The Threshold Non-linear Model for the Effects of Cumulative Stressors and Traumas: A Chained Cusp Catastrophe Analysis

Studying the macro-dynamics of traumatization should significantly advance psychological science. The current article tested the validity of the non-linear threshold model of the effects of cumulative stressors and traumas (CST). We used a combined data set of adults and adolescents from Egypt, Kuwait, and the UK (N = 968) that measured CST, internalizing, externalizing and thought disorders. We used the cusp catastrophe analysis that can identify the non-linear threshold dynamics. The non-linear cusp (threshold) model accounted for a much higher variance than the linear model, indicating the presence of threshold effects of CST on internalizing, externalizing and thought disorders. Interventions should target the multiply traumatized who are at risk for the cusp effects and not only those who are victims of the cusp. The study provided evidence for the utility of a paradigm shift to study the macro-dynamics of stressors and traumas, and not only their micro-dynamics.


Clinical Vignette That Exemplify the Impact of Cumulative Stressors and Traumas
Mrs. XX was a 55-year-old Iraqi refugee woman who has nine living children. In Iraq, she lost two brothers and two older sons who had been killed by Saddam's regime. She had to flee, walking in the desert for days with her family to Saudi Arabia. She spent 4 years in a refugee camp in the desert. When she arrived in the United States, she encountered discrimination and the bullying of her children in school. When she discovered her husband's infidelity, she divorced him. All of her family members described her as the heroine of the family, whose strength and resilience helped her and they survive. The clinician was treating one of her sons. Several years later, she was involved in a moderate car accident that resulted in some bruises. After this accident, she developed intense symptoms of fear, panic attacks, auditory and visual hallucinations (e.g., seeing blood everywhere in the house; some type of thought disorder), and nightmares related to the terror she had previously experienced. Further, she called her ex-husband to beg for forgiveness, for divorcing him, and for violating his rights as a man. (Kira & Tummala-Narra, 2015). Authors consider traumatic events as one type of stressors that are acute and are an intricate part of the general theory of stressors. Stressors, in this context, mean all kinds of acute (traumatic), chronic and non-chronic stressors. Studying the cumulative total impact and the macro-dynamics of the total impact of stressors will significantly advance clinical science. In this clinical vignette, the client survived a series of chained severe and chronic stressors over years that have a cumulative effect, and the last, which probably may have been one of the least severe, became the ''the last straw that broke the camel's back.'', with the sudden emergence of thought and internalizing disorders. While micro-dynamics of traumatization focus more on studying the impact of a single trauma or distinct trauma type which is the current dominant trend in stress and trauma research, traumatization macro-dynamics focus on the impact of total exposure and the interaction between different traumas and stressors and their total cumulative impact on the person's mental health. Cumulative stressors and traumas (CST) has consistently been a powerful linear predictor of the severity, complexity, and comorbidity of psychopathology (e.g., Cloitre et al., 2009;Kira et al., 2008;Kira et al., 2013c;Kira, Omidy, & Ashby, 2014;Martin, Cromer, DePrince, & Freyd, 2013). Macro-dynamics investigate also the impact of early traumas, e.g., of attachment and oppression on subsequent traumas or the mechanisms of trauma proliferation (Kira et. al., 2018a). Examples of the proliferated trauma dynamics are the proliferation of attachment and early childhood traumas to other subsequent traumas (Ferrajão, Badoud, & Oliveira, 2017;Kanninen, Punamäki, & Qouta, 2003;Mikulincer, Solomon, Shaver, & Ein-Dor, 2014; for meta-analysis see Madigan, Brumariu, Villani, Atkinson, & Lyons-Ruth, 2016;and Woodhouse, Ayers, & Field, 2015). This might include the proliferation of childhood poly-victimization (Collings, Penning, & Valjee, 2014;Ford, Elhai, Connor, & Frueh, 2010). It may also include the proliferation of early adolescence systemic I. Kira et al. trauma such as discrimination leading to subsequent traumas (Al-Ibraheem, Kira, Aljakoub, & Al-Ibraheem, 2017;Kira, Alawneh, Aboumediene, Lewandowski, & Laddis, 2014;Szymanski & Balsam, 2011). The inclusion of social identity systemic traumas (collective identity traumas) that are continuous and intersecting, such as different discrimination and oppression (e.g., refugees, minority groups experience, gender discrimination), have been identified as severe trauma types that proliferate to subsequent traumas (Holmes, Facemire, and DaFonseca, 2016;Reisner et al., 2016;Kira et al., 2018a).
