Mixed Models with Emphasis on Large Data Sets

Abstract

In many contexts, hierarchical, multilevel, or clustered data are collected. Examples are longitudinal studies in which subjects are measured repeatedly at various time points (measurements within subject), surveys in which all members of a sample of families are questioned (members within families), educational data in which students from various schools are tested (students within schools), etc. From a statistical perspective, the challenge is to account for the fact that the measurements within clusters are not necessarily independent anymore, implying that standard models such as linear regression or generalized linear regression are no longer applicable. Mixed models are currently amongst the most flexible models for the analysis of such data. They can be interpreted as standard linear, generalized linear, or non-linear models, with cluster-specific random effects shared by all measurements within the cluster, hereby implicitly accounting for within-cluster associations. In this chapter, mixed models will be introduced with special attention for the correct interpretation of the parameters in the models. Also, examples will be given of situations in which results obtained from fitting mixed models are incorrectly interpreted. Many commercial software packages nowadays include mixed model procedures. However, when (extremely) large data sets are to be analyzed, standard likelihood based inference is no longer feasible. Examples include data sets with crossed random effects, with many clusters, with many observations per cluster, or contexts where mixed models are used to build a joint model for high-dimensional multivariate responses. In such cases, pseudo-likelihood techniques provide good alternatives. Various versions will be presented and illustrated. All concepts will be introduced and extensively illustrated using data sets from various contexts.