Logistic Cusp Catastrophe Regression for Binary Outcome: Method Development and Empirical Testing

Abstract

Cusp catastrophe models are unique to advance life sciences, psychology and behavioral studies. Extensive progresses have been made to utilize this modeling technique for continuous outcome and there is no development for binary data. To fill this gap, this chapter is then aimed to develop a cusp catastrophe modelling method for binary outcome. Building upon our previous research on the nonlinear regression cusp (RegCusp) catastrophe model for continuous outcome, we propose a logistic cusp catastrophe regression (LogisticCusp). LogisticCusp is based on the principles of logistic regression for binary outcome variable y (yes/no) being expressed as a latent binary variable Y through a logit link. This latent regression provides a mathematical connection between an observed outcome variable as a binomially distributed random variable and the deterministic cusp catastrophe at its equilibrium. By connecting the two, Y in the LogisticCusp is considered as one of the true roots of the deterministic cusp catastrophe model determined using the Maxwell or Delay conventions. We validate the method using a 5-step Monte-Carlo simulation with two predictors and three parameters for both bifurcation and asymmetry control variables. We further tested the method with binge drinking behavior in youth with data from the Monitoring the Future Study. Results from 5000 Monte-Carlo simulations indicate that the parameter estimates obtained through LogisticCusp are unbiased and efficient using maximum likelihood estimation with quasi-Newton numerical search algorithm. Results from empirical testing with real data are consistent with those estimated using other methods. LogisticCusp adds a new tool for researchers to examine many issues in psychology, life sciences, and behavioral studies, particularly, issues in medicine and public health with the powerful cusp catastrophe modeling for binary outcome.