  Does having firearms in the home increase suicide risk? To test this
hypothesis, a matched case-control study can be performed, in which suicide
case subjects are compared to living controls who are similar in observed
covariates in terms of their retrospective exposure to firearms at home. In
this application, cases can be defined using a broad case definition (suicide)
or a narrow case definition (suicide occurred at home). The broad case
definition offers a larger number of cases but the narrow case definition may
offer a larger effect size. Moreover, restricting to the narrow case definition
may introduce selection bias (i.e., bias due to selecting samples based on
characteristics affected by the treatment) because exposure to firearms in the
home may affect the location of suicide and thus the type of a case a subject
is. We propose a new sensitivity analysis framework for combining broad and
narrow case definitions in matched case-control studies, that considers the
unmeasured confounding bias and selection bias simultaneously. We develop a
valid randomization-based testing procedure using only the narrow case matched
sets when the effect of the unmeasured confounder on receiving treatment and
the effect of the treatment on case definition among the always-cases are
controlled by sensitivity parameters. We then use the Bonferroni method to
combine the testing procedures using the broad and narrow case definitions.
With the proposed methods, we find robust evidence that having firearms at home
increases suicide risk.
