RE: “MEASUREMENT OF VACCINE DIRECT EFFECTS UNDER THE TEST-NEGATIVE DESIGN”

In a recent article, Lewnard et al. (1) presented a detailed examination of the test-negative design odds ratio. In the test-negative design, vaccine effectiveness is estimated as 1 minus the odds ratio. The authors considered 2 key vaccine protection models: “all-or-nothing,” where a proportion of the vaccinated population is fully protected and the remaining population is fully susceptible, and “leaky,” where the vaccine reduces the rate of infection in all vaccinated individuals. A central conclusion of the authors is that the test-negative design odds ratio is unable to recover an unbiased estimate of vaccine effectiveness under the leaky model.

An underappreciated fact about the test-negative design is that selection of test-negative controls naturally parallels density sampling (2). Controls emerge from the population at risk for the test-positive disease, and, given the central assumption that vaccine has no effect on other etiologies, these controls reflect the underlying distribution of exposure to vaccine. As a result, the odds ratio directly estimates the incidence rate ratio without requiring a rare disease assumption (3).

For density sampling to hold in a test-negative design, persons in the at-risk population must be able to repeatedly test negative (i.e., test-negative infections are not immunizing), and people must be censored from the at-risk population once they test positive. In the article by Lewnard et al. (1), the authors account for the first feature in their derivations but do not capture the second. Adopting the authors’ notation, assuming that the vaccine is leaky with incidence rate ratio θ, the expected cumulative numbers of test-negative cases in vaccinated and unvaccinated participants, respectively, are

$$C_{VN}\left(t\right)={\mathrm{\lambda }}_N{\mathrm{\pi }}_N{\mathrm{\mu }}_{V}vP\frac{\left(1-e^{-{\mathrm{\theta }\mathrm{\lambda }}_{I}t}\right)}{{\mathrm{\theta }\mathrm{\lambda }}_I}$$

and

$$C_{UN}\left(t\right)={\mathrm{\lambda }}_N{\mathrm{\pi }}_N{\mathrm{\mu }}_U(1-v)P\frac{\left(1-e^{-{\mathrm{\lambda }}_{I}t}\right)}{{\mathrm{\lambda }}_I}.$$

Here, t is replaced by the expected person-time at risk (<t), reflecting that persons in a density sampling design would be censored if they tested positive before time t (see Appendix). Importantly, these terms capture differential depletion of vaccinated and unvaccinated persons from the pool at risk for the test-negative disease. The test-negative design odds ratio (OR) is then

$$\begin{array}{ll}\mathrm{OR}\left(t\right)& =\frac{C_{VI}\left(t\right)C_{UN}\left(t\right)}{C_{UI}\left(t\right)C_{VN}(t)}\\ & =\frac{\left[{\mathrm{\pi }}_I{\mathrm{\mu }}_V\left(1-e^{-{\mathrm{\theta }\mathrm{\lambda }}_{I}t}\right)vP\right]\left[{\mathrm{\lambda }}_N{\mathrm{\pi }}_N{\mathrm{\mu }}_U(1-v)P\frac{\left(1-e^{-{\mathrm{\lambda }}_{I}t}\right)}{{\mathrm{\lambda }}_I}\right]}{\left[{\mathrm{\pi }}_I{\mathrm{\mu }}_U\left(1-e^{-{\mathrm{\lambda }}_{I}t}\right)\left(1-v\right)P\right]\left[{\mathrm{\lambda }}_N{\mathrm{\pi }}_N{\mathrm{\mu }}_{V}vP\frac{\left(1-e^{-{\mathrm{\theta }\mathrm{\lambda }}_{I}t}\right)}{{\mathrm{\theta }\mathrm{\lambda }}_I}\right]}\\ & =\mathrm{\theta }.\end{array}$$

Thus, the test-negative design odds ratio is unbiased for a leaky vaccine when the study meets the criteria for density sampling. Practically, this means that participants 1) must be able to repeatedly test negative, 2) must be able to test negative and later test positive, and 3) must not be able to test negative after testing positive. This censoring structure, though simple, differs from previously recommended approaches (4). Besides yielding an unbiased estimator for vaccine effectiveness, this structure has other advantages, such as naturally accommodating changes in vaccine coverage over time, as would occur during an outbreak response.

APPENDIX

Let $T_I~\mathrm{Exponential}({\mathrm{\lambda }}_I)$ denote the random test-positive time for an unvaccinated individual. At study time t, individuals who have not tested positive will have person-time at risk t. Otherwise, individuals will have person-time at risk $T_I$, where $T_I\le t$. The expected value of $T_I$ conditional on $T_I\le t$ is calculated as follows:

$$\begin{array}{ll}E\left[T_I|T_I\le t\right]& =\frac{1}{1-e^{-{\mathrm{\lambda }}_{I}t}}{\int }_0^t\mathit{u}{\mathrm{\lambda }}_I{e}^{-{\mathrm{\lambda }}_{I}u}du\\ & =\frac{1}{1-e^{-{\mathrm{\lambda }}_{I}t}}[-u{e}^{-{\mathrm{\lambda }}_{I}u}-\frac{1}{{\mathrm{\lambda }}_I}{e}^{-{\mathrm{\lambda }}_{I}u}{|}_{u=0}^{u=t}\\ & =\frac{1}{1-e^{-{\mathrm{\lambda }}_{I}t}}\left[-t{e}^{-{\mathrm{\lambda }}_{I}t}-\frac{1}{{\mathrm{\lambda }}_I}{e}^{-{\mathrm{\lambda }}_{I}t}+\frac{1}{{\mathrm{\lambda }}_I}\right]\\ & =\frac{1}{{\mathrm{\lambda }}_I}-\frac{t{e}^{-{\mathrm{\lambda }}_{I}t}}{1-e^{-{\mathrm{\lambda }}_{I}t}}.\end{array}$$

Thus, the expected person-time at risk for each unvaccinated individual by study time t is

$$\begin{array}{l}E\left[T_I|T_I\le t\right]\mathrm{Pr}\left(T_I\le t\right)+t\mathrm{Pr}\left(T_I>t\right)\\ =\left[\frac{1}{{\mathrm{\lambda }}_I}-\frac{t{e}^{-{\mathrm{\lambda }}_{I}t}}{1-e^{-{\mathrm{\lambda }}_{I}t}}\right]\left(1-e^{-{\mathrm{\lambda }}_{I}t}\right)+t{e}^{-{\mathrm{\lambda }}_{I}t}\\ =\frac{\left(1-e^{-{\mathrm{\lambda }}_{I}t}\right)}{{\mathrm{\lambda }}_I}-t{e}^{-{\mathrm{\lambda }}_{I}t}+t{e}^{-{\mathrm{\lambda }}_{I}t}\\ =\frac{\left(1-e^{-{\mathrm{\lambda }}_{I}t}\right)}{{\mathrm{\lambda }}_I}.\end{array}$$

A parallel derivation is used for vaccinated individuals.