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**Maximum likelihood computations with repeated measures: Application of the EM algorithm.**
*(English)*
Zbl 0613.62063

The purpose of this article is to consider the use of the EM algorithm [A. P. Dempster, N. M. Laird and D. B. Rubin, J. R. Stat. Soc., Ser. B 39, 1-38 (1977; Zbl 0364.62022)] for both maximum likelihood and restricted maximum likelihood estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure. Several models and methods of analysis have been proposed for repeated measures data. One application of the EM algorithm arises naturally when we use mixed models to analyze serial measurements. In this setting, the incomplete data (or missing data) are modeled quite differently. The missing data would not be viewed as data in the traditional sense. This article shows that this approach is more general and encompasses the missing-data approach as a special case.

This result has several important applications. First, it means that EM algorithms encoded for models with random effects can also be used for multivariate normal models with arbitrary covariance structure and missing data. Second, this approach avoids specification of covariates for missing observations. Finally, use of the general formulation means that closed-form solutions for the complete data maximization will exist for a much broader class of models, enabling one to avoid use of a generalized EM or iterations within each M step. Formulas for the closed- form solutions for a certain class of multivariate growth curve models with random effects structure are given that are applicable whenever the solution is not on the boundary. The choice of starting values for the EM iterations is important. Several possibilities for starting values are given. The actual speed of convergence in two data examples is shown to depend heavily on both the actual data set and the assumed structure for the covariance matrix. The authors discuss two methods for accelerating convergence of the EM algorithm.

This result has several important applications. First, it means that EM algorithms encoded for models with random effects can also be used for multivariate normal models with arbitrary covariance structure and missing data. Second, this approach avoids specification of covariates for missing observations. Finally, use of the general formulation means that closed-form solutions for the complete data maximization will exist for a much broader class of models, enabling one to avoid use of a generalized EM or iterations within each M step. Formulas for the closed- form solutions for a certain class of multivariate growth curve models with random effects structure are given that are applicable whenever the solution is not on the boundary. The choice of starting values for the EM iterations is important. Several possibilities for starting values are given. The actual speed of convergence in two data examples is shown to depend heavily on both the actual data set and the assumed structure for the covariance matrix. The authors discuss two methods for accelerating convergence of the EM algorithm.

Reviewer: U.B.Paik