Reduced sensitivity to hidden bias at upper quantiles in observational studies with dilated treatment effects

When a treatment has a dilated effect, with larger effects when responses a re higher, there can be much less sensitivity to bias at upper quantiles th an at lower quantiles; i.e., small, plausible hidden biases might explain t he ostensible effect of the treatment for many subjects, and yet only quite large hidden biases could explain the effect on a few subjects having dram atically elevated responses. An example concerning kidney function of cadmi um workers is discussed in detail. In that example, the treatment effect is far from additive: It is plausibly zero at the lower quartile of responses to control, and it is large and fairly insensitive to bias at the upper qu artile.