A network model of stress contagion: evidence from the vocational classroom

Purpose: Our study focuses on stress contagion in vocational school classes, examining how students’ stress experiences affect their spatial classmates. For this purpose, we apply a novel formal network model that allows us to differentiate between endogenous and exogenous peer effects in the stress contagion process. Using the network model, we investigate effects on students’ stress levels, considering the stress and coping experiences of spatial peers as well as didactic-methodological context factors. Methods: We test our statistical model using secondary data collected in a study involving 53 students from two classes at a public German vocational training school. The students’ experiences of stress (time pressure, pressure to succeed) and coping strategies (understanding of the subject matter, self-confidence) were captured using the experience sampling method, while classroom characteristics (e.g., teacher instruction, cooperative work) were recorded through video-based analysis of lessons. Utilizing the panel data, we employ maximum likelihood estimation to assess the spatial peer effects model for both classrooms. Results: Among other findings, all model specifications revealed significant peer effects for both stress measures, indicating that the higher the stress experience of immediate peers in the classroom, the higher the individual stress experienced by the students. Concerning the considered context factors, we found, for example, that increased cooperative work leads to higher levels of stress experience. Conclusion: From a substantive perspective, our results underscore the role of peer-to-peer contagion in the vocational classroom and thus suggest a nuanced examination of cooperative practices. From a research methodology perspective, our approach illustrates how various methods (such as experience sampling, video-based classroom observation, and spatial network information) complement and enrich each other, highlighting the added value of our network analytical approach


Introduction
In various professional fields such as business and administration, technical engineering, or healthcare, cooperation willingness is demanded, and collaboration ability is considered a central component of professional competence (e.g., Paeßens and Winther 2021;OECD 2024;Rausch et al. 2021;Spliethoff et al. 2021;Wesselborg 2021;Weyland and Reiber 2022;Wittmann et al. 2024).There is no need to doubt the high value of cooperative practices, even if one doubts that the undifferentiated postulate for a general "more" of cooperation and collaboration (OECD 2023) easily obscures the fact that such practices do not necessarily have to have a constructive and productive effect per se (e.g., Tenenbaum et al. 2020).These include phenomena such as the groupthink described by Janis (1991) and associated dysfunctional group dynamics that undermine their own proclaimed values and norms, such as pressure to reach consensus and conformity or suboptimal results of collective decision-making processes (Brown 2021;Mullen et al. 1991).Spieß (2004) characterizes this as the paradox of cooperation, suggesting that the intended expansion of decision-making and action spaces through cooperation can also imply their restriction, laying the groundwork for conflict and strain.In this regard, arguments can be found both for supportive and inhibiting relationships with individual stress and resource assessments, whereby the individual conditions and dynamics of specific collaborative interactions must be taken into account.Collaboration, depending on factors such as individual skills and dispositions of team members, group or team composition, or work organizational conditions, can equally act as increasing stress (e.g., due to additional work and coordination efforts) and reducing stress (e.g., by sharing the workload instead of completing demanding tasks individually) (Schadt et al. 2022).Given the shortage of skilled workers (e.g., Peichl et al. 2022), vocational training dropouts (e.g., Krötz and Deutscher 2022), or the steadily increasing work absences and sick leave due to stress-related reasons (e.g., Badura et al. 2023), there is an increasing focus on socio-emotional learning and working conditions and "non-cognitive" results of learning and work processes (e.g., Baumert et al. 2024;Sembill et al. 2013).In this context, there has been a trend in vocational education and training (VET) research towards increased attention to topics such as health promotion (e.g., Solberg et al. 2023), resilience (e.g., Schmid and Haukedal 2022), coping, and the promotion of dealing with occupational stress (e.g., Lang et al. 2019;Warwas et al. 2023), which can be interpreted as a reaction to the aforementioned developments.
In vocational education settings, the outlined matters become relevant because students are already supposed to learn to collaborate constructively and cooperate with each other (e.g., Mikkonen et al. 2017;Polkowski et al. 2020).Studies show that stress arising from social interaction with classmates can be a central stress factor for students, which can be compounded by stressors such as high workloads or time pressure (Kouzma and Kennedy 2004).Thus, exploring the influence of classmates on one's learning experiences is particularly intriguing, given their pivotal role in the learning and achievement process at school (e.g., Dishion and Tipsord 2011;Fortuin et al. 2016;Gaševic et al. 2013;Helm et al. 2017;King 2020;King and Datu 2017;King and Mendoza 2020).
With a few exceptions (e.g., Burgess et al. 2018;Dishion and Tipsord 2011), research on school stress has largely overlooked the role of stress contagion, especially in terms of peer-to-peer effects.Empirical evidence suggests that peers significantly influence each other's moods and overall experiences during class.While peer influence typically implies pressure to conform to specific behaviors, social contagion among peers reflects broader, often subconscious dynamics (Dishion and Dodge 2005).Previous empirical investigations into stress contagion have predominantly focused on regression models (e.g., Oberle and Schonert-Reichl 2016), which, however, by their very nature as reduced form specifications, are incapable of disentangling the sources of similar behavior and experience.Therefore, we advocate for a structural network approach to explore the contagion of stress in the classroom, enabling the separation of individual stress sources generated by spatial peers.Utilizing network information allows us to differentiate between endogenous and exogenous peer effects in the stress contagion process.The differentiation between different effects is crucial, because stress contagion is likely to have a multiplier effect, where the direct impact of an external stress-triggering event is amplified by a stress effect due to classmates' stress levels.Understanding the mechanisms of stress contagion in the classroom, particularly investigating how a student's experience of stress affects his or her spatial classmates, remains challenging and empirically elusive.To achieve this, we draw upon statistical approaches considering complex interdependencies between observational units, as popularized in the spatial econometric literature (e.g., Anselin 2003;Elhorst 2014), as well as in the literature on network peer effects (e.g., Bramoullé et al. 2009Bramoullé et al. , 2020)).Our methodological framework builts on both the peer effects literature and the spatial econometric literature.The local neighbors are taken as members of the peer group with equal weights, leading to a sparse network of stress contagion.In contrast to this, the spatial literature builds on spatial weight matrix which is more or less arbitrarily computed by as a function of a geographical distance measure.
Against this background, our contribution pursues two objectives.Firstly, we introduce a novel approach that enables us to identify and estimate parameters of students' stress contagion.Secondly, we test various substantive hypotheses using our statistical approach, which can contribute to elucidating the stress contagion phenomenon.Our empirical results derive from a secondary data analysis (Kärner 2015).The dataset encompasses information from two vocational classrooms, with seating plans used to construct networks of spatial peers.In addition, students were asked to report their stress experiences, including time pressure and pressure to succeed, and report how they were coping with classroom situations.Furthermore, video recordings allow us to create contextual variables that reflect the characteristics of the classroom environment at each moment of time.
The article is structured as follows: section Stress contagion in the classroom defines stress, coping, and stress contagion and provides an overview of the psychological mechanisms involved in contagion processes, accompanied by a discussion of previous empirical approaches used to assess stress contagion.Section Modelling stress contagion: a network econometric approach introduces our novel network econometric approach and outlines its application in characterizing context-related stress contagion among students.In section Study design and data, we present a description of the study design and data underpinning our empirical study, along with some descriptive statistics that characterize the sample-specific variables.Section Estimations of the spatial peer effects models is dedicated to presenting our empirical findings, while section Discussion serves as the discussion of our findings and includes an outlook on potential future research directions.Further information on maximum likelihood estimation and the Python code can be found in the appendix and online supplementary material.

