Biostatistics Advance Access published online on January 20, 2006
Biostatistics, doi:10.1093/biostatistics/kxj019
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1 Harvard School of Public Health and Brigham & Women's Hospital, Boston, U.S.A.
* To whom correspondence should be addressed. In many observational studies individuals are measured repeatedly over time, although not necessarily at a set of pre-specified occasions. Instead, individuals may be measured at irregular intervals, with those having a history of poorer health outcomes being measured with somewhat greater frequency and regularity. In this paper we consider likelihood-based estimation of the regression parameters in marginal models for longitudinal binary data when the follow-up times are not fixed by design, but can depend on previous outcomes. In particular, we consider assumptions regarding the follow-up time process that result in the likelihood function separating into two components: one for the follow-up time process, the other for the outcome measurement process. The practical implication of this separation is that the follow-up time process can be ignored when making likelihood-based inferences about the marginal regression model parameters. That is, maximum likelihood (ML) estimation of the regression parameters relating the probability of success at a given time to covariates does not require that a model for the distribution of follow-up times be specified. However, to obtain consistent parameter estimates, the multinomial distribution for the vector of repeated binary outcomes must be correctly specified. In general, ML estimation requires specification of all higher-order moments and the likelihood for a marginal model can be intractable except in cases where the number of repeated measurements is relatively small. To circumvent these difficulties we propose a pseudo-likelihood for estimation of the marginal model parameters. The pseudo-likelihood uses a linear approximation for the conditional distribution of the response at any occasion, given the history of previous responses. The appeal of this approximation is that the conditional distributions are functions of the first two moments of the binary responses only. When the follow-up times depend only on the previous outcome, the pseudo-likelihood requires correct specification of the conditional distribution of the current outcome given the outcome at the previous occasion only. Results from a simulation study and a study of asymptotic bias are presented. Finally, we illustrate the main results using data from a longitudinal observational study that explored the cardiotoxic effects of doxorubicin chemotherapy for the treatment of acute lymphoblastic leukemia in children.
Received May 29, 2003
Revised January 4, 2006
Accepted January 18, 2006
Article
Estimation in regression models for longitudinal binary data with outcome-dependent follow-up
Garrett M. Fitzmaurice 1 *,
Stuart R. Lipsitz 2,
Joseph G. Ibrahim 3,
Richard Gelber 4,
and
Steven Lipshultz 5
2 Medical University of South Carolina, U.S.A.
3 School of Public Health, University of North Carolina, U.S.A.
4 Dana Farber Cancer Institute, Boston, U.S.A.
5 University of Rochester Medical Center, Rochester, New York U.S.A
Garrett M. Fitzmaurice, E-mail: fitzmaur{at}hsph.harvard.edu
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