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Handbook of Multilevel Analysis by Jan de Leeuw, Erik Meijer (auth.), Jan de Leeuw, Erik Meijer

By Jan de Leeuw, Erik Meijer (auth.), Jan de Leeuw, Erik Meijer (eds.)

Multilevel research is the statistical research of hierarchically and non-hierarchically nested information. the best instance is clustered facts, comparable to a pattern of scholars clustered inside colleges. Multilevel information are in particular regular within the social and behavioral sciences and within the bio-medical sciences. The versions used for this kind of information are linear and nonlinear regression types that account for saw and unobserved heterogeneity on the a number of degrees within the info.

This e-book provides the state-of-the-art in multilevel research, with an emphasis on extra complicated themes. those issues are mentioned conceptually, analyzed mathematically, and illustrated by means of empirical examples. The authors of the chapters are the top specialists within the field.

Given the omnipresence of multilevel information within the social, behavioral, and biomedical sciences, this ebook comes in handy for empirical researchers in those fields. past wisdom of multilevel research isn't required, yet a easy wisdom of regression research, (asymptotic) information, and matrix algebra is assumed.

Jan de Leeuw is exclusive Professor of data and Chair of the dep. of data, collage of California at l. a.. he's former president of the Psychometric Society, former editor of the Journal of academic and Behavioral Statistics, founding editor of the Journal of Statistical Software, and editor of the Journal of Multivariate Analysis. he's coauthor (with Ita Kreft) of Introducing Multilevel Modeling and a member of the Albert Gifi workforce who wrote Nonlinear Multivariate Analysis.

Erik Meijer is Economist on the RAND company and Assistant Professor of Econometrics on the college of Groningen. he's coauthor (with Tom Wansbeek) of the hugely acclaimed ebook Measurement errors and Latent Variables in Econometrics.

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Example text

2 Maximum Likelihood Except for some special cases, explicit closed-form expressions for the maximum likelihood estimators are not available. The loglikelihood function has to be optimized by using some kind of numerical algorithm. This section discusses several of the available algorithms. We can distinguish, on the one hand, generic numerical optimization techniques that can be used for any wellbehaved function and, on the other hand, algorithms that are more specific to the problem at hand. Let f (θ) be a loss function of a parameter vector θ.

We can also arrive at these bias adjustments in a slightly different way, which allows us to continue to use the log-likelihood. Suppose we compute the likelihood of the deviations of the mean, or in the more general case the likelihood of the observed regression residuals. These residuals have a singular multivariate normal distribution, and the maximum likelihood estimate of the variance turns out to be precisely the bias-adjusted estimate. Thus, in 22 J. de Leeuw, E. Meijer these simple cases, residual maximum likelihood (REML; also frequently called restricted maximum likelihood ) estimates can actually be computed from full information maximum likelihood estimates by a simple multiplicative bias adjustment.

There are also some differences in the more advanced options or less frequently used model specifications, so users with specific desires may prefer one over the other for this reason. Originally, VARCL [75] was also one of the major packages, but development of this package has been terminated. There are many packages that focus on more specific multilevel models, options, or other aspects. These tend to be research software, with fewer options and less user-friendly interfaces, and development of these progresses faster if the authors are working on new directions in their research that requires additions to the programs.

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