Stress and coping
Stress at school has been a widely discussed topic, not only since the pandemic-related school closures (e.g., Bujard et al. 2021), as it can lead to mental health issues and learning difficulties (e.g., Achtenhagen 1978;Auerbach et al. 2018;Vogel and Schwabe 2016).In general, stress is considered as a specific relationship between the individual and the environment, particularly as an interaction between situational demands and personality characteristics.Internal or external demands that strain or exceed a person's adaptive resources can lead to psychological and somatic stress reactions (Lazarus 1966 et passim).Specifically, in the vocational context, Bakker and Demerouti (2007) describe demands as physical, psychological, social, or organizational aspects of the learning or work environment that require sustained physical and/or psychological efforts or abilities and are therefore associated with certain physiological and/or psychological costs.School-related factors, for example identified by Kouzma and Kennedy (2004) as the primary sources of student stress, include examinations, high workloads, pressure to perform, time pressure, and issues arising from social interaction with classmates such as social comparison processes and pressure to conform.Coping resources, on the other hand, refer to aspects that are either functional for achieving goals, reduce demands, or promote personal growth, learning, and development (Bakker and Demerouti 2007).Such resources can be individual (e.g., professional expertise, work-related selfcare behavior) or social (e.g., social support and dyadic coping) in nature (e.g., Kärner et al. 2018Kärner et al. , 2021)).Specifically on strategies students use to cope with academic stressors, Skinner and Saxton (2019) conducted an extensive literature review.Among other things, the authors identify different ways of coping in the academic domain.These include subject-related aspects, such as information seeking in order to understand the subject matter, and psychological aspects, such as self-reliance, i.e. attempts to regulate one's own behavior or emotions by strengthening confidence and optimism (ibid.).

