Abstract
With an increase in complexity of latent growth curve models (LGCMs), comes an increase in problems estimating the models. This research first proposes new growth models to address the perennial problems of almost all longitudinal research, namely, missing data. Different non-ignorable missingness models are formulated. These models include the latent coefficient (intercept or slope)-dependent missingness, in which the missing data rates vary across different latent individual initial levels or slopes; and the potential outcome-dependent missingness, in which the missing data rates on each occasion depend on potential outcomes. Second, this study proposes a full Bayesian approach to estimate the proposed LGCMs with non-ignorable missing data through data augmentation algorithm and Gibbs sampling procedure. And third, model selecting criteria are proposed in a Bayesian context to identify the best-fit model.Simulation studies were conducted. Conclusions include the proposed method can accurately recover model parameters, the mis-specified missingness may result in severely misleading conclusions, and almost all the model selection criteria can correctly identify the true model with high certainty. The application of the model and the method are illustrated with a longitudinal data set showing growth in mathematical ability. Finally, related implications of the approach and future research directions are discussed.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Akaike, H. (1974). A new look at the statistical model identification. IEEE Transactions on Automatic Control, 1919(6), 716–723.
Baraldi, A. N., & Enders, C. K. (2010). An introduction to modern missing data analyses. Journal of School Psychology, 48, 5–37.
Bartholomew, D. J., & Knott, M. (1999). Latent variable models and factor analysis: Kendall’s library of statistics (2nd ed., Vol. 7). New York, NY: Edward Arnold.
Bollen, K., & Curran, P. (2006). Latent curve models: A structural equation perspective. Hoboken, NJ: Wiley.
Box, G. E. P., & Tiao, G. C. (1973). Bayesian inference in statistical analysis. Hoboken, NJ: Wiley.
Bozdogan, H. (1987). Model selection and Akaike’s Information Criterion (AIC): The general theory and its analytical extensions. Psychometrika, 52, 345–370.
Bureau of Labor Statistics, U.S. Department of Labor. (1997). National longitudinal survey of youth 1997 cohort, 1997–2003 (rounds 1–7). [computer file]. 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, OH, 2005. Available from http://www.bls.gov/nls/nlsy97.htm
Celeux, G., Forbes, F., Robert, C., & Titterington, D. (2006). Deviance information criteria for missing data models. Bayesian Analysis, 4, 651–674.
Dempster, A. (1974). The direct use of likelihood for significance testing. In Proceedings of Conference on Foundational Questions in Statistical Inference (pp. 335–352). University of Aarhus: Aarhus.
Enders, C. K. (2011). Missing not at random models for latent growth curve analyses. Psychological Methods, 16, 1–16.
Fitzmaurice, G., Davidian, M., Verbeke, G., & Molenberghs, G. (Eds.). (2008). Longitudinal data analysis. Boca Raton, FL: Chapman & Hall.
Fitzmaurice, G. M., Laird, N. M., & Ware, J. H. (2004). Applied longitudinal analysis. Hoboken, NJ: Wiley.
Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.
Geweke, J. (1992). Evaluating the accuracy of sampling-based approaches to calculating posterior moments. In J. M. Bernado, J. O. Berger, A. P. Dawid, & A. F. M. Smith (Eds.), Bayesian Statistics 4 (pp. 169–193). Oxford, UK: Clarendon.
Glynn, R. J., Laird, N. M., & Rubin, D. B. (1986). Drawing inferences from self-selected samples. In H. Wainer (Ed.), (pp. 115–142). New York: Springer.
Huber, P. (1996). Robust statisticalprocedures (2nd ed.). Philadelphia: SIAM.
Janssen, R., & De Boeck, P. (1999). Confirmatory analyses of componential test structure using multidimensional item response theory. Multivariate Behavioral Research, 34, 245–268.
Jelicic, H., Phelps, E., & Lerner, R. M. (2009). Use of missing data methods in longitudinal studies: The persistence of bad practices in developmental psychology. Developmental Psychology, 45, 1195–1199.
Lee, S. Y. (2007). Structural equation modeling: A Bayesian approach. Chichester, UK: Wiley.
Little, R. J. A., & Rubin, D. B. (1987). Statistical analysis with missing data. New York, NY: Wiley.
Little, R. J. A., & Rubin, D. B. (2002). Statistical analysis with missing data (2nd ed.). New York: Wiley-Interscience.
Lu, Z., Zhang, Z., & Lubke, G. (2011). Bayesian inference for growth mixture models with latent-class-dependent missing data. Multivariate Behavioral Research, 46, 567–597.
McLachlan, G. J., & Peel, D. (2000). Finite mixture models. New York, NY: Wiley.
Micceri, T. (1989). The unicorn, the normal curve and the other improbable creatures. Psychological Bulletin, 105, 156–166.
Muthén, B., & Asparouhov, T. (2012). Bayesian SEM: A more flexible representation of substantive theory. Psychological Methods, 17(3), 313–335.
Oldmeadow, C., & Keith, J. M. (2011). Model selection in Bayesian segmentation of multiple DNA alignments. Bioinformatics, 27, 604–610.
Rissanen, J. (1978). Modeling by shortest data description. Automatica, 14, 465–471.
Robert, C. P., & Casella, G. (2004). Monte Carlostatistical methods. New York, NY: Springer.
Roth, P. L. (1994). Missing data: A conceptual review for applied psychologists. Personnel Psychology, 47, 537–560.
Schafer, J. L. (1997). Analysisof incomplete multivariate data. Boca Raton, FL: Chapman & Hall/CRC.
Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics, 6 (2), 461–464.
Sclove, L. S. (1987). Application of mode-selection criteria to some problems in multivariate analysis. Psychometrics, 52, 333–343.
Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change and event occurrence. New York, NY: Oxford University Press.
Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & Van Der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 64(4), 583–639.
Spiegelhalter, D. J., Thomas, A., Best, N., & Lunn, D. (2003). WinBUGS manual Version 1.4. (Cambridge CB2 2SR, UK: MRC Biostatistics Unit, Institute of Public Health, Robinson Way. http://www.mrc-bsu.cam.ac.uk/bugs)
Sturtz, S., Ligges, U., & Gelman, A. (2005). R2WinBUGS: A package for running WinBUGS from R. Journal of Statistical Software, 12, 1–16.
Tanner, M. A., & Wong, W. H. (1987). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82, 528–540.
Yuan, K.- H., & Lu, Z. (2008). SEM with missing data and unknown population using two-stage ML: Theory and its application. Multivariate Behavioral Research, 43, 621–652.
Zhang, Z., Hamagami, F., Wang, L., Grimm, K. J., & Nesselroade, J. R. (2007). Bayesian analysis of longitudinal data using growth curve models. International Journal of Behavioral Development, 31(4), 374–383.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this paper
Cite this paper
Lu, Z.(., Zhang, Z., Cohen, A. (2013). Bayesian Methods and Model Selection for Latent Growth Curve Models with Missing Data. In: Millsap, R.E., van der Ark, L.A., Bolt, D.M., Woods, C.M. (eds) New Developments in Quantitative Psychology. Springer Proceedings in Mathematics & Statistics, vol 66. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9348-8_18
Download citation
DOI: https://doi.org/10.1007/978-1-4614-9348-8_18
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-9347-1
Online ISBN: 978-1-4614-9348-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)