Investigations of sharp bounds for causal effects under selection bias

Selection bias is a common type of bias, and depending on the causal estimand of interest and the structure of the selection variable, it can be a threat to both external and internal validity. One way to quantify the maximum magnitude of potential selection bias is to calculate bounds for the causal estimand. Here, we consider previously proposed bounds for selection bias, which require the specification of certain sensitivity parameters. First, we show that the sensitivity parameters are variation independent. Second, we show that the bounds are sharp under certain conditions. Furthermore, we derive improved bounds that are based on the same sensitivity parameters. Depending on the causal estimand, these bounds require additional information regarding the selection probabilities. We illustrate the improved bounds in an empirical example where the effect of breakfast eating on overweight is estimated. Lastly, the performance of the bounds are investigated in a numerical experiment for sharp and non-sharp cases.