Statistical Modelling 1 (2001), 233–234
The focus of the meeting was on the many and recent developments in Mixed Models and Mixture Models, and included both computational and theoretical aspects. Mixed Models are of great use if data are collected cluster-wise so that stochastic independence assumptions do not hold. For instance, a cluster could be an individual on which a number of observations are taken over time. In Mixed Models, for each cluster, a cluster-specific random effect is included in the model. This is the simplest basic structure of a Mixed Model. The complexity of the model can easily be increased by allowing for multivariate random effects, or by modelling multi-clustered or hierarchical data structures. Besides this structural framework of Mixed Models distributional assumptions are made for the random components. Different assumptions lead to different estimation routines. Restricting attention to normal response variables and assuming normality for the random effects gives the basis of the well developed field of Linear Mixed Models. Generalizations to non-normal responses lead to Generalized Linear Mixed Models (GLMM). Moreover, the normality assumption for the random effects can be circumvented by utilizing a discrete approximation to the random effects distribution, which leads to Nonparametric Maximum Likelihood (NPML) estimation. All approaches have their advantages, disadvantages and their limitations. The workshop provided a forum for an extensive and open discussion of these ideas and an airing of many controversial issues.
The six articles included in this special issue broadly cover the various areas of current research in Mixed Models. Molenberghs and Verbeke give an overview of recent developments in Linear Mixed Models. Special focus is given to the analysis of longitudinal data, where the influence of drop outs is of particular interest. A competing class of Mixed Models are Marginal Models where cross-sectional effects are of interest. Crouchley and Davies discuss the differences between these two model classes and provide an interesting contribution to a longstanding debate. If the random effects distribution is not specified but approximated and estimated by a discrete distribution defined on some masspoints, this leads to NPML estimation. Aitkin's paper contrasts NPML estimation with Bayesian approaches based on complex hierarchical modellling and Markov chain Monte Carlo routines. The focus here is on the investigation of the amount of information available from the data compared to that from the prior specifications required in a Bayesian framework. Karlis continues the discussion on NPML estimation by pointing out special features of the method for mixed Poisson regression models. If data are observed multi-clusterwise, models become more complex and estimation can be burdensome. The last two papers show how to handle estimation in complex Mixed Models. Ecochard and Clayton propose a procedure based on block-wise Gibbs sampling, which is feasible with standard software. Their approach is illustrated by an interesting but complex data set. Finally, Booth, Hobert and Jank discuss stochastic approximations to the likelihood function in Mixed Models and the relative merits of simulated maximum likelihood or Monte Carlo EM.
For us, these six articles represent the wide and interesting range of work on Mixed Models discussed at the workshop. The workshop itself profited from its small size and the open, co-operative atmosphere. Each session was followed by a discussion forum which gave participants an informal way of exchanging ideas and knowledge. The organisers of the workshop maintain a web page where further information about the event is provided, including minutes of the discussion groups. For details see
http://www.stat.uni-muenchen.de/euroworkshop/2000.html
Last but not least, we would like to thank the authors in this issue for their carefully prepared and interesting contributions. We are also greatful to a number of referees for their scientific service.
Herwig Friedl
Göran Kaumermann
Gainesville/Graz and Glasgow