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Growth Curve Modeling for Nonnormal Data: A Two-Stage Robust Approach Versus a Semiparametric Bayesian Approach

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Quantitative Psychology Research

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 167))

Abstract

Growth curve models are often used to investigate growth and change phenomena in social, behavioral, and educational sciences and are one of the fundamental tools for dealing with longitudinal data. Many studies have demonstrated that normally distributed data in practice are rather an exception, especially when data are collected longitudinally. Estimating a model without considering the nonnormality of data may lead to inefficient or even incorrect parameter estimates, or misleading statistical inferences. Therefore, robust methods become important in growth curve modeling. Among the existing robust methods, the two-stage robust approach from the frequentist perspective and the semiparametric Bayesian approach from the Bayesian perspective are promising. We propose to use these two approaches for growth curve modeling when the nonnormality is suspected. An example about the development of mathematical abilities is used to illustrate the application of the two approaches, using school children’s Peabody Individual Achievement Test mathematical test scores from the National Longitudinal Survey of Youth 1997 Cohort.

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References

  • Bureau of Labor Statistics, U.S. Department of Labor. (2005). National Longitudinal Survey of Youth 1997 cohort, 1997–2003 (rounds 1–7) [computer file]. OSU, Produced by the National Opinion Research Center, the University of Chicago and distributed by the Center for Human Resource Research, The Ohio State University, Columbus, Ohio.

    Google Scholar 

  • Ferguson, T. (1973) A bayesian analysis of some nonparametric problems. The Annals of Statistics, 1, 209–230.

    Article  MathSciNet  MATH  Google Scholar 

  • Ferguson, T. (1974) Prior distributions on spaces of probability measures. The Annals of Statistics, 2, 615–629.

    Article  MathSciNet  MATH  Google Scholar 

  • Filzmoser, P. (2005) Identification of multivariate outliers: A performance study. Austrian Journal of Statistics, 34, 127–138.

    Google Scholar 

  • Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J., & Stahel, W. A. (1986) Robust statistics: The approach based on influence functions. New York: Wiley.

    Google Scholar 

  • Hardin, J., & Rocke, D. M. (2005) The distribution of robust distances. Journal of Computational and Graphical Statistics, 14, 928–946.

    Google Scholar 

  • Huber, P. J. (1981) Robust statistics. New York: Wiley.

    Google Scholar 

  • Ishwaran, H. (2000) Inference for the random effects in bayesian generalized linear mixed models. In American Statistical Association, (ed.), ASA Proceedings of the Bayesian Statistical Science Section (pp. 1–10).

    Google Scholar 

  • Lange, K. L., Little, R. J. A., & Taylor, J. M. G. (1989) Robust statistical modeling uisng the t distribution. Journal of the Americal Statistical Association, 84(408), 881–896.

    Google Scholar 

  • Lunn, D., Jackson, C., Best, N., Thomas, A., & Spiegelhalter, D. (2013) The BUGS book: A practical introduction to Bayesian analysis. Boca Raton, FL: CRC Press.

    Google Scholar 

  • Maronna, R. A., Martin, R. D., & Yohai, V. J. (2006) Robust statistics: Theory and methods. New York: Wiley.

    Google Scholar 

  • McArdle, J. J. (1988) Dynamic but structural equation modeling of repeated measures data. In J. R. Nesselroade, & R. B. Cattell, (eds.), Handbook of Multivariate Experimental Psychology (2nd ed., pp. 561–614). New York: Plenum Press.

    Google Scholar 

  • Meredith, W., & Tisak, J. (1990) Latent curve analysis. Psychometrika, 55, 107–122.

    Article  Google Scholar 

  • Micceri, T. (1989) The unicorn, the normal curve, and other improbable creatures. Psychological Bulletin, 105(1), 156–166.

    Google Scholar 

  • MĂĽller, P., & Quintana, F. A. (2004) Nonparametric bayesian data analysis. Statistical Science, 19, 95–110.

    Google Scholar 

  • MuthĂ©n, B., & Shedden, K. (1999) Finite mixture modeling with mixture outcomes using the em algorithm. Biometrics, 55(2), 463–469.

    Google Scholar 

  • Peña, D., & Prieto, F. J. (2001) Multivariate outlier detection and robust covariance matrix estimation (with discussion). Technometrics, 43, 286–310.

    Google Scholar 

  • Pendergast, J. F., & Broffitt, J. D. (1985) Robust estimation in growth curve models. Communications in Statistics: Theory and Methods, 14, 1919–1939.

