Prediction of Functional Status for the Elderly Based on a New Ordinal Regression Model

Mid-quantile regression for discrete responses

We develop quantile regression methods for discrete responses by extending Parzen’s definition of marginal mid-quantiles. As opposed to existing approaches, which are based on either jittering or latent constructs, we use interpolation and define the conditional mid-quantile function as the inverse of the conditional mid-distribution function. We propose a two-step estimator whereby, in the first step, conditional mid-probabilities are obtained nonparametrically and, in the second step, regression coefficients are estimated by solving an implicit equation. When constraining the quantile index to a data-driven admissible range, the second-step estimating equation has a least-squares type, closed-form solution. The proposed estimator is shown to be strongly consistent and asymptotically normal. A simulation study shows that our estimator performs satisfactorily and has an advantage over a competing alternative based on jittering. Our methods can be applied to a large variety of discrete responses, including binary, ordinal, and count variables. We show an application using data on prescription drugs in the United States and discuss two key findings. First, our analysis suggests a possible differential medical treatment that worsens the gender inequality among the most fragile segment of the population. Second, obesity is a strong driver of the number of prescription drugs and is stronger for more frequent medications users. The proposed methods are implemented in the R package Qtools. Supplemental materials for this article, including a brief R tutorial, are available as an online supplement.

Tackling Ordinal Regression Problem for Heterogeneous Data: Sparse and Deep Multi-Task Learning Approaches

Many real-world datasets are labeled with natural orders, i.e., ordinal labels. Ordinal regression is a method to predict ordinal labels that finds a wide range of applications in data-rich domains, such as natural, health and social sciences. Most existing ordinal regression approaches work well for independent and identically distributed (IID) instances via formulating a single ordinal regression task. However, for heterogeneous non-IID instances with well-defined local geometric structures, e.g., subpopulation groups, multi-task learning (MTL) provides a promising framework to encode task (subgroup) relatedness, bridge data from all tasks, and simultaneously learn multiple related tasks in efforts to improve generalization performance. Even though MTL methods have been extensively studied, there is barely existing work investigating MTL for heterogeneous data with ordinal labels. We tackle this important problem via sparse and deep multi-task approaches. Specifically, we develop a regularized multi-task ordinal regression (MTOR) model for smaller datasets and a deep neural networks based MTOR model for large-scale datasets. We evaluate the performance using three real-world healthcare datasets with applications to multi-stage disease progression diagnosis. Our experiments indicate that the proposed MTOR models markedly improve the prediction performance comparing with single-task ordinal regression models.

Quantile regression for survival data in modern cancer research: expanding statistical tools for precision medicine

Quantile regression links the whole distribution of an outcome to the covariates of interest and has become an important alternative to commonly used regression models. However, the presence of censored data such as survival time, often the main endpoint in cancer studies, has hampered the use of quantile regression techniques because of the incompleteness of data. With the advent of the precision medicine era and availability of high throughput data, quantile regression with high-dimensional predictors has attracted much attention and provided added insight compared to traditional regression approaches. This paper provides a practical guide for using quantile regression for right censored outcome data with covariates of low- or high-dimensionality. We frame our discussion using a dataset from the Boston Lung Cancer Survivor Cohort, a hospital-based prospective cohort study, with the goals of broadening the scope of cancer research, maximizing the utility of collected data, and offering useful statistical alternatives. We use quantile regression to identify clinical and molecular predictors, for example CpG methylation sites, associated with high-risk lung cancer patients, for example those with short survival.

Residuals and Diagnostics for Ordinal Regression Models: A Surrogate Approach

Ordinal outcomes are common in scientific research and everyday practice, and we often rely on regression models to make inference. A long-standing problem with such regression analyses is the lack of effective diagnostic tools for validating model assumptions. The difficulty arises from the fact that an ordinal variable has discrete values that are labeled with, but not, numerical values. The values merely represent ordered categories. In this paper, we propose a surrogate approach to defining residuals for an ordinal outcome Y. The idea is to define a continuous variable S as a "surrogate" of Y and then obtain residuals based on S. For the general class of cumulative link regression models, we study the residual's theoretical and graphical properties. We show that the residual has null properties similar to those of the common residuals for continuous outcomes. Our numerical studies demonstrate that the residual has power to detect misspecification with respect to 1) mean structures; 2) link functions; 3) heteroscedasticity; 4) proportionality; and 5) mixed populations. The proposed residual also enables us to develop numeric measures for goodness-of-fit using classical distance notions. Our results suggest that compared to a previously defined residual, our residual can reveal deeper insights into model diagnostics. We stress that this work focuses on residual analysis, rather than hypothesis testing. The latter has limited utility as it only provides a single p-value, whereas our residual can reveal what components of the model are misspecified and advise how to make improvements.