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Likelihood Ratio Testing under Nonidentifiability with Applications to Biomedical Studies

Chongzhi Di, PhD Candidate, Johns Hopkins Department of Biostatistics

This dissertation contains statistical research in two main areas. The first area is likelihood ratio testing when one of the key regularity conditions -- identifiability -- does not hold under the null hypothesis. The second area is statistical analysis of multilevel functional data. The work on both topics is motivated by and applied to public health and biomedical studies.

The first main part of the dissertation considers likelihood ratio testing under nonidentifiability. In particular, we focus on two classes of hypothesis testing problems in which one or more parameters are present only under the alternative. In Class 1, the null hypothesis is specified via the parameter of interest while a nuisance parameter is not identifiable under the null. It has been established that the LRT statistic converges to the supremum of a squared Gaussian process. We characterize conditions under which such limiting distributions simplify to chi-square. When these conditions are not satisfied, we also provide efficient computational algorithms to approximate p values based on the principal component decomposition of Gaussian processes. These are illustrated by Andersonís stereotype models for ordinal response data. In Class 2, the null hypothesis can be specified equivalently via each of the two parameters, and under either specification, the other parameter is not identifiable. Motivating examples for this class include testing homogeneity in admixture models and testing linearity versus a nonlinear trend in generalized linear models. This class has received relatively less attention in the literature, except for the special case of mixture models. We derive the limiting distribution of the LRT statistic in this situation. We also present a penalized likelihood ratio test that has a simple chi-square limiting distribution under the null, based on previous work in mixture models (Chen et al. 2001). These approaches are compared through statistical power and illustrated in a genetic linkage study of schizophrenia that is subject to genetic heterogeneity.

The second part involves development of new statistical methods in functional data analysis, motivated by the Sleep Heart Health Study (SHHS), a comprehensive landmark study of sleep and its impacts on health outcomes. The SHHS data contains quasi-continuous electroencephalographic (EEG) signals for each subject, at two visits. The volume and importance of this data presents enormous challenges for analysis. To address these challenges, we introduce multilevel functional principal component analysis, a novel statistical methodology designed to extract intra- and inter-subject geometric components of multilevel functional data. The proposed methodology is generally applicable to many modern scientific studies of hierarchical functional data.

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