# Random-Effects Models for Longitudinal Data

**Nan M. Laird****James H. Ware**

CiteWeb id: 19820000012

CiteWeb score: 7112

DOI: 10.2307/2529876

SUMMARY Models for the analysis of longitudinal data must recogrlize the relationship between serial observations on the same unit. Multivariate models with general covariance structure are often difficult to apply to highly unbalanced data, whereas two-stage random-effects models can be used easily. In two-stage models, the probability distributions for the response vectors of different individuals belong to a single family, but some random-effects parameters vary across individuals, with a distribution specified at the second stage. A general family of models is discussed, which includes both growth models and repeated-measures models as special cases. A unified approach to fitting these models, based on a combination of empirical Bayes and maximum likelihood estimation of model parameters and using the EM algorithm, is discussed. Two examples are taken from a current epidemiological study of the health effects of air pollution. Many longitudinal studies are designed to investigate changes over time in a characteristic which is measured repeatedly for each study participant. In medical studies, the measurement might be blood pressure, cholesterol level, lung volume, or serum glucose. Multiple measurements are obtained from each individual, at different times and possibly under changing experimental conditions. Often, we cannot fully control the circumstances under which the measurements are taken, and there may be considerable variation among individuals in the number and timing of observations. The resulting unbalanced data sets are typically not amenable to analysis using a general multivariate model with unrestricted covariance structure. Statisticians have often analyzed data of this form using some variant of a two-stage model. In this formulation, the probability distribution for the multiple measurements has the same form for each individual, but the parameters of that distribution vary over individuals. The distribution of these parameters, or 'random effects', in the population constitutes the second stage of the model. In a study of changes in lung volume during childhood, for instance, it may be reasonable to assume that the relationship between lung volume and the cube of height is linear for each child, but with linear regression parameters that vary among children. If we assume that the usual linear regression model applies for each child, conditional on the child's individual parameters, and that the regression parameters have a bivariate normal distribution in the population, the marginal distribution of the serial measurements is multivariate normal with a special covariance structure.

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Nan M. Laird, James H. Ware,

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