When observations are made repeatedly over time on the same experimental units, unbalanced patterns of observations are a common occurrence. This complication makes standard analyses more difficult or inappropriate to implement, means loss of efficiency, and may introduce bias into the results as well. Some possible approaches to dealing with missing data include complete case analyses, univariate analyses with adjustments for variance estimates, two-step analyses, and likelihood based approaches. Likelihood approaches can be further categorized as to whether or not an explicit model is introduced for the non-response mechanism. This paper will review the use of likelihood based analyses for longitudinal data with missing responses, both from the point of view of ease of implementation and appropriateness in view of the non-response mechanism. Models for both measured and dichotomous outcome data will be discussed. The appropriateness of some non-likelihood based analyses is briefly considered.

Missing data in longitudinal studies

The purpose of this article is to consider the use of the EM algorithm (Dempster, Laird, and Rubin 1977) for both maximum likelihood (ML) and restricted maximum likelihood (REML) estimation in a general repeated measures setting using a multivariate normal data model with linear mean and covariance structure (Anderson 1973). Several models and methods of analysis have been proposed in recent years for repeated measures data; Ware (1985) presented an overview. Because the EM algorithm is a general-purpose, iterative method for computing ML estimates with incomplete data, it has often been used in this particular setting (Dempster et al. 1977; Andrade and Helms 1984; Jennrich and Schluchter 1985). There are two apparently different approaches to using the EM algorithm in this setting. In one application, each experimental unit is observed under a standard protocol specifying measurements at each of n occasions (or under n conditions), and incompleteness implies that the number of measurements actua...

Maximum likelihood computations with repeated measures: application of the EM algorithm

A flexible method is introduced to model the structure of a covariance matrix C and study the dependence of the covariances on explanatory variables by observing that for any real symmetric matrix A, the matrix exponential transformation, C = exp (A), is a positive definite matrix. Because there is no constraint on the possible values of the upper triangular elements on A, any possible structure of interest can be imposed on them. The method presented here is not intended to replace the existing special models available for a covariance matrix, but rather to provide a broad range of further structures that supplements existing methodology. Maximum likelihood estimation procedures are used to estimate the parameters, and the large-sample asymptotic properties are obtained. A simulation study and two real-life examples are given to illustrate the method introduced.

The Matrix-Logarithmic Covariance Model

A survey is given of papers which have influenced or have been influenced by the Growth Curve Model due to Potthoff & Roy (1964). The review covers, among others, methods of estimating parameters, the canonical version of the model, tests, extensions, incomplete data, Bayesian approaches and covariance structures.

The growth curve model: a review

We propose a possible statistical model for both contextual analysis and slopes as outcomes analysis. These techniques have been used in multilevel analysis for quite some time, but a precise specification of the regression models has not been given before. We formalize them by proposing

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Random coefficient models for multilevel analysis

The maximum likelihood equations for a multivariate normal model with structured mean and structured covariance matrix may not have an explicit solution. In some cases the model's error term may be decomposed as the sum of two independent error terms, each having a patterned covariance matrix, such that if one of the unobservable error terms is artificially treated as "missing data", the EM algorithm can be used to compute the maximum likelihood estimates for the original problem. Some decompositions produce likelihood equations which do not have an explicit solution at each iteration of the EM algorithm, but within-iteration explicit solutions are shown for two general classes of models including covariance component models used for analysis of longitudinal data.

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Dal ton F Andrade

Papers

Maximum likelihood estimates in the multivariate normal with patterned mean and covariance via the em algorithm