  Longitudinal studies of a binary outcome are common in the health, social,
and behavioral sciences. In general, a feature of random effects logistic
regression models for longitudinal binary data is that the marginal functional
form, when integrated over the distribution of the random effects, is no longer
of logistic form. Recently, Wang and Louis [Biometrika 90 (2003) 765--775]
proposed a random intercept model in the clustered binary data setting where
the marginal model has a logistic form. An acknowledged limitation of their
model is that it allows only a single random effect that varies from cluster to
cluster. In this paper we propose a modification of their model to handle
longitudinal data, allowing separate, but correlated, random intercepts at each
measurement occasion. The proposed model allows for a flexible correlation
structure among the random intercepts, where the correlations can be
interpreted in terms of Kendall's $\tau$. For example, the marginal
correlations among the repeated binary outcomes can decline with increasing
time separation, while the model retains the property of having matching
conditional and marginal logit link functions. Finally, the proposed method is
used to analyze data from a longitudinal study designed to monitor cardiac
abnormalities in children born to HIV-infected women.
