Fitbit wristband is used in a setting
8 that requires strong criterion validity (e.g., a clinical trial in which Fitbit-assessed MVPA represents the
9 clinical outcome), it represents a minor threat here since the tendency to overestimate is systematic and
10 not hypothesized to influence the evaluation of MVPA patterns of change over time.

11 Data analysis

12 **Time series characteristics**

13 The length of the time series was set to 6 months (182 days) for all participants. The median
14 number of missing days per participant was 3 (range = 0 - 33 days) in analytical group 1, and 5 days (range
15 = 0 - 36 days) in analytical group 2. Missing days were imputed using the Kalman Filter method which is
16 recommended for univariate time series (Gómez & Maravall, 1994; Moritz & Bartz-Beielstein, 2017).

17 **Analytical approach**

18 We compared how MVPA changed over time – the trajectory – in relation to the subsample-
19 specific physical activity thresholds. In doing so, we adopted an idiographic, person-specific approach
20 and employed a statistical method previously utilized in the literature on physical activity to identify and
21 characterize changes in the trajectory of time series data for step count (Chevance, Baretta, et al., 2021).
22 Specifically, we used a recursive partitioning algorithm to characterize within-person behavioral changes
23 over the 6-month observation period. This analysis was conducted using regression trees from the *rpart*
24 package (Therneau et al., 2022). Regression trees enable the identification of nodes (i.e., the decision
points where the dataset is divided, akin to forks in the tree's branches) and partition a continuous dependent variable in recursive splits based on different predictors, creating a hierarchical structure. In the present study, we partitioned the MVPA variable (calculated as a weekly, 7-day rolling sum) based on the progressive day of the study as predictor. As a result, each time series was split into separate *phases* characterized by significantly different MVPA means. For each participant, we compared the MVPA mean of each phase with the corresponding target threshold according to the subsample. Subsequently, we recoded this information into a new binary outcome that specified whether the participant was active above or below the threshold during a given phase.

To compare the physical activity trajectory with the target thresholds at different timescales, we repeated this analytical procedure four times. Each time, we modified the 'minbucket' parameter within the 'rpart' function, altering the minimum length of the recursive split in the model, which represents the minimum duration of each phase. Specifically, we set the 'minbucket' values to 7, 14, 28, and 56 days (corresponding to 4%, 8%, 15%, and 31% of the total length of the time series). For example, a 'minbucket' value of 7 days imposed a minimum requirement of seven days to detect a phase transition, while a 'minbucket' value of 56 days required 56 days. These values were chosen as they represent progressive multiples of a 7-day window, corresponding to the timescale of the physical activity goal (e.g., engaging in more than 150 minutes of MVPA per week).

Ultimately, this adjustment allowed us to systematically vary the first boundary condition (i.e., the timescale of change), enabling us to scrutinize the temporal dynamics between the mean MVPA of each phase and the target threshold at different timescales. Finally, we aggregated individual statistics on phase transitions (i.e., moving from below the threshold to above or vice versa) for each subsample to examine whether it was possible to detect robust data pattern indicating that physical activity was consistently performed above threshold. We then compared these statistics across the various timescales with the visual
1 support of histograms and heatmaps.

2 Statistical analyses were performed in R version 4.3.0. The data and code for the statistical analyses

3 used in the present study are available on OSF (https://osf.io/6kjwg/).

4 **Results**

5 *Participants characteristics*

6 Subsample “BL-Inactive” comprised 101 participants (59% female) with a mean age of 25.5 years
7 (range: 18–35 years) and a mean BMI at baseline of 30 kg/m2 (range: 25–40 kg/m2). At baseline,
8 participants had a weekly mean of 56 minutes of MVPA, measured through the GPAQ (Min = 0, Max =
9 149, Median = 50), while the weekly mean of physical activity during the study across participants was
10 277 minutes of MVPA measured with the Fitbit (Min = 0, Max = 1721, Median = 229).

