A method for automated parameter estimation and testing of fit of nonstandard models for mean vectors and covariance matrices is described. Nonlinear equality and inequality constraints on the parameters of the model are allowed for. All the user will need to provide are subroutines to evaluate the mean vector and covariance matrix according to the model and, if required, the constraint functions. Subroutines for derivatives need not be provided. Some applications are described.

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Browne and Du Toit's contribution to this volume is unusual. Chapter 4 allows the reader to compare three different statistical treatments of a single empirical data set. Rarely does the reader have such an opportunity. The data are measures of learning obtained six times in a sample of 137 adults. The three models on which the statistical treatments are based are described in a concise overview by Harlow in her comment at the end of this chapter. Harlow's figures are particularly useful in indicating the nature of the analyses; in fact, the reader might well turn to Harlow's comment before tackling this chapter.The three models can be briefly described as follows: The first model, the time series model, is a first-order, autoregressive moving average time series. In such a model the measures at each time are a function of the measures at the just-previous time and the residual of that previous timethe part not predicted from the measures before that. In the second model, the stochastic parameter learning curve model, growth is approximated by means of a monotonic nonlinear equation with an estimate for each person of asymptote (highest level reached), increase from the first to the last trial, and rate of change (growth). In the third model, the latent curve model, We are indebted to Robert Cudeck for regenerating our interest in this topic with his enthusiasm and helpfulness, and to Ruth Kanfer and Phillip Ackerman for providing us with details about their work and making their carefully collected data available to us. We are also indebted to the editors, Linda Collins and John Horn, for their particularly detailed and helpful comments, which have had a substantial influence on this chapter.

Summary This paper discusses marginalization of the regression parameters in mixed models for correlated binary outcomes. As is well known, the regression parameters in such models have the “subject-specific” (SS) or conditional interpretation, in contrast to the “population-averaged” (PA) or marginal estimates that represent the unconditional covariate effects. We describe an approach using numerical quadrature to obtain PA estimates from their SS counterparts in models with multiple random effects. Standard errors for the PA estimates are derived using the delta method. We illustrate our proposed method using data from a smoking cessation study in which a binary outcome (smoking, Y/N) was measured longitudinally. We compare our estimates to those obtained using GEE and marginalized multilevel models, and present results from a simulation study.

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