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 are higher, there can be much less sensitivity to bias at upper quantiles than at lower quantiles; i.e., small, plausible hidden biases might explain the ostensible effect of the treatment for many subjects, and yet only quite large hidden biases could explain the effect on a few subjects having dramatically elevated responses. An example concerning kidney function of cadmium 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 quartile.