11 Subsample “BL-Active” included 251 participants (56% female) with a mean age of 22.8 years
12 (range: 18–34 years) and a mean BMI at baseline of 30 kg/m2 (range: 25–40 kg/m2). The weekly mean
13 of physical activity at baseline measured through the GPAQ was 541 minutes of MVPA (Min = 150, Max
14 = 5150, Median = 390), while the weekly mean of physical activity during the study across participants
15 was 374 minutes of MVPA measured with the Fitbit (Min = 0, Max = 2315, Median = 321 minutes).

16 *Timescales of change in physical activity*

17 Figure 3 (Panel A) visually represents how frequently physical activity changed over time in
18 relation to the target thresholds (i.e., phase transitions) at different timescales. Histograms depict the
19 distribution of participants based on the number of phase transitions, revealing a decreasing count of phase
20 transitions as the minbucket parameter increases. In subsample 1 "BL-Inactive", with target threshold set
21 to 150 MVPA/week, the mean number of phase transitions exhibited the following results: in the 7-day
22 minbucket condition, the mean was 3.05 (SD = 2.88, Min = 0, Max = 12), indicating that, on average,
23 participants moved from below to above the threshold or vice versa three times. With the minbucket set
24 to 14 days, the mean reduced to 1.49 (SD = 1.49, Min = 0, Max = 5). For a 28-day minbucket, the mean
further decreased to 0.64 (SD = 0.83, Min = 0, Max = 3), and with a 56-day minbucket, the mean was 0.20
(SD = 0.40, Min = 0, Max = 1). In subsample "BL-Active", with target threshold set to baseline level of
MVPA/week, the results were as follows: in the 7-day minbucket condition, the mean number of phase
transitions was 2.55 (SD = 2.50, Min = 0, Max = 10). Setting the minbucket to 14 days resulted in a mean
of 1.36 (SD = 1.57, Min = 0, Max = 6). With a 28-day minbucket, the mean was 0.59 (SD = 0.87, Min =
0, Max = 3), and with a 56-day minbucket, the mean was 0.24 (SD = 0.42, Min = 0, Max = 1).

In summary, when inspecting the relation between the physical activity trajectory and the target thresholds with a shorter timescale (7 days, left part of Figure 3), we frequently observed recurring phase transitions within participants. For instance, with a minimum phase duration of seven days, 64% of participants in subsample 1 and 59% in subsample 2 experienced at least two-phase transitions (e.g., transitioning from below the threshold to above and then returning below the threshold, or vice versa). Extending the timescale to a minimum of 14 days reduced the percentages of participants experiencing at least two-phase transitions to 49% in subsample 1 and 41% in subsample 2. As we increased the timescale to 28 or 56 days, the results revealed more stable trajectories that often settled either below or above the threshold. For example, with a minimum phase duration of 28 days, 79% of participants in subsample 1 and 81% in subsample 2 experienced a maximum of one phase transition (e.g., transitioning from below the threshold to above or vice versa). Extending the phase duration to at least 56 days, none of the participants from both subsamples had more than one phase transition.

Panels B (subsample 1 "BL-Inactive") and C (subsample 2 "BL-Active") in Figure 3 provide a more detailed analysis of phase transitions by considering the temporal dimension. Each row represents a unique participant, while each column signifies progressive days since the intervention started. The color scale distinguishes between MVPA means above (blue) and below (red) the target thresholds. Visual inspection of the heatmaps further confirms a consistent trend in both subsamples: as the minbucket
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parameter increases, indicating a coarse-grained timescale, predicted MVPA values stabilize and converge into stable distinctive patterns that differentiate between phases above or below a particular threshold.

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Finally, the current analysis enables us to identify participants who show a consistent pattern of sustained physical activity (Table 1). In subsample “BL-Inactive”, the focal phenomenon of sustained physical activity was observed in 27% of participants at a 7-day timescale, 36% at 14 days, 52% at 28 days, and 71% at 56 days. In subsample “BL-Active”, sustained physical activity was observed in 5% of participants at a 7-day timescale, 10% at 14 days, 21% at 28 days, and 26% at 56 days. The remaining participants either exhibited fluctuations around the thresholds or consistently remained below it.

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## Discussion

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In the introduction of this paper, we highlighted the essential role of exploring possible boundary definitions of focal phenomena to constrain the space of theories and contribute to their further development (Borsboom et al., 2021; Busse et al., 2017; Haig, 2013). The current results provided valuable insights in this direction by examining various formulations of key boundary conditions (i.e., *timescale* and *threshold*) that could be used to better understand the phenomenon of sustained physical activity, contributing to advance existing theorization as well as intervention development.

