The cusp catastrophe model is proposed to model system outcomes which can incorporate the linear model with extension to nonlinear model along with discontinuous transitions in equilibrium states as control variables vary. According to the catastrophe systems theory [1] [18] - [20] , the dynamics for a cusp system outcome is expressed by the time derivative of its state variable (often called behavioral variable within the context of catastrophe theory) to the potential function: The first derivative of will consist of the equilibrium plane of the cusp catastrophe:

where is called asymmetry or normal control variable and is called bifurcation or splitting control variable. In the model, the two control variables and co-vary to determine the behavior outcome variable. Figure 1 depicts the equilibrium plane which reflects the response surface of the outcome measure at various combinations of and. It can be seen from the figure that the dynamic changes in a behavior measure has two stable regions (attractors), the lower area in the front left and the upper areas in the front right. Beyond these two regions, behavior becomes unstable. This characteristic can be further revealed by projecting the unstable region to the and control plane as a cusp region. The cusp region is characterized by two lines, line O-Q (the ascending threshold) and line O-R (the descending threshold) of the equilibrium surface. In this region, the outcome measure becomes highly unstable, and sudden change or jumping in behavior status will occur, because a very small change in or or both will lead to cross either the threshold line O-Q or O-R.

Furthermore, the paths A, B, and C in Figure 1 depict three typical but different pathways of change in the

outcome measure. Path A shows that in any situations where, there is a smooth relation between outcome measure and the asymmetry variable; path B shows that in any situations where, if the asymmetry variable increases to reach and pass the ascending threshold link O-Q, outcome measure will increase suddenly from the low stable region to the upper stable region of the equilibrium plane; and Path C shows a sudden drop in outcome measure as declines to reach and pass the descending threshold line O-R .

From the affirmative description, it is clearly that a cusp model differs from a linear model in that: 1) A cusp model allows the forward and backward progression follows different paths in the outcome measure and both processes can be modeled simultaneously (see Paths B and C in Figure 1) while a linear model only permits one type of relationship; 2) A cusp model covers both a discrete component and a continuous component of a behavior change while a linear model covers on continuous process (Path A). In this case a linear model can be considered as a special case of the cusp model; 3) A cusp model consists of two stable regions and two thresholds for sudden and discrete changes. Therefore, the application of the cusp modeling will advance the linear approach and better assist researchers to describe the behavior data while evidence obtained from such analysis, in turn, can be used to advance theories and models to better explain a behavior.

To operationalize the cusp catastrophe model for behavior research, Guastello [6] [7] developed the polynomial regression approach to implement the concept of cusp model. Since the first publication of this method, it has been widely used in analyzing real data as we described in the Introduction. In this study, we referred the method as Gastello’s polynomial cusp regression. According to Gustello, this approach is derived by inserting regression coefficients into the Equation (1), with change scores (the differences in the measurement scores of a behavior assessed at time 1 and time 2) as a numerical approximation of:

where is the intercept and is the normally distributed error term. Two additional term and are added to capture potential deviations of the data from the equilibrium plane. When conducting modeling analysis, a cusp is indicated ONLY if the estimated for the cubic term, plus (for the interaction term) or (for control variable x) in Equation (2) are statistically significant.

To demonstrate the efficiency of the polynomial regression approach in describing behavioral changes that are cusp, Guastelly [7] recommended a comparative approach. In this approach, two types, four alternative linear models can be constructed and used in modeling the same variables:

1) Change scores linear models

2) Pre-and post-linear models

These alternative linear models add another analytical strategy to strength the polynomial regression method. A better data-model fitting (or a larger) of the cusp model (2) than the alternative linear models (3) through (6) is often used as additional evidence supporting the hypothesis that the dynamics of a study behavior follows the cusp catastrophe. Fitting Guastello’s cusp regression model and the four alternative models can all be conducted with commonly available statistical software, including SAS, SPSS, STATA and. More recent discussions and applications of the cusp catastrophe modeling methods can be found in [21] .

In statistics, power is defined as the probability of correctly rejecting the null hypothesis. Stated in common language, power is the fraction of the times that the specified null-hypothesis value will be rejected from statistical tests. Operationally based on this definition, if we specify an alternative hypothesis, a desired type-I error rate, and a desired power, then we can calculate the required sample size. Alternatively, we can calculate the statistical power as a function of sample size under a specified alternative hypothesis and a desired type-I error rate. There are extensive literatures on sample size calculation as well as statistical power analysis, see the seminal books from [15] - [17] for power analysis for behavioral sciences.

As detailed in Chapter 7 in [17] , five factors related to research design interplay with each other to determine the statistical power and sample size for a simple t-test: 1) the rate of type-I error; 2) the desired statistical power, 3) the expected treatment effect size of, 4) the standard error for the expected effect size, and 5) the sample size. The mathematical formula can then be derived as. Therefore, to determine the required sample size, we would need to provide data for four of the five design characteristics. Typically, the type-I error is set at 0.05 and the desired power is chosen to be 0.85 (or 0.80). The other two will be treatment effect size and its standard error. Depending on actual research questions, different values are often selected for these two characteristics.

