  The conventional model for assessing insensitivity to hidden bias in paired
observational studies constructs a worst-case distribution for treatment
assignments subject to bounds on the maximal bias to which any given pair is
subjected. In studies where rare cases of extreme hidden bias are suspected,
the maximal bias may be substantially larger than the typical bias across
pairs, such that a correctly specified bound on the maximal bias would yield an
unduly pessimistic perception of the study's robustness to hidden bias. We
present an extended sensitivity analysis which allows researchers to
simultaneously bound the maximal and typical bias perturbing the pairs under
investigation while maintaining the desired Type I error rate. We motivate and
illustrate our method with two sibling studies on the impact of schooling on
earnings, one containing information of cognitive ability of siblings and the
other not. Cognitive ability, clearly influential of both earnings and degree
of schooling, is likely similar between members of most sibling pairs yet
could, conceivably, vary drastically for some siblings. The method is
straightforward to implement, simply requiring the solution to a quadratic
program.
