Cusp Catastrophe Model - MIDUS

1 Cusp Catastrophe ModelA Nonlinear Model for Health Outcomes in Nursing ResearchDing-Geng (Din) Chen Feng Lin Xinguang (Jim) Chen Wan Tang Harriet KitzmanBackground:Although health outcomes may havefundamentally nonlinear relationships with relevant behavioral, psychological,cognitively, or biological predictors, most analytical models assume a linear relationship. Furthermore, some healthoutcomes may have multimodal distributions, but most statistical models in common use assume a unimodal, normaldistribution. Suitable nonlinear models should be developed to explain health :The aim of this study is to provide an overview of a cusp Catastrophe Model for examining health outcomes and topresent an example using grip strength as an indicator of a physical functioning outcome to illustrate how the technique maybe used.

2 Results using linear regression, nonlinear logistic Model , and the cusp Catastrophe Model were :Data from 935 participants from the Survey of Midlife Development in the United States ( MIDUS ) were outcome was grip strength; executive function and the inflammatory cytokine interleukin-6 were predictor :Grip strength was bimodally distributed. On the basis of fit and Model selection criteria, the cusp Model wassuperior to the linear Model and the nonlinear logistic regression Model . The cusp Catastrophe Model identifiedinterleukin-6 as a significant asymmetry factor and executive function as a significant bifurcation :The cusp Catastrophe Model is a useful alternative for explaining the nonlinear relationships commonly seenbetween health outcome and its predictors.

3 Considerations for the use of cusp Catastrophe Model in nursing researchare discussed and Words:cusp Catastrophe Model health outcomes stochastic nonlinear modelThe statistical Model used to examine a health outcomein nursing research is typically based on a linear re-gression approach. However, the influence of environ-mental, behavioral, psychological, or biological factors on healthoutcomes are often complicated and nonlinear (Ray, 1998).Small and inconsequential changes in predictive factors maylead to abrupt changes in health outcomes. Under these con-ditions, the linear approach would seriously limit knowing theeffects of factors hypothesized to be relevant to a health out-come. Other natural extensions of the linear regression toincorporate nonlinearity are nonparametric regression methods,such as the kernel regression or regression/smoothing splinesin low-dimensional scenarios.

4 For high-dimensional data, tech-niques such as the additive models, multivariate adaptiveregression splines, random forests, neural networks, and supportvector machine, etc., which have been discussed extensively inFaraway (2006), are available. However, these nonparametricregressions do not have the mechanisms to identify and incor-porate cusp jumps, which are the fundamental advantages ofthe cusp Catastrophe cusp Catastrophe Model is capable of handling complexlinear and nonlinear relationships simultaneously using a high-order probability density function that has the advantage ofbeing able to incorporate sudden behavioral jumps (Zeeman,1976). Historically, the cusp Catastrophe Model has been ap-plied to prediction of health behaviors or system quality andsafety, such as attitudes and social behavior (Flay, 1978), therapyand program evaluation (Guastello, 1982), accident processes(Guastello, 1989), anxiety and performance (Hardy & Parfitt,1991), cognitive development (van der Maas & Molenaar,1992), selection of target behaviors (Bosch & Fuqua, 2001), ad-olescent alcohol use (Clair, 1998), changes in adolescent sub-stance use (Mazanov & Byrne, 2006), complexity of drinkingrelapse (Witkiewitz & Marlatt, 2007), binge drinking amongcollege students (Guastello, Aruka, Doyle, & Smerz,, 2008), earlyDing-Geng (Din)

5 Chen, PhD,isProfessor, SchoolofNursing and Departmentof Biostatistics and Computational Biology, University of Rochester MedicalCenter, New York, and Tianjin International Joint Academy of Biotechnologyand Medicine, Lin, PhD, RN,is Assistant Professor, School of Nursing and Depart-ment of Psychiatry, School of Medicine and Dentistry, University of RochesterMedical Center, New (Jim) Chen, PhD,is Professor, Department of Epidemiology,College of Public Health and Health Professions and College of Medicine,University of Florida, Tang, PhD,is Associate Professor, Department of Biostatistics andComputational Biology, University of Rochester Medical Center, New Kitzman, PhD, RN, FAAN,is Professor, School of Nursing, Univer-sity of Rochester Medical Center, New : 2014 Wolters Kluwer Health | Lippincott Williams & Wilkins.

