Cusp Catastrophe Polynomial Model: Power and Sample Size Estimation
Author(s)
Ding-Geng Chen1,2,3,
Xinguang Chen4,5,
Feng Lin1,6,
Wan Tang2,
Yuhlong Lio7,
Yuanyuan Guo8
Affiliation(s)
1 Center of Research, School of Nursing, University of Rochester Medical Center, Rochester, NY, USA. 2 Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester, NY, USA. 3 Institute of Data Sciences, University of Rochester, Rochester, NY, USA. 4 Department of Epidemiology, University of Florida, Gainesville, FL, USA. 5 School of Public Health, Wuhan University, Wuhan, China. 6 AD-CARE, Department of Psychiatry, University of Rochester Medical Center, Rochester, NY, USA. 7 Department of Mathematical Sciences, University of South Dakota, Vermillion, SD, USA. 8 Department of Statistics, Central South University, Changsha, China.
1 Center of Research, School of Nursing, University of Rochester Medical Center, Rochester, NY, USA. 2 Department of Biostatistics and Computational Biology, University of Rochester Medical Center, Rochester, NY, USA. 3 Institute of Data Sciences, University of Rochester, Rochester, NY, USA. 4 Department of Epidemiology, University of Florida, Gainesville, FL, USA. 5 School of Public Health, Wuhan University, Wuhan, China. 6 AD-CARE, Department of Psychiatry, University of Rochester Medical Center, Rochester, NY, USA. 7 Department of Mathematical Sciences, University of South Dakota, Vermillion, SD, USA. 8 Department of Statistics, Central South University, Changsha, China.
ABSTRACT
Guastello’s polynomial regression method for solving cusp catastrophe model has been widely applied to analyze nonlinear behavior outcomes. However, no statistical power analysis for this modeling approach has been reported probably due to the complex nature of the cusp catastrophe model. Since statistical power analysis is essential for research design, we propose a novel method in this paper to fill in the gap. The method is simulation-based and can be used to calculate statistical power and sample size when Guastello’s polynomial regression method is used to do cusp catastrophe modeling analysis. With this novel approach, a power curve is produced first to depict the relationship between statistical power and samples size under different model specifications. This power curve is then used to determine sample size required for specified statistical power. We verify the method first through four scenarios generated through Monte Carlo simulations, and followed by an application of the method with real published data in modeling early sexual initiation among young adolescents. Findings of our study suggest that this simulation-based power analysis method can be used to estimate sample size and statistical power for Guastello’s polynomial regression method in cusp catastrophe modeling.
Guastello’s polynomial regression method for solving cusp catastrophe model has been widely applied to analyze nonlinear behavior outcomes. However, no statistical power analysis for this modeling approach has been reported probably due to the complex nature of the cusp catastrophe model. Since statistical power analysis is essential for research design, we propose a novel method in this paper to fill in the gap. The method is simulation-based and can be used to calculate statistical power and sample size when Guastello’s polynomial regression method is used to do cusp catastrophe modeling analysis. With this novel approach, a power curve is produced first to depict the relationship between statistical power and samples size under different model specifications. This power curve is then used to determine sample size required for specified statistical power. We verify the method first through four scenarios generated through Monte Carlo simulations, and followed by an application of the method with real published data in modeling early sexual initiation among young adolescents. Findings of our study suggest that this simulation-based power analysis method can be used to estimate sample size and statistical power for Guastello’s polynomial regression method in cusp catastrophe modeling.
KEYWORDS
Cusp Catastrophe Model, Polynomial Regression Method, Statistical Power Analysis, Sample Size Determination
Cusp Catastrophe Model, Polynomial Regression Method, Statistical Power Analysis, Sample Size Determination
Cite this paper
Chen, D. , Chen, X. , Lin, F. , Tang, W. , Lio, Y. , Guo, Y. , (2014) Cusp Catastrophe Polynomial Model: Power and Sample Size Estimation. Open Journal of Statistics, 4, 803-813. doi: 10.4236/ojs.2014.410076.
