Variance estimation in inverse probability weighted Cox models

Inverse probability weighted Cox models can be used to estimate marginal hazard ratios under different point treatments in observational studies. To obtain variance estimates, the robust sandwich variance estimator is often recommended to account for the induced correlation among weighted observations. However, this estimator does not incorporate the uncertainty in estimating the weights and tends to overestimate the variance, leading to inefficient inference. Here we propose a new variance estimator that combines the estimation procedures for the hazard ratio and weights using stacked estimating equations, with additional adjustments for the sum of terms that are not independently and identically distributed in a Cox partial likelihood score equation. We prove analytically that the robust sandwich variance estimator is conservative and establish the asymptotic equivalence between the proposed variance estimator and one obtained through linearization by Hajage et al. in 2018. In addition, we extend our proposed variance estimator to accommodate clustered data. We compare the finite sample performance of the proposed method with alternative methods through simulation studies. We illustrate these different variance methods in both independent and clustered data settings, using a bariatric surgery dataset and a multiple readmission dataset, respectively. To facilitate implementation of the proposed method, we have developed an $R$ package ipwCoxCSV.