Definition and conditions of stress contagion
From the general concepts of stress and coping, we proceed to the conceptual analysis of stress contagion.In general, stress contagion refers to the transfer of stressors from one individual to others.In the classroom, a shared demand can trigger school-related stress, which, when transmitted, elicits various reactions and adaptations in other classmates (Wethington 2000).Demanding teaching conditions experienced by an individual can lead to shared and even amplified stress among others.Therefore, understanding the underlying mechanisms of stress contagion among peers is crucial as they are closely associated with learning, academic performance, and overall well-being.Westman and Vinokur (1998) discusses three potential mechanisms through which stress contagion can occur: (i) both interaction partners are exposed to the same stressor because they share the common social environment, (ii) direct transfer of stress from one person to the other due to an empathetic reaction on behalf of the receiver, and (iii) indirect transfer via behavioral interaction (see also Härtel and Page 2009).These three mechanisms have their formal counterparts in the econometric literature on peer effects modeling, which distinguishes between different sources of similar behavior among individuals.Specifically, (i) corresponds to exogenous peer effects and unobserved correlated effects, (ii) is related to the endogenous peer effect, and (iii) to the social multiplier effect resulting from all interactions (see Manski 1993, and section Modelling stress contagion: a network econometric approach for a more formal definition within a peer effects network model).Bakker et al. (2009) describe specific situational conditions under which processes of stress contagion are most likely to occur.According to the authors, factors such as the frequency of interactions and spatial and personal proximity or distance can influence the probability of stress contagion.Corresponding peer effects are more likely among students when classmates frequently interact, such as during cooperative tasks.Physical proximity between peers also plays a decisive role; the closer students sit together over a longer period of time, the more likely peer-to-peer effects become (e.g., Faur and Laursen 2022;Tsai et al. 2011).Within the framework of our peer-effects model, we will categorize conditions for stress contagion in the classroom into two primary categories: (1) peer-related factors (peers as stressors and peers as resources), and (2) contextrelated factors, which will be elaborated upon in the following paragraphs and which form the basis for deriving our research hypotheses.
Peer-related factors: peers as stressors.On the one hand, peers can serve as sources of stress, particularly in classroom situations that require coordination, communication, and mutual feedback.Inadequate group work is often associated with reduced group effectiveness, individual disengagement, and negative performance evaluations by fellow group members (e.g., Arvey and Murphy 1998;Druskat and Wolff 1999;Schadt et al. 2022).Due to stress contagion processes, feelings of time pressure, uncertainty, or pressure to succeed can permeate the team or group (e.g., Westman and Vinokur 1998).Previous studies have demonstrated that stress can be induced by group assessments, stemming from differing efforts and grading expectations among group members and a lack of individual control over the quality of joint group work outcomes (Pitt et al. 2018).Moreover, learning environments that overly challenge individual students, leading to social comparisons with more skilled peers, can create stressful experiences accompanied by unfavorable emotions such as shame or anxiety, potentially impacting the stress levels of multiple students involved in the group (Pekrun 2006).Additionally, social comparison with similar others has been found to influence stress contagion effects, with emotional exhaustion being more pronounced in individuals who perceive that their peers are also stressed (Bakker and Schaufeli 2000).These considerations lead us to the first hypothesis: • Hypothesis 1: The higher the stress experience of immediate classmates in the classroom, the higher an individual student's stress experience should be ("peers as stressors").
Peer-related factors: peers as a resource.On the other hand, peers can also be a valuable resource for coping with stress under certain circumstances.The presence of other students can make the collaborative classroom setting less stressful, offering potential sources of social support.This support has positive effects on psychological adjustment as collaboration fosters knowledge exchange and the distribution of cognitive effort among group members (Kirschner et al. 2009;Sweller et al. 2011).Moreover, a positive climate within teams or groups can mitigate the effects of contagion processes by providing social support to group members during stressful periods (Cohen and Wills 1985).In a randomized experimental field study, Minkley et al. (2017) examined the effects of the work setting on stress, finding that conducting experiments alone in biology classes carries the risk of selfattributed failure, leading to elevated stress.In contrast, conducting experiments in a group is less stressful because peers serve as a source of social support.Similarly, Kärner et al. (2021) demonstrate that support from socially similar individuals, i.e., peers, is perceived as particularly effective and helpful.From these considerations arises the second hypothesis: • Hypothesis 2: The higher the coping experience of immediate classmates in the classroom, the lower an individual student's stress experience should be ("peers as resource").
Context-related factors.In addition to peer-related factors contributing to stress contagion, context-related factors play a significant role as interaction partners share the same social environment and are exposed to the same stressful conditions (Westman and Vinokur 1998).In a classroom setting, where all students undergo the same "educational treatment, " lessons can be characterized by the proportions of teacher instruction, individual work, and collaborative work among students.Stress contagion processes may be more likely in educational situations where students collaborate in teams or groups, compared to situations where they solely listen to the teacher without engaging in peer interaction.Additionally, the stress experienced by the teacher influences how lessons are perceived as stressful by students, as the teacher's stress can impact the pace of teaching due to their own time pressure (Kärner 2015).Furthermore, the complexity of the learning content may affect the stress experience of students.The complexity of instruction in schools can be understood through different didactic and methodological approaches.De Kock et al. (2004), for example, refers to teaching-learning activities and cognitive processes in learning as possible criteria for classifying learning environments in the classroom.Generally, a distinction can be made between more passive, teacher-centered instruction and active, constructive, and interactive student-centered instruction.The latter places higher demands on learners' cognitive processes compared to the former (Chi 2009;De Kock et al. 2004).Learners in student-centered instruction typically bear greater responsibility for their own learning outcomes, contend with higher content complexity, and navigate uncertainty and ambiguity associated with instructional content.At the cognitive level, this entails taxing the information system, particularly working memory and executive functions (Sweller 1994).Existing knowledge must be activated, new information processed, and integrated into existing knowledge structures, often achieved through cooperative efforts involving discussions with peers to arrive at common solutions (Chi 2009;Sembill et al. 2002).This leads us to the last two hypotheses: • Hypothesis 3: The more group and individual work (i.e. the more the students have to work themselves), the more complex the learning content in class and the more stressed the teacher is, the greater the impact of peer effects on the students' stress experience should be ("stress-reinforcing contextual conditions"). 1 1 With regard to the assumed effect of individual work, this may seem counterintuitive at first glance, as students do not obviously work together in individual work as they do in cooperative work.The literature does not provide clear evidence on this matter.However, our argument for the corresponding hypothesis would be that students can also exchange information with their immediate classmates in phases of individual work, e.g. through eye contact or short conversations, or they can get an impression of how far they have already progressed with their work by looking at their classmate's worksheet.Furthermore, it can be assumed that individual work per se leads to more stressful experiences, as students work independently on more or less complex tasks compared to situations of teacher instruction, where students generally only have to listen.
• Hypothesis 4: The higher the degree of instruction by the teacher in the classroom, i.e. the less the students have to work themselves, the lower the influence of peer effects on the students' stress experience should be ("stress-reducing contextual conditions").

Assessing stress contagion: previous approaches
In addition to the psychological-pedagogical aspects of stress contagion in the classroom just described, we focus on the statistical-empirical modeling.In this section, we examine the methods used previously to measure stress contagion.Processes of stress contagion have been extensively examined in diverse contexts, including working environments (e.g., Bakker et al. 2009), romantic relationships (e.g., Song et al. 2011), or mother-infant relationships (e.g., Waters et al. 2014).Contagion processes have also been explored in psychological and educational sciences.For instance, research among principals and teachers (e.g., Westman and Etzion 1999) and between teachers and students (e.g., Frenzel et al. 2009Frenzel et al. , 2018;;Warwas and Helm 2017).In the context of teacherstudent relationships, research has delved into the transmission of personal experiences (Bakker 2005) and enjoyment (Frenzel et al. 2018), as well as associations between teachers' burnout and students' motivation (Shen et al. 2015).Empirical approaches for studying contagion processes include experimental variations in facial expression conditions (e.g., Hatfield et al. 2014), the analysis of dyadic relationships using the actorpartner interdependence model (Cook and Kenny 2005), and the use of physiological measures and behavioral observation methods (e.g., Dimitroff et al. 2017;Donker et al. 2018;Erkens et al. 2019).Studies on contagion effects within dyadic relationships, utilizing observational data, typically rely on linear regression approaches, structural equation modeling, and multilevel modeling (Cook and Kenny 2005).For instance, Becker et al. (2014) investigated the relationship between teachers' stress emotions (e.g., anger and anxiety), their instructional behavior, and students' experiences in the classroom.
The authors employed an experience sampling approach and modeled intraindividual variability through multilevel regression analyses, with repeated measures nested within individuals who were, in turn, nested within classes.The studies mentioned above and their respective methodological approaches have in common that they are, to varying degrees, reduced-form approaches, meaning that they do not explicitly account for the network aspects of the contagion of stress in the classroom.However, observed similarities in stress reactions among students can result from different mechanisms, all leading to the same or similar reduced-form scenarios.Therefore, empirical studies based on reduced-form setups leave room for a wide range of interpretations (see Manski 2000).Specifically, reduced-form setups do not address how stress experiences are transmitted between students through peer behavior.In contrast, our structural, network-based approach enables us to disentangle various factors that explain similarities in stress experiences among students.Notably, we can identify stress contagion due to endogenous peer behavior and its multiplier effect.In the next section, we describe our approach in more detail.

