Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloguing-in-Publication DataGoldstein, Harvey.Multilevel statistical models / Harvey Goldstein. -4th ed. p. cm. Includes bibliographical references and index. Set in 10/12 Times by Aptara.

In the light of an increasing interest in the accountability of public institutions, this paper sets out the statistical issues involved in making quantitative comparisons between institutions in the areas of health and education. We deal in detail with the need to take account of model-based uncertainty in making comparisons. We discuss the need to establish appropriate measures of institutional 'outcomes' and base-line measures and the need to exercise care and sensitivity when interpreting apparent differences. The paper emphasizes that statistical methods exist which can contribute to an understanding of the extent and possible reasons for differences between institutions. It also urges caution by discussing the limitations of such methods.

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

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.

. (1970). Archives of Disease in Childhood, 45, 755. Standards for children's height at ages 2-9 years allowing for height of parents. Charts* are presented which give centile standards for boys' and girls' heights at ages 2 to 9 when parents' height is allowed for. Mid-parent height is used (i.e. the average of father's and mother's height).A comparison is made with results from the existing 'parent-unknown' British standard charts. A child at the 3rd centile on the parent-unknown charts is (i) at the 20th centile on the new charts if his parents are small enough to average 3rd centile for adults, (ii) at about the 1st centile if his parents average the 97th centile. Conversely a child with 97th centile parents has only to be at the 25th centile for the population in the parent-unknown charts to be at the conventional 3rd centile limit of normal when parental height is allowed for. Thus the new standards result in considerably increased precision.Examples are given of normal boys with small parents who piotted outside the 3rd centile on the conventional charts but inside on the present charts. The differential diagnosis of genetic small stature is made considerably more straightforward by the use of these charts.The correlation coefficients are given at successive ages, from 1 month to 9 years, for child's supine length or height with mid-parent height and for mother-daughter, mother-son, father-daughter, and father-son relationships.Current standards for the height attained by a child at a given age (e.g. Tanner, Whitehouse, and Takaishi, 1966) make no alowance for the height of his parents. We know, however, that tall parents in general produce tall children and short parents short children. If a child is at the 5th centile for height, therefore, it makes a considerable difference whether his parents are themselves 5th centile persons (in which case he is probably normal) or whether they are 95th centile persons (in which case he is almost certainly pathologicaUy small). This paper gives standards which allow for parental height. They apply at present only to children aged 2-0 to 9 0 years, since earlier and later ages require separate treatment. These standards are more powerful than the 'parentunknown' standards in the sense of being able to

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.

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.

A common application of multilevel models is to apportion the variance in the response according to the different levels of the data. Whereas partitioning variances is straightforward in models with a continuous response variable with a normal error distribution at each level, the extension of this partitioning to models with binary responses or to proportions or counts is less obvious. We describe methodology due to Goldstein and co-workers for apportioning variance that is attributable to higher levels in multilevel binomial logistic models. This partitioning they referred to as the variance partition coefficient. We consider extending the variance partition coefficient concept to data sets when the response is a proportion and where the binomial assumption may not be appropriate owing to overdispersion in the response variable. Using the literacy data from the 1991 Indian census we estimate simple and complex variance partition coefficients at multiple levels of geography in models with significant overdispersion and thereby establish the relative importance of different geographic levels that influence educational disparities in India. Copyright 2005 Royal Statistical Society.