# Cusp Catastrophe Regression and Its Application in Public Health and Behavioral Research

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## Abstract

**:**

## 1. Introduction

## 2. An Overview of the Cusp Catastrophe Model

#### 2.1. Cusp Catastrophe Model

#### 2.2. Implementation of Cusp Catastrophe Model

^{2}.

- (1)
- Negative log-likelihood values and the associated likelihood-ratio (LR) ${\chi}^{2}$-test where the smaller negative log-likelihood values, the better cusp model than the alternative models. If the LR ${\chi}^{2}$-test is used, the p-value < 0.05 would indicate a better fit of the cusp model.
- (2)
- R
^{2}defined as R^{2}= 1 − (error variance/variance of y) where the larger the R^{2}, the better fit of the cusp model. Note that this is a pseudo-R^{2}since real R^{2}cannot be calculated for a cusp model. - (3)
- (4)
- At least 10% (α, β) of the data pairs are located within the cusp region [28,29,30], or a nonlinear least squares regression with the logistic curve proposed by Hartelman [29], Hartelman et al. [31], and van der Maas and colleagues [32]:$${y}_{i}=\frac{1}{1+{e}^{-{\alpha}_{i/{\beta}_{i}^{2}}}}+{\epsilon}_{i}$$

## 3. Cusp Catastrophe Regression

#### 3.1. Introduction to the Cusp Catastrophe Regression Model

_{i}, plus measurement errors characterized by ${\epsilon}_{i}$. The ${\epsilon}_{i}$ are assumed to be normally distributed as ${\u03f5}_{i}~N\left(0,{\sigma}^{2}\right)$. The true Y

_{i}in Equation (6) is then one of the real roots from the cusp catastrophe Equation (1) under equilibrium:

_{i}are converted from a canonical deterministic to a stochastic and probabilistic process. Consequently, sudden shifts in Y

_{i}are no longer restricted to the edge of the two threshold lines, but occur more frequently as Y

_{i}approaches toward the thresholds. Likewise, the stable regions based on the deterministic concept are also relaxed to allow for measurements of frequency or likelihood. In another word, a stable region means that Y

_{i}for any subject i has a large probability to be located in that area on the equilibrium plane of the cusp model.

#### 3.2. Implementation Strategy

#### 3.3. Special Properties of the Cusp Model

- (a)
- if $\alpha =\beta =\mathrm{\Delta}=0,$ all the three roots would be the same corresponding to the cusp point (labeled O in Figure 1);
- (b)
- if $\mathrm{\Delta}=0,$ but $\alpha \ne 0or\beta \ne $0, two of the three roots are equal which are corresponding to the two threshold lines OQ and OR in Figure 1 to characterize the boundary of the cusp region; and
- (c)
- if $\mathrm{\Delta}<0,$ and $\alpha \ne 0or\beta \ne $0, all three roots are different which form the cusp region between OQ and OR in Figure 1.

- (1)
- The delay convention is to select the root from the $\frac{dV\left(Y;\alpha ,\beta \right)}{dY}=0$ in Equation (1) that are close to the observed y;
- (2)
- The Maxwell convention is to select the root from the $\frac{dV\left(Y;\alpha ,\beta \right)}{dY}=0$ in Equation (1) that corresponds to the minimum of $V\left(Y;\alpha ,\beta \right)=\alpha Y+\frac{1}{2}\beta {Y}^{2}-\frac{1}{4}{Y}^{4}$.

#### 3.4. Monte-Carlo Simulations

_{1}and x

_{2}are generated from the standard normal distribution with true regression parameter vectors

**a**= (2, 2, 0),

**b**= (2, 0, 2) using Equation (8) with a

_{2}= 0 (implying that x

_{1}is an asymmetry variable) and Equation (9) with b

_{1}= 0 (implying that x

_{2}is a bifurcation variable). This parameterization is used to assess if the RegCusp can correctly identify the role of the two variables while detecting the cusp catastrophe from the simulated data.

- Step 1: randomly simulate n = 300 observations for x
_{1}and x_{2}from the standard normal distribution and for the error term of ${\epsilon}_{i}$ from normal distribution with mean 0 and standard deviation $\sigma $ = 0.5; - Step 2: calculate ${\alpha}_{i}$ and ${\beta}_{i}$ with Equations (8) and (9) using
**a**= (2, 2, 0) and**b**= (2, 0, 2) as well as the x_{1}and x_{2}from Step 1; - Step 3: for each set of ${\alpha}_{i}$ and ${\beta}_{i}$ from Step 2, we solve Equation (7) to obtain Y
_{i}. Based on the discussions in Section 3.2, we select one root corresponding to the Maxwell convention which is to minimize V (Y_{i}, ${\alpha}_{i}$, ${\beta}_{i}$); - Step 4: using Y
_{i}from Step 3 and ${\epsilon}_{i}$ from Step 1, calculate y_{i}based on the Equation (6); - Step 5: with the data from Steps 1 to 4, estimate
**a**and**b**using the maximum likelihood estimation in Equation (11).

