Abstract
In spite of the fact that probability weighting is widely used in statistics to correct for
unequal sampling, control for confounding, and handle missing data, it has two main limitations.
First, statistical inferences may be inefficient in the presence of extreme probability
weights. Second, probability weighting-based methods are highly sensitive to model mis-specifications. The aim of this Ph.D. thesis work was to develop novel methods, based
on mathematical programming techniques, for optimal probability weighting. Specifically,
in Paper I, we proposed a method that estimates optimal probability weights, which are
obtained as the solution to a constrained optimization problem that minimizes the Euclidean
distance from the target (original/design) weights among all sets of weights that
satisfy a constraint on the precision of the resulting weighted estimator. In Paper II, we
extended optimal probability weights to estimate the causal effect of a time-varying treatment
on a survival outcome. Optimal probability weights were obtained as the solution to
a constrained optimization problem which constrained the variance of the weights, rather
than the standard error of the resulting weighted estimator, as in Paper I. In Paper III,
we proposed Kernel Optimal Weighting (KOW), to obtain weights that optimally balance
time-dependent confounders while controlling for the precision of the resulting marginal
structural model estimate by directly minimizing the error in estimation. This error is
expressed as an operator derived from the g-computation formula and KOW minimizes its
operator norm with respect to a reproducing kernel Hilbert spaces by solving a quadratic
optimization problem. KOW mitigates the e ects of possible misspecification of the treatment
model by directly balancing covariates and control for precision by penalizing extreme
weights. In Paper IV, we evaluated the e ect of treatment switch on time to second-line
HIV treatment failure using data from the Swedish InfCare HIV registry. This Ph.D. thesis
provided methods that will likely help to (1) extend the use of probability weighting in
medicine, epidemiology, and economics, (2) extend knowledge on how mathematical programming
and machine learning could be used to conduct robust analyses for improved
decision-making, and, (3) provide powerful, strong, and robust results to clinicians and
policy-makers.