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### Boundary conditions of sustained physical activity

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#### First boundary condition: timescale of change

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The current analysis of physical activity time series revealed that varying the timescale of change would offer different plausible options to choose from when seeking to define, as a replicable phenomenon, what is commonly called, “physical activity maintenance”. Within the study context (participants in a weight loss intervention who intended to increase their physical activity), these ranged from a highly dynamic scenario marked by repeated fluctuations to a scenario characterized by stable physical activity trajectories. At a finer-grained timescale (7-14 days), our findings aligns with previous literature, suggesting that physical activity is a volatile behavior, often fluctuating around a target
threshold over time (Chevance, Baretta, et al., 2021; Shang et al., 2018). This phenomenon is increasingly
recognized as a distinctive feature of physical activity behavior. However, in the past, it has often been
neglected in favor of more easily accessible dichotomous categorizations (e.g., individuals meeting vs.
not meeting the threshold) (Shang et al., 2018). At the same time, finer grained timescales result in fewer
observed instances of sustained physical activity. At coarse-grained timescales of change (28-56 days),
the data pattern tends to indicate a polarization between scenarios where individuals' physical activity is
consistently performed above a given threshold and those where it falls below it.

Critically, these contrasting phenomena (i.e., fluctuating vs. stable trajectory) represent distinct facets of physical activity, each resulting from different boundary condition specifications. For example, a person may meet a threshold over the longer term while exhibiting significant short-term fluctuations. Thus, if the goal is to see whether a person is active in the long-term, we would argue that building statistical models able to identify short-term fluctuations can be misleading. On the other hand, modeling the data at coarse-grained timescales filters out a considerable amount of variance that might be otherwise useful in understanding how physical activity unfolds over time. Despite the intuitive nature of these findings from a statistical perspective, to the best of our knowledge, this is the first empirical study to systematically address this topic and establish specific, replicable timescale boundaries for distinguishing between stable and fluctuating physical activity phenomena.

## Second boundary condition: physical activity threshold

Relevant insights also came from comparing patterns of sustained physical activity based on different operational definitions of threshold (i.e., comparison between the two subsamples). The results show that at each timescale the distribution of participants transitioning from below to above the intended threshold and vice versa was similar in the two subsamples (see Figure 3, panel A). One difference was that the trajectories of participants with a lower threshold were more frequently above it (subsample “BL-
1 Inactive”; Figure 3, panel B), while it was the opposite for participants with a higher threshold (subsample
2 “BL-Active”; Figure 3, panel C). In other words, the absolute level of the threshold appears to influence
3 the likelihood of instances where the physical activity trajectory is above the threshold, thus affecting the
4 detection of phenomena of sustained physical activity, independent of the timescale (see Table 1). It is
5 important to underscore that in subsample “BL-Active”, the threshold was set equal to the baseline level
6 of physical activity, which was already very high, with a median of 390 minutes of MVPA/week.

7 *Implications*

8 **<u>Theoretical implications</u>**

9 Research on theory construction methods has positioned the identification of phenomena as a
10 fundamental first step in theory development (Borsboom et al., 2021; Haig, 2013), with the study of
11 boundary conditions playing a critical role by contributing to the production of stable descriptions of these
12 phenomena (Busse et al., 2017). In this context, this study advances the theoretical development of
13 behavior change theories concerned with the phenomenon of sustained physical activity. A logical next
14 step would be to evaluate such theories based on their ability to explain the identified phenomena and, if
15 necessary, update and refine them. For example, theories should be assessed for their capacity to account
16 for phenomena such as short-term variability nested within longer-term stable trajectories, or the effects
17 of threshold difficulties on sustained physical activity. In practice, a recommended approach for assessing
18 whether a theory can adequately account for the observed phenomena is the development of a formal
19 model of the theory itself (Borsboom et al., 2021). Formal modeling plays a critical role in theory
20 construction by ensuring that the resulting theories remain tightly connected to the phenomena they aim
21 to explain (Hekler et al., 2016; Perski et al., 2024; Robinaugh et al., 2021).