However, these macro-dynamics are poorly understood. While most of the studies of micro-dynamics focus on the dose-response linear models, Macro-dynamics involve more the non-linear traumatization dynamics or the total impact of cumulative stressors and traumas (CST). Previous, concurrent and subsequent life events, impact the individual conjointly linearly and non-linearly. We emphasize that in evaluating the impact of life events, we cannot separate the impact of the exposure to chronic stressors, major life stressors and different trauma types in real life and real-time. Focusing solely on PTSD Criterion "A" trauma types, which was and still the dominant trend in trauma and PTSD research, can be misleading. A recent study found that adding the non-criterion "A" traumas of attachment and collective identity trauma types (a non-criterion "A" stressors and traumas) resulted in an increase in the incremental predictive validity of criterion "A" over six-fold (Kira et al., 2019b).
The total influence of cumulative and proliferated stressors, chronic and acute, on mental health outcomes, as we observed in the clinical vignette, is often complicated and nonlinear. Relatively small and inconsequential changes in predictive factors may lead to abrupt changes in behavior. Traditional statistical methods assume relationships linearity between variables and have provided a wealth of information about correlates of cumulative stressors and traumas (CST). However, complex behavior macro-dynamics may be poorly represented by linear models (e.g., Ehlers, 1995). Non-linear dynamic systems model provides a framework for exploring and evaluating complex systems and behaviors wherein the effects of independent variables such as CST might follow a threshold non-linear path.
Studies of cumulative risk found to follow a non-linear path in causing difficulties and distress (Oldfield, Humphrey, & Hebron, 2015). In this dynamic non-linear threshold model of causality (Kira & Wrobel, 2016), the accumulation of stressors' impact reaches a threshold where the last stressor becomes the "straw that breaks the camel's back" breaching the threshold of distress tolerance.
This non-linear threshold model presumes that individuals have different breaking points. Even for people with higher distress tolerance, like the mother in the clinical vignette, the added stressor can cause the person to "break". These nonlinear cusp shifts from one state to another can happen upon exposure to cumulative and proliferated external and internal pressures/stressors (Guastello, & Liebovitch, 2009;Kira & Wroble, 2016;Zeeman, 1976) and the most recent stres-sor (or chain of stressors) can falsely appear as the direct cause of the disorder. Within this dynamic system model, including linear and non-linear causal chains and loops, the relatively recent single or chain traumas can be the stressor that finally triggers a pathological response, which is not the actual cause of the symptom presentation.
A complex dynamic systems perspective of macro dynamics may be more helpful than a traditional linear model in understanding the etiology of psychopathology. For example, the non-linear models of the relationship between trauma and PTSD explained over three times the variance explained by the linear model (Kira et al., 2019b). Armey and Crowther (2008), comparing a linear versus a non-linear model of aversive self-awareness, dissociation, and non-suicidal self-injury, found, as they predicted, that the non-linear model evidenced a better fit to the data, accounting for 6 times the variance (66%) than the linear model (9% -10%). Similar results were observed in the effects of torture on mental health (Kira, 2017), and the role of identity salience in the mental health of Syrian refugees (Kira, Shuwiekh, Al Ibraheem, & Aljakoub, 2018b). Such a complex dynamic systems perspective of macro dynamics may be more helpful than a traditional linear model in understanding the etiology of different forms of psychopathology (i.e., internalizing, externalizing and thought disorders).
Cumulative stressors and traumas (CST) has consistently been a powerful linear predictor of the severity and comorbidity of mental health disorders (e.g., Cloitre et al., 2009;Kira et al., 2008;Kira et al., 2013c;Kira, Omidy & Ashby, 2014;Martin, Cromer, DePrince, & Freyd, 2013). The impact of CST dynamics is related to various processes including distress tolerance (Leyro, Zvolensky, & Bernstein, 2010), stress sensitization, the kindling process (Post, Weiss, & Smith, 1995), and the diathesis-stress model (Stein, Jang, Taylor, Vernon, & Livesley, 2002). All assumed a stress tolerance and buffer that breaks upon reaching a threshold that may be different from person to person. There were previous attempts to address physical and mental health issues as cusp functions of age and stress in an occupational setting (Guastello, 1992(Guastello, , 1995. The linear statistical approach would seriously limit knowing the effects of factors hypothesized to be relevant to the effects of different factors that influence behavior (i.e., cumulative stress, genetic, epigenetic and environmental factors). The nonparametric regressions do not have the mechanisms to identify and incorporate "cusp jumps", which are the fundamental advantages of the cusp catastrophe models. The statistical methods used to examine mental health variables in psychological research are typically based on a linear regression approach. While there are different non-linear statistical models (e.g., quadratic, cubic) that identify different non-linear trajectories, they do not identify if these non-linear trajectories inherent in the relationship between variables or follow a threshold breaking points. Catastrophe cusp theory (Chen & Chen, 2015;Thom, 1975) is one application of non-linear dynamics that can be used to verify the threshold model and to describe the factors contributing to the development of DOI: 10.4236/psych.2020.113025 I. Kira et al. psychopathology following the reach of a certain cumulative intensity value of CST.