**a**= (2, 2, 0) and

**b**= (2, 0, 2). The simulation results suggest the unbiasedness from RegCusp.

**a**and

**b**were calculated from the 5000 simulations using the gradient Hessian matrix from the likelihood function. The empirical coverage probability (column “ECP”) was also presented in the table. Statistically, the sampling variance (i.e., “EmpV”) should be a consistent estimate of the true variance of the parameters and the estimated variance (i.e., “EstV”) should be close to “EmpV” so that the “ECP” should be close to 95%. A close assessment of the results in Table 1 reveals that the estimates of ECP are less than 95% for all of the parameters, indicating the lack of efficiency of the Hessian matrix for variance estimation. As shown in Figure 2, the distribution of the estimated parameters

**a**and

**b**from the 5000 simulations are highly leptokurtic which indicates that the variance estimation in RegCusp is inadequate and should be re-calibrated. To solve this problem, we resorted to a bootstrapping procedure.

#### 3.5. Boostrapping Re-Sampling for Variance Estimation

- Step 1 to Step 5, the same as in Section 3.4;
- Step 6: bootstrap the data from Step 4 and re-run the estimation Step 5 for B = 500 times to generate a bootstrapping sample.

- (1)
- to estimate the variances from these 500 samples. With the estimated bootstrapping variance, we can then construct the 95% CI for the parameter. Then the CIs are used to calculate the coverage probability (denoted by “ECP1”).
- (2)
- to use the B = 500 samples to direct construct the 95% confidence intervals (CI) for each estimate and then the associated coverage probability (denoted by “ECP2”).

## 4. Real Data Analysis

#### 4.1. Data Source

_{1}(mean = 14.29, SD = 2.39). Participants were asked 18 true/false questions regarding their knowledge on HIV transmission and prevention. An example question read, “A woman can get HIV if she has anal sex with a man who has HIV”. A participant received one point for each correct answer. Participants with higher total scores were considered more knowledgeable about HIV/AIDS. The score ranged from 0 (no HIV knowledge) to 18 (fully knowledgeable).

_{2}(mean score = 4.36, SD = 0.88). Response efficacy measures the perceived effectiveness of condom use in preventing HIV infection. This variable was measured using three items (Cronbach alpha = 0.80). An example item read: “Condoms are an important way to prevent you from getting a sexually transmitted disease (STD).” Items were assessed using the 5-point Likert scale with 1 (strongly disagree) and 5 (strongly agree).

_{1}and the outcome variable condom-use self-efficacy y. Different from conventional regression analysis, in cusp catastrophe model, the continuous relationship between x

_{1}and y can be bifurcated by x

_{2}, the response efficacy or perceived effectiveness of condom use. When x

_{2}is located behind the bifurcation point O, the positive relationship between x

_{1}and y will be continuous; however, when the bifurcation variable x

_{2}is located in front of point O, changes in y will become discrete with two y values distributed at all (x

_{1}, x

_{2}) points corresponding to the cusp equilibrium surface.

#### 4.2. Linear Regression Analysis

_{1}(HIV knowledge, β = 0.0080, p < 0.01) and x

_{2}(the perceived condom efficacy, β = 0.2033, p < 0.01) with y (condom-use self-efficacy). The positive relationships are supported by the behavioral theory. However, the R

^{2}estimated from the linear regression, including the adjusted R

^{2}was less than 8%, rather small.

#### 4.3. SDECusp Analysis

_{1}(HIV knowledge, a = 0.1760, p < 0.01) and the bifurcation variable x

_{2}(perceived condom efficacy, b = 0.2147, p < 0.01) were highly significant in predicting the outcome variable y (condom use self-efficacy), as observed from the linear regression analysis.

^{2}for the cusp modeling was 34%, much greater than 7.8%, the R

^{2}for linear regression model. This result indicates the superiority of cusp catastrophe model over linear regression model in quantifying the relationship between the predictor and the outcome variable. Chi-square test also demonstrated that the data could be better modeled with a cusp catastrophe than a linear regression model (χ

^{2}= 1110, df = 2, p < 0.0001).

_{1}for the asymmetry variable and b

_{1}for the bifurcation variable were much smaller from cusp catastrophe model (0.1760 and 0.2147) than from the linear regression model (0.0080 and 0.2033).

_{0}, a

_{1}, b

_{0}and b

_{1}) from Table 4 and the standardized A and B to calculate the standardized asymmetry control factor α and the bifurcation control factor β. These standardized factors are then transformed back to the original scales as seen in Figure 3. All of the data points (n = 1790) are plotted in Figure 3.

#### 4.4. Analysis with RegCusp Modeling Method

_{0}= −0.083, a

_{1}= 0.094 (p < 0.01 for both). Likewise, the estimated two parameters for the bifurcation variable were b

_{0}= 1.568; b

_{1}= 0.672 (p < 0.01).

_{1}(HIV knowledge) and x

_{2}(response efficacy) both significantly and positively predicted the outcome variable y (condom use self-efficacy). The results were consistent with those obtained from linear regression (Section 4.2) as well as the SDECusp modeling analysis (Section 4.3) with regard to the prediction direction. Furthermore, the estimated coefficients for the two control variables in RegCusp were closer to each other as compared to those estimated with the SDECusp modeling method.