Chen, D. , Chen, X. , Lin, F. , Tang, W. , Lio, Y. , Guo, Y. , (2014) Cusp Catastrophe Polynomial Model: Power and Sample Size Estimation. Open Journal of Statistics, 4, 803-813. doi: 10.4236/ojs.2014.410076.
References
[1] Thom, R. (1975) Structural Stability and Morphogenesis. Benjamin-Addison-Wesley, New York. [2] Thom, R. and Fowler, D.H. (1975) Structural Stability and Morphogenesis: An Outline of a General Theory of Models. W. A. Benjamin, Michigan. [3] Cobb, L. and Ragade, R.K. (1978) Applications of Catastrophe Theory in the Behavioral and Life Sciences. Behavioral Science, 23, 291-419.
http://dx.doi.org/10.1002/bs.3830230511 [4] Cobb, L. and Watson, B. (1980) Statistical Catastrophe Theory: An Overview. Mathematical Modelling, 1, 311-317.
http://dx.doi.org/10.1016/0270-0255(80)90041-X [5] Cobb, L. and Zacks, S. (1985) Applications of Catastrophe Theory for Statistical Modeling in the Biosciences. Journal of the American Statistical Association, 80, 793-802.
http://dx.doi.org/10.1080/01621459.1985.10478184 [6] Guastello, S.J. (1982). Moderator Regression and the Cusp Catastrophe: Application of Two-Stage Personnel Selection, Training, Therapy and Program Evaluation. Behavioral Science, 27, 259-272.
http://dx.doi.org/10.1002/bs.3830270305 [7] Guastello, S.J. (1989) Catastrophe Modeling of the Accident Processes: Evaluation of an Accident Reduction Program Using the Occupational Hazards Survey. Accident Analysis and Prevention, 21, 61-77.
http://dx.doi.org/10.1016/0001-4575(89)90049-3 [8] Grasman, R.P., van der Mass, H.L. and Wagenmakers, E. (2009) Fitting the Cusp Catastrophe in R: A Cusp Package Primer. Journal of Statistical Software, 32, 1-27. [9] Clair, S. (1998) A Cusp Catastrophe Model for Adolescent Alcohol Use: An Empirical Test. Nonlinear Dynamics, Psychology, and Life Sciences, 2, 217-241.
http://dx.doi.org/10.1023/A:1022376002167 [10] Mazanov, J. and Byrne, D.G. (2006) A Cusp Catastrophe Model Analysis of Changes in Adolescent Substance Use: Assessment of Behavioural Intention as a Bifurcation Variable. Nonlinear Dynamics, Psychology, and Life Sciences, 10, 445-470. [11] Guastello, S.J., Aruka, Y., Doyle, M. and Smerz, K.E. (2008) Cross-Cultural Generalizability of a Cusp Catastrophe Model for Binge Drinking among College Students. Nonlinear Dynamics, Psychology and Life Sciences, 12, 397-407. [12] Chen, X., Lunn, S., Harris, C., Li, X., Deveaux, L., Marshall, S., et al. (2010) Modeling Early Sexual Initiation among Young Adolescents Using Quantum and Continuous Behavior Change Methods: Implications for HIV Prevention. Nonlinear Dynamics, Psychology and Life Sciences, 14, 491-509. [13] Wagner, C.M. (2010) Predicting Nursing Turnover with Catastrophe Theory. Journal of Advanced Nursing, 66, 2071-2084. [14] Chen, X., Lunn, S., Deveaus, L., Li, X., Brathwaite, N., Cottrell, L. and Stanton, B. (2008) A Cluster Randomized Controlled Trial of an Adolescent HIV Prevention Program among Bahamian Youth: Effect at 12 Months Post-Intervention. AIDS and Behavior, 13, 495-508. [15] Cohen, J. (1988) Statistical Power Analysis for the Behavioral Sciences. 2nd Edition, Lawrence Berbaum Associates, Hillsdale. [16] Chow, S., Shao, J. and Wang, H. (2008) Sample Size Calculations in Clinical Research. 2nd Edition, Chapman and Hall/CRC, Boca Raton. [17] Chen, D.G. and Peace, K.E. (2011) Clinical Trial Data Analysis Using R. Chapman and Hall/CRC, Boca Raton. [18] Saunders, P.T. (1980) An Introduction to Catastrophe Theory. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9781139171533 [19] Hartelman, A.I. (1997) Stochastic Catastrophe Theory. University of Amsterdam, Amsterdam. [20] Iacus, S.M. (2008) Simulation and Inference for Stochastic Differential Equations with R Examples. Springer, Berlin.
