Bayesian Inference for Growth Mixture Models with Latent Class Dependent Missing Data

REFERENCES

• Agresti , A. 2002 . Categorical data analysis, , 2nd ed. Hoboken, NJ : Wiley. .
   Google Scholar
• Akaike , H. 1974 . A new look at the statistical model identification . IEEE Transactions on Automatic Control , 1919 : 716 – 723 .
   Web of Science ®Google Scholar
• Anderson , T. W. and Bahadur , R. R. 1962 . Classification into two multivariate normal distributions with different covariance matrices . The Annals of Mathematical Statistics , 33 : 420 – 431 .
   Google Scholar
• Ansari , A. , Jedidi , K. and Jagpal , S. 2000 . A hierarchical bayesian methodology for treating heterogeneity in structural equation models . Marketing Science , 19 : 328 – 347 .
   Web of Science ®Google Scholar
• Baltes , P. B. and Nesselroade , J. R. 1979 . “ History and rationale of longitudinal research. ” . In Longitudinal research in the study of behavior and development Edited by: Nesselroade , J. R. and Baltes , P. B. 1 – 39 . New York, NY : Academic Press. .
   Google Scholar
• Barnard , J. , Frangakis , C. E. , Hill , J. L. and Rubin , D. B. 2003 . Principal stratification approach to broken randomized experiments: A case study of school choice vouchers in New York City . Journal of the American Statistical Association , 98
   PubMed Web of Science ®Google Scholar
• Bartholomew , D. J. and Knott , M. 1999 . Latent variable models and factor analysis: Kendall's library of statistics 7. New York, NY : Edward Arnold. .
   Google Scholar
• Berger , J. O. 1985 . Statistical decision theory and Bayesian analysis, , 2nd ed. New York, NY : Springer-Verlag. .
   Google Scholar
• Bollen , K. A. 1989 . Structural equations with latent variables New York, NY : Wiley. .
   Google Scholar
• Box , G. E. P. and Tiao , G. C. 1973 . Bayesian inference in statistical analysis Hoboken, NJ : John Wiley & Sons. .
   Google Scholar
• Bureau of Labor Statistics, U.S. Department of Labor . 1997 . National longitudinal survey of youth 1997 cohort, 1997–2003 (Rounds 1–7) [Computer file] Columbus, OH : The Ohio State University. . Retrieved from http://www.bls.gov/nls/nlsy97.htmProduced by the National Opinion Research Center, the University of Chicago, and distributed by the Center for Human Resource Research, 2005
   Google Scholar
• Cai , J. H. and Song , X. Y. 2010 . A Bayesian analysis of mixtures in structural equation models with nonignorable missing data . British Journal of Mathematical and Statistical Psychology , 63 : 491 – 508 .
   PubMed Web of Science ®Google Scholar
• Cai , J. H. , Song , X. Y. and Hser , Y. I. 2010 . A Bayesian analysis of mixture structural equation models with non-ignorable missing responses and covariates . Statistic in Medicine , 29 : 1861 – 1874 .
   PubMed Web of Science ®Google Scholar
• Casella , G. and George , E. I. 1992 . Explaining the Gibbs sampler . The American Statistican , 46 ( 3 ) : 167 – 174 .
   Web of Science ®Google Scholar
• Celeux , G. , Forbes , F. , Robert , C. and Titterington , D. 2006 . Deviance information criteria for missing data models . Bayesian Analysis , 4 : 651 – 674 .
   Web of Science ®Google Scholar
• Congdon , P. 2003 . Applied Bayesian modelling New York, NY : Wiley. .
   Google Scholar
• Coronary Drug Project Research Group . 1980 . Influence of adherence to treatment and response of cholesterol on mortality in the coronary drug project . New England Journal of Medicine , 303 : 1038 – 1041 .
   PubMed Web of Science ®Google Scholar
• Demidenko , E. 2004 . Mixed models: Theory and applications New York, NY : Wiley. .
   Google Scholar
• Diggle , P. and Kenward , M. G. 1994 . Informative drop-out in longitudinal data analysis . Journal of the Royal Statistical Society, Series C (Applied Statistics) , 43 : 49 – 93 .
   Web of Science ®Google Scholar
• Dunson , D. B. 2000 . Bayesian latent variable models for clustered mixed outcomes . Journal of the Royal Statistical Society, B , 62 : 355 – 366 .
