What is Multilevel Modeling?
Multilevel modeling (MM) is a family of statistical procedures that try to come to terms with influences that are located on different, well, levels. So naturally the question arises what is meant by "level".
One way to think about it is as follows: People do not live entirely on their own, but rather embedded in social units. Even though today, in a globalized world, we may say that people have relationships with other people all over the world, most people have some relationships that are more special than others. People who are linked together via special relationships frequently communicate among each other, and thus the possibility rises that the people you are linked to influence your views. So we may think about the individuals as one (the lowest) level and their network (whether it consists of people that are met in person or of people communication with whom may take place only via artifical media) as a next (higher) level.
(Note that " low" and " high" are just names; we may well think about things the other way round. "High" just means something like "aggregate"; that is, several individuals – entities on the "low" level – are seen as agglutinated).
A second way: Opportunities structure the behaviour of individuals, and as many people
select their opportunities by local proximity, the region in which a person lives may enhance or restrict his or her opportunity. For instance, if a person lives in a region with high unemployment, this may influence his or her behaviour about acceptable wage levels when looking for a new job.
A third way: Often people, be it voluntarily or not, are subject to common external influences. Take, for instance, a university professor. All the students that come to him or her are subject to her or his way of teaching. Could be that this way of teaching influences these students (even though this certainly – if sometimes fortunately – happens less frequently than we professors might desire).Therefore, again we may think of a multitude of professors as the "higher" level units and of their many students as the "lower" level units.
For a variety of reasons, data referring to more than one level often cannot be analyzed by conventional statistical models. For instance, classical OLS regression analysis requires that residuals from individual observations are not correlated. This requirement becomes doubtful if these individual observations are subject to the same influences or are related to each other in other ways.
After so many words, this page – for the time being – does very little: It provides a few links to MM related pages, and it also provides selected references to the literature, with short comments.
Regrettably, updates of this page are quite limited:
- The section with links is checked from time to time concerning changes in URLs; if I encounter interesting sites, they will be added. However, during the recent years I have only rarely looked for other sites.
- Only the first part of the literature section, dealing with introductory texts and textbooks, is updated (i.e. augmented) occasionally. The remaining parts of this section were written in 1999 and have not changed since then.
- The Centre for Multilevel Modeling at Bristol – the follow-up of the former Multilevel Models Project at London
- Software Review – a subsite of the Centre for Multilevel Modeling of tremendous help for users, therefore mentioned specifically here.
- The Multilevel Modelling Portal at the UCLA.
Note that this site is no longer maintained. Is there some sort of Murphy's Law for websites? It would go like this: The probability of a website to disappear or to be abandoned increases with its usefulness.
- Note specifically this very helpful paper by Judith Singer. The paper is about how to use SAS for estimatation of multilevel models; furthermore, this page demonstrates how this can be achieved with specialized (HLM or MLwiN) as well as multi-purpose (SPSS, S-Plus) software.
- Tom Snijders' Page
- Wolfgang Langer's Multilevel Page
- G. David Garson's Page on Linear Mixed Models (another term to denote multilevel models).
This is more or less an exhaustive introduction to multilevel modelling. Includes examples for SPSS and HML, a bibliography, and many other interesting features.
- Paper on multilevel analysis software by Jan de Leeuw and Ita G.G. Kreft.
Even though not very up to date, this paper provides a very useful overview about essential features of a variety of multilevel analysis programs and packages.
Dear reader, the following remarks are intended to serve as a guide to the growing literature on multilevel modelling. They cover most of the textbooks and many of the "classical" papers by those statisticians who did most of the pathbreaking work.
Regrettably my overview covers only the years up to 1999 (even though I will try to stay up to date as far as textbooks are concerned)! Of course, since that time many useful new papers have appeared, but even though I take an occasional glance at some of them, I do not find the time to include them here. Please accept my apologies!
