Statistical methods for robust inference in causal and missing data models

In many observational studies or randomized trials, N i.i.d data O≡ON≡ Oi,i=1&ldots;N are observed from a model MQ= F˙;q,q∈ Q, with the objective to make valid inference on a functional psi (theta). In general, psi (theta) can be infinite dimensional but this dissertation only considers the finite dimensional case. The analytic challenge facing the biostatistician is to make his or her inference while making little assumptions about the part of theta which is not of scientific interest. Semi/nonparametric theory offers a framework to conduct such robust inference in the presence of high dimensional data O. This dissertation makes several contributions to the development of novel semiparametric robust methods with important applications in causal inference and complex missing data problem. In the first chapter of the thesis, we construct doubly robust (dr) estimators for the parameters of a Marginal Structural Cox Proportional Hazards Model in the presence of censoring. In a Cox MSM, an estimator is doubly robust if it remains consistent and asymptotically normal when either (1) a model for the treatment assignment mechanism or (2) a model restricting the partial likelihood of the observed data not involving the treatment mechanism is correctly specified. This work was done in collaboration with James Robins. The following three chapters are concerned with the theory of higher order influence functions and its application to problems in causal inference and missing data models. This work was done in collaboration with James Robins, Lingling li and Aad van der Vaart. In the second chapter, we thoroughly discuss the modern theory of higher order estimation influence functions. At the start of this chapter, we state and prove two key theorems: a higher order influence function "Extended Information Equality-Theorem" and an "Efficient Influence Function Theorem". The former may be thought of as an extension of the first order influence function "information equality theorem" and it serves as a motivation for why higher order influence functions are useful for deriving point estimators of psi(theta) with small bias and for deriving valid (1-alpha) confidence interval estimators centered on an estimate of psi(theta). The second theorem further extends several results from first order semiparametric theory to their corresponding higher order generalization. (Abstract shortened by UMI.)...