State space models and filtering methods in longitudinal studies

Abstract

The main objective of this dissertation is to study the use of state space models and filtering methods in tackling several fundamental issues in longitudinal studies involving multiple subjects. These include serial dependence of a subject's responses that come naturally from time, inter-subject heterogeneity, missing values and measurement errors in subjects' responses, and non-stationary process drifts. We consider both repeated measure problems and problems involving event histories, and in particular, recurrent events. Several classes of models are introduced and filtering methods developed to implement parameter estimation. Properties of the models and methods are examined. We consider two sets of data for illustrations: a dataset from automobile manufacturing (repeated multivariate measurements), and a set of small bowel motility data (recurrent events).

We consider a class of general state space models and give a review of some common sub-models and the available tools for statistical inference. We point out the need for more efficient estimation for handling missing values and measurement errors, a careful understanding of different types of random effect models, and a tractable likelihood inference procedure.

We first discuss methods of estimating the variation in product quality characteristics measured at several stages in a manufacturing process. By determining which stages contribute most to variation one can focus variation reduction activities more effectively. A multivariate Gaussian Markov process is used to model the variation in characteristics. Methods that deal with measurement error and missing data are introduced through a state space formulation.

Then, we differentiate random effects models for recurrent events into autocorrelated and dynamic random effects models. Their similarities and key differences are discussed in the case of Gaussian models. Numerical comparisons are provided by using the small bowel motility data and cases when the models might be used are discussed.

Thirdly, we study a dynamic proportional hazards random effects model for recurrent events with non-informative right censoring. Subject heterogeneity and potential non-stationary process drifts are handled by repeatedly updating an initial frailty as more recurrence times are observed. An arbitrary baseline hazard together with an external time-dependent covariate process are allowed . The full model is actually a non-Gaussian state space model with a multiplicative state transition process. Parametric inference on hyperparameters is carried out by maximizing the likelihood function, which can be shown to be numerically tractable. A simulation study is conducted for further insight into the model.

Finally, we conclude this dissertation with some general remarks and point to some potential future research directions.