Latent variable models (LVMs) are commonly used in the scenario where the outcome of the main interest is an unobservable measure, associated with multiple observed surrogate outcomes, and affected by potential risk factors. This thesis develops an approach of efficient handling missing surrogate outcomes and covariates in two- and three-level latent variable models. However, corresponding statistical methodologies and computational software are lacking efficiently analyzing the LVMs given surrogate outcomes and covariates subject to missingness in the LVMs. We analyze the two-level LVMs for longitudinal data from the National Growth of Health Study where surrogate outcomes and covariates are subject to missingness at any of the levels. A conventional method for efficient handling of missing data is to reexpress the desired model as a joint distribution of variables, including the surrogate outcomes that are subject to missingness conditional on all of the covariates that are completely observable, and estimate the joint model by maximum likelihood, which is then transformed to the desired model. The joint model, however, identifies more parameters than desired, in general. The over-identified joint model produces biased estimates of LVMs so that it is most necessary to describe how to impose constraints on the joint model so that it has a one-to-one correspondence with the desired model for unbiased estimation. The constrained joint model handles missing data efficiently under the assumption of ignorable missing data and is estimated by a modified application of the expectation-maximization (EM) algorithm.

Indiana University-Purdue University Indianapolis (IUPUI) / Useful medical indices pose important roles in predicting medical outcomes. Medical indices, such as the well-known Body Mass Index (BMI), Charleson Comorbidity Index, etc., have been used extensively in research and clinical practice, for the quantification of risks in individual patients. However, the development of these indices is challenged; and primarily based on heuristic arguments. Statistically, most medical indices can be expressed as a function of a linear combination of individual variables and fitted by single-index model. Single-index model represents a way to retain latent nonlinear features of the data without the usual complications that come with increased dimensionality. In my dissertation, I propose a single-index model approach to analytically derive indices from observed data; the resulted index inherently correlates with specific health outcomes of interest. The first part of this dissertation discusses the derivation of an index function for the prediction of one outcome using longitudinal data. A cubic-spline estimation scheme for partially linear single-index mixed effect model is proposed to incorporate the within-subject correlations among outcome measures contributed by the same subject. A recursive algorithm based on the optimization of penalized least square estimation equation is derived and is shown to work well in both simulated data and derivation of a new body mass measure for the assessment of hypertension risk in children. The second part of this dissertation extends the single-index model to a multivariate setting. Specifically, a multivariate version of single-index model for longitudinal data is presented. An important feature of the proposed model is the accommodation of both correlations among multivariate outcomes and among the repeated measurements from the same subject via random effects that link the outcomes in a unified modeling structure. A new body mass index measure that simultaneously predicts systolic and diastolic blood pressure in children is illustrated. The final part of this dissertation shows existence, root-n strong consistency and asymptotic normality of the estimators in multivariate single-index model under suitable conditions. These asymptotic results are assessed in finite sample simulation and permit joint inference for all parameters.