  Marginal structural models (MSMs) estimate the causal effect of a
time-varying treatment in the presence of time-dependent confounding via
weighted regression. The standard approach of using inverse probability of
treatment weighting (IPTW) can lead to high-variance estimates due to extreme
weights and be sensitive to model misspecification. Various methods have been
proposed to partially address this, including truncation and stabilized-IPTW to
temper extreme weights and covariate balancing propensity score (CBPS) to
address treatment model misspecification. In this paper, we present Kernel
Optimal Weighting (KOW), a convex-optimization-based approach that finds
weights for fitting the MSM that optimally balance time-dependent confounders
while simultaneously controlling for precision, directly addressing the above
limitations. KOW directly minimizes the error in estimation due to
time-dependent confounding via a new decomposition as a functional. We further
extend KOW to control for informative censoring. We evaluate the performance of
KOW in a simulation study, comparing it with IPTW, stabilized-IPTW, and CBPS.
We demonstrate the use of KOW in studying the effect of treatment initiation on
time-to-death among people living with HIV and the effect of negative
advertising on elections in the United States.