Extending the same concept described above for Guastello’s polynomial cusp regression, we would need to specify the corresponding parameter effect size for all in Equation (2), the standard deviation of the error term. In addition, we need to specify the distribution of the two control variables, the asymmetry and the bifurcation; and the distribution of the outcome variable at time 1 (i.e.). With these parameters and variables being specified, the required sample size for a significant Cuastello’s cusp regression model can be determined and statistical power can be analyzed.

Power analysis and sample size determination can be developed for specific purpose. Typically, it is developed to detect treatment effect as in clinical trials or to detect the effect of specific risk factor as in regression. Similar development can be done to Guastello’s cusp regression model for specific repressor in asymmetry variable or the bifurcation variable if they are linked to multiple regressors or even to the overall goodness- of-fit index of. However, we aim to tackle a more complicated problem to determine whether we can detect a significant overall cusp model. The complexity of cusp catastrophe model makes it rather challenging, if not impossible to derive an analytical formula to determine the statistical power for Guastello’s cusp regression. To deal with this difficult, we propose a Monte-Carlo simulation-based approach. In this method the statistical power is calculated as the fraction of the times that the specified null-hypothesis of “no cusp” is rejected at the given level of type I error. Stated in another way, if there is a cusp, the statistical power will be, among 100 simulations, how many times can we detect the cusp given the sample size and type I error? The detailed steps of the simulation-based approach are outlined as follows:

1) Simulate data with sample size (i.e. the number of observations for Guastello’s cusp regression modeling) for the asymmetry variable, bifurcation variable and outcome variable at time 1 (i.e.). Data are generated under required specifications for desired study, such as normal distribution with specific means and standard deviations. Guastello’s cusp regression requires that all variables be standardized before data analysis and modeling. In this case, the standard normal distribution can be used to generate data for, and;

2) Specify model parameter effect size and the standard deviation of the error term of (Equation (2)) obtained from prior knowledge;

3) Calculate using the data obtained in the previous two Steps. Also generate;

4) Fit the Guastello’s cusp regression model (Equation (2)) with least squares method using the data generated for, , , and. After model fitting, a significant test is conducted to determine whether the data fit Guastello’s cusp regression model satisfactorily according to the decision rules proposed by Guastello (1982): 1) the estimated for the cubic term and 2) (for the and interaction term) or (for control variable) must be are statistically significant;

5) Repeat Steps 1 to 4 a large number of times (typically 1000) and calculate the proportion of simulations which satisfy the Guastello’s decision rules. This proportion then provides an estimate of the statistical power for the pre-specified sample size and the study specifications given in Steps 1 and 2;

6) With the above established five steps for power assessment, sample size is then determined to reach a pre-specified level of statistical power. This is carried out by running Steps 1 to 5 with a range of sample sizes first to obtain the corresponding values of statistical power. Then a statistical power curve is constructed for these ranges of sample sizes. With this power curve, the sample size is determined through back-calculation for a pre-specified power, such as power = 0.85.

The simulation-based approach described above is implemented in free package and the computer program is available up request from the authors.

The best approach to demonstrate the validity of the simulation approach would be to test it with observed data. To use our approach, we need two sets of data from any reported study: parameter estimates as effect size and estimated mean error of model fitting. However, we experienced difficulties in finding such data from all the studies we accessed in the published literature database. For example, all coefficients were reported by all studies but was not; furthermore, data-model fitting error fitting was never reported in any of the published studies using Guastelle’s cusp polynomial regression method. Fortunately, one author of this paper [12] published a study that modeled early sexual initiation among young adolescents using this polynomial regression approach.

Briefly, in Chen’s study participants were 469 virgins in the control group for a randomized controlled trial to assess the effect of an HIV behavioral prevention intervention program [22] [23] . The participants in grade 6 in the Bahamian public schools were randomly assigned to receive either intervention or control conditions. They were followed every 6 months up to 24 months at the time when the analysis was conducted. A participant was categorized as having initiated sex if he or she had the first penile-vagina sexual intercourse during the follow-up period. In addition to sexual initiation, the likelihood to initiate sex was also assessed using a 5-point rating scale with 1 = very unlikely to have sex in the next 6 months and 5 = very likely to have sex. A sexual progression index (SPI) was thus created as the dependent variable for modeling analysis was defined as the first time. SPI = 1 for participants who never had sex and reported very unlikely to have sex; SPI = 2 for participants who never had sex but unsure if they are going to have sex in the next 6 months; SPI = 3 for participants who never had sex but reported very like to have sex in the next 6 months; and SPI = 4 for participants who initiated sex. In addition to SPI, age was used as the asymmetry variable, and self-efficacy not to have sex (scale score based on 5 items) was used as the bifurcation.

To verify the simulation-based method, the parameter effect size estimates were obtained from the paper with, and the data-model fitting error was obtained by accessing to the original computing records. With these estimates, the simulation-based approach in Section 3.2 is applied. Figure 4 presents the sample size-power curve. From the figure it can be seen that the estimated sample size is 153 to achieve 85% power. This sample size is much smaller than the sample in the original study.