Bayesian Modeling Using Latent Structures
This dissertation is devoted to modeling complex data from the
Bayesian perspective via constructing priors with latent structures.
There are three major contexts in which this is done -- strategies for
the analysis of dynamic longitudinal data, estimating
shape-constrained functions, and identifying subgroups. The
methodology is illustrated in three different
interdisciplinary contexts: (1) adaptive measurement testing in
education; (2) emulation of computer models for vehicle crashworthiness; and (3) subgroup analyses based on biomarkers.
Chapter 1 presents an overview of the utilized latent structured
priors and an overview of the remainder of the thesis. Chapter 2 is
motivated by the problem of analyzing dichotomous longitudinal data
observed at variable and irregular time points for adaptive
measurement testing in education. One of its main contributions lies
in developing a new class of Dynamic Item Response (DIR) models via
specifying a novel dynamic structure on the prior of the latent
trait. The Bayesian inference for DIR models is undertaken, which
permits borrowing strength from different individuals, allows the
retrospective analysis of an individual's changing ability, and
allows for online prediction of one's ability changes. Proof of
posterior propriety is presented, ensuring that the objective
Bayesian analysis is rigorous.
Chapter 3 deals with nonparametric function estimation under
shape constraints, such as monotonicity, convexity or concavity. A
motivating illustration is to generate an emulator to approximate a computer
model for vehicle crashworthiness. Although Gaussian processes are
very flexible and widely used in function estimation, they are not
naturally amenable to incorporation of such constraints. Gaussian
processes with the squared exponential correlation function have the
interesting property that their derivative processes are also
Gaussian processes and are jointly Gaussian processes with the
original Gaussian process. This allows one to impose shape constraints
through the derivative process. Two alternative ways of incorporating derivative
information into Gaussian processes priors are proposed, with one
focusing on scenarios (important in emulation of computer
models) in which the function may have flat regions.
Chapter 4 introduces a Bayesian method to control for multiplicity
in subgroup analyses through tree-based models that limit the
subgroups under consideration to those that are a priori plausible.
Once the prior modeling of the tree is accomplished, each tree will
yield a statistical model; Bayesian model selection analyses then
complete the statistical computation for any quantity of interest,
resulting in multiplicity-controlled inferences. This research is
motivated by a problem of biomarker and subgroup identification to
develop tailored therapeutics. Chapter 5 presents conclusions and
some directions for future research.
Item Response Theory
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