  Test-negative designs with added controls have recently been proposed to
study COVID-19. An individual is test-positive or test-negative accordingly if
they took a test for a disease but tested positive or tested negative. Adding a
control group to a comparison of test-positives vs test-negatives is useful
since additional comparison of test-positives vs controls can have potential
biases different from the first comparison. Bonferroni correction ensures
necessary type-I error control for these two comparisons done simultaneously.
We propose two new methods for inference which have better interpretability and
higher statistical power for these designs. These methods add a third
comparison that is essentially independent of the first comparison, but our
proposed second method often pays much less for these three comparisons than
what a Bonferroni correction would pay for the two comparisons.
