3 Monte-Carlo Simulation-Based Power Analysis Approach

3.1 Statistical Power and Sample Size Determination

Sample size determination is a essential step in planning and designing a study. Sample size is usually associated with statistical power. In statistics, power is the probability of correctly rejecting the null hypothesis (i.e. no cusp catastrophe) which in common sense is the fraction of the times that the specified null-hypothesis will be rejected from statistical tests. The sample size and power calculation for the stochastic cusp catastrophe model has never been done to the best of our knowledge which probably due to the theoretical complexity of this model from the high-order density function as well as to comply with the three guidelines from Section 2.3. This theoretical derivation of a power function might be impossible.

3.2 Monte-Carlo Simulation-Based Approach

To overcome this difficult, we propose a Monte-Carlo simulation-based approach to calculate the statistical power for a series of specified sample size (n, i.e. the number of observations for the stochastic cusp catastrophe modeling) to generate a sample size-power curve. With this curve, the sample size can be then reverse-determined for specific statistical power (say 80% or 85% as typically chosen). Specifically, for a pre-specified sample size (n), the following steps are needed to calculate the statistical power:

1) Specify α, β and w from Equation (1) to be detected based on prior knowledge;

2) Simulate data from q predictor variables Xi = (Xi1, Xi2,..., Xiq) with prespecified distributions and then calculate the corresponding asymmetry (αi) and bifurcation (βi) variables from the last two equations of Equation (1);

3) Simulate the p dependent variables Yi = (Yi1, Yi2,..., Yip) and calculate the corresponding state measure (yi) from the first equation of Equation (1);

4) Fit the stochastic cusp catastrophe model using the maximum-likelihood as outlined in Equation (2) with the data generated from Steps 2) and 3); and make conclusion on whether there is a significant cusp catastrophe based on the guidelines in Section 2.3;

5) Repeat Steps 2) to 4) a large number of time (say 1,000 times) and calculate the proportion of simulations which satisfy the decision rules. This proportion is then the statistical power for the pre-specified n and the cusp parameters given in Step 1);

6) Sample size determination can be carried out by running Steps 1) to 5) with a series of ns to produce a power curve and then back-calculate the sample size required for pre-specified power, such as power at 0.8(or 0.85) in typical study design.

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