Note that the Centre for Multilevel Modeling now has a similar website that introduces you particularly to newer, and also to more specialised, books. (However, it does not treat the texts in German that are covered below).
Introductory texts (elementary and advanced) (updated every now and then)
- Bickel, Rorbert (2007): Multilevel Analysis für Applied Research. It's Just Regression!, New York, London: Guilford Press
The subtitle runs through the book from the beginning. Bickel of course is right. But in my view the book sometimes is cumbersome to read, as it is rather wordy, which Bickel admits himself. On the other hand, if you like long and wordy explanations, this may be the book of your choice. What's really nice is the short overwiew of useful resources at the end of each section. Another great feat, even though I think this is not the effect intended by Bickel: The examples of how to do multilevel analysis via SPSS's menu system that are scattered over the entire book make it clear how any understanding of what you're doing is utterly obfuscated by doing statistics this way (i.e. using menus instead of commands written in plain text).
- Bressoux, Pascal, Coustère, Paul & Leroy-Audouin, Christine (1997): Les modèles multiniveau dans l'analyse écologique: le case de la recherche en éducation, in: Revue française de sociologie, 38, 67-96.
Some like it French: An introductory paper on multilevel analysis of educational achievement, with good data. No maths.
- Bryk, A. S. & Raudenbush, S. W. (1992): Hierarchical Linear Models. Applications and
Data Analysis Methods. Newbury Park, CA: Sage.
Very good for those who wish to arrive at an advanced understanding.
- DiPrete, Thomas A. & Forristal, Jerry D. (1994): Multilevel Models: Methods and
Substance, in: Annual Review of Sociology, 20, 331-357.
A good overview of the basic ideas and of applications, with most emphasis on random-coefficient models.
- Ditton, Hartmut (1998): Mehrebenenanalyse. Grundlagen und Anwendungen des
Hierarchisch Linearen Modells. Weinheim und München: Juventa.
This book is specifically useful for those who want to analyze school data (and have to resort to books in German). However, readers of chapter 3 (dealing largely with
centering) should consult chapter 5.2 in Kreft/de Leeuw 1998 and the paper by Kreft/de Leeuw/Aiken 1995.
- Engel, Uwe (1998): Einführung in die Mehrebenenanalyse. Opladen: Westdeutscher
Verlag (WV Studium 182).
The wide coverage of this book has much to recommend it - for users who already have acquired an elementary understanding.
- Gelman, Andrew/Hill, Jennifer: Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge: Cambridge University Press, 2007.
A newer book that I can strongly recommend. Uses elements of Bayesian analysis.
- Goldstein, Harvey (2010): Multilevel Statistical Models. London: Arnold (fourth edition).
A more advanced introduction by one of the " fathers" of multilevel
modeling. The first edition appeared in 1987. The manuscript of the second edition can be downloaded here for free. Some material concerning the third edition, not least some errata, may be found here.
- Hox, Joop (2002; 2nd edtion 2010): Multilevel Analysis. Techniques and Applications. Mahwah, NJ: Erlbaum.
This book is the expanded and updated version of an earlier book, Multilevel Analysis, Amsterdam: TT Publishers, 1995, which is downloadable for free (attention, this is a PDF file with a size of several MB).
- Hox, Joop J. & Kreft, Ita G. G. (1994): Multilevel Analysis Methods, in: Sociological
Methods & Research, 22, 283-299.
This introduction to a special issue of SMR gives a brief overview of problems
resulting from using traditional approaches for multilevel data, of the basic structure of the random coefficient model, of available software, and of current problems and possible future developments.
- Kreft, Ita G. G. (1991): Using Hierarchically Linear Models to Analyse Multilevel Data. In: ZUMA-Nachrichten, 29, 44-56.
A brief introduction that should be found in most German sociology libraries, but
(naturally) cannot replace a textbook.
- Kreft, Ita & de Leeuw, Jan (1998): Introducing Multilevel Modeling, London: Sage.
