  In estimating the average treatment effect in observational studies, the
influence of confounders should be appropriately addressed. To this end, the
propensity score is widely used. If the propensity scores are known for all the
subjects, bias due to confounders can be adjusted by using the inverse
probability weighting (IPW) by the propensity score. Since the propensity score
is unknown in general, it is usually estimated by the parametric logistic
regression model with unknown parameters estimated by solving the score
equation under the strongly ignorable treatment assignment (SITA) assumption.
Violation of the SITA assumption and/or misspecification of the propensity
score model can cause serious bias in estimating the average treatment effect.
To relax the SITA assumption, the IPW estimator based on the outcome-dependent
propensity score has been successfully introduced. However, it still depends on
the correctly specified parametric model and its identification. In this paper,
we propose a simple sensitivity analysis method for unmeasured confounders. In
the standard practice, the estimating equation is used to estimate the unknown
parameters in the parametric propensity score model. Our idea is to make
inference on the average causal effect by removing restrictive parametric model
assumptions while still utilizing the estimating equation. Using estimating
equations as constraints, which the true propensity scores asymptotically
satisfy, we construct the worst-case bounds for the average treatment effect
with linear programming. Different from the existing sensitivity analysis
methods, we construct the worst-case bounds with minimal assumptions. We
illustrate our proposal by simulation studies and a real-world example.