   Google Scholar
• Efron , B. and Tibshirani , R. 1993 . An introduction to the bootstrap New York, NY : CRC Press. .
   Google Scholar
• Elliott , M. R. , Gallo , J. J. , Have , T. R. T. , Bogner , H. R. and Katz , I. R. 2005 . Using a Bayesian latent growth curve model to identify trajectories of positive affect and negative events following myocardial infarction . Biostatistics , 6 : 119 – 143 .
   PubMed Web of Science ®Google Scholar
• Fitzmaurice , G. M. , Laird , N. M. and Ware , J. H. 2004 . Applied longitudinal analysis Hoboken, NJ : Wiley. .
   Google Scholar
• Frangakis , C. E. and Rubin , D. B. 1999 . Addressing complications of intention-to-treat analysis in the combined presence of all-or-none treatment-noncompliance and subsequent missing outcomes . Biometrika , 86 : 365 – 379 .
   Web of Science ®Google Scholar
• Fruhwirth-Schnatter , S. 2001 . MCMC estimation of classical and dynamic switching and mixture models . Journal of the American Statistical Association , 96 : 194 – 209 .
   Web of Science ®Google Scholar
• Gelman , A. , Carlin , J. B. , Stern , H. S. and Rubin , D. B. 2003 . Bayesian data analysis, , 2nd ed. Boca Raton, FL : Chapman & Hall/CRC. .
   Google Scholar
• Gelman , A. and Meng , X.-L. 1998 . Simulating normalizing constants: From importance sampling to bridge sampling to path sampling . Statistical Science , 13 : 163 – 185 .
   Web of Science ®Google Scholar
• Geman , S. and Geman , D. 1984 . Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images . IEEE Transactions on Pattern Analysis and Machine Intelligence , 6 : 721 – 741 .
   PubMed Web of Science ®Google Scholar
• Geweke , J. 1992 . “ Evaluating the accuracy of sampling-based approaches to calculating posterior moments. ” . In Bayesian statistics 4 Edited by: Bernado , J. M. , Berger , J. O. , Dawid , A. P. and Smith , A. F. M. 169 – 193 . Oxford, , UK : Clarendon Press. .
   Google Scholar
• Gill , J. 2002 . Bayesian methods: A social and behavioral sciences approach Boca Raton, FL : CRC Press. .
   Google Scholar
• Green , P. J. 1995 . Reversible jump Markov chain Monte Carlo computation and Bayesian model determination . Biometrika , 82 : 711 – 732 .
   Web of Science ®Google Scholar
• Hastings , W. K. 1970 . Monte Carlo sampling methods using Markov chains and their applications . Biometrika , 57 : 97 – 109 .
   Web of Science ®Google Scholar
• Heckman , J. 1976 . The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models . Annals of Economic and Social Measurement , 5 : 475 – 492 .
   Google Scholar
• Heckman , J. and Robb , R. 1986 . “ Alternative methods for solving the problem of selection bias in evaluating the impact of treatments on outcomes. ” . In Drawing inferences from self-selected samples Edited by: Wainer , H. 63 – 107 . New York, NY : Springer. .
   Google Scholar
• Hedeker , D. and Gibbons , R. D. 2006 . Longitudinal data analysis Hoboken, NJ : Wiley. .
   Google Scholar
• Hedges , L. V. 1994 . “ Fixed effects models. ” . In The handbook of research synthesis Edited by: Cooper , H. and Hedges , L. V. 285 – 299 . New York, NY : Russell Sage Foundation. .
   Google Scholar
• Jordan , M. I. and Xu , L. 1995 . Convergence results for the em approach to mixtures of experts architectures . Neural Networks , 8 : 1409 – 1431 .
   Web of Science ®Google Scholar
• Kass , R. E. and Raftery , A. E. 1995 . Bayes factors . Journal of the American Statistical Association , 90 : 773 – 795 .
   Web of Science ®Google Scholar
• Lee , S. Y. 1981 . A Bayesian approach to confirmatory factor analysis . Psychometrika , 46 : 153 – 160 .
   Web of Science ®Google Scholar
• Lee , S. Y. 2007 . Structural equation modeling: A Bayesian approach Chinchester, , UK : John Wiley & Sons. .
   Google Scholar
• Lee , S. Y. and Shi , J. Q. 2000 . Joint Bayesian analysis of factor scores and structural parameters in the factor analysis model . Annals of the Institute of Statistical Mathematics , 52 : 722 – 736 .
