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Continuous Random effects in non-hierarchical multilevel models

LEMMA developed several extensions to cross-classified models.

Non-independence between classifications sharing units: In general, correlations between random effects from different classifications cannot be modelled. One important exception is where classifications share units. We might have a neighbourhood of residence classification and a neighbourhood of employment classification here we can estimate the correlation between employment and residential neighbourhood random effects. Correlated classifications also arise in social network analyses (Snijders and Kenny, 1999) where network members have roles both as actors and partners.

Non-additivity between classifications not sharing units: Where two classifications are crossed and do not share units we can explore non-additivity between the two classifications by fitting an interaction classification. Significant variation across the interaction classification tells us that the two classifications are not acting additively. Again this may be of importance in many geographical situations and in the analysis of networks

Non-independence within sets of random effects: Several models need to relax the assumption that random effects are uncorrelated across the units of a classification, including time-series models, spatial models and genetic models

Researchers: Rasbash, Goldstein

Modelling between unit variability using latent discrete effects

Growth trajectory or mixture models: The usual distributional assumption of Normality for higher-level random effects is over-restrictive. Nagin (1999) describes a formulation for growth curves where a discrete set of latent groups is posited and each individual has a membership distribution across the groups. Separate growth trajectories are modelled for each group. LEMMA generalised these models so that continuous and/or discrete latent effects can be fitted to any classification.
Discrete latent effects for “mover/stayer” models: In binary response and event history models, many higher level units have response patterns of all zero or all one. This can make the assumption of an overall Normal distribution of random effects untenable. LEMMA developed the class of discrete latent-effects mixture models, extending existing implementations, such as SABRE, to handle mixtures at any level and to allow complex models for group membership.

Researchers: Rasbash, Goldstein, Jones

Geography of school effects

This project addressed the substantively and politically important question - the relationship between school effectiveness, school choice and parental relocation, thereby addressing current debates concerning the effects of quasi-markets in education. This was tackled by using Pupil Level Annual School Census data and also ALSPAC data drawn from the Bristol region. Such data have a highly complex structure including multiple membership and crossed classifications: repeated measures on individuals within areas; movement of individuals between areas and schools, repeated measures on individuals within primary year cohort within primary school and repeated measures within secondary year cohort within schools. Spatial models will also be used in modelling ‘competition’ between the higher level units, such as schools with overlapping catchments, differentiated by school type.

Researchers: Thomas, Jones, Goldstein, Hepple, Johnston, Wilson, Harris

Voting choice

The substantive issue focuses on the individual, household and neighbourhood determinants of voting abstention and party choice. Using the BHPS we have repeated binary measures on voting intention for individuals within households within areas at a variety of scales. Normally distributed individual level random effects are unrealistic as part of the mover/stayer problem as are Normal household effects. LEMMA compared the autocorrelation approach of Goldstein and Barbosa (2000) and the discrete latent-effects model, in particular a doubly-nested model, with latent classifications at the individual and household level.

Researchers: Johnston, Jones, Rasbash

Mental health and psychosocial development

This research used measures on mental well-being from two data sets. The first was the BHPS, where the structure is repeated measures on individuals, within households within neighbourhoods. LEMMA compared the use of Normal random effects and discrete latent effects to describe between individual variations in patterns of change over time with a multiple-membership model to take account of changes in household composition. The second dataset was the Avon Brothers and Sisters Study with repeated measures on psychosocial adjustment, for multiple children within families. LEMMA explored the use of continuous and discrete latent effects to describe between individual patterns of variation in psycho-social development.

Researchers: Rasbash, Lewis, Propper, Jones

Modelling group diversity

Traditional statistical models have concentrated on the modelling of mean effects as functions of predictor variables. Multilevel models allow us to model the variation for any classification as a function of further variables. For example Goldstein and Noden (2003) modeled the between-school and between-LEA variation in the percentage of pupils eligible for free school meals as functions of LEA characteristics. Such models, applied for example to measures of poverty or service delivery, are highly relevant to debates about diversity since they avoid certain arbitrary features of traditional index measures, and provide efficient and objective estimates of between-unit variation. LEMMA applied this approach to modeling international data on childhood poverty and ethnicity (the latter through the PLASC data). Another development was to models that use such estimates of diversity, for example estimated for each unit in a classification, as predictors in a further model where outcomes such as individual religious attitudes are a function of area level measures of diversity.

Researchers: Goldstein, Burgess, Gordon