Empirical likelihood (EL) is a non-parametric analog of standard parametric likelihood, which has gained in popularity since its introduction in the late 1980s. Versions of EL have been shown to apply to linear models, generalized linear models, survival data and more. Furthermore, EL inherits many of the asymptotic properties and behaviors of ordinary likelihoods and is also connected to other popular procedures such as the bootstrap. All of these facts make it an attractive alternative to parametric methods. It is more robust, since it makes fewer assumptions (typically only assumptions on the first two moments or on the specific functionals that are being estimated), yet the theoretical performance is equal to that of standard techniques which posit distributional assumptions on the observations. To date, there has been no extensive work exploring the usefulness of EL in the context of longitudinal data. This, in spite of the fact that longitudinal data are prevalent in medicine, psychology, epidemiology, and social sciences more broadly. A popular family of models for longitudinal data is that of generalized estimating equations (GEE), which can also be fit into the framework of EL, creating a natural extension that would combine these two constructs. Comparison with other widespread approaches, for instance the generalized method of moments and the quadratic inference
function, is then possible. In the current work the investigator builds EL for longitudinal data settings, compares its performance to existing methods, and develops diagnostics for evaluating the sensitivity of the inference to the specific sample on which it is based.
Longitudinal data - collected on the same individuals over time - are important in many medical and biological applications. For instance, in studying the effectiveness of a new treatment for cancer, medical
researchers need to follow patients over time and compare their survival rates to the survival rates of patients receiving an existing treatment. It is therefore critical to have good statistical models for longitudinal data, models that will take best advantage of the information available from these types of studies. In this project, the researcher develops a class of models based on "empirical likelihood." Empirical likelihood models make minimal assumptions about the data and are therefore very flexible, general, and robust. At the same time, they inherit many of the important mathematical properties of standard statistical techniques, which make much stronger assumptions that are unlikely to be met in many real-life situations.
The investigator also compares the class of empirical likelihood models to existing methods for the analysis of longitudinal data and establishes conditions under which the new models are expected to perform better.