6 Unauthorized reproduction of this article is initiation among young adolescents (Chen et al.,2010), nursing turnover (Wagner, 2010), and HIV prevention(Chen, Stanton, Chen, & Li, 2013). The cusp Catastrophe Model ,though, has seldom been applied to the understanding ofhealth outcomes, such as the incidence of a disease or changesin a health condition where the nature can be extremely com-plicated and dynamic. The goal of this article is to provide anoverview of the cusp Catastrophe Model , focusing on its appli-cation in the examination of health outcomes. Such a methodcan assist nurse researchers in taking the next steps in under-standing the multifaceted nonlinear impact of different predictorson health outcomes in a new way.

7 Findings based on the cuspcatastrophe Model may guide evaluation of outcomes frominterventions more effectively than other of the Cusp Catastrophe ModelPopularized in the 1970s by Thom (1975), Catastrophe theorywas originally proposed to explain complicated sets of behaviorsthat include both continuous changes and sudden discontinuousor catastrophical changes. Theoretically, five elements calledcatastrophe flags define the presence of Catastrophe (Gilmore,1993): (a) bimodality, where two distinctly different modes ex-ist in the distribution of the outcome; (b) sudden jump, wherethe outcome changes abruptly between the modes even withslight changes in the predictors; (c) inaccessibility, where an out-come value in the area between the modes is unlikely; (d) hys-teresis, where change from one mode to the other cannot bedetermined by the same values for control factors; and (e) di-vergence, where a slight change in the control factors can leadto substantial change in the outcome and deviation from thelinear Model .

8 In summary, a cusp Catastrophe Model would beparticularly appropriate if am outcome measure has theproperties of a bimodal distribution (bimodality)withspurts(sudden jump) along with a middle inaccessible region be-tween two modes (inaccessibility) with delay between tran-sitions (hysteresis) and deviation from a linear relationshipbetween the response outcome measure and the predictors(divergence). Further definition andexplanations are summa-rized in Table cusp Catastrophe models have been well establishedtheoretically and extensively applied to physical sciences, cuspcatastrophe models were criticized in the early 1970s in appli-cations in social and behavior sciences partially because math-ematics were misused, models were based on unreasonableassumptions, and predictions were thought to be vague or im-possible to test experimentally ( , Sussmann & Zahler, 1978,p.)

9 118), charges that were later reconsidered by Rosser (2007),who argued for utility of the cusp modeling approach for prob-lems with dynamic discontinuities in outcomes. It is of interestin nursing in part because it is associated with theories pro-posed by Rogers (1971).The deterministic cusp Catastrophe Model is specified us-ing three components: two control factors ( ,xandy)andone outcome variable ( ,z). This Model is defined by a differ-ential equations-based dynamic system:dzdt dV z;x;y dz(1)where the potential function isVz;x;y 14z4 12z2y the functionV,theargumentxis called asymmetry ornormal control factor where the outcomezchanges asymmet-rically from one mode to the other eventually asxincreases,yis called bifurcation or splitting control factor, which causesthe outcome surface to split and bifurcate from smooth changesto sudden jumps as y increases.

10 Bothxandyare linked to de-termine the outcome variablezin a three-dimensional outcomeresponse surface. When the right side of Equation 1 movestoward 0, the outcomezwill not change with time. Such statusis called equilibrium; this assumption is needed to interpret cuspmodels based on cross-sectional data. In general, the behaviorof the outcomez, that is, how it changes with timet, is in generalcomplicated, but each subject will move toward an equilib-rium status. Figure 1 graphically depicts the equilibrium plane,which reflects the response surface of the outcome measure (z)at various combinations of asymmetry control factor (x)andbifurcation control factor (y).Cusp Catastrophe models can be assessed qualitatively andquantitatively.