  Studies of vaccine efficacy often record both the incidence of
vaccine-targeted virus strains (primary outcome) and the incidence of
non-targeted strains (secondary outcome). However, standard estimates of
vaccine efficacy on targeted strains ignore the data on non-targeted strains.
Assuming non-targeted strains are unaffected by vaccination, we regard the
secondary outcome as a negative control outcome and show how using such data
can (i) increase the precision of the estimated vaccine efficacy against
targeted strains in randomized trials, and (ii) reduce confounding bias of that
same estimate in observational studies. For objective (i), we augment the
primary outcome estimating equation with a function of the secondary outcome
that is unbiased for zero. For objective (ii), we jointly estimate the
treatment effects on the primary and secondary outcomes. If the bias induced by
the unmeasured confounders is similar for both types of outcomes, as is
plausible for factors that influence the general risk of infection, then we can
use the estimated efficacy against the secondary outcomes to remove the bias
from estimated efficacy against the primary outcome. We demonstrate the utility
of these approaches in studies of HPV vaccines that only target a few highly
carcinogenic strains. In this example, using non-targeted strains increased
precision in randomized trials modestly but reduced bias in observational
studies substantially.
