Rank-Based Semiparametric Methods: Covariate-Adjusted Spearman's Correlation with Probability-Scale Residuals and Cumulative Probability Models

Abstract

In this dissertation, we develop semiparametric rank-based methods. These types of methods are particularly useful with skewed data, nonlinear relationships, and truncated measurements. Semiparametric rank-based methods can achieve a good balance between robustness and efficiency.

The first part of this dissertation develops new estimators for covariate-adjusted Spearman's rank correlation, both partial and conditional, using probability-scale residuals (PSRs). These estimators are consistent for natural extensions of the population parameter of Spearman's rank correlation in the presence of covariates and are general for both continuous and discrete variables. We evaluate their performance with simulations and illustrate their application in two examples. To preserve the rank-based nature of Spearman's correlation, we obtain PSRs from ordinal cumulative probability models for both discrete and continuous variables.

Cumulative probability models were first invented to handle discrete ordinal outcomes, and their potential utility for the analysis of continuous outcomes has been largely unrecognized. This motivates the second part of this dissertation: an in-depth study of the application of cumulative probability models to continuous outcomes. When applied to continuous outcomes, these models can be viewed as semiparametric transformation models. We present a latent variable motivation for these models; describe estimation, inference, assumptions, and model diagnostics; conduct extensive simulations to investigate the finite sample performance of these models with and without proper link function specification; and illustrate their application in an HIV study.

Finally, we developed an R package, PResiduals, to compute PSRs, to incorporate them into conditional tests of association, and to implement our covariate-adjusted Spearman's rank correlation. The third part of this dissertation contains a vignette for this package, in which we illustrate its usage with a publicly available dataset.