Limits for the Magnitude of M-bias and Certain Other Types of Structural Selection Bias

Structural selection bias and confounding are key threats to validity of causal effect estimation. Here, we consider M-bias, a type of selection bias, described by Hernán et al as a situation wherein bias is caused by selecting on a variable that is caused by two other variables, one a cause of the exposure, the other a cause of the outcome. Our goals are to derive a bound for (the maximum) M-bias, explore through examples the magnitude of M-bias, illustrate how to apply the bound for other types of selection bias, and provide a program for directly calculating M-bias and the bound.

We derive a bound for selection bias assuming specific, causal relationships that characterize M-bias and further evaluate it using simulations.

Through examples, we show that, in many plausible situations, M-bias will tend to be small. In some examples, the bias is not small-but plausibility of the examples, ultimately to be judged by the researcher, may be low. The examples also show how the M-bias bound yields bounds for other types of selection bias and also for confounding. The latter illustrates how Lee's bound for confounding can arise as a limiting case of ours.

We have derived a new bound for M-bias. Examples illustrate how to apply it with other types of selection bias. They also show that it can yield tighter bounds in certain situations than a previously published bound for M-bias. Our examples suggest that M-bias may often, but not uniformly, be small.