22 **<u>Intervention implications</u>**

23 Current results yield important implications also for the design and evaluation of physical activity
1 interventions. Notably, echoing previous recommendations (Scholz, 2019; Spruijt-Metz et al., 2015), they
2 emphasize the need to understand the functional implications of short-term behavioral variability for
3 longer-term behavioral maintenance. This understanding is crucial for better comprehending how
4 behavior changes over time and, consequently, for intervening more effectively. Modeling fluctuations in
5 physical activity at a finer-grained timescale offers insights into possible short-term behavioral variability
6 (e.g., lapses) that could be meaningful to guide the design of just-in-time adaptive interventions (Nahum-
7 Shani et al., 2018). This type of intervention often targets momentary changes (from hour to hour) in
8 physical activity, such as prompting individuals to stand up and take a walk after a predefined period of
9 sedentary behavior (Hardeman et al., 2019). However, they tend to overlook longer-term changes (e.g., 7-
10 day timescale), that our results revealed to be quite frequent.

11 In contrast, short-term behavioral variability, such as a 7-day lapse, does not necessarily result in
12 a longer-lasting trend below the threshold. For instance, we observed multiple instances of rapid
13 behavioral lapses that 'disappeared' when we zoomed out and inspected the fluctuations at a coarse-grained
14 timescale (see Figure 3, panel B and C). In this scenario, when the overarching trend (long term variability)
15 indicates that a person meets the target threshold over time, it becomes crucial to carefully assess whether
16 intervening on short-term lapses is advantageous or detrimental. Therefore, future research should aim at
17 developing approaches that allow distinguishing and predicting whether instances of short-term behavioral
18 variability are just descriptive of rapid, erratic fluctuations or will result in long-term trajectories that
19 stabilize below the threshold. These two scenarios carry distinct levels of vulnerability for individuals who
20 might benefit from receiving different types of intervention content based on the prospective evolution of
21 short-term behavioral variability. It is important to differentiate between these two scenarios, as the goal
22 of physical activity interventions is not simply to change the shorter-term state of the target system (e.g.,
23 the person), but to bring it to a new equilibrium that persists over time and generates a positive impact on
1 health outcomes (e.g., reducing the risk of cardiovascular diseases).

2 *Strengths and limitations*

3 One major strength of this study is the integration of two approaches grounded in fundamental
4 principles of the philosophy of science and essential for theory construction: the phenomena-driven
5 research approach and the systematic exploration of boundary conditions of focal phenomena (Borsboom
6 et al., 2021; Busse et al., 2017; Eronen & Bringmann, 2021). Specifically, we provided various
7 formulations for the boundary conditions of timescale of change and threshold and examined how such
8 formulations impacted the detection of phenomena of sustained physical activity. Additionally, this study
9 discussed the pros and cons of the various boundary condition options, contributing to a more cogent
10 perspective on creating repeatable, interrelated phenomena that could be leveraged in combination. The
11 phenomenon-driven approach was made possible thanks to the use of intensive longitudinal data, which
12 are necessary to investigate phenomena that unfold over time and are therefore integral for research on
13 behavioral maintenance (Chevance, Perski, et al., 2021; Dunton et al., 2021). To the best of our
14 knowledge, this is the first empirical study to systematically address this topic and purposefully examine
15 different formulations of boundary conditions for identifying phenomena of sustained physical activity.

16 While this study offers valuable insights, it also has limitations. First, participants' physical activity
17 at baseline was assessed using the GPAQ, which is susceptible to typical biases inherent in self-report
18 measures in health-care research, such as recall bias and social desirability bias, among others (Althubaiti,
19 2016). However, we preferred to adopt this approach instead of other potential solutions to quantify the
20 baseline level (e.g., averaging the first two weeks of Fitbit data) to avoid calculating the baseline level
21 from a period during which the participants were already exposed to the intervention content.