http://dx.doi.org/10.1007/978-0-387-75839-8 [21] Guastello, S.J. and Gregson, A.M. (2011) Nonlinear Dynamic Systems Analysis for the Behavioral Sciences Using Real Data. CPC Press, Boca Raton. [22] Gong, J., Stanton, B., Lunn, S., Devearus, L., Li, X., Marshall, S., Brathwaite, N.V., Cottrell, L., Harris, C. and Chen, X. (2009) Effects through 24 Months of an HIV/AIDS Prevention Intervention Program Based on Protection Motivation Theory among Preadolescents in the Bahamas. Pediatrics, 123, 917-928.
http://dx.doi.org/10.1542/peds.2008-2363 [23] Chen, X., Stanton, S., Chen, D.G. and Li, X. (2013) Is Intention to Use Condom a Linear Process? Cusp Modeling and Evaluation of an HIV Prevention Intervention Trial. Nonlinear Dynamics, Psychology and Life Sciences, 17, 385-403. [24] Bolker, B. (2008) Ecological Models and Data in R. Princeton University Press, Princeton. [25] Mazanov, J. and Byrne, D.G. (2008) Modeling Change in Adolescent Smoking Behavior: Stability of Predictors across Analytic Models. British Journal of Health Psychology, 13, 361-379.
http://dx.doi.org/10.1348/135910707X202490 [26] West, R. and Sohal, T. (2006) “Catastrophic” Pathways to Smoking Cessation: Findings from National Survey. British Medical Journal, 332, 458-460.
http://dx.doi.org/10.1136/bmj.38723.573866.AE
[1] Thom, R. (1975) Structural Stability and Morphogenesis. Benjamin-Addison-Wesley, New York. [2] Thom, R. and Fowler, D.H. (1975) Structural Stability and Morphogenesis: An Outline of a General Theory of Models. W. A. Benjamin, Michigan. [3] Cobb, L. and Ragade, R.K. (1978) Applications of Catastrophe Theory in the Behavioral and Life Sciences. Behavioral Science, 23, 291-419.
http://dx.doi.org/10.1002/bs.3830230511 [4] Cobb, L. and Watson, B. (1980) Statistical Catastrophe Theory: An Overview. Mathematical Modelling, 1, 311-317.
http://dx.doi.org/10.1016/0270-0255(80)90041-X [5] Cobb, L. and Zacks, S. (1985) Applications of Catastrophe Theory for Statistical Modeling in the Biosciences. Journal of the American Statistical Association, 80, 793-802.
http://dx.doi.org/10.1080/01621459.1985.10478184 [6] Guastello, S.J. (1982). Moderator Regression and the Cusp Catastrophe: Application of Two-Stage Personnel Selection, Training, Therapy and Program Evaluation. Behavioral Science, 27, 259-272.
http://dx.doi.org/10.1002/bs.3830270305 [7] Guastello, S.J. (1989) Catastrophe Modeling of the Accident Processes: Evaluation of an Accident Reduction Program Using the Occupational Hazards Survey. Accident Analysis and Prevention, 21, 61-77.
http://dx.doi.org/10.1016/0001-4575(89)90049-3 [8] Grasman, R.P., van der Mass, H.L. and Wagenmakers, E. (2009) Fitting the Cusp Catastrophe in R: A Cusp Package Primer. Journal of Statistical Software, 32, 1-27. [9] Clair, S. (1998) A Cusp Catastrophe Model for Adolescent Alcohol Use: An Empirical Test. Nonlinear Dynamics, Psychology, and Life Sciences, 2, 217-241.
