Reflection on modern methods: combining weights for confounding and missing data.

Inverse probability weights are increasingly used in epidemiological analysis, and estimation and application of weights to address a single bias are well discussed in the literature. Weights to address multiple biases simultaneously (i.e. a combination of weights) have almost exclusively been discussed related to marginal structural models in longitudinal settings where treatment weights (estimated first) are combined with censoring weights (estimated second). In this work, we examine two examples of combined weights for confounding and missingness in a time-fixed setting in which outcome or confounder data are missing, and the estimand is the marginal expectation of the outcome under a time-fixed treatment. We discuss the identification conditions, construction of combined weights and how assumptions of the missing data mechanisms affect this construction. We use a simulation to illustrate the estimation and application of the weights in the two examples. Notably, when only outcome data are missing, construction of combined weights is straightforward; however, when confounder data are missing, we show that in general we must follow a specific estimation procedure which entails first estimating missingness weights and then estimating treatment probabilities from data with missingness weights applied. However, if treatment and missingness are conditionally independent, then treatment probabilities can be estimated among the complete cases.