Multilevel models with multivariate mixed response types

We build upon the existing literature to formulate a class of models for multivariate mixtures of Gaussian, ordered or unordered categorical responses and continuous distributions that are not Gaussian, each of which can be defined at any level of a multilevel data hierarchy. We describe a Markov chain Monte Carlo algorithm for fitting such models. We show how this unifies a number of disparate problems, including partially observed data and missing data in generalized linear modelling. The two-level model is considered in detail with worked examples of applications to a prediction problem and to multiple imputation for missing data. We conclude with a discussion outlining possible extensions and connections in the literature. Software for estimating the models is freely available.

[1]  John Geweke,et al.  Efficient Simulation from the Multivariate Normal and Student-t Distributions Subject to Linear Constraints and the Evaluation of Constraint Probabilities , 1991 .

[2]  William J Browne,et al.  MCMC Estimation in MLwiN (Version 2.13) Centre for Multilevel Modelling, University of Bristol , 2009 .

[3]  Harvey Goldstein,et al.  Multilevel Structural Equation Models for the Analysis of Comparative Data on Educational Performance , 2007 .

[4]  Harvey Goldstein,et al.  Multilevel multivariate modelling of childhood growth, numbers of growth measurements and adult characteristics , 2009 .

[5]  W. Callaghan,et al.  Effects of different data-editing methods on trends in race-specific preterm delivery rates, United States, 1990-2002. , 2007, Paediatric and perinatal epidemiology.

[6]  Michael G Kenward,et al.  Multiple imputation: current perspectives , 2007, Statistical methods in medical research.

[7]  B. Muthén,et al.  Multilevel Mixture Models , 2006 .

[8]  Pickles A SkrondalA. Rabe-HeskethS.  GLLAMM: A general class of multilevel models and a STATA programme , 2001 .

[9]  Harvey Goldstein,et al.  Multilevel factor analysis modelling using Markov Chain Monte Carlo (MCMC) estimation , 2002 .

[10]  MULTILEVEL MODELLING NEWSLETTER , 1995 .

[11]  William J. Browne,et al.  Bayesian and likelihood-based methods in multilevel modeling 1 A comparison of Bayesian and likelihood-based methods for fitting multilevel models , 2006 .

[12]  Harvey Goldstein,et al.  Multilevel Factor Analysis Models for Continuous and Discrete Data , 2005 .

[13]  D. Rubin.  Multiple imputation for nonresponse in surveys , 1989 .

[14]  Xiao-Li Meng,et al.  The Art of Data Augmentation , 2001 .

[15]  V. Carstairs,et al.  Deprivation and health in Scotland. , 1990, Health bulletin.

[16]  S. van Buuren.  Multiple imputation of discrete and continuous data by fully conditional specification , 2007, Statistical methods in medical research.

[17]  Fritz Scheuren,et al.  Regression Analysis of Data Files that Are Computer Matched , 1993 .

[18]  Recai M. Yucel,et al.  Multiple imputation inference for multivariate multilevel continuous data with ignorable non-response , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  Joseph L Schafer,et al.  Analysis of Incomplete Multivariate Data , 1997 .

[20]  D. Cox,et al.  An Analysis of Transformations , 1964 .

[21]  S. Chib,et al.  Analysis of multivariate probit models , 1998 .

[22]  S. N. Gabhainn,et al.  Health behaviour in school-aged children , 2000 .

[23]  Ian H. Langford,et al.  A User’s Guide to MLwiN, Version 2.10 , 2000 .

[24]  William J. Browne,et al.  MCMC Estimation in MlwiN , 2002 .

[25]  John Aitchison,et al.  Polychotomous quantal response by maximum indicant , 1970 .

[26]  Harvey Goldstein,et al.  Likelihood methods for fitting multilevel models with complex level-1 variation , 2002 .

[27]  Bradley P. Carlin,et al.  Bayesian measures of model complexity and fit , 2002 .

[28]  D. Dunson,et al.  Bayesian latent variable models for clustered mixed outcomes , 2000 .

[29]  D. Rubin,et al.  Ignorability and Coarse Data , 1991 .

[30]  H. Goldstein.  Multilevel Statistical Models , 2006 .

[31]  D. V. Dyk,et al.  A Bayesian analysis of the multinomial probit model using marginal data augmentation , 2005 .

[32]  S. Chib,et al.  Bayesian analysis of binary and polychotomous response data , 1993 .

[33]  Harvey Goldstein,et al.  MODELS FOR MULTILEVEL RESPONSE VARIABLES WITH AN APPLICATION TO GROWTH CURVES , 1989 .

[34]  S. Rabe-Hesketh,et al.  Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects , 2005 .

[35]  M. Pitt,et al.  Efficient Bayesian inference for Gaussian copula regression models , 2006 .

[36]  R. Darrell Bock,et al.  Multilevel analysis of educational data , 1989 .

[37]  Aki Vehtari.  Discussion to "Bayesian measures of model complexity and fit" by Spiegelhalter, D.J., Best, N.G., Carlin, B.P., and van der Linde, A. , 2002 .