  We discuss observational studies that test many causal hypotheses, either
hypotheses about many outcomes or many treatments. To be credible an
observational study that tests many causal hypotheses must demonstrate that its
conclusions are neither artifacts of multiple testing nor of small biases from
nonrandom treatment assignment. In a sense that needs to be defined carefully,
hidden within a sensitivity analysis for nonrandom assignment is an enormous
correction for multiple testing: in the absence of bias, it is extremely
improbable that multiple testing alone would create an association insensitive
to moderate biases. We propose a new strategy called "cross-screening",
different from but motivated by recent work of Bogomolov and Heller on
replicability. Cross-screening splits the data in half at random, uses the
first half to plan a study carried out on the second half, then uses the second
half to plan a study carried out on the first half, and reports the more
favorable conclusions of the two studies correcting using the Bonferroni
inequality for having done two studies. If the two studies happen to concur,
then they achieve Bogomolov-Heller replicability; however, importantly,
replicability is not required for strong control of the family-wise error rate,
and either study alone suffices for firm conclusions. In randomized studies
with a few hypotheses, cross-split screening is not an attractive method when
compared with conventional methods of multiplicity control, but it can become
attractive when hundreds or thousands of hypotheses are subjected to
sensitivity analyses in an observational study. We illustrate the technique by
comparing 46 biomarkers in individuals who consume large quantities of fish
versus little or no fish.