This is certainly the most helpful textbook for those with a strong dislike of maths, formal derivations and the like. The reader is carefully guided through a number of examples. However, one should be aware that there are many advanced topics that are not dealt with in this book.
- Langer, Wolfgang (2004): Mehrebenenanalyse: Eine Einführung für Forschung und Praxis. Wiesbaden: VS-Verlag.
This should definitely be the source for beginners who need a text in German. What I like particularly is the author's approach to start with detailed exploratory analyses and only then to proceed to more advanced modeling strategies. Complex mathematical detail (for instance, concerning estimation methods) is omitted, but the practical implications are dealt with in a fairly comprehensive manner. Examples use MLA, a free software for estimating multilevel models.
A website provides data sets and command files used in this book as well as links to software and other sources.
- Longford, N. (1993): Random coefficient models. Oxford: Oxford University Press.
In this book, statistical reasoning is paramount, but it is also applied to several datasets with helpful discussions of the results.
- Luke, Douglas A. (2004): Multilevel Modeling. Thousand Oaks, CA: Sage (Sage university papers series: Quantitative applications in the social sciences, 143).
Like all other volumes in this series, this is a nice and accessible introduction which, due to the usual limitations of space, cannot discuss all the complications that may arise during a particular analysis.
- O'Connell, Ann A. & McCoach, Betsy (Eds.) (2008): Multilevel Modeling of Educational Data. Charlotte, NC: IAP - Information Age Publishing.
Part I gives an overview of multilevel modeling, with special chapters for growth curve models, cross-classified random effects models and models for categorical dependent variables. Part II deals with assessment of model fit and discusses issues of power, sample size and design. Part III discusses extensions of the multilevel framework to issues like meta-analysis and measurement modeling. Part IV, finally, presents chapters about how to report results, about available software and about estimation procedures. All chapters are written in an accessible style.
- Ohr, Dieter (1999): Modellierung von Kontexteffekten: Voraussetzungen, Verfahren und
eine empirische Anwendung am Beispiel des politischen Informationsverhaltens, in: ZA-Information 44, 39-63.
A fine brief introduction, but I feel that it is somewhat infortunate that MM is
introduced via an example where the gains from using a MM approach are almost nil
(statistically speaking, i.e. the MM analysis barely differs from OLS regression results).
- Singer, J. D. & Willett, J. B. (2003): Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence. New York: Oxford University Press.
Although focussing on one specific application of multilevel modelling, i.e. within-individual change, this book certainly is worthwhile reading. – There is also a website for this book with data sets, errata and other stuff.
- Snijders, Tom & Bosker, Roel (1999): Multilevel Analysis. An introduction to basic and advanced multilevel modeling. London, Thousand Oaks: SAGE.
This is a very thorough introduction that requires quite some effort on part of the student - but it pays. Especially chapters 8 and 9 should be studied carefully, as I have found no comparable discussion of heteroscedascity and the basic assumptions (and how to check them) of multilevel modeling. There is a website with some material pertaining to this book, not least some additional remarks and some corrections.
The second edition appeared in 2011, with material for this edition to be found here.
Other papers on multilevel random coefficient (or variance components) models (not updated since 1999)
Getting to the basics
- Aitkin, M. & Longford, N. (1986): Statistical Modelling Issues in School Effectiveness
Studies, in: Journal of the Royal Statistical Society, Series A 149, 1-43.
This paper introduces a variance component (or "random intercept" and "random
slope") model based on a Maximum Likelihood algorithm. Results of this model on a school
data set are contrasted to some individual level models (including one with
context effects) and to an aggregate level analysis.
- Bock, Darrell R. (Ed.) (1989): Multilevel Analysis of Educational Data. San Diego:
See statisticians work. This book is largely about how to assess school effectiveness. Some chapters give you a good idea of what Empirical Bayes estimation is about.
