Mixed Models

A mixed-effects model, or just mixed model, is a statistical model in which the set of predictor variables includes both fixed and random effects. A common application of mixed models is to longitudinal, or repeated measures, data in which the data consist of multiple observations on each subject. Here, observations on different subjects can be treated as independent, but observations on the same subject cannot. The inclusion of random effects in the model induces a correlation among repeated measures on a given subject. More generally, mixed-effects models are potentially useful any time a practitioner is faced with grouped data, in which observations are correlated within groups but independent across groups.

Whether a predictor term is to be treated as a fixed or random effect depends on the desired scope of inference, as well as the mechanism by which factor levels were chosen for inclusion in the study. If the investigator is concerned only with the factor levels included in the data set, then that factor should be treated as a fixed effect. If the investigator wishes to draw inference about the population from which the observed levels were drawn, the factor should be modeled as a random effect. Alternatively, the analyst can ask, If the experiment were to be repeated, would the observed levels of this factor be the same or possibly different? If the same, then the factor is a fixed effect; otherwise, it should be treated as a random effect.

The mixed model most commonly encountered in applications is the linear mixed-effects (LME) model with normally distributed random effects. A mathematical formulation of the model is given here. Let yi denote the mi-dimensional vector of responses for the ith subject in the study, for i going from 1 to M, the total number of subjects. Under the LME model, we have

where Xi and Zi are known matrices of dimension mi by p and mi by q respectively, β is the p-dimensional vector of unknown fixed effects, bi is the q-dimensional vector of random effects associated with the ith subject, and ei is a random error term. The random vectors bi and ei, for i going from 1 to M, are assumed to be mutually independent and multivariate normally distributed with a mean of zero. The unknown parameters in this model consist of the vector of coefficients β and whatever parameters determine the covariance matrices of the random vectors bi and ei. The usual statistical inference consists of the estimation of unknown parameters and prediction of the unobserved random effects.

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Following are two examples of the general LME model. Both data sets are included in the “nlme” library within the free statistical software package R.

Example 1: An Ergometric Experiment

This experiment involved nine human subjects and four types of stools. Each subject was asked to arise from each of the stools, and the effort exerted (Borg scale) was recorded. The data are available with the R package, as the object “ergoStool” in the “nlme” library.

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