Quantile calculus and censored regression

Quantile regression has been advocated in survival analysis to assess evolving covariate effects. However, challenges arise when the censoring time is not always observed and may be covariate-dependent, particularly in the presence of continuously-distributed covariates. In spite of several recent advances, existing methods either involve algorithmic complications or impose a probability grid. The former leads to difficulties in the implementation and asymptotics, whereas the latter introduces undesirable grid dependence. To resolve these issues, we develop fundamental and general quantile calculus on cumulative probability scale in this article, upon recognizing that probability and time scales do not always have a one-to-one mapping given a survival distribution. These results give rise to a novel estimation procedure for censored quantile regression, based on estimating integral equations. A numerically reliable and efficient Progressive Localized Minimization (PLMIN) algorithm is proposed for the computation. This procedure reduces exactly to the Kaplan--Meier method in the $k$-sample problem, and to standard uncensored quantile regression in the absence of censoring. Under regularity conditions, the proposed quantile coefficient estimator is uniformly consistent and converges weakly to a Gaussian process. Simulations show good statistical and algorithmic performance. The proposal is illustrated in the application to a clinical study.Comment: Published in at http://dx.doi.org/10.1214/09-AOS771 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

On testing the change-point in the longitudinal bent line quantile regression model

The problem of detecting changes has been receiving considerable attention in various fields. In general, the change-point problem is to identify the location(s) in an ordered sequence that divides this sequence into groups, which follow different models. This dissertation considers the change-point problem in quantile regression for observational or clinical studies involving correlated data (e.g. longitudinal studies) . Our research is motivated by the lack of ideal inference procedures for such models. Our contributions are two-fold. First, we extend the previously reported work on the bent line quantile regression model [Li et al. (2011)] to a longitudinal framework. Second, we propose a score-type test for hypothesis testing of the change-point problem using rank-based inference. The proposed test in this thesis has several advantages over the existing inferential approaches. Most importantly, it circumvents the difficulties of estimating nuisance parameters (e.g. density function of unspecified error) as required for the Wald test in previous works and thus is more reliable in finite sample performance. Furthermore, we demonstrate, through a series of simulations, that the proposed methods also outperform the extensively used bootstrap methods by providing more accurate and computationally efficient confidence intervals. To illustrate the usage of our methods, we apply them to two datasets from real studies: the Finnish Longitudinal Growth Study and an AIDS clinical trial. In each case, the proposed approach sheds light on the response pattern by providing an estimated location of abrupt change along with its 95% confidence interval at any quantile of interest "” a key parameter with clinical implications. The proposed methods allow for different change-points at different quantile levels of the outcome. In this way, they offer a more comprehensive picture of the covariate effects on the response variable than is provided by other change-point models targeted exclusively on the conditional mean. We conclude that our framework and proposed methodology are valuable for studying the change-point problem involving longitudinal data

Resampling methods for longitudinal data analysis

Ph.DDOCTOR OF PHILOSOPH

Uncovering Gender Differences in the Effects of Early Intervention: A Reevaluation of the Abecedarian, Perry Preschool, and Early Training Projects

The view that the returns to public educational investments are highest for early childhood interventions stems primarily from several influential randomized trials - Abecedarian, Perry, and the Early Training Project - that point to super-normal returns to preschool interventions. This paper presents a de novo analysis of these experiments, focusing on core issues that have received little attention in previous analyses: treatment effect heterogeneity, over-rejection of the null hypothesis due to multiple inference, and robustness of the findings to attrition and deviations from the experimental protocol. The primary finding of this reanalysis is that girls garnered substantial short- and long-term benefits from the interventions, particularly in the domain of total years of education. However, there were no significant long-term benefits for boys. These conclusions change little when allowance is made for attrition and possible violations of random assignment.preschool early intervention human capital education treatment effects

A shared-parameter continuous-time hidden Markov and survival model for longitudinal data with informative dropout

A shared-parameter approach for jointly modeling longitudinal and survival data is proposed. With respect to available approaches, it allows for time-varying random effects that affect both the longitudinal and the survival processes. The distribution of these random effects is modeled according to a continuous-time hidden Markov chain so that transitions may occur at any time point. For maximum likelihood estimation, we propose an algorithm based on a discretization of time until censoring in an arbitrary number of time windows. The observed information matrix is used to obtain standard errors. We illustrate the approach by simulation, even with respect to the effect of the number of time windows on the precision of the estimates, and by an application to data about patients suffering from mildly dilated cardiomyopathy

Network and panel quantile effects via distribution regression

This paper provides a method to construct simultaneous con fidence bands for quantile functions and quantile effects in nonlinear network and panel models with unobserved two-way effects, strictly exogenous covariates, and possibly discrete outcome variables. The method is based upon projection of simultaneous confi dence bands for distribution functions constructed from fixed effects distribution regression estimators. These fi xed effects estimators are bias corrected to deal with the incidental parameter problem. Under asymptotic sequences where both dimensions of the data set grow at the same rate, the confi dence bands for the quantile functions and effects have correct joint coverage in large samples. An empirical application to gravity models of trade illustrates the applicability of the methods to network data.https://arxiv.org/abs/1803.08154First author draf

Quantile Correlations and Quantile Autoregressive Modeling

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