Modelling stress contagion: a network econometric approach
In the following, we consider the classroom as a network, treating seating neighbors as spatial peers, indicating that the stress experience levels of seating neighbors may influence each other.Thus, we hypothesize that the specific seating plan of a classroom implies a distinct network structure, thereby determining how and to what extent individual stress spills over to other classmates.
In our econometric model, the specific seating plan of a classroom with N students is algebraically represented by the adjacency matrix A = [A ij ] N ×N .If students i and j are direct seating neighbors (spatial peers), A ij is 1, and 0 indicates the absence of a direct neighborhood.By definition, the diagonal elements of A are 0, indicating that students do not interact with themselves.In the following, we encode the seating plan as an undirected network.In this case, A is symmetric, ensuring that stress contagion between seating neighbors operates in both directions.
In our empirical study, we consider two classrooms with different seating plans as given in Fig. 1.Students of class A study in a classroom with a block-wise seating plan, while students of class B study in a classroom with learning groups of three to five students. 2ased on the two seating plans we construct corresponding adjacency matrices for the classroom networks.For instance, taking a look at classroom A in Fig. 1, we consider as spatial peers the students sitting next to each person.So, the peers of A 06 are A 12 and A 20 , meaning that the elements of the adjacency matrix A, A 06,12 , A 06,20 , A 12,06 , A 20,06 are 1.For classroom B we consider as spatial peers the students sitting at the same table. 3n our econometric model, the dependent variable y it represents the stress level reported by student i at time t.We formulate the individual stress level as a linear function of the sum of the stress levels of the i-th student's spatial peers ("linear in sums model"): Student i-th stress level is also driven by other exogenous factors x it and and by the exogenous factors of the spatial peers.
Considering the seating arrangements, classrooms are partitioned into groups, where every individual interacts with all others within their group (excluding self-interaction).To ensure proper identification in this context, it is essential to have diverse group sizes (refer to Bramoullé et al. 2009, p. 45).Notably, in both classrooms, there is variability in the number of students sharing a table .(1) Our model distinguishes between three different mechanisms (Manski 1993;Manski 2000) that may generate similarities in the stress experiences of students: the endogenous peer effect, the exogenous (or contextual) peer effect, and correlated effects (see also section Definition and conditions of stress contagion).Of particular interest in the context of stress contagion is the endogenous peer effect, represented by the parameter δ t .Its presence leads to stress contagion, wherein the stress level of student i varies with the stress level of their peers.
To illustrate the econometric model, the causal graph shown in Fig. 2 sketches the various channels through which the stress levels of two seating neighbors are affected.The stress level of Student 1 (without index) is denoted by node Y, while the stress level of the second student is captured by node Y2.As a seating neighbor, Student 2 is a member of the peer group of Student 1, depicted by node Y_p .The endogenous peer group effect δ t of Eq. 1 is represented by the edge between Y_p and Y.Moreover, the stress experience of Student 1 can also be transmitted by his/her own factors (edge from node X to Y) and the exogenous factors of his/her peer group node (edge from node X_p to Y; note that X and X2 can be the same sources of stress).Stress enhancement for the two students occurs through the multiplier effect, where an increase in the stress level Y impacts the peer group of Y 2_p , which then passes on to Y2 and feeds back via Y 1_p onto Y. There- fore, stress contagion and stress enhancement depend crucially on the size of the parameter δ t .
In our model, we assume complementary behavior, meaning that the stress level of student i depends on the aggregate levels of stress experience of their spatial peers, such that more spatial peers generate more stress experience.In the network peer effects literature, such a model specification is referred to as a local aggregate model (e.g., Calvó-Armengol et al. 2009).An alternative specification is the local average model, where a student's stress level is determined by the average stress level of his/her peer group.In game-theoretical models, the local average effect can be derived as a result of norm or conformity behavior, where a student's well-being (the student's "utility" in microeconomic terms) decreases with the deviation of his/her behavior from the social norm represented by the average stress level of the peer group.Such a game-theoretic interpretation, however, requires that the student's stress level is at least partly under his/her Please note that the presence of an endogenous peer effect results in stress contagion, influenced by the specific form of the network and the magnitude of δ t .Let Y t = (y 1t , y 2t , . . ., y nt ) ′ be the vector representing the levels of stress experience of the students at time t.Let c 0 denote the intervention level without an additional interven- tion, and c 0 + c denote the intervention level with an additional intensity of c .The change in stress levels due to the increased intervention c is then given by: Here, v(A, δ t ) represents the Katz-Bonacich centrality vector, with δ t determining the decaying influence of higher-order neighbors. 4While we previously assumed that the intervention level is increased by the same amount for all students, its impact on the stress level generally varies depending on the centrality of the student.Moreover, even if only a fraction of the students undergoes the intervention, every student is influenced by it through the network.
By construction the regressor n j=1 A ij y jt capturing endogenous stress contagion gen- erates an endogeneity problem as the error term ε it is correlated with this regressor.Thus an ordinary least squares regression yields biased and inconsistent parameter estimates.
A novel element of our model is to allow for a time varying endogenous contagion effect.We model δ t as a time varying parameter depending on the contextual factor Z t . (2) (3) In network studies, statistics that capture importance or popularity of an individual are crucial.Centrality, one of these measures, is calculated using various methods, one of which is the Katz-Bonacich vector.This vector consists of centralities assigned to individuals within the network.What distinguishes this measure is its consideration of both first-order links (direct peers) and higher-order connections (peers of peers) of the individual.Additionally, the influence of higherorder peers diminishes according to a decreasing function determined by the peer effects parameter (Katz 1953).
For an easier interpretation we scale the context variable by min-max scaling as Z t = Zt − min{ Zt } / max{ Zt } − min{ Zt } , where Zt is the unscaled variable.Thus δ t = δ 0 is the size of the endogenous contagion effect when the corresponding context variable takes it lowest value, while δ t = δ 0 + δ z gives the endogenous peer effect at the maximum level of Z t .
In the exogenous peer effect term represented in (1), we adopt a local average representation.Here, the adjacency matrix G = [G ij ] N ×N is the row-normalized version of A. Consequently, the term j = 1 N G ij x jt captures changes in the stress level for student i resulting from alterations in the contextual factors of their spatial peers.
Finally, shared stress experiences between student i and their peers may also arise from shared unobservable exogenous factors, which are captured by the individual effect µ i and/or the error term.Unlike the endogenous contagion effect, the exogenous peer effect, and the correlated effects do not lead to social multiplication.As suggested in the literature, stress contagion is influenced not only by peer and contextual factors but also by individual characteristics (e.g., empathy, gender; see, e.g., Bakker et al. 2009).The individual effect µ i accounts for these individual factors.The panel structure of our data enables us to eliminate the individual effect µ i through within-transformation.

