  An important endpoint variable in a cocaine rehabilitation study is the time
to first relapse of a patient after the treatment. We propose a joint modeling
approach based on functional data analysis to study the relationship between
the baseline longitudinal cocaine-use pattern and the interval censored time to
first relapse. For the baseline cocaine-use pattern, we consider both
self-reported cocaine-use amount trajectories and dichotomized use
trajectories. Variations within the generalized longitudinal trajectories are
modeled through a latent Gaussian process, which is characterized by a few
leading functional principal components. The association between the baseline
longitudinal trajectories and the time to first relapse is built upon the
latent principal component scores. The mean and the eigenfunctions of the
latent Gaussian process as well as the hazard function of time to first relapse
are modeled nonparametrically using penalized splines, and the parameters in
the joint model are estimated by a Monte Carlo EM algorithm based on
Metropolis-Hastings steps. An Akaike information criterion (AIC) based on
effective degrees of freedom is proposed to choose the tuning parameters, and a
modified empirical information is proposed to estimate the variance-covariance
matrix of the estimators.
