Harvey Goldstein

Partitioning Variation in Multilevel Models

In multilevel modeling 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, makes sense only in simple variance components, Normal response, models where it is often referred to as the intra-unit correlation. In this article 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.

Smoking in Pregnancy and Subsequent Child Development

A national sample of several thousand children has been followed longitudinally from birth. At the ages of 7 and 11 years physical and mental retardation due to smoking in pregnancy has been found, and this deficit increases with the number of cigarettes smoked after the fourth month of pregnancy. Children of mothers who smoked 10 or more cigarettes a day are on average 1·0 cm shorter and between three and five months retarded on reading, mathematics, and general ability compared with the offspring of non-smokers, after allowing for associated social and biological factors.

New systems for dental maturity based on seven and four teeth

An updated system for estimating dental maturity is presented. It extends the original system (Demirjian et al., 1973) based on radiographs of 7 teeth by including two extra stages, and by enlarging the standardizing sample to include 2407 boys and 2349 girls. Percentile standards from ages 2-5 to 17-0 years are presented separately for boys and girls. Scoring systems and percentile standards are presented for two different sets of 4 teeth and a comparison of all three systems is made. It is suggested that these systems may measure somewhat different aspects of dental maturity.

Cigarette Smoking in Pregnancy: Its Influence on Birth Weight and Perinatal Mortality

In a British population cigarette smoking during pregnancy increased the late fetal plus neonatal mortality rate by 28% and reduced birth weight by 170 g, and these differences persist even after allowing for a number of “mediating” maternal and social variables. A change in smoking habit by the end of the fourth month of pregnancy places a mother in the risk category appropriate to her changed habit. This evidence should have important implications for health education aimed at getting pregnant mothers to give up smoking.

The Graphical Presentation of a Collection of Means

SUMMARY When a study produces estimates for many units or categories a common problem is that end-users will wish to make their own comparisons among a subset of these units. This problem will occur, for example, when estimates of school performance are produced for all schools. The paper proposes a procedure, based on the graphical presentation of confidence intervals, which enables such comparisons to be carried out while maintaining an average required type I error rate. When the means of two independent samples are to be presented graphically, it is a common practice to accompany the two points by error bars giving the 95% confidence intervals for each mean. As a visual aid, these bars are not very effective in assessing the statistical significance of the quantity of interest, which is the difference between the means. It is a common statistical misconception to suppose that two quantities whose 95% confidence intervals just fail to overlap are signif- icantly different at the 5% level. Clearly, however, it is possible to adjust the confidence level so that the required significance level is achieved by the non-overlap criterion. With equal known standard errors, and assuming normality, the width of the intervals to achieve a 5% significance level should be - 1.39u. The problem is more acute and difficult when several means are to be presented from a large study which is of interest to a variety of consumers. The results reported are likely to be used by different individuals for their own purposes and any two out of the set of means may need to be compared. This can occur, for instance, in the publication of results from population surveys, where estimates of a characteristic for each geographical unit are available. In the simplest case each individual will be interested only in a single comparison: in this situation multiple- comparison considerations do not arise. We are concerned to provide a simple presentation which will allow the results of a statistical analysis to be properly appreciated by a reader with little statistical sophistication. Our proposal is that the means presented graphically should be accompanied by error bars corresponding to confidence intervals at a level (3, drawn so that the non- overlap significance level averaged over all possible pairs is equal to the required value.

Weighting for unequal selection probabilities in multilevel models

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.

Differential School Effectiveness

Abstract Studies of school effectiveness are briefly reviewed, pointing to the need to study effectiveness for sub-groups within each school as well as overall. The results of a multilevel analysis of a large dataset covering the years 1985, 1986 and 1987 and using examination performance as the outcome measure are presented, revealing substantial differences between ethnic groups. The findings also show that the effectiveness of a school varies along several dimensions, and that there is also variation over time. The implications of these findings are discussed.

Handbook of multilevel analysis

to Multilevel Analysis.- Bayesian Multilevel Analysis and MCMC.- Diagnostic Checks for Multilevel Models.- Optimal Designs for Multilevel Studies.- Many Small Groups.- Multilevel Models for Ordinal and Nominal Variables.- Multilevel and Related Models for Longitudinal Data.- Non-Hierarchical Multilevel Models.- Multilevel Generalized Linear Models.- Missing Data.- Resampling Multilevel Models.- Multilevel Structural Equation Modeling.

A general model for the analysis of multilevel data

A general model is developed for the analysis of multivariate multilevel data structures. Special cases of the model include repeated measures designs, multiple matrix samples, multilevel latent variable models, multiple time series, and variance and covariance component models.

Methods in School Effectiveness Research

This paper discusses the methodological requirements for valid inferences from school effectiveness research studies. The requirements include long term longitudinal data and proper statistical modelling of hierarchical data structures. The paper outlines the appropriate multilevel statistical models and shows how these can model the complexities of school, class and student level data.