Study design and data
We test our statistical model using secondary data initially analyzed by Kärner (2015).Overall 53 students (18 males, 35 females; mean age = 19.53 ± 4.76 years) from two school classes attending a public German vocational training school were investigated during nine school lessons focused on the subject of "economic business processes." All participants were industrial clerks in training and in tenth grade at the time of the study.The sample is a convenience sample, meaning that the classes or students were not randomly selected.Both classes are comparable in terms of student composition, for example in terms of gender distribution (approximately 30% males in class A, 40% males in class B) and age (average age of nearly 20 years in class A, average age of nearly 19 years in class B).Although both classes were taught by different teachers, the thematic context regarding the subject matter, as well as the duration of learning, were kept constant for both classes.The study received approval from the Bavarian Ministry of Education and Cultural Affairs and was conducted in accordance with the Declaration of Helsinki.Consent to participate in the study was obtained from all participants of legal age.For participants not of legal age at the time of the study, parental consent was obtained.
The students' stress experience, situational coping, and the time pressure experienced by the teachers were sampled at 10-min intervals using mobile handheld computers (Palm Tungsten E2).In this study, participants had a maximum of 38 measurement points each, distributed over three days of observation.Participants could rate their current experience on a continuous scale ranging from 0 ("I fully disagree") to 100 ("I fully agree").A total of 1932 measurements were collected, which corresponds to a very good utilization rate of over 95% of the maximum possible 2014 (53 students x 38) measurements.The following four student experience variables were recorded: The students' stress experience was measured with two items: "I am under time pressure" (Students' time pressure) and "I am under pressure to succeed" (Students' pressure to succeed).Situational coping of students was assessed using two items: "I can cope with the current demands" (Students' self-confidence) and "I understand the subject matter" (Students' understanding) (Kärner 2015).With experienced time pressure and pressure to succeed, as well as subject-related self-confidence and understanding, central stressors and resources were selected.These can be characterized as frequently occurring and essential for students (e.g., Kouzma and Kennedy 2004;Skinner and Saxton 2019).Additionally, teachers were asked every 10 min about their own experience of stress related to time pressure using the item "I am under time pressure" (Teachers' time pressure).In terms of content, the established experience sampling items from Sembill et al. (2008) (see also Seifried and Sembill 2005) were the basis for the items adapted and used by Kärner (2015) in this study.In general, the experience sampling method has the advantage of offering a high degree of ecological (external) validity due to the situational context of data collection (e.g., Csikszentmihalyi and Larson 1987;Seifried and Rausch 2022;Sembill et al. 2008).We deliberately chose to use single items because of the high-frequency experience sampling, enabling a short, user-friendly survey.This decision is based on a thorough consideration of scientific recommendations regarding the use of single items (e.g., Allen et al. 2022;Matthews et al. 2022).
In addition to the participants' self-reports, the lessons were recorded for subsequent video-based analysis of classroom characteristics.Context-related factors within the classrooms were assessed through video-based time-sampling analysis using the observational scheme proposed by Seidel et al., (2001).Time intervals of 15 s each were coded and then aggregated into 10-min intervals using sum scores, aligning contextual conditions with individual-related data (students' stress experience and situational coping, as mentioned above).To ensure coding reliability, a third of the videos were independently coded by two raters, resulting in a satisfactory Cohen's kappa of 0.73.The coded instruction-related variables included teacher instruction (where the teacher explains and lectures to the class), individual work of students (where learners work on tasks alone), and cooperative work of students (where learners work on tasks in teams or groups).Additionally, to assess the complexity of the learning content during the lesson, 1-min time intervals were coded.A four-point Likert scale based on Bloom's taxonomy was employed to measure the complexity of the learning content and tasks, with increasing levels of difficulty (0 = "apply already known content, " 1 = "analyze content, " 2 = "synthesize complex ideas, " 3 = "evaluate problems").The between-coder correlation, serving as a reliability measure for this assessment, was 0.82 (Kärner 2015).
Table 1 provides a summary of the experience sampling data collected from students during the lessons.Our data reveal that students in class B experience significantly more pressure to succeed ( η 2 = 0.245, p = 0.002) and time pressure ( η 2 = 0.135, p = 0.028).In terms of understanding ( η 2 = 0.011, p = 0.543) and self-confidence ( η 2 = 0.076, p = 0.104), students in class A exhibit marginally higher values, but the differences are not significant. 5able 2 presents a summary of the contextual factors, separately for the two classes.The descriptive data clearly indicate that, in comparison to class A, class B's lessons are characterized by a higher level of content complexity ( η 2 = 0.451, p < 0.001), reduced teacher instruction ( η 2 = 0.168, p < 0.001), diminished individual work ( η 2 = 0.098, p = 0.006), and an increased emphasis on cooperative work ( η 2 = 0.238, p < 0.001).Addi- tionally, the teacher in class B reports significantly higher levels of time pressure during class ( η 2 = 0.705, p < 0.001).
Table 3 summarizes the correlations between the context factors for class A and class B. For class A, it can be seen that the complexity of the learning content correlates significantly negatively with teacher instruction and significantly positively with individual work.This means that the learning content tends to be easier in phases of teacher instruction and somewhat more difficult in phases of individual work.Furthermore, there are significant negative correlations between teacher instruction and individual work as well as cooperative work for class A. For class B, the correlations show that a high complexity of the learning content is more likely to be observed in the context