Catastrophe theory allows for the modeling of large, "catastrophic", non-linear changes in behavior that result from small changes in continuous predictor variables (as the observable sudden change in behavior in the clinical vignette).
Nonparametric regressions do not have the mechanisms to identify and incorporate "cusp jumps", which are the fundamental advantages of the cusp catastrophe models. Theory and methodology from nonlinear dynamical systems (NDS) provided a considerable advantage to the study of psychopathology. Catastrophe cusp theory (Chen & Chen, 2015;Thom, 1975) is one application of non-linear dynamics that can be used to verify the threshold model and to describe the factors contributing to the development of psychopathology following the reach of a certain cumulative intensity value of CST. Catastrophe theory allows for the modeling of large, "catastrophic", non-linear changes in behavior that result from small changes in the continuous predictor. This type of catastrophe model is referred to as the "cusp catastrophe" model (Gilmore, 1993) since a sudden behavioral change is exhibited once a predictor variables cross the "cusp" threshold. These nonlinear cusp shifts from one state to another can happen upon exposure to cumulative and proliferated external and internal pressures/stressors (Guastello & Liebovitch, 2009;Kira & Wroble, 2016;Zeeman, 1976) and the most recent stressor (or chain of stressors) can falsely appear as the direct cause of psychopathological response. Within this dynamic system model, including linear and non-linear causal chains and loops, the relatively recent single or chain of stressors/traumas can be the event/s that finally triggers a psychopathological response and not the actual cause of the emerging symptom.
Catastrophe theory is a branch of bifurcation theory in the study of dynamical systems to study phenomena characterized by sudden shifts in behavior from small changes in circumstances. The most commonly used catastrophe model in health science investigations thus far has been the cusp, which has two control parameters: a) asymmetry, which has a smooth and linear relationship with the system output, and b) bifurcation, which induces a threshold effect on system output. The cusp catastrophe model is capable of handling complex linear and nonlinear relationships simultaneously using a high-order probability density function that has the advantage of being able to incorporate sudden behavioral jumps (Zeeman, 1976). Historically, the cusp catastrophe model has been applied to the prediction of health behaviors (Flay, 1978). For psychopathology, we propose two control factors to control the outcome response where personal characteristics as a function of his/her age maturity is the asymmetry control factor that controls the outcome changes asymmetrically from one mode to the another mode eventually as it increases, while the stressors and traumas accumulation is the bifurcation control factor that controls the outcome to split and bifurcate from smooth changes to sudden jumps as it increases.
To date, a dynamic systems approach has not yet been used to investigate the propositions of the non-linear cumulative stressors impact on psychopathology.
Most of the previous studies focused on personal characteristics, genetic, epigenetic and environmental factors that lead to psychopathology, however, the non-linear impact of cumulative stressors rarely been examined. A recent study tested the cusp model on suicidal phenomena and found that cumulative stressors and traumas are significant bifurcation factor and age as asymmetrical factor and the cusp non-linear model accounted for much higher variance than the linear model (Kira, Barger, Shuwiekh, Kucharska, & Al-Huwailah, 2019a). The primary aim of the present study was to examine the cumulative non-linear dynamics that lead to psychopathology and provide evidence that the non-linear model explains more variance than linear models of psychopathology.
Recent advanced analysis of the structure of psychopathology, using both the diagnostic and continuous models identified a single general factor with three basic components of psychopathology: internalizing, externalizing and thought disorder (psychoticism) (e.g. Caspi et al., 2014;Laceulle, Volleberge, & Ormel, 2015). It is important to verify the threshold model of the effects of CTS on Psychopathology in Western and non-Western cultures since most of the clinical studies have been conducted on samples from Western cultures.
The goal of the current study is to explore the validity of the cusp (threshold) model of the effects of cumulative stressors (chronic and traumatic) as bifurcation controlling factor ( with age as asymmetrical controlling factor) on the emergence of the different types of psychopathology (i.e., internalizing, externalizing and thought disorder) in different age groups and Western and non-Western participants.