http://dx.doi.org/10.1023/A:1022376002167 [10] Mazanov, J. and Byrne, D.G. (2006) A Cusp Catastrophe Model Analysis of Changes in Adolescent Substance Use: Assessment of Behavioural Intention as a Bifurcation Variable. Nonlinear Dynamics, Psychology, and Life Sciences, 10, 445-470. [11] Guastello, S.J., Aruka, Y., Doyle, M. and Smerz, K.E. (2008) Cross-Cultural Generalizability of a Cusp Catastrophe Model for Binge Drinking among College Students. Nonlinear Dynamics, Psychology and Life Sciences, 12, 397-407. [12] Chen, X., Lunn, S., Harris, C., Li, X., Deveaux, L., Marshall, S., et al. (2010) Modeling Early Sexual Initiation among Young Adolescents Using Quantum and Continuous Behavior Change Methods: Implications for HIV Prevention. Nonlinear Dynamics, Psychology and Life Sciences, 14, 491-509. [13] Wagner, C.M. (2010) Predicting Nursing Turnover with Catastrophe Theory. Journal of Advanced Nursing, 66, 2071-2084. [14] Chen, X., Lunn, S., Deveaus, L., Li, X., Brathwaite, N., Cottrell, L. and Stanton, B. (2008) A Cluster Randomized Controlled Trial of an Adolescent HIV Prevention Program among Bahamian Youth: Effect at 12 Months Post-Intervention. AIDS and Behavior, 13, 495-508. [15] Cohen, J. (1988) Statistical Power Analysis for the Behavioral Sciences. 2nd Edition, Lawrence Berbaum Associates, Hillsdale. [16] Chow, S., Shao, J. and Wang, H. (2008) Sample Size Calculations in Clinical Research. 2nd Edition, Chapman and Hall/CRC, Boca Raton. [17] Chen, D.G. and Peace, K.E. (2011) Clinical Trial Data Analysis Using R. Chapman and Hall/CRC, Boca Raton. [18] Saunders, P.T. (1980) An Introduction to Catastrophe Theory. Cambridge University Press, Cambridge.
http://dx.doi.org/10.1017/CBO9781139171533 [19] Hartelman, A.I. (1997) Stochastic Catastrophe Theory. University of Amsterdam, Amsterdam. [20] Iacus, S.M. (2008) Simulation and Inference for Stochastic Differential Equations with R Examples. Springer, Berlin.
http://dx.doi.org/10.1007/978-0-387-75839-8 [21] Guastello, S.J. and Gregson, A.M. (2011) Nonlinear Dynamic Systems Analysis for the Behavioral Sciences Using Real Data. CPC Press, Boca Raton. [22] Gong, J., Stanton, B., Lunn, S., Devearus, L., Li, X., Marshall, S., Brathwaite, N.V., Cottrell, L., Harris, C. and Chen, X. (2009) Effects through 24 Months of an HIV/AIDS Prevention Intervention Program Based on Protection Motivation Theory among Preadolescents in the Bahamas. Pediatrics, 123, 917-928.
http://dx.doi.org/10.1542/peds.2008-2363 [23] Chen, X., Stanton, S., Chen, D.G. and Li, X. (2013) Is Intention to Use Condom a Linear Process? Cusp Modeling and Evaluation of an HIV Prevention Intervention Trial. Nonlinear Dynamics, Psychology and Life Sciences, 17, 385-403. [24] Bolker, B. (2008) Ecological Models and Data in R. Princeton University Press, Princeton. [25] Mazanov, J. and Byrne, D.G. (2008) Modeling Change in Adolescent Smoking Behavior: Stability of Predictors across Analytic Models. British Journal of Health Psychology, 13, 361-379.
http://dx.doi.org/10.1348/135910707X202490 [26] West, R. and Sohal, T. (2006) “Catastrophic” Pathways to Smoking Cessation: Findings from National Survey. British Medical Journal, 332, 458-460.
http://dx.doi.org/10.1136/bmj.38723.573866.AE