Multilevel models for censored and latent responses.

Multilevel models were originally developed to allow linear regression or ANOVA models to be applied to observations that are not mutually independent. This lack of independence commonly arises due to clustering of the units of observations into 'higher level units' such as patients in hospitals. In linear mixed models, the within-cluster correlations are modelled by including random effects in a linear model. In this paper, we discuss generalizations of linear mixed models suitable for responses subject to systematic and random measurement error and interval censoring. The first example uses data from two cross-sectional surveys of schoolchildren to investigate risk factors for early first experimentation with cigarettes. Here the recalled times of the children's first cigarette are likely to be subject to both systematic and random measurement errors as well as being interval censored. We describe multilevel models for interval censored survival times as special cases of generalized linear mixed models and discuss methods of estimating systematic recall bias. The second example is a longitudinal study of mental health problems of patients nested in clinics. Here the outcome is measured by multiple questionnaires allowing the measurement errors to be modelled within a linear latent growth curve model. The resulting model is a multilevel structural equation model. We briefly discuss such models both as extensions of linear mixed models and as extensions of structural equation models. Several different model structures are examined. An important goal of the paper is to place a number of methods that readers may have considered as being distinct within a single overall modelling framework.