Marginal and Random Intercepts Models for Longitudinal Binary Data With Examples From Criminology

Abstract

Two models for the analysis of longitudinal binary data are discussed: the marginal model and the random intercepts model. In contrast to the linear mixed model (LMM), the two models for binary data are not subsumed under a single hierarchical model. The marginal model provides group-level information whereas the random intercepts model provides individual-level information including information about heterogeneity of growth. It is shown how a type of numerical averaging can be used with the random intercepts model to obtain group-level information, thus approximating individual and marginal aspects of the LMM. The types of inferences associated with each model are illustrated with longitudinal criminal offending data based on N = 506 males followed over a 22-year period. Violent offending indexed by official records and self-report were analyzed, with the marginal model estimated using generalized estimating equations and the random intercepts model estimated using maximum likelihood. The results show that the numerical averaging based on the random intercepts can produce prediction curves almost identical to those obtained directly from the marginal model parameter estimates. The results provide a basis for contrasting the models and the estimation procedures and key features are discussed to aid in selecting a method for empirical analysis.

ACKNOWLEDGMENTS

The work on this article was supported by Grant 96-MU-FX-0012 from the Office of Juvenile Justice and Delinquency Prevention, Grant 50778 from the National Institute of Mental Health, and Grant 411018 from the National Institute of Drug Abuse. Points of view or opinions in this document are those of the authors and do not necessarily represent the official position or policies of the U.S. Department of Justice, the National Institute of Mental Health, and the National Institute of Drug Abuse.

Notes

¹ The expectation is analogous to an average if there are no missing data and a weighted average if there are missing data.

² The marginal model is also known as the population-averaged model.

³ A number of quadrature points were tried and the parameter estimates appeared to become stable when greater than 10 point, were used. The value 50 was arbitrarily chosen to provide a high degree of accuracy in the results.

† One-tailed p = .44; all other Z values p < .05.

⁴ If the working correlation matrix is an identity matrix (not just diagonal), then GEE is identical to the traditional logistic regression model estimated with ML (Fitzmaurice et al., 2004, chap. 10).

⁵ Some ad hoc procedures have been suggested; see, for example, Pan (2001).

⁶ Statistics based on aggregates are, of course, used extensively to summarize information at the group level regardless if they represent empirical impossibilities (e.g., 3.5 children per household).

₇ The age at peak can be determined by an inspection of the predicted prevalence values, but it is not clear if the instantaneous rate of change can be computed.