   Web of Science ®Google Scholar
• Lee , S. Y. and Song , X. Y. 2003 . Bayesian model selection for mixtures of structural equation models with an unknown number of components . British Journal of Mathematical and Statistical Psychology , 56 : 145 – 165 .
   PubMed Web of Science ®Google Scholar
• Lee , S. Y. and Tang , N. S. 2006 . Bayesian analysis of nonlinear structural equation models with nonignorable missing data . Psychometrika , 71 : 541 – 564 .
   Web of Science ®Google Scholar
• Little , R. J. A. and Rubin , D. B. 2002 . Statistical analysis with missing data, , 2nd ed. New York, NY : Wiley-Interscience. .
   Google Scholar
• Long , J. S. 1997 . Regression models for categorical and limited dependent variables. Thousand Oaks, CA : Sage. .
   Google Scholar
• Lubke , G. H. and Neale , M. C. 2006 . Distinguishing between latent classes and continuous factors: Resolution by maximum likelihood? . Multivariate Behavioral Research , 41 : 499 – 532 .
   PubMed Web of Science ®Google Scholar
• Lubke , G. H. and Neale , M. C. 2008 . Distinguishing between latent classes and continuous factors with categorical outcomes: Class invariance of parameters of factor mixture models . Multivariate Behavioral Research , 43 : 592 – 620 .
   PubMed Web of Science ®Google Scholar
• Luke , D. A. 2004 . Multilevel modeling (quantitative applications in the social sciences) Thousand Oaks, CA : Sage Publication, Inc. .
   Google Scholar
• Lunn , D. , Spiegelhalter , D. , Thomas , A. and Best , N. 2009 . The BUGS project: Evolution, critique and future directions (with discussion) . Statistics in Medicine , 28 : 3049 – 3082 .
   PubMed Web of Science ®Google Scholar
• McCullagh , P. and Nelder , J. 1989 . Generalized linear models, , 2nd ed. Boca Raton, FL : Chapman & Hall/CRC. .
   Google Scholar
• McLachlan , G. and Peel , D. 2000 . Finite mixture models. New York, NY : John Wiley & Sons. .
   Google Scholar
• McLachlan , G. J. 1987 . On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture . Applied Statistics , 36 : 318 – 324 .
   Web of Science ®Google Scholar
• Metropolis , N. , Rosenbluth , A. , Rosenbluth , M. , Teller , A. and Teller , E. 1953 . Equations of state calculations by fast computing machines . Journal of Chemical Physics , 21 : 1087 – 1092 .
   Web of Science ®Google Scholar
• Muthén , B. and Brown , C. H. 2001 . Non-ignorable missing data in a general latent variable modeling framework. Unpublished draft
   Google Scholar
• Muthén , B. , Jo , B. and Brown , C. H. 2003 . Principal stratification approach to broken randomized experiments: A case study of school choice vouchers in New York City (with comment) . Journal of the American Statistical Association , 98 : 311 – 314 .
   Web of Science ®Google Scholar
• Muthén , B. and Shedden , K. 1999 . Finite mixture modeling with mixture outcomes using the EM algorithm . Biometrics , 55 : 463 – 469 .
   PubMed Web of Science ®Google Scholar
• Ntzoufras , I. 2009 . Bayesian modeling using WinBUGS Hoboken, NJ : John Wiley & Sons. .
   Google Scholar
• Preacher , K. J. , Wichman , A. L. , MacCallum , R. C. and Briggs , N. E. 2008 . Latent growth curve modeling Thousand Oaks, CA : Sage. .
   Google Scholar
• Richardson , S. and Green , P. J. 1997 . On Bayesian analysis of mixtures with unknown number of components (with discussion) . Journal of the Royal Statistical Society, Series B , 59 : 731 – 792 .
   Google Scholar
• Robert , C. P. and Casella , G. 2004 . Monte Carlo statistical methods New York, NY : Springer Science & Business Media Inc. .
   Google Scholar
• Roeder , K. and Wasserman , L. 1997 . Practical bayesian density estimation using mixtures of mormals . Journal of the American Statistical Association , 92 : 894 – 902 .
   Web of Science ®Google Scholar
• Roy , J. 2003 . Modeling longitudinal data with nonignorable dropouts using a latent dropout class model . Biometrics , 59 : 829 – 836 .
