# Table 4 Estimates of vaccine effectiveness ($${\varvec{V}}{\varvec{E}}$$) under non-differentially imperfect sensitivity, small cumulative risk of infection and low rate of false-positive events

True Estimation adjusted for $${se}_{0}={se}_{1}=0.04$$$$\kappa ={10}^{-6}$$ Naïve estimation
$$VE$$ $$\widehat{VE}$$ $$\widehat{SE}$$ $${SE}_{\widehat{VE}}$$ $${\sqrt{MSE}}_{\widehat{VE}}$$ Bias Cov $$\widehat{VE}$$ $$\widehat{SE}$$ $${SE}_{\widehat{VE}}$$ $${\sqrt{MSE}}_{\widehat{VE}}$$ Bias Cov
Cohort of 50,000 individuals (30% vaccinated at season onset)
10% 10% 0.09 0.11 0.11  ± 0 92% 8% 0.09 0.09 0.09 − 2 95%
30% 30% 0.08 0.09 0.09  ± 0 92% 26% 0.08 0.08 0.09 − 4 95%
50% 50% 0.06 0.07 0.07  ± 0 93% 46% 0.07 0.07 0.08 − 4 93%
70% 69% 0.05 0.05 0.05 − 1 94% 66% 0.05 0.05 0.07 − 4 90%
90% 89% 0.03 0.03 0.03 − 1 96% 87% 0.03 0.03 0.04 − 3 88%
Cohort of 1,000,000 individuals (50% vaccinated at season onset)
10% 10% 0.02 0.02 0.02  ± 0 92% 9% 0.02 0.02 0.02 − 1 89%
30% 30% 0.02 0.02 0.02  ± 0 91% 27% 0.02 0.02 0.04 − 3 44%
50% 50% 0.01 0.01 0.01  ± 0 93% 46% 0.01 0.01 0.05 − 4 7%
70% 70% 0.01 0.01 0.01  ± 0 94% 66% 0.01 0.01 0.04 − 4 0%
90% 90% 0.00 0.00 0.00  ± 0 94% 87% 0.01 0.01 0.03 − 3 0%
1. Mean of the vaccine effectiveness estimates ($$\widehat{VE}$$), mean of the standard error estimates ($$\widehat{SE}$$), standard error of the vaccine effectiveness estimates ($${SE}_{\widehat{VE}}$$), root-mean-squared error of the vaccine effectiveness estimates ($${\sqrt{MSE}}_{\widehat{VE}}$$), bias in percentage points, and empirical coverage probability (Cov) of the 95% confidence intervals when estimating vaccine effectiveness from 104 repeated data sets given non-differential sensitivity ($${se}_{0}={se}_{1}$