Relating the classical covariance adjustment techniques of multivariate growth curve models to modern univariate mixed effects models

The relationship between the modern univariate mixed model for analyzing lo ngitudinal data, popularized by Laird and Ware (1982, Biometrics 38, 963-97 4), and its predecessor, the classical multivariate growth curve model, sum marized by Grizzle and Alien (1969, Biometrics 25, 357-381), has:never been clearly established. Here, the link between the two methodologies is deriv ed, and balanced polynomial and cosinor examples cited in the literature ar e analyzed with both approaches. Relating the two models demonstrates that classical covariance adjustment for higher-order terms is analogous to incl uding them as random effects in the mixed model. The polynomial example cle arly illustrates the relationship between the methodologies and shows their equivalence when all matrices are properly defined. The cosinor example de monstrates how results from each method may differ when the total variance- covariance matrix is positive definite, but that the between-subjects compo nent of that matrix is not so constrained by the growth curve approach. Add itionally, advocates of each approach tend to consider different covariance structures. Modern mixed model analysts consider only those terms in a mod el's expectation (or linear combinations), and preferably the most parsimon ious subset, as candidates for random effects. Classical growth curve analy sts automatically consider all terms in a model's expectation as random eff ects and then investigate whether "covariance adjusting" for higher-order t erms improves the model. We apply mixed model techniques to cosinor analyse s of a large, unbalanced data set to demonstrate the relevance of classical covariance structures that were previously conceived for use only with com pletely balanced data.