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Sample Size Determination to Detect Cusp Catastrophe in Stochastic Cusp Catastrophe Model: A Monte-Carlo Simulation-Based Approach

Abstract. Stochastic cusp catastrophe model has been utilized extensively to model the nonlinear social and behavioral outcomes to detect the exisitance of cusp catastrophe. However the foundamental question on sample size needed to detect the cusp catastrophe from the study design point of view has never been investigated. This is probably due to the complexity of the cusp model. This paper is aimed at filling the gap. In this paper, we propose a novel Monte-Carlo simulation-based approach to calculate the statistical power for stochastic cusp catastrophe model so the sample size can be determined. With this approach, a power curve can be produced to depict the relationship between its statistical power and samples size under different specifications. With this power curve, researchers can estimate sample size required for specified power in design and analysis data from stochastic cusp catastrophe model. The implementation of this novel approach is illustrated with data from Zeeman's cusp machine.

Keywords: Stochastic cusp catastrophe model, power analysis, sample size de-termination, Monte-Carlo simulations.

1 Introduction

Study should be well-designed. An important aspect of good design of study is to determine the number of study subjects (i.e. sample size) required to adequately statistically power the study to address the research questions or objectives. From this perspective, statistical power analysis is in fact an essential component of any valid study. By definition, statistical power analysis is to calculate the (frequentist) statistical power which is the probability of failing to reject the null hypothesis when it is false. The formal statistical basis for sample size determination requires: (i) the questions or objectives of the study to be defined; (ii) the most relevant outcome measures reflecting the objective to be identified; (iii) the specification of the effect size (which embodies the research question) in the study that can be detected; (iv) specification of the magnitudes of the Type-I and Type-II decision errors; and (v) estimates of the mean and variability of the endpoint. Statistical power analysis is specific to different data type and dependent on the distribution as well as the statistical model.

Recently, cusp catastrophe model has been used extensively [1]. However, to the best of our knowledge, there is no research and publications on determining the sample size and calculating the statistical power for this model. This paper is then aimed to investigate this gap.

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