Inference for a test-negative case-control study with added controls

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.