Estimations of the spatial peer effects models
For further analysis and interpretation of the measures, several considerations must be taken into account.Firstly, the 10-min experience measurements and the observation data aggregated to 10 min each were chronologically synchronized, ensuring that each experience measurement corresponds to the previous 10 min of instruction.Secondly, in our analysis, we employ two alternative yet related dependent variables: Students' pressure to succeed and Students' time pressure, each capturing distinct sources of stress.
In order to avoid multicollinearity, we combined the two coping items Students' selfconfidence and Students' understanding through averaging.Furthermore, the assumed dependence of students' stress experiences on the context factors must be considered, including (Teacher instruction, Complexity of the learning content, Teachers' time pressure, Individual work, and Cooperative work).To avoid overfitting and to ensure robustness in our analysis, separate models are estimated for each context factor.Lastly, given the considerable differences in teaching characteristics and mean experience values between the two classes, as illustrated in previous tables, we estimated the spatial peer effects model separately for each class.All analyses were conducted as Maximum Likelihood estimates and programmed in Python. 6e estimate the spatial peer effects model by maximum likelihood, with contextspecific contagion of stress as given by equations ( 1) and ( 3).In all our estimated specifications, we discover compelling evidence of stress contagion effects in both classrooms.With the exception of one case (where the dependent variable is Students' time pressure and the context factor is Cooperative work), contagion of stress is more pronounced in classroom B compared to classroom A. When considering the contagion parameter evaluated at the sample mean, denoted as δ( Z) , we observe a more pronounced difference in the contagion of stress between classroom B and classroom A for the measure Students' pressure to succeed than for the measure Students' time pressure.In addition, stress contagion in classroom B is significantly influenced by the context factor regardless of the context factor used, whereas this is not always the case in classroom A. The following paragraphs describe the results for each context factor-specific model.
Table 4 presents the estimation results for the context factor Teacher instruction.When examining the stress measure Students' pressure to succeed, we observe that the direct spatial peer effect, evaluated at the mean level of the context variable teacher instruction, is 0.076 for classroom B compared to 0.043 for classroom A. Consequently, students in classroom B are influenced almost twice as much by the stress level of their spatial peers.Conversely, when delving into the details of the stress measure Students' time pressure (columns 3 and 4 in the panel of Table 4), the difference in stress contagion between the two classrooms is only marginal.
Continuing to examine the role of the teacher, we consider the time pressure experienced by the teacher as a significant factor in stress contagion among peers.Table 5 presents the estimation results for the context factor Teachers' time pressure.This context factor is only relevant in classroom B, and the contagion parameter, evaluated at the mean Teachers' time pressure, is larger in class B compared to class A.
Table 6 provides a summary of the estimation results for the context factor Complexity of the learning content.This factor appears to be a strong driving force for contagion of stress in class B, whereas it does not play a significant role in class A.
Table 7 illustrates the effects for the context factor Cooperative work.The results indicate that phases where students collaborate in class B are associated with a notable rise in stress contagion among peers seated in close proximity.
Table 8 provides a summary of the estimation results when the determinant of the peer effect parameter is Individual work.Neither in class A nor in class B does individual work exhibit a significant effect on any of the stress measures.
Regarding coping, it is noteworthy that at the individual level, it significantly contributes to a reduction in stress experience in all models, regardless of the stress parameters considered (Tables 4,5,6,7,8; coping as an individual characteristic).The negative sign aligns with the intuition that better Students' understanding and higher Students' selfconfidence lead to less stress contagion (note that the two coping items were combined for the analyses through averaging).Furthermore, in all models, it becomes evident that the coping skills of peers contribute significantly to an increase in individual stress experience (Tables 4, 5, 6, 7, 8; coping as an exogenous peer effect; see below for a discussion of this finding).
Table 4 Maximum Likelihood Estimation results for the context factor Teacher instruction N = 28 students in class A, N = 25 students in class B, T = 38 measurement time points for both classrooms.δ( Z) (time varying endogenous peer effect), σ 2 (variance), log-lik (log-likelihood).Standard errors in brackets.* denotes significance at 5 % and ** at 1 % significance level.Standard errors were computed by the inverse of the Hessian