22 A second limitation pertains to the accuracy of Fitbit Charge devices in measuring MVPA. Prior
23 research has revealed a tendency for these devices to overestimate MVPA (Matlary et al., 2022). As
1 previously noted, this overestimation could pose challenges in settings requiring high criterion validity,
2 such as clinical trials where Fitbit-assessed MVPA serves as a direct clinical outcome. However, in our
3 context, this issue represents a minor concern. The systematic nature of this overestimation did not hinder
4 our examination of the boundary conditions of sustained physical activity. At most, it may have led to an
5 overestimation of the total number of phases during which participants were active above the threshold.

6 Another limitation concerns the assessment of participants' intentions to increase their physical
7 activity level. According to the SMART 2.0 protocol, this information was collected at baseline and during
8 the 6, 12, 18, and 24-month follow-up assessments, rather than using more frequent measurements
9 (Mansour-Assi et al., 2022). As a result, we were unable to evaluate whether changes in participants'
10 intentions corresponded with changes in their physical activity levels. Additionally, we analyzed physical
11 activity trajectories using time series data spanning 180 days. It is possible that we did not capture certain
12 patterns in the physical activity data (e.g., disengagement) which might require longer time series to unfold
13 (e.g., between 6- and 24-months post-baseline). Last, this study did not disentangle the impact of short-
14 and long-term variability on target health outcomes (e.g., body mass index, cardiovascular fitness). Future
15 research should address this gap, which would help determine the range of boundary conditions of
16 sustained physical activity that could be used to predict health outcomes.

## 17 Conclusions

18 In this study, we provided initial insights into the boundary conditions that determine the
19 observability of phenomena of sustained physical activity, a critical aspect for the theorization of physical
20 activity maintenance and enhancing intervention development. Our findings demonstrate that various
21 formulation of boundary conditions significantly affect the nature of detectable phenomena. This paper
22 invites further empirical research in the field, encouraging the testing of current findings across different
23 contexts and exploring various formulations of boundary conditions.
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21
<table><caption>Table 1. Frequency of participants matching specific physical activity phenomena at the varying of the boundary conditions of *timescale* and *threshold*.</caption><thead><tr><th rowspan="2">Subsample</th><th rowspan="2">Baseline (median)</th><th rowspan="2">Threshold</th><th rowspan="2">Phenomenon</th><th colspan="4">Timescale</th></tr><tr><th>7 days</th><th>14 days</th><th>28 days</th><th>56 days</th></tr></thead><tbody><tr><td rowspan="3">"BL-Inactive"</td><td rowspan="3">50 mins MVPA/wee k</td><td rowspan="3">150 mins MVPA/wee k</td><td>Sustained physical activity</td><td>27 (27%)</td><td>36 (36%)</td><td>52 (51%)</td><td>71 (70%)</td></tr><tr><td>Fluctuating behavior</td><td>71 (70%)</td><td>61 (60%)</td><td>43 (43%)</td><td>20 (20%)</td></tr><tr><td>Below threshold</td><td>3 (3%)</td><td>4 (4%)</td><td>6 (6%)</td><td>10 (10%)</td></tr><tr><td rowspan="3">"BL-Active"</td><td rowspan="3">390 mins MVPA/wee k</td><td rowspan="3">Person-specific baseline level MVPA/wee k</td><td>Sustained physical activity</td><td>13 (5%)</td><td>26 (10%)</td><td>52 (21%)</td><td>65 (26%)</td></tr><tr><td>Fluctuating behavior</td><td>167 (67%)</td><td>135 (54%)</td><td>92 (37%)</td><td>59 (23%)</td></tr><tr><td>Below threshold</td><td>71 (28%)</td><td>90 (36%)</td><td>107 (43%)</td><td>127 (50%)</td></tr></tbody></table>

3

4
1 Figure 1. General structure of relations between theories, data, and phenomena as described in
2 and adapted from Borsboom et al., 2021.

<figure><img src="image_2.png" /></figure>

3 *Note*.: Generalization: data substantiate the existence of phenomena; Prediction: phenomena predict specific patterns in the data; Explanation: theories try to explain phenomena; and Abduction: theories imply the existence of phenomena.

7
<figure><img src="image_3.png" /><figcaption>1 **Figure 2. Participant selection flow for the current study sample.**</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3. Histograms and heatmaps showing phase transitions across the maintenance thresholds in the two subsamples.</figcaption></figure>