_{1}(HIV knowledge) = 14.55, and x

_{2}(response efficacy) = 2.33. It is clear that these two values are completely within the range of the original data for the two predictor variables. Furthermore, the two values of the cusp point O were practically meaningful. According to the estimated cusp point, to have a sudden change in condom-use self-efficacy, a student should have adequate knowledge with s knowledge score of at least 14.55 on the 18-item HIV Knowledge Scale. To achieve a sudden change, a student should have a minimum score of 2.33 on the Response Efficacy measure within the original score range of 1 (strongly disagree) and 5 (strongly agree). Therefore, when compared to the results from SDECusp/R “cusp” package, the results from the RegCusp was more convincing.

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Cusp catastrophe model showing continuous and discrete changes in outcome (Y) on the equilibrium plane as the asymmetry variable α and the bifurcation variable β changes.

**Figure 2.**Sampling distributions of 5000 simulations for the parameters in RegCusp model. The x-axis denotes the range of the simulated parameters and y-axis denote the range of the associated probability density.

Parameter | Mean | Med | EmpV | EstV | ECP |
---|---|---|---|---|---|

a_{0} | 2.0094 | 2.0035 | 0.0079 | 0.9525 | 0.3323 |

a_{1} | 2.0106 | 2.0062 | 0.0134 | 1.2496 | 0.2558 |

a_{2} | −0.0014 | −0.0009 | 0.0082 | 0.3232 | 0.2502 |

b_{0} | 2.0038 | 2.0016 | 0.0048 | 0.3240 | 0.3093 |

b_{1} | −0.0069 | −0.0029 | 0.0102 | 0.7649 | 0.2483 |

b_{2} | 2.0115 | 2.0057 | 0.0169 | 1.4016 | 0.2246 |

Parameter | Mean | Med | EmpV | EstV | ECP1 | ECP2 |
---|---|---|---|---|---|---|

a_{0} | 2.023 | 2.014 | 0.0370 | 0.0369 | 0.949 | 0.953 |

a_{1} | 2.031 | 2.009 | 0.0614 | 0.0615 | 0.951 | 0.949 |

a_{2} | −0.014 | 0.002 | 0.0363 | 0.0365 | 0.952 | 0.950 |

b_{0} | 2.005 | 2.009 | 0.0195 | 0.0194 | 0.948 | 0.951 |

b_{1} | −0.023 | −0.009 | 0.0467 | 0.0466 | 0.949 | 0.953 |

b_{2} | 2.027 | 2.010 | 0.0787 | 0.0786 | 0.949 | 0.948 |

Parameter | Estimate | Standard Error | t Value | p Value |
---|---|---|---|---|

(Intercept) | 2.877 | 0.119 | 24.245 | <0.0001 |

x_{1} | 0.047 | 0.008 | 5.935 | <0.0001 |

x_{2} | 0.203 | 0.020 | 10.065 | <0.0001 |

^{2}= 0.07986, Adjusted R

^{2}= 0.07894; F = 86.45 (2, 1992), p ≤ 0.01.

Parameter | Estimate | Std. Error | z Value | Pr (>|t|) | |
---|---|---|---|---|---|

A (Intercept, a_{0}) | 1.076 | 0.049 | 21.967 | <0.0001 | |

A (Slope, a_{1}) | 0.176 | 0.026 | 6.839 | <0.0001 | |

B (Intercept, b_{0}) | 2.243 | 0.082 | 27.332 | <0.0001 | |

B (Slope, b_{1}) | 0.215 | 0.035 | 6.073 | <0.0001 | |

Y (Intercept, w_{0}) | 1.359 | 0.021 | 64.199 | <0.0001 | |

Y (Slope, w_{1}) | 0.798 | 0.013 | 62.038 | <0.0001 | |

R.Squared | logLik npar | AIC | AICc | BIC | |

Linear model | 0.0798 | −2747.254 | 5502.51 | 5502.53 | 5524.91 |

Cusp model | 0.3381 | −2192.024 6 | 4396.05 | 4396.09 | 4429.64 |

^{2}= 1110, df = 2, p < 0.000.

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**MDPI and ACS Style**

Chen, D.-G.; Chen, X. Cusp Catastrophe Regression and Its Application in Public Health and Behavioral Research. *Int. J. Environ. Res. Public Health* **2017**, *14*, 1220.
https://doi.org/10.3390/ijerph14101220

**AMA Style**

Chen D-G, Chen X. Cusp Catastrophe Regression and Its Application in Public Health and Behavioral Research. *International Journal of Environmental Research and Public Health*. 2017; 14(10):1220.
https://doi.org/10.3390/ijerph14101220

**Chicago/Turabian Style**

Chen, Ding-Geng, and Xinguang Chen. 2017. "Cusp Catastrophe Regression and Its Application in Public Health and Behavioral Research" *International Journal of Environmental Research and Public Health* 14, no. 10: 1220.
https://doi.org/10.3390/ijerph14101220