Summary of the results in the light of our hypotheses
Our study aim was to apply a novel formal network model and examine the extent to which spatial peers and context factors contribute to the contagion of stress among students in the vocational classroom.The dataset for our analyses included information from two school classes, from which we derived seating plans to establish networks comprising spatial peers.We utilized self-report data collected from students regarding their experiences of stress and coping, self-report data from teachers regarding their experiences of time pressure, and video observation data to delineate the context factors.
The following findings can be summarized to emphasize the advantages of networkbased approaches for investigating contagion processes in the classroom.
In Hypothesis 1, we assumed that the higher the stress experience of immediate classmates in the classroom, the higher an individual student's stress experience should be ("peers as stressors hypothesis").In all models, we observed significant peer effects for both stress experience indicators, suggesting that higher stress experiences, in terms of perceived time pressure and pressure to succeed among immediate peers in the classroom, are associated with higher individual stress levels experienced by the students.In the majority of the estimated models, the contagion of stress is more pronounced in classroom B compared to classroom A. In terms of interpreting students' stress experiences, it's noteworthy that the perceived stress in class B (averaging just under 26 points out of 100 for both stress measures) is indeed higher than in class A (averaging just under 12 points out of 100 for both stress measures) (see Table 1).However, the reported stress values in both classes are clearly within the lower third of the scale range on average.Additionally, when considering coping experiences, which range between 70 and 80 scale points out of 100 possible points for both measures in both classes, it can be summarized that the students in the sample are generally experiencing relatively low levels of stress and perceive their coping abilities as above average.
In Hypothesis 2, we assumed that the higher the coping experience of immediate classmates in the classroom, the lower an individual student's stress experience should be ("peers as resource hypothesis").In all estimated models, higher levels of personal coping experiences were associated with lower individual stress experiences.However, a counterintuitive effect was discovered when considering the coping experiences of immediate peers in the classroom.Surprisingly, the higher the coping experiences of these peers, the higher (rather than lower) the stress experienced by the students.The positive exogenous peer effect may initially seem counterintuitive but can be plausibly explained by the concept of social comparison processes (e.g., Pekrun 2006).Social comparison can induce stress itself when a student, after comparing themselves to their peers, perceives from the verbal or behavioral expressions of peers that they understand the subject matter better or are better equipped to handle class-related demands.For example, the theory of the so-called big-fish-little-pond respective little fish-big pond effect (Marsh et al. 2008) emphasizes the importance of classroom composition and peer effects for learning in class (Hattie 2002).According to this theory, social comparison processes and frames of reference for comparisons are decisive for self-perception.Marsh et al. (2008) point out that the same objective performance can lead to different self-perceptions depending on the frame of reference or comparative standard the individual uses for their self-assessment, and these self-beliefs have important implications for future performance and behavior.The frame of reference in the classroom can be characterized by different attributes, such as the general learning climate, which can be focused more on cooperation or competition.Furthermore, the reference norms used for performance assessments play an important role, which can promote social comparison and competition between peers or emphasize individual improvements measured against criterial reference standards (Marsh et al. 2008).Thus, from an educational perspective, it is important to consider interventions that facilitate peers functioning as resources rather than stressors.One approach could involve modifying the emphasis of reference norms, shifting from social reference norms to individual or criterion-related reference norms, or transitioning from a general competition dispositive to a cooperative learning climate, in which the individual characteristics of the learners are valued (e.g., Dickhäuser et al. 2002;Marsh et al. 2008).
In Hypothesis 3, we assumed that the more cooperative and individual work, the more complex the learning content in class, and the more stressed the teacher is, the greater the impact of peer effects on the students' stress experience should be ("stress-reinforcing contextual conditions hypothesis").The results of the analysis show the following: The assumption that increased cooperative work leads to higher levels of stress is valid in class B but not in class A, where the relevant effect is not significant.This is because students in class B work collaboratively and independently on complex tasks and content, and they are accountable for their work results to both the teacher and their peers.With increased communication and exchange, there is a more prominent channel for transmitting stress experience.Regarding the complexity of the learning content, the assumption is only met for class B in the case of the variable Students' pressure to succeed, where a significant positive effect on stress contagion is observed.For Students' time pressure, there is a significant negative effect in class B, likely due to students in this class having ample time to work independently on their tasks.Thus, the complexity of the learning content mitigates stress contagion in terms of time pressure, meaning that students are less affected by the time pressure of their peers when the course material is complex.This stands in contrast to the effect of the complexity of the learning content on the transmission of pressure to succeed.The opposite signs observed for the complexity of the learning content indicate that the two endogenous measures of stress experience capture different phenomena.For the context factor Individual work, neither in class A nor in class B does individual work exhibit a significant effect on any of the stress measures.One notable finding is the counterintuitive negative effect of teachers' time pressure, observed exclusively in class B for both stress measures.This can be explained by the fact that the teacher is under time pressure during the instructional phase, which means that during this time, the students are not actively challenged, and consequently, they are less susceptible to stress contagion (e.g., Dreyer 2011).
In Hypothesis 4, we assumed that the higher the degree of instruction by the teacher in the classroom, i.e., the less the students have to work themselves, the lower the influence of peer effects on the students' stress experience should be ("stress-reducing contextual conditions hypothesis").The results show, that the effect of teacher instruction is negative and significant in class B for both indicators of stress experience, indicating that the teacher's interventions have a more substantial impact in that class.This can be explained by the fact that students in class B must work more independently and take on greater responsibility for complex tasks during class.The teacher's intervention may contribute to reducing their levels of stress, for example, through the use of group-or whole-class scaffolding strategies (e.g., Hermkes et al. 2018Hermkes et al. , 2022)).

Limitations of the study
The study's design reveals both strengths and limitations.The realistic classroom and the high frequency of data collection (every 10 min) thus increase external validity by capturing the situational nature of the data, but with the major limitation that the results are not generalizable due to the selective and small sample.Specific limitations may therefore be associated with the sample (person-related confounders), the teaching contexts examined (context-related confounders) and the experience sampling method used (method-specific confounders).
Regarding potential person-related confounders, it is worth noting, as previously described, that we have taken measures to ensure that person-related characteristics (e.g., empathy, gender) do not confound our findings regarding stress contagion through within-transformation.
With regard to possible context-related confounders, it cannot be ruled out that context-specific unobserved confounding factors may have influenced the results (e.g."experimenter effects" in the sense that both classes were taught by different teachers).This means that strict control of the study conditions, as is the case with experiments, for example, is usually not possible in field studies at the expense of internal validity.Furthermore, in our study, we treat the stress experienced by the teacher, measured by teachers' time pressure, as an exogenous factor.Future research on contagion processes in the classroom should also incorporate the teacher's behavior as an endogenous factor.In other words, stress contagion in the classroom could be treated as a joint and dynamic process, wherein stress spills over from the teacher to the students and vice versa.
With regard to possible method-specific confounders, it should first be noted that we set fixed time intervals (10 min) within our experience sampling survey for "ergonomic" reasons.This was done on the one hand to facilitate the integration of the measurements with the observation data, and on the other hand to minimize disruptions during instruction and to keep the disruptions constant respectively to standardize them.An alternative to fixed survey intervals would have been randomized time intervals, known to reduce habituation effects, but at the expense of the aforementioned reasons, which, in our view, favored fixed collection intervals.Another possible limitation arises from the use of single-item measures.Although these are particularly useful in highfrequency experience sampling studies for reasons of survey economy and face validity (see Allen et al. 2022 for a compact discussion of the advantages and disadvantages of single-item measures and corresponding validity aspects), modeling using latent variables with several indicator items would possibly improve the measurements and cover a broader spectrum of possible behavioral and experience indicators.Another methodspecific limitation results from the fact that although we captured various context factors and associated them with the experience of stress, we lack additional information necessary for a comprehensive interpretation of the findings.To illustrate this point with an example: As described in the second section, we assume that instructional situations characterized by cooperative work evoke more stress among peers compared to situations where the teacher, for example, explains or gives a monologue.However, this assumption is based solely on the fact that students have more opportunities for exchange and collaborate more intensively during cooperative work phases than during teacher instruction phases.From a content perspective, we cannot ascertain how, for example, the interaction in the learning groups actually transpired, which could significantly influence whether peers are perceived more as stressors or as resources (for effects of learning group compositions and group dynamic processes see e.g., Schumacher 2002).Furthermore, we cannot make assertions regarding what we described above in the context of the discussion of Hypothesis 2; for example, regarding the general learning and working climate in the classes (e.g., degree of competition and cooperation orientation) or the reference norms established in the classes for performance evaluation.