   PubMed Web of Science ®Google Scholar
• Schafer , J. L. 1997 . Analysis of incomplete multivariate data Boca Raton, FL : Chapman & Hall/CRC. .
   Google Scholar
• Scheines , R. , Hoijtink , H. and Boomsma , A. 1999 . Bayesian estimation and testing of structural equation models . Psychometrika , 64 : 37 – 52 .
   Web of Science ®Google Scholar
• Schwarz , G. E. 1978 . Estimating the dimension of a model . Annals of Statistics , 6 : 461 – 464 .
   Web of Science ®Google Scholar
• Singer , J. D. and Willett , J. B. 2003 . Applied longitudinal data analysis: Modeling change and event occurrence New York, NY : Oxford University Press. .
   Google Scholar
• Song , X. Y. and Lee , S. Y. 2007 . Bayesian analysis of latent variable models with nonignorable missing outcomes from exponential family . Statistics in Medicine , 26 : 681 – 693 .
   PubMed Web of Science ®Google Scholar
• Spiegelhalter , D. J. , Best , N. G. , Carlin , B. P. and v. d. Linde , A. 2002 . Bayesian measures of model complexity and fit . Journal of the Royal Statistical Society, Series B (Statistical Methodology) , 64 : 583 – 639 .
   Web of Science ®Google Scholar
• Spiegelhalter , D. J. , Thomas , A. , Best , N. and Lunn , D. 2003 . WinBUGS manual, , Version 1.4 Cambridge, , UK : MRC Biostatistics Unit, Institute of Public Health. . Retrieved from http://www.mrc-bsu.cam.ac.uk/bugs
   Google Scholar
• Tanner , M. A. and Wong , W. H. 1987 . The calculation of posterior distributions by data augmentation . Journal of the American Statistical Association , 82 : 528 – 540 .
   Web of Science ®Google Scholar
• Taylor , L. and Zhou , X. H. February 2009 . Relaxing latent ignorability in the ITT analysis of randomized studies with missing data and noncompliance. February , Seattle, WA : UW Biostatistics Working Paper Series. .
   Google Scholar
• Thomas , A. , O'Hara , B. , Ligges , U. and Sturtz , S. 2006 . Making BUGS open . R News , 6 : 12 – 17 .
   Google Scholar
• Tueller , S. , Drotar , S. and Lubke , G. 2011 . Addressing the problem of switched class labels in latent variable mixture model simulation studies . Structural Equation Modeling: A Multidisciplinary Journal , 18 : 110 – 131 .
   Web of Science ®Google Scholar
• van Dyk , D. A. and Meng , X.-L. 2001 . The art of data augmentation . Journal of Computational and Graphical Statistics , 10 : 1 – 50 .
   Web of Science ®Google Scholar
• Willett , J. and Sayer , A. 1994 . Using covariance structure analysis to detect correlates and predictors of individual change over time . Psychological Bulletin , 116 : 363 – 381 .
   Web of Science ®Google Scholar
• Wu , M. C. and Bailey , K. R. 1989 . Estimation and comparison of changes in the presence of informative right censoring: Conditional linear model . Biometrics , 45 : 939 – 955 .
   PubMed Web of Science ®Google Scholar
• Wu , M. C. and Carroll , R. J. 1988 . Estimation and comparison of changes in the presence of informative right censoring by modeling the censoring process . Biometrics , 44 : 175 – 188 .
   Web of Science ®Google Scholar
• Yung , Y. F. 1997 . Finite mixtures in confirmatory factor-analysis models . Psychometrika , 62 : 297 – 330 .
   Web of Science ®Google Scholar
• Zhang , Z. , Hamagami , F. , Wang , L. , Grimm , K. J. and Nesselroade , J. R. 2007 . Bayesian analysis of longitudinal data using growth curve models . International Journal of Behavioral Development , 31 : 374 – 383 .
   Web of Science ®Google Scholar
• Zhang , Z. , McArdle , J. J. , Wang , L. and Hamagami , F. 2008 . An SAS interface for Bayesian analysis with WinBUGS . Structural Equation Modeling , 15 : 705 – 728 .
   Web of Science ®Google Scholar
• Zhu , H. T. and Lee , S. Y. 2001 . A Bayesian analysis of finite mixtures in the LISREL model . Psychometrika , 66 : 133 – 152 .
   Web of Science ®Google Scholar