Bounds on the average causal effects in randomized trials with noncompliance by covariate adjustment

In randomized trials with noncompliance, causal effects cannot be identified without strong assumptions. Therefore, several authors have considered bounds on the causal effects. Applying an idea of VanderWeele (), Chiba () gave bounds on the average causal effects in randomized trials with noncompliance using the information on the randomized assignment, the treatment received and the outcome under monotonicity assumptions about covariates. But he did not consider any observed covariates. If there are some observed covariates such as age, gender, and race in a trial, we propose new bounds using the observed covariate information under some monotonicity assumptions similar to those of VanderWeele and Chiba. And we compare the three bounds in a real example.