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Multilevel Model Prediction

Published online by Cambridge University Press:  01 January 2025

Edward W. Frees  and
Jee-Seon Kim
Edward W. Frees
Affiliation:
University of Wisconsin, Madison
Jee-Seon Kim*
Affiliation:
University of Wisconsin, Madison
*
Requests for reprints should be sent to Jee-Seon Kim, Department of Educational Psychology, University of Wisconsin, 1025 West Johnson Street, Madison, WI 53706, USA. E-mail: jeeseonkim@wisc.edu
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Abstract

Multilevel models are proven tools in social research for modeling complex, hierarchical systems. In multilevel modeling, statistical inference is based largely on quantification of random variables. This paper distinguishes among three types of random variables in multilevel modeling—model disturbances, random coefficients, and future response outcomes—and provides a unified procedure for predicting them. These predictors are best linear unbiased and are commonly known via the acronym BLUP; they are optimal in the sense of minimizing mean square error and are Bayesian under a diffuse prior.

For parameter estimation purposes, a multilevel model can be written as a linear mixed-effects model. In this way, parameters of the many equations can be estimated simultaneously and hence efficiently. For prediction purposes, we show that it is more convenient to retain the multiple equation feature of multilevel models. In this way, the efficient BLUPs are easy to compute and retain their intuitively appealing recursive form. We also derive explicit equations for standard errors of these different types of predictors.

Prediction in multilevel modeling is important in a wide range of applications. To demonstrate the applicability of our results, this paper discusses prediction in the context of a study of school effectiveness.

Keywords

best linear unbiased predictorslinear mixed-effects modelshierarchical linear modelsrandom coefficientsschool effectiveness
Type
Original Paper
Information
Psychometrika , Volume 71 , Issue 1 , March 2006 , pp. 79 - 104
Copyright
Copyright © 2006 The Psychometric Society

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Footnotes

This research was supported by a grant from the Graduate School at the University of Wisconsin at Madision and the National Science Foundation, Grant number SES-0436274. We are grateful to Norman Webb at Wisconsin Center for Education Research for making available the data used in the reported application.

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