|Title||Semiparametric models for joint analysis of longitudinal data and counting processes|
In this dissertation, we study statistical methodology for joint modeling that correctly controls for the interplay among longitudinal and counting processes and makes the most efficient use of data. Three types of joint modeling approaches are proposed based on three different purposes of studies. In the first topic, we develop a method for joint modeling of longitudinal data and recurrent events in the presence of an informative terminal event. We focus on data from patients who experience the same type of event at multiple times, such as multiple infection episodes or recurrent strokes, have longitudinal biomarkers, and may be subject to an event, for example death, that makes further observations impossible. To analyze such complicated data, we propose joint models based on a likelihood approach. A broad class of transformation models for the cumulative intensity of recurrent events and the cumulative hazard of the terminal event is considered. We propose to estimate all the parameters using nonparametric maximum likelihood estimators NPMLE), and we provide computationally efficient EM algorithms to implement the proposed inference procedure. Asymptotic properties of the estimators are shown to be asymptotically normal and semiparametrically efficient. Finally, we evaluate the performance of the proposed method through extensive simulations and application to real data. In the second topic, we develop a method for joint modeling of longitudinal and cure-survival data. By cure-survival data, we mean time-to-event data in which a certain proportion of patients never have any event during a sufficiently long follow-up period. These patients are believed to have been cured by treatment, such as radiation therapy or an initial surgery, and are often the source of heavy tail probabilities in survival curves. To take into account the possibility of patients being cured, we propose to model time-to-event through a transformed promotion time cure model, jointly with a linear mixed effects model for longitudinal data. Due to transformations applied to the promotion time cure model, the proposed method is able to be used in cases where the proportionality assumption does not hold. All the parameters are estimated using NPMLEs, and inference procedures are implemented via a simple EM algorithm. Asymptotic properties of the proposed NPMLEs are derived based on empirical process theory. Simulation studies are conducted and the method is applied to the ARIC data in order to demonstrate the small-sample performance of the proposed method. In the third topic, we develop a partially linear model for longitudinal data with informative censoring, where the main interest is in making inferences about the individuals trajectory of longitudinal responses, which may be informatively censored. Since a fully parameterized mean structure may be insufficient to capture the underlying patterns of longitudinal and event processes, we propose to use a partially linear model for longitudinal responses, where an unspecified underlying function is formulated along with linear covariate effects, and a transformation model is used for informative censoring times. We employ a sieve estimation for the nonparametric trajectory of longitudinal responses, where the unknown trajectory is approximated by cubic B-spline basis functions. All parameters are estimated based on a likelihood approach, and inference procedures are implemented via the EM algorithm. We also investigate a reliable way to select the number of knots and the best transformation. Through empirical process theory, asymptotic properties of the proposed estimators are shown to provide desirable properties. The validity of the proposed method is confirmed by simulated and real data examples.
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