Abstract.In multilevel modelling, the residual variation in a response variable is split into component parts that are attributed to various levels. In applied work, much use is made of the percentage of variation that is attributable to the higher-level sources of variation. Such a measure however only makes sense in simple variance components, Normal response, models where it is often referred to as the 'intra-unit correlation'. In this paper we describe how similar measures can be found for both more complex random variation in Normal response models and models with discrete responses. In these cases the variance partitions are dependent on predictors associated with the individual observation. We compare several computational techniques to compute the variance partitions
When multilevel models are estimated from survey data derived using multistage sampling, unequal selection probabilities at any stage of sampling may induce bias in standard estimators, unless the sources of the unequal probabilities are fully controlled for in the covariates. This paper proposes alternative ways of weighting the estimation of a two-level model by using the reciprocals of the selection probabilities at each stage of sampling. Consistent estimators are obtained when both the sample number of level 2 units and the sample number of level 1 units within sampled level 2 units increase. Scaling of the weights is proposed to improve the properties of the estimators and to simplify computation. Variance estimators are also proposed. In a limited simulation study the scaled weighted estimators are found to perform well, although non-negligible bias starts to arise for informative designs when the sample number of level 1 units becomes small. The variance estimators perform extremely well. The procedures are illustrated using data from the survey of psychiatric morbidity.
This tutorial presents an overview of multilevel or hierarchical data modelling and its applications in medicine. A description of the basic model for nested data is given and it is shown how this can be extended to fit flexible models for repeated measures data and more complex structures involving cross-classifications and multiple membership patterns within the software package MLwiN. A variety of response types are covered and both frequentist and Bayesian estimation methods are described.
In the social and other sciences many data are collected with a known but complex underlying structure. Over the past two decades there has been an increase in the use of multilevel modelling techniques that account for nested data structures. Often however the underlying data structures are more complex and cannot be fitted into a nested structure. First, there are cross-classified models where the classifications in the data are not nested. Secondly, we consider multiple membership models where an observation does not belong simply to one member of a classification. These two extensions when combined allow us to fit models to a large array of underlying structures. Existing frequentist modelling approaches to fitting such data have some important computational limitations. In this paper we consider ways of overcoming such limitations using Bayesian methods, since Bayesian model fitting is easily accomplished using Monte Carlo Markov chain (MCMC) techniques. In examples where we have been able to make direct comparisons, Bayesian methods in conjunction with suitable ‘diffuse’ prior distributions lead to similar inferences to existing frequentist techniques. In this paper we illustrate our techniques with examples in the fields of education, veterinary epidemiology, demography, and public health illustrating the diversity of models that fit into our framework.
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