By Jan de Leeuw, Erik Meijer, H. Goldstein

ISBN-10: 0387731830

ISBN-13: 9780387731834

ISBN-10: 0387731865

ISBN-13: 9780387731865

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

This ebook offers the state-of-the-art in multilevel research, with an emphasis on extra complex issues. those subject matters are mentioned conceptually, analyzed mathematically, and illustrated by means of empirical examples. The authors of the chapters are the best specialists within the field.

Given the omnipresence of multilevel information within the social, behavioral, and biomedical sciences, this publication turns out to be useful for empirical researchers in those fields. past wisdom of multilevel research isn't required, yet a uncomplicated wisdom of regression research, (asymptotic) records, and matrix algebra is assumed.

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On the other hand, consistency of s2 only requires n → ∞, which is obviously much weaker. However, the latter also requires the much stronger assumption that all residual variances are equal. 1 Introduction to Multilevel Analysis 29 Observing that Ω = Cov(β j ) = E (β j − Zj γ)(β j − Zj γ)′ , a simple estimator of Ω is obtained by inserting the least squares estimators of β j and γ in this expression: ˆ= 1 Ω m m j=1 ˆ )(bj − Zj γ ˆ )′ , (bj − Zj γ ˆ is or perhaps with m − 1 instead of m in the denominator, and where γ the one-step or two-step OLS estimator.

18 J. de Leeuw, E. Meijer a distribution with heavy tails or a mixture distribution. Moreover, the normal distribution has positive density for both positive and negative values, whereas in many cases, theory or common sense (which often coincide) says that a coefficient should have a specific sign. For example, in economics, a higher price should decrease (indirect) utility, and in education, higher intelligence should lead to higher scores on school tests. In economics, marketing, and transportation, the lognormal distribution has been proposed as a convenient alternative distribution for random coefficients in discrete choice models, perhaps after changing the sign of the explanatory variable.

34], Laird and Ware [70], Jennrich and Schluchter [63], Laird et al. [69], Lindstrom and Bates [73], and Raudenbush and Bryk [101, Chapter 14]. Further Numerical and Computational Issues As we have seen, most formulas for computing estimates for multilevel models can be expressed in different ways. Some of these are clearly computationally inefficient, whereas others use the structure of the problem in better ways. This pertains to usage of memory, sizes of inverses needed, and other ways to compute the same expressions.

### Handbook of Multilevel Analysis by Jan de Leeuw, Erik Meijer, H. Goldstein

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