- Bryk, Anthony S. & Raudenbush, Stephen (1987): Application of Hierarchical Linear
Models to Assessing Change, in: Psychological Bulletin, 101, 147-158.
A fine paper with a very clear structure, demonstrating the possibilities of multilevel modeling in the analysis of longitudinal data.
- de Leeuw, Jan & Kreft, Ita G. G. (1986): Random Coefficient Models for Multilevel Analysis, in: Journal of Educational Statistics 11, 57-85.
Discusses nicely some of the limitations of traditional contextual analysis and then proceeds to elaborate at some length the statistical properties of different estimation methods for RC models. These methods eventually are applied to a Dutch school example.
- de Leeuw, Jan & Kreft, Ita G. G. (1995): Questioning Multilevel Models, in: Journal of
Educational and Behavioral Statistics 20, 171-189.
Illuminating discussion of some implications of multilevel modeling.
- Draper, D. (1995): Inference and hierarchical modeling in the social sciences, in: Journal of Educational and Behavioral Statistics 20(2), 115-147.
Technical discussion (but without any maths) of inferential issues in multilevel modelling; pleas for more frequent use of Markov-Chain Monte Carlo techniques.
- Goldstein, Harvey & Rasbash, J. (1996): Improved approximations for multilevel models with binary responses, in: Journal of the Royal Statistical Society, Series A, 159, 505-513.
This paper develops a penalized quasi-likelihood (PQL) method for the estimation of multilevel models with a dichotomous dependent variable. It demonstrates that this method is superior to the earlier marginal quasi-likelihood approach, especially in the case of extreme data constellations such as a very skewed dependent variable.
- Goldstein, Harvey, Rasbash, Jon, Yang, Min, Woodhouse, Geoffrey, Pan, Huiqi, Nuttall, Desmond & Thomas, Sally (1993): A Multilevel Analysis of School Examination Results, in: Oxford Review of Education 19(4), 425-433.
This paper highlights some of the problems of comparing schools; see also Goldstein & Spiegelhalter (1996) for more extensive discussion of these topics
- Goldstein, Harvey & Spiegelhalter, David J. (1996): League tables and their limitations: statistical issues in comparisons of institutional performance, in: Journal of the Royal Statistical Society, Series A, 159(3), 385-443.
- Raudenbush, Stephen W. (1988): Educational Applications of Hierarchical Linear
Models: A Review, in: Journal of Educational Statistics 13, 85-116.
Despite its title, this paper is basically a rather formal exposition of the estimation theory in multilevel models, accessible only to readers with substantial statistical training.
Specific Issues in Multilevel Modelling
- Enders, Craig K., & Tofighi, Davood (2007). Centering predictor variables in cross-sectional multilevel models: A new look at an old issue. Psychological Methods, 12 (2), 121-138.
A newer paper on centering.
- Kreft, G. G. & de Leeuw, E. D. (1988): The See-Saw Effect: a multilevel problem?, in:Quality & Quantity 22, 127-137.
The see-saw effect is a difference -- perhaps even an (apparent) contradiction --
between individual and aggregate level effects, as for instance when a variable (say, sex) has a positive slope on the individual level, but the same attribute measured on the aggregate level (the proportion of girls in a class) has a zero or even negative slope. This paper discusses the conditions under which such effects may occur and in addition propagates random coefficient models as procedures to get appropriate estimates of standard errors, appropriate specifically when compared to the "Cronbach procedure" that is discussed as a contrast.
- Kreft, Ita G. G., de Leew, Jan & Aiken, Leona S. (1995): The Effects of Different Forms of Centering in Hierarchical Linear Models, in: Multivariate Behavioral Research, 30, 1-21.
Mandatory reading for all who want to know more about centering of variables.
- Heath, Anthony, Yang, M. & Goldstein, Harvey (1996): Multilevel analysis of the changing relationship between class and party in Britain 1964-1992, in: Quality & Quantity 30, 389-404.
