![[PDF]](/pb-assets/Icons/adobe-pdf-icon-1498649896243.png){kind=icon}
Abstract
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.
Keywords Box–Cox transformation, data augmentation, data coarsening, latent Gaussian model, maximum indicant model, MCMC, missing data, mixed response models, multilevel, multiple imputation, multivariate, normalising transformations, partially known values, prediction, prior-informed imputation, probit model
|
Aitchison J
and
Bennett JA
(1970)
Polychotomous quantal response by maximum indicant
. Biometrika, 57,
253–62
. Google Scholar | Crossref | ISI | |
|
Asparouhov T
and
Muthen B
(2007) Multilevel mixture models. In
Hancock GR
and
Samuelson KM
(eds). Advances in latent mixture models Charlotte, NC: Information Age Publishing, Inc
., 27–51. Google Scholar | |
|
Box GEP
and
Cox DR
(1964)
An analysis of transformations (with discussion)
. Journal of the Royal Statistical Society, B, 26,
211–52
. Google Scholar | |
|
Browne WJ
(2009) MCMC estimation in MLwiN.
Bristol: University of Bristol
. Google Scholar | |
|
Browne WJ
and
Draper D
(2006)
A comparison of Bayesian and likelihood based methods for fitting multilevel models
. Bayesian Analysis, 1,
473–514
. Google Scholar | Crossref | |
|
Carpenter J
and
Goldstein H
(2004)
Multiple imputation using MLwiN
. Multilevel Modelling Newsletter, 16,
9–18
. Google Scholar | |
|
Carstairs V
and
Morris R
(1991) Deprivation and health in Scotland.
Aberdeen, Scotland: Aberdeen University Press
. Google Scholar | |
|
Currie C
,
Levin K
, and
Todd J
(2008) Health behaviour in school-aged children: findings from the 2006 HBSC survey in Scotland.
Child and Adolescent Health Research Unit, University of Edinburgh
. Google Scholar | |
|
Dunson DB
(2000)
Bayesian latent variable models for clustered mixed outcomes
. Journal of the Royal Statistical Society, Series B, 62,
355–66
. Google Scholar | Crossref | |
|
Geweke J
(1991)
Efficient simulation from the multivariate normal and student-t distributions subject to linear constraints
. Computing science and statistics: Proceedings of the 23rd symposium on the interface. Fairfa Station, VA: Interface foundation of North America. Google Scholar | |
|
Goldstein H
(1989) Models for multilevel response variables with an application to Growth Curves. In
Bock RD
(ed). Multilevel analysis of educational data.
New York: Academic Press
, 107–25. Google Scholar | Crossref | |
|
Goldstein H
(2003) Multilevel statistical models. Third edition.
London: Edward Arnold
. Google Scholar | |
| Goldstein H (2009) The analysis of survival and event history data using a latent normal model. (in press). Google Scholar | |
|
Goldstein H
,
Bonnet G
, and
Rocher T
(2007)
Multilevel structural equation models for the analysis of comparative data on educational performance
. Journal of Educational and behavioural Statistics, 32,
252–86
. Google Scholar | SAGE Journals | |
|
Goldstein H
and
Browne W
(2005) Multilevel factor analysis models for continuous and discrete data. In
Olivares A
and
McArdle JJ
(eds). Contemporary psychometrics. A Festschrift to Roderick P. McDonald.
Mahwah, NJ: Lawrence Erlbaum
. Google Scholar | |
|
Goldstein H
and
Kounali D
(2009)
Multivariate multilevel modelling of childhood growth, members of growth measurements and adult characteristics
. Journal of the Royal Statistical Society, A, 172,
599–613
. Google Scholar | Crossref | |
|
Heitjan DF
and
Rubin DB
(1991)
Ignorability and coarse data
. Annals of Statistics, 19,
2244–53
. Google Scholar | Crossref | ISI | |
|
Imai K
and
van Dyk DA
(2005)
A Bayesian analysis of the multinomial probit model using marginal data augmentation
. Journal of Econometrics, 124,
311–34
. Google Scholar | Crossref | |
|
Kenward M
and
Carpenter J
(2007)
Multiple imputation: current perspectives
. Statistical Methods in Medical Research, 16,
199–218
. Google Scholar | SAGE Journals | |
| Mathworks (2004) Matlab. Available at http://www.mathworks.co.uk. Google Scholar | |
|
Muthen LK
and
Muthen BO
(2004) MPLUS users guide version 5.
Los Angeles: University of California, Graduate School of Education
. Google Scholar | |
|
Pitt M
,
Chan D
, and
Kohn R
(2006)
Efficient Bayesian inference for Guassian copula regression models
. Biometrika, 93,
537–54
. Google Scholar | Crossref | |
|
Qin C
,
Dietz PM
,
England LJ
,
Martin JA
,
et al.
(2007)
Effects of different data-editing methods on trends in race-specific delivery rates, United States, 1990–2002
. Pediatric and Perinatal Epidemiology, 21,
41–49
. Google Scholar | Crossref | Medline | |
|
Rabe-Hesketh S
,
Pickles A
, and
Skrondal A
(2001)
GLLAMM: a general class of multilevel models and a STATA program
. Multilevel Modelling Newsletter, 13,
17–23
. Google Scholar | |
|
Rabe-Hesketh S
,
Skrondal A
, and
Pickles A
(2005)
Maximum likelihood estimation of limited and discrete dependent variable models with nested random effects
. Journal of Econometrics, 128,
301–23
. Google Scholar | Crossref | ISI | |
|
Rasbash J
,
Steele F
,
Browne W
and
Goldstein H
(2009). A user’s guide to MLwiN version 2.10.
Bristol, Centre for Multilevel Modelling, University of Bristol
. Google Scholar | |
|
Rubin DB
(1987) Multiple imputation for non response in surveys.
Chichester: Wiley
. Google Scholar | Crossref | |
|
Schafer JL
(1997) Analysis of incomplete multivariate data.
London: Chapman & Hall
. Google Scholar | Crossref | |
|
Scheuren F
and
Winkler WE
(1993)
Regression analysis of data files that are computer matched
. Survey Methodology, 19,
35–38
. Google Scholar | |
| Scottish Executive (2003) Hungry for success: a whole school approach to school meals in Scotland.
Edinburgh: The Stationary Office
. Google Scholar | |
| Scottish Health Promoting Unit (2004) Being well–doing well: a framework for health promoting schools in Scotland 2004.
Dundee: SHPSU
. Google Scholar | |
|
Spiegelhalter D
,
Best N
,
Carlin BP
, and
Van der Linde A
(2002)
Bayesian measures of model complexity and fit (with discussion)
. Journal of the Royal Statistical Society, B, 64,
583–640
. Google Scholar | Crossref | |
|
Spiegelhalter DJ
,
Thomas A
, and
Best NG
(1999) WINBUGS version 1.2, user manual.
Cambridge: MRC Biostatistics Unit
. Google Scholar | |
|
Van Buuren S
(2007)
Multiple imputation of discrete and continuous data by fully conditional specification
. Statistical Methods in Medical Research, 16,
219–42
. Google Scholar | SAGE Journals | ISI | |
|
Van Dyk D
and
Meng X
(2001)
The art of data augmentation
. Journal of Computational and Graphical Statistics, 10,
1–30
. Google Scholar | Crossref | ISI | |
|
Yucel R
(2008)
Multiple imputation for multilevel continuous data. Philosophical transactions of the Royal Society, A
, 2,
2389–403
. Google Scholar |
