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2 Overview of the Cusp Catastrophe Model

2.1 Deterministic Cusp Catastrophe Model

To overcome the limitation of linear analytical approach, cusp catastrophe model is proposed to model any system outcomes which can incorporate both linear and nonlinear along with discontinuous transitions in equilibrium states as control variables are varied. According to the catastrophe systems theory [2], the dynamics for system outcome is modeled as -V(y; a, /3) = ay + 1 /3y2 1 y4 with dynamical system in the form of ay = av(y,a,{3), where α is called asymmetry control variable and β is called bifurcation control variable which are linked to determine the health outcome variable y.

2.2 Stochastic Cusp Catastrophe Model

This cusp catastrophe model is fundamentally a deterministic. In order to use this cusp model for real-life applications which are stochastic nature, Cobb and his colleagues [2-3] casted this model into a stochastic differential equation as follows dy = av(y,a,{3) dt + dW(t), where dW(t) is a white noise Wiener process with variance σ2. With this SDE, the probability distribution of the health behavior measure (y) under equilibrium can be expressed as f(y) = lJ1 exp a(y-,.1)+2{3(y-,.

where ψ is a normalizing constant and λ is to determine the origin of y. With this

density function, the theory of maximum likelihood can be employed for estimating parameters and statistical inference which is implemented in R Package "Cusp" [1], 2009). Specifically for data from n study subjects, we denote the observed p dependent variables as Yi = (Yi1, Yi2,...,Yip), q predictor variables Xi = (Xi1, Xi2,..., Xiq), the behavior measure yi and the control variables αi and βi are modeled as linear combinations of the X and Z as follows:

yi = w 0 +w1Yi1 +w 2 Yi 2 + ... + w p Yip= w ´ Yi

a i = a 0 + a 1 X i1 + a 2 X i 2 + ... + a q X iq = a ´ X I (1)

b i =b 0 +b 1 X i1 +b 2 X i 2+ ... +b q X iq= b ´ X i

Then the log-likelihood function for these n observations is as follows:

hich is then maximized to estimate model parameters for associated statistical inference. A likelihood-ratio Chi-square test is used for model comparison and to test the goodness-of-fit between cusp model to the linear regression model. Other model selection criteria are also calculated, such as the R2 =1-(error variance/variance of y) as well as the model selection information indices of AIC and BIC [5-6].

2.3 Detection of Cusp Catastrophe

In order to establish the presence of a cusp catastrophe, three guidelines are proposed by Cobb [5]. First, the cusp model should be substantially better than linear model which can be evaluated by the likelihood ratio test. Second, any of the coefficients w1, ..., wp should be statistically significant (w0 does not have to) and at least one of the αi's or βi's should be statistically significant. Thirdly, at least 10% of the (αi, βi) pairs should lie within the bifurcation region. This 10% guideline of Cobb was modified by Hartelman [6] to compare the cusp model to the non-linear logistic regression:aiyi = 1/ 1 + e {3i +Ei (i = 1,… , n) with better AIC and BIC for cusp model than for the logistic regression.

 
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