Hypotheses
Hypothesis 1: The non-linear models (quadratic and cubic) will explain more variance than the linear models in estimating the association between CST and internalizing, externalizing, and thought disorders.
Hypothesis 2: Cusp Model will explain a higher variance than the linear model in the association between CST as bifurcation control factor and age as asymmetry control factor and internalizing, externalizing, and thought disorders as outcome Cusp variables.
The following describes in detail each of the three sub-samples.

Procedures for the Second Sample: The Kuwait Data (N = 300)
The research team consisted of 4 doctoral students with a professor in Psychology as the project leader. The team members have been trained in conducting tools and interviewing. The sample designed was a purposeful quota sample to represent different age groups, students, and community (50% of students and 50% community). The questionnaire was administered to college students using group class format, while the community participants were interviewed personally.

Procedures for the Third Sample: The UK Data (N = 178)
Participants were recruited using two strategies. The first was by utilizing a crowdsourcing website "prolific.ac", where the participants received £2 for completing the survey. It was specified in the settings that they have to be students, survey. It was specified in the settings that they have to be students, age range 18 -40, and nationality, country of birth and current country of residence: UK.
Further, the link to the online survey was sent to a student's university organization in London. Table 1 details the demographics of each of three combined samples.
Test-retest using an independent sample of 35 males with four weeks interval yielded excellent stability coefficients (0.995 for cumulative trauma frequency, and 0.997 for cumulative trauma appraisal). The alpha for the main scale of occurrence was 0.89 in current data.
Psychopathology Screening Measure (Kira, Shuwiekh, & Kucharska, 2017) Adapted GAIN Short Screener (GAIN-SS) (Dennis, Chan, & Funk, 2006) is a screener, that quickly identifies clients (adults and adolescents) who are likely to have mental health disorders, issues with crime/violence, and issues with substance use. The participant is asked to indicate if the behavior (or feeling) happened in the past month (scored 4), or happened in the last 2 -3 months (scored 3), or in the last 3 -12 months (scored 2), or the last year or more (scored 1), or never happened (scored 0). The original measure includes three parts: Internalizing, Externalizing, and substance abuse sections. High scores indicate potentially higher symptoms in these areas. The measure was adapted to include a section for psychoticism and dissociation, adding items from psychoticism/dissociation subscale of cumulative trauma disorder scale (Kira et al., 2012). Further, items were added to internalizing that are related to PTSD symptoms. The original version did not include different PTSD symptoms. The goal of its adaptation was to include the three basic components of psychopathology: Internalizing, Externa-

Data Analysis
Descriptive statistics (frequency, rate, mean and standard deviation) were used to describe the sample characteristics. The linear and non-linear (quadratic and cubic) relationships among the predictor variable: cumulative Stressors and Traumas (CST) and the outcome variables: internalizing, externalizing and thought disorder, were examined first using the conventional curve-estimation regression.
We used SPSS-IBM 22 in data analysis, at this stage. The chained cusp quantum models were tested employing a cusp catastrophe modeling method to test three models. Cusp catastrophe models involve asymmetry controlling factors (e.g., age) and bifurcation factors (e.g., CST) that contribute to the cusp, or critical shift in behavior, or a dependent variable (e.g., internalizing, externalizing, and thought disorders). We used age as an asymmetry variable and CST as the bifurcation variable to explore the potential Cusp change in the effects of CST on internalizing, externalizing and thought disorders. The three cusp models were used to assess the fit of each of the three cusp models of CST cusp impact across age (asymmetry) and the bifurcating impact of CST on internalizing, externalizing, and thought disorders. The appropriateness of a cusp catastrophe model is evaluated based on the comparison with both linear and cusp models by using an R 2 value, the Akaike information criterion (AIC), and the Bayesian information criterion (BIC) statistics (Browne, 2000;Zucchini, 2000). The model with the highest

Conclusion
Current findings propose a paradigm shift in clinical research to investigate the trauma macro-dynamics such as cumulative and proliferation impact of stressors (chronic and traumatic), instead, or in addition to the microdynamics of studying single trauma which is the current focus of most clinical science research.    The current study, while provides significant potential contributions to understanding the dynamics leading to internalizing, externalizing and thought disorders, has several limitations. One of the limitations was using age as a function of the person's characteristics at a certain point of time as the asymmetrical factor in the Cusp dynamics. Age can provide a limited representation of a person's characteristics. A variety of alternative asymmetrical factors may be possible.
Another limitation is that the study used mostly convenient samples that may have limited and biased representation. We recommend more studies that use more representative samples. Another limitation is that the measures we used are based on participants' self-reports, which could be subject to under-or over-reporting of events due to current symptoms, embarrassment, shame, or social desirability.
Another limitation is that we utilized a cross-sectional design in testing our model; cross-sectional data can produce biased estimates (Maxwell & Cole, 2007).
Cross-sectional design does not provide deterministic hard science models, only probabilistic relationships. Deterministic causal relationships can be obtained using experimental designs if feasible.