UC San Diego

In this dissertation, we aim to utilize modern statistical frameworks to perform power and probability calculations in longitudinal outcome measures and image analysis. Chapter 1 and Chapter 2 primarily focus on longitudinal outcome measures data. In practice, two different analysis plans are commonly applied to these data: linear mixed effects model and repeated measures analysis. In Chapter 1, we are interested in generalizing current power formulas for linear mixed effects models to accommodate missing data due to study subject attrition, and unequal sample size and variance parameters across groups.

For repeated measures analysis, a covariance structure needs to be specified when modeling the data and computing type I error and power. In Chapter 2, we describe a parsimonious covariance structure for repeated measures analysis that is useful for modeling longitudinal repeated measures of chronic progressive conditions and derive the power calculation formulas.

In image analysis, finding the peak height distribution and power for peak detection are known to be challenging due to the spatial aspect of the data. In Chapter 3, we propose a novel way to approximate the power for peak detection using Gaussian random field theory (RFT) and demonstrate scenarios where the approximation works well. We also apply our formulas to 2D and 3D simulated datasets, and the 3D data is induced by real fMRI data to measure performance in a realistic setting.

The main limitation of RFT-based image analysis is the model assumptions, and these assumptions are known to be difficult to check and even not appropriate in many application settings. In Chapter 4, we seek to relax the stationarity assumption and study the peak height distribution of non-stationary Gaussian random fields. The explicit formula for the peak height distribution is derived for 1D smooth Gaussian random fields and efficient numerical algorithms are proposed as a general solution for computing the peak height distribution in applications.