PhD Defense Alexander Mangulad Christgau
Title: Model-free Methods for Event History Analysis and Efficient Adjustment
Abstract:
This thesis contains a series of independent contributions to mathematical statistics, unified by a model-free perspective. The first chapter elaborates on how a model-free perspective can be used to formulate flexible methods that leverage prediction techniques from machine learning. Mathematical insights are obtained from concrete examples, and these insights are generalized to principles that permeate the rest of the thesis.
The second chapter studies the concept of local independence, which describes whether the evolution of one stochastic process is directly influenced by another. To test local independence, we define a model-free parameter called the Local Covariance Measure (LCM). We formulate an estimator for the LCM, from which a test of local independence is proposed. We discuss how the size and power of the proposed test can be controlled uniformly and investigate the test in a simulation study.
The third chapter focuses on covariate adjustment, a method used to estimate the effect of a treatment by accounting for observed confounding. We formulate a general framework that facilitates adjustment for any subset of covariate information. We identify the optimal covariate information for adjustment and, based on this, introduce the Debiased Outcome-adapted Propensity Estimator (DOPE) for efficient estimation of treatment effects. An instance of DOPE is implemented using neural networks, and we demonstrate its performance on both simulated and real data.
The fourth and final chapter introduces a model-free measure of the conditional association between an exposure and a time-to-event, which we call the Aalen Covariance Measure (ACM). The ACM serves as an assumption-lean generalization of the exposure coefficient in the Aalen additive hazards model. We develop a model-free estimation method and show that it is doubly robust, ensuring √n-consistency provided that the nuisance functions can be estimated with modest rates. A simulation study demonstrates the use of our estimator in several settings.
Supervisor: Professor Niels Richard Hansen, University of Copenhagen
Assessment Committee:
Associate Professor Sebastian Weichwald (chair), University of Copenhagen
Professor Ingrid Van Keilegom, KU Leuven
Professor Stijn Vansteelandt, Ghent University