    Google Scholar 

  • Pinheiro, J. C., Liu, C., & Wu, Y. N. (2001) Efficient algorithms for robust estimation in linear mixed-effects models using the multivariate t distribution. Journal of Computational and Graphical Statistics, 10(2), 249–276.

    Google Scholar 

  • Sethuraman, J. (1994) A constructive definition of dirichlet priors. Statistica Sinica, 4, 639–650.

    Google Scholar 

  • Silvapulle, M. J. (1992) On m-methods in growth curve analysis with asymmetric errors. Journal of Statistical Planning and Inference, 32(3), 303–309.

    Google Scholar 

  • Singer, J. M., & Sen, P. K. (1986) M-methods in growth curve analysis. Journal of Statistical Planning and Inference, 13, 251–261.

    Google Scholar 

  • Tong, X. (2014) Robust semiparametric bayesian methods in growth curve modeling. Unpublished doctoral dissertation, University of Notre Dame, Notre Dame.

    Google Scholar 

  • Tong, X., & Zhang, Z. (2012) Diagnostics of robust growth curve modeling using student’s t distribution. Multivariate Behavioral Research, 47, 493–518.

    Google Scholar 

  • Yuan, K.-H., & Bentler, P. M. (1998) Structural equation modeling with robust covariances. Sociological Methodology, 28, 363–396.

    Google Scholar 

  • Yuan, K.-H., & Bentler, P. M. (2001) Effect of outliers on estimators and tests in covariance structure analysis. British Journal of Mathematical and Statistical Psychology, 54, 161–175.

    Google Scholar 

  • Yuan, K.-H., Bentler, P. M., & Chan, W. (2004) Structural equation modeling with heavy tailed distributions. Psychometrika, 69, 421—436.

    Google Scholar 

  • Yuan, K.-H., & Zhang, Z. (2012a) Structural equation modeling diagnostics using r package semdiag and eqs. Structural Equation Modeling, 19, 683–702.

    Google Scholar 

  • Yuan, K.-H., & Zhang, Z. (2012b) Robust structural equation modeling with missing data and auxiliary variables. Psychometrika, 77, 803–826.

    Google Scholar 

  • Zhang, Z., Davis, H. P., Salthouse, T. A., & Tucker-Drob, E. A. (2007) Correlates of individual, and age-related, differences in short-term learning. Learning and Individual Differences, 17(3), 231–240.

    Google Scholar 

  • Zhang, Z., Lai, K., Lu, Z., & Tong, X. (2013) Bayesian inference and application of robust growth curve models using student’s t distribution. Structural Equation Modeling, 20, 47–78.

    Google Scholar 

  • Zhong, X., & Yuan, K.-H. (2010) Weights. In N. J. Salkind (ed.), Encyclopedia of research design (pp. 1617–1620). Thousand Oaks: Sage.

    Google Scholar 

  • Zhong, X., & Yuan, K.-H. (2011) Bias and efficiency in structural equation modeling: Maximum likelihood versus robust methods. Multivariate Behavioral Research, 46, 229–265.

    Google Scholar 

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Author information

Authors and Affiliations

  1. University of Virginia, Charlottesville, VA, 22904, USA

    Xin Tong

  2. Sun Yat-Sen University, Guangzhou, Guangdong, 510275, China

    Zijun Ke

Authors
  1. Xin Tong
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  2. Zijun Ke
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Correspondence to Xin Tong .

Editor information

Editors and Affiliations

  1. University of Amsterdam, Amsterdam, The Netherlands

    L. Andries van der Ark

  2. University of Wisconsin, Madison, Wisconsin, USA

    Daniel M. Bolt

  3. Education University of Hong Kong, Hong Kong, China

    Wen-Chung Wang

  4. University of Illinois, Champaign, Illinois, USA

    Jeffrey A. Douglas

  5. UmeĂĄ University, UmeĂĄ, Sweden

    Marie Wiberg

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Tong, X., Ke, Z. (2016). Growth Curve Modeling for Nonnormal Data: A Two-Stage Robust Approach Versus a Semiparametric Bayesian Approach. In: van der Ark, L., Bolt, D., Wang, WC., Douglas, J., Wiberg, M. (eds) Quantitative Psychology Research. Springer Proceedings in Mathematics & Statistics, vol 167. Springer, Cham. https://doi.org/10.1007/978-3-319-38759-8_17

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