Doubly weighted estimating equations and weighted multiple imputation for causal inference with an incomplete subgroup variable

Health research often aims to investigate whether the effect of an exposure variable is common across different subgroups of individuals, but sometimes the variable defining subgroups is not recorded in all individuals. We propose and evaluate two methods for estimation of the marginal causal effect of an exposure variable within subgroups in the observational setting where the subgroup variable is incompletely observed. The first approach involves doubly weighted estimating functions with one weight based on a propensity score for exposure and a second weight addressing the selection bias when analyses are restricted to individuals with complete data. The second approach uses the inverse probability of exposure weights in conjunction with multiple imputation for the incomplete subgroup variable. The resulting estimators are consistent when the auxiliary models are correctly specified; we assess the finite sample performance via simulation. An illustrative analysis is provided involving patients with psoriatic arthritis treated with biologic therapy where interest lies in the effect of therapy according to the presence or absence of the human leukocyte antigen marker HLA-B27 which is incompletely observed.