Applications of multilevel models to continuous outcomes nearly always assume constant residual variance and constant random effects variances and covariances. However, modeling heterogeneity of variance can prove a useful indicator of model misspecification, and in some educational and behavioral studies, it may even be of direct substantive interest. The purpose of this article is to review, describe, and illustrate a set of recent extensions to two-level models that allow the residual and random effects variance–covariance components to be specified as functions of predictors. These predictors can then be entered with random coefficients to allow the Level-1 heteroscedastic relationships to vary across Level-2 units. We demonstrate by simulation that ignoring Level-2 variability in residual variances leads the Level-1 variance function regression coefficients to be estimated with spurious precision. We discuss software options for fitting these extensions, and we illustrate them by reanalyzing the classic High School and Beyond data and two-level school effects models presented by Raudenbush and Bryk.

Longitudinal observations on known individuals are an important source of data with which to test evolutionary theory within natural populations, in particular, the evolution and maintenance of life-history traits. In this paper, we concentrate on the reproductive behaviour and survival of a small passerine bird, the great tit (Parus major). The dataset we consider is taken from the long-term study of great tits in Wytham Woods in Oxfordshire. The models we consider are designed to relate variation in several phenotypic response variables that are linked to evolutionary fitness, alongside the correlations between them, to both general environmental and individual-specific factors. We fit multivariate cross-classified random effects models using a Markov chain Monte Carlo (MCMC) estimation algorithm described in the paper. Our results show for which traits variability is influenced by environmental factors and for which traits individual bird factors are more important. The partitioning of correlations is particularly illuminating, producing some pairs of ‘antagonistic’ correlations which are biologically meaningful.

In this article, we discuss the effect of removing the independence assumptions between the residuals in two-level random effect models. We first consider removing the independence between the Level 2 residuals and instead assume that the vector of all residuals at the cluster level follows a general multivariate normal distribution. We demonstrate how this assumption can allow us to fit higher levels of clustering and school competition effects via an example from education. We then consider removing the assumption of independence between Level 1 residuals within clusters. We show how this extension can allow time series type models. Both normal and binary responses are considered.

We analysed two consecutive inspection reports for each of 136 British zoos made by government-appointed inspectors between 2005 and 2011 to assess how well British zoos were complying with minimum animal welfare standards; median interval between inspections was 1,107 days. There was no conclusive evidence for overall improvements in the levels of compliance by British zoos. Having the same zoo inspector at both inspections affected the outcome of an inspection; animal welfare criteria were more likely to be assessed as unchanged if the same inspector was present on both inspections. This, and erratic decisions as to whether a criterion applied to a particular zoo, suggest inconsistency in assessments between inspectors. Zoos that were members of a professional association (BIAZA) did not differ significantly from non-members in the overall number of criteria assessed as substandard at the second inspection but were more likely to meet the standards on both inspections and less likely to have criteria remaining substandard. Lack of consistency between inspectors, and the high proportion of zoos failing to meet minimum animal welfare standards nearly thirty years after the Zoo Licensing Act came into force, suggest that the current system of licensing and inspection is not meeting key objectives and requires revision.

Periodontal disease is one of the most common conditions affecting dogs worldwide and is reported to be particularly prevalent in racing greyhounds. A range of potential risk factors have been hypothesised. Previous research has suggested that regular tooth brushing can reduce both calculus and gingivitis, but the frequency required is unclear. Here, we report a controlled blinded in situ trial, in which kennel staff brushed 160 racing greyhounds’ teeth (living at six kennel establishments), either weekly, daily or never over a two-month period. All of the visible teeth were scored for calculus and gingivitis, using previously validated scales. We calculated average scores for each of the three teeth groups and overall whole mouth scores, averaging the teeth groups. Changes were compared to the baseline. After two months, the total calculus scores (controlling for baseline) were significantly different in the three treatment groups, (F_(2,129) = 10.76, p < 0.001) with both weekly and daily brushing resulting in significant reductions. Gingivitis was also significantly different between groups (F_(2,128) = 4.57, p = 0.012), but in this case, only daily brushing resulted in a significant reduction. Although the dogs in different kennels varied significantly in their levels of both calculus (F_(5,129) = 8.64, p < 0.001) and gingivitis (F_(5,128) = 3.51 p = 0.005), the intervention was similarly effective in all of the establishments. The teeth groups varied, and the incisors were not significantly affected by the treatment. Since the trainers implementing the routine, reported a minimal time commitment and positive experiences, we suggest that daily brushing is recommended for racing greyhounds, and that any instructions or demonstrations should include attention to all teeth groups including the incisors. Similar trials need to be conducted with retired greyhounds since these have been shown to present particularly high levels of periodontal disease.