Conclusions
To sum up and conclude, prior studies on contagion processes have predominantly relied on conventional regression models, structural equation modeling, or multilevel modeling as statistical approaches.While there are a few exceptions that employed network-based approaches, to the best of our knowledge, no prior study has investigated the contagion of stress among students in real classroom settings using a network econometric approach.Therefore, our study adopts a peer effects network model, advancing methodologically beyond previous approaches to analyzing contagion processes.It also illustrates how different methods (e.g., experience sampling, video-based classroom observation, and spatial network information) can complement and enrich each other within the context of process-based teaching-learning research in VET.
From a substantive perspective, our results suggest that the undifferentiated assumption of a general "more" cooperation and collaboration discussed in the introduction should be critically reconsidered.However, this does not necessarily mean that the high value of interpersonal exchange and cooperation must be fundamentally questioned.Instead, it necessitates a differentiated examination and consideration of specific personal and contextual conditions that must be present for collaborative learning and working to be effective and fruitful regarding the intended goals.This is particularly important because there is no collaboration per se that is free from content and context; rather, it always has specific content, specific goals, and a specific form.Consequently, the definition of the content, objectives, and form of collaboration, in accordance with democratic practice, requires the involvement of the individuals collaborating for cooperative purposes.

Appendix A: Estimation
Let xt contain the exogenous individual characteristics x t and the peers' characteristics Gx t and β = [β x , δ x ] ′ .The log-likelihood of the model (1) under normality of the error term is given by: with e t = y t − µ − δ t Ay t − xt β , where µ is the vector of fixed effects, which can be removed by within-transformation to yield the corresponding log-likelihood: The log-likelihood is computed as the sum of the log likelihoods for each day assuming independence across time and time homogeneity.No specific funding sources were used for this study.

Availability of data and materials
The data set cannot be made publicly accessible due to data protection guidelines.Interested persons should contact the correspondence author.The Python code is available in the online supplementary material.Use of generative AI and AI-assisted technologies in the writing process: During the preparation of this work the authors used ChatGPT and DeepL in order to polish the writing in native speaker tone and to achieve grammatical correctness.After using this tool/service, the authors reviewed and edited the content as needed and takes full responsibility for the content of the publication.

Fig. 1
Fig. 1 Seating plans of class A and class B

Fig. 2
Fig. 2 Causal graph of the econometric model illustrating the mutual influence of two students where e * t = y * t − δ t Gy * t − x * t β , where * denotes within transformation:and Taking the derivative with respect to σ and replacing the estimator σ 2 = 1

Table 1
Summary statistics of the measures of stress and coping experience

Table 2
Summary statistics of the context factorsExcept for Teachers' time pressure for class B ( T = 36 ), for all variables T = 38 measurement time points

Table 3
Correlations between the context factors of cooperative work and less in phases of teacher instruction and individual work.The teacher from class B is more likely to be under time pressure in phases of teacher instruction and less in phases in which the students work cooperatively.
Pearson correlation coefficients.Correlations for class A above the diagonal, correlations for class B below the diagonal.* denotes significance at 5 % and ** at 1 % significance level

Table 5
Maximum Likelihood Estimation results for the context factor Teachers' time pressure N = 28 students in class A, N = 25 students in class B, T = 38 measurement time points for class A, T = 36 measurement time points for class B. δ( Z) (time varying endogenous peer effect), σ 2 (variance), log-lik (log-likelihood).Standard errors in brackets.* denotes significance at 5 % and ** at 1 % significance level.Standard errors were computed by the inverse of the Hessian

Table 6
Maximum Likelihood Estimation results for the context factor Complexity of the learning content N = 28 students in class A, N = 25 students in class B, T = 38 measurement time points for both classrooms.δ( Z) (time varying endogenous peer effect), σ 2 (variance), log-lik (log-likelihood).Standard errors in brackets.* denotes significance at 5 % and ** at 1 % significance level.Standard errors were computed by the inverse of the Hessian

Table 7
Maximum Likelihood Estimation results for the context factor Cooperative work N = 28 students in class A, N = 25 students in class B, T = 38 measurement time points for both classrooms.δ( Z) (time varying endogenous peer effect), σ 2 (variance), log-lik (log-likelihood).Standard errors in brackets.* denotes significance at 5 % and ** at 1 % significance level.Standard errors were computed by the inverse of the Hessian

Table 8
Maximum likelihood estimation results for the context factor Individual work % significance level.Standard errors were computed by the inverse of the Hessian N = 28 students in class A, N = 25 students in class B, T = 38 measurement time points for both classrooms.δ( Z) (time varying endogenous peer effect), σ 2 (variance), log-lik (log-likelihood).Standard errors in brackets.* denotes significance at 5 % and ** at 1