# Linear Mixed Effects Models

I have commonly heard that LME models are more sound in the analysis of accuracy data (i.e., in psychology experiments), in that they can work with binomial and other non-normal distributions that traditional approaches (e.g., ANOVA) can't.

What is the mathematical basis of LME models that allow them to incorporate these other distributions, and what are some not-overly-technical papers describing this?

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One major benefit of mixed-effects models is that they don't assume independence amongst observations, and there can be a correlated observations within a unit or cluster.

This is covered concisely in "Modern Applied Statistics with S" (MASS) in the first section of chapter 10 on "Random and Mixed Effects". V&R walk through an example with gasoline data comparing ANOVA and lme in that section, so it's a good overview. The R function to be used in lme in the nlme package.

The model formulation is based on Laird and Ware (1982), so you can refer to that as a primary source although it's certainly not good for an introduction.

• Laird, N.M. and Ware, J.H. (1982) "Random-Effects Models for Longitudinal Data", Biometrics, 38, 963–974.
• Venables, W.N. and Ripley, B.D. (2002) "Modern Applied Statistics with S", 4th Edition, Springer-Verlag.

You can also have a look at the "Linear Mixed Models" (PDF) appendix to John Fox's "An R and S-PLUS Companion to Applied Regression". And this lecture by Roger Levy (PDF) discusses mixed effects models w.r.t. a multivariate normal distribution.

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Wow, Shane, thanks for the very fast response! I'll look into those references! –  Mike Wong Jul 27 '10 at 14:02

A very good article explaining the general approach of LMMs and their advantage over ANOVA is:

Linear mixed-effects models (LMMs) generalize regression models to have residual-like components, random effects, at the level of, e.g., people or items and not only at the level of individual observations. The models are very flexible, for instance allowing the modeling of varying slopes and intercepts.

LMMs work by using a likelihood function of some kind, the probability of your data given some parameter, and a method for maximizing this (Maximum Likelihood Estimation; MLE) by fiddling around with the parameters. MLE is a very general technique allowing lots of different models, e.g., those for binary and count data, to be fitted to data, and is explained in a number of places, e.g.,

• Agresti, A. (2007). An Introduction to Categorical Data Analysis (2nd Edition). John Wiley & Sons.

LMMs, however, can't deal with non-Gaussian data like binary data or counts; for that you need Generalized Linear Mixed-effects Models (GLMMs). One way to understand these is first to look into GLMs; also see Agresti (2007).

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