This thesis contains three projects that propose novel methods for studying mechanisms that explain statistical relationships. The ultimate goal of each of these methods is to help researchers describe how or why complex relationships between observed variables exist.
The first project proposes and studies a method for recovering mediation structure in high dimensions. We take a dimension reduction approach that generalizes the ``product of coefficients'' concept for univariate mediation analysis through the optimization of a loss function. We devise an efficient algorithm for optimizing the product-of-coefficients inspired loss function. Through extensive simulation studies, we show that the method is capable of consistently identifying mediation structure. Finally, two case studies are presented that demonstrate how the method can be used to conduct multivariate mediation analysis.
The second project uses tools from conditional inference to improve the calibration of tests of univariate mediation hypotheses. The key insight of the project is that the non-Euclidean geometry of the null parameter space causes the test statistic’s sampling distribution to depend on a nuisance parameter. After identifying a statistic that is both sufficient for the nuisance parameter and approximately ancillary for the parameter of interest, we derive the test statistic’s limiting conditional sampling distribution. We additionally develop a non-standard bootstrap procedure for calibration in finite samples. We demonstrate through simulation studies that improved evidence calibration leads to substantial power increases over existing methods. This project suggests that conditional inference might be a useful tool in evidence calibration for other non-standard or otherwise challenging problems.
In the last project, we present a methodological contribution to a pharmaceutical science study of {em in vivo} ibuprofen pharmacokinetics. We demonstrate how model misspecification in a first-principles analysis can be addressed by augmenting the model to include a term corresponding to an omitted source of variation. In previously used first-principles models, gastric emptying, which is pulsatile and stochastic, is modeled as first-order diffusion for simplicity. However, analyses suggest that the actual gastric emptying process is expected to be a unimodal smooth function, with phase and amplitude varying by subject. Therefore, we adopt a flexible approach in which a highly idealized parametric version of gastric emptying is combined with a Gaussian process to capture deviations from the idealized form. These functions are characterized by their distributions, which allows us to learn their common and unique features across subjects despite that these features are not directly observed. Through simulation studies, we show that the proposed approach is able to identify certain features of latent function distributions.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163026/1/josephdi_1.pd