A sociological example with a binary dependent variable. Focus is not on statistical basics, but on substantive meaning of multilevel vs. single-level modelling.
- Hox, Joop J. (1994): Hierarchical Regression Models for Interviewer and Respondent Effects, in: Sociological Methods & Research, 22, 300-318.
A useful substantive application, especially for those who are tired of school studies
- Kreft, Ita G. G. (1993): Using Multilevel Analysis to Assess School Effectiveness: A
Study of Dutch Secondary Schools, in: Sociology of Education 66, 104-129.
This is an applied paper dealing mainly with the substantive results of multilevel
(random coefficient) modeling.
- Kreft, Ita G. G. & de Leew, Jan (1994): The Gender Gap in Earning. A Two-Way Nested
Multiple Regression Analysis with Random Effects, in: Sociological Methods & Research,
Another substantive application that does not focus on school data.
- Nuttall, D. L., Goldstein, Harvey, Prosser, R. & Rasbash, J. (1989): Differential school effectiveness, in: International Journal of Educational Research, 13, 769-776.
Focusses on substantive implications; no maths.
- Raudenbush, Stephen & Bryk, Anthony S. (1986): A Hierarchical Model for Studying School Effects, in: Sociology of Education 59, 1-17.
A short presentation of a random coefficient model with applications on a school data set to demonstrate the different variance components and cross-level interaction.
Literature with emphasis on the "pre-random coefficients" stage in thinking about multilevel models
- Blalock, H. M. (1984): Contextual-Effects Models: Theoretical and Methdological Issues, in: Annual Review of Sociology 10, 353-372.
Main focus here is on substantive interpretation and discussion of theoretical implications.
- Burstein, L. (1980): The Analysis of Multilevel Data in Educational Research and Evaluation, in: Review of Research in Education 8, 158-223.
This paper first discusses the problems of drawing inferences from aggregate level analysis at individual level relationships (cross level inference). Next, discussions about the appropriate level of analysis (individual or aggregate) are summed up. Another section discusses cross-level effects, that is, possible effects of aggregate level on individual level characteristics. Finally, some statistical procedures for a multilevel analysis are discussed, such as the "slope-as-outcomes" strategy developped by Burstein et al. in other papers. This paper is quite thorough, but probably of interest mainly to those with a strong interest in the developments that occurred during the 1970s.
- Burstein, Leigh, Linn, Robert L. & Capell, Frank J. (1978): Analyzing Multilevel Data in the Presence of Heterogeneous Within-Class Regressions, in: Journal of Educational Statistics, 3, 347-383.
Compares three strategies of analysis, (1) a Cronbach analysis (see Cronbach & Webb 1976), (2) a Slopes-as-Outcomes analysis, and (3) a Keesling-Wiley procedure. Concludes that all three procedures can give misleading estimates of class effects on mean class outcomes, but that they may also indicate sources of model mis-specification.
- Cheung, K. C., Keeves, J. P., Sellin, N. & Tsoi, S. C. (1990): The analysis of multilevel data in educational research: Studies of problems and their solutions, in: International Journal of Educational Research 14, S. 215-319.
This is sort of a "transitional" document. It discusses and criticizes at great length the "context analysis" of the 1970s and earlier 1980s, and then proceeds to present the more recent approaches by Goldstein, Bryk/Raudenbush, or Longford, which at that time were in an early stage, especially computer-wise.
- Iversen, G. (1991): Contextual Analysis, Newbury Park: Sage (Sage University Paper Series on Quantitative Applications in the Social Sciences).
This small volume is a careful and elementary introduction to modeling individual and aggregate level effects, but only in the final chapter random coefficient models are mentioned briefly and the problems inherent in the older approach are not really discussed. Iversen seems to imply (even though I am not sure whether he says so explicitly) that smaller standard errors for estimated coefficients are superior to larger standard errors (see p. 58 et seq.), but of course this is only true if the smaller errors are indeed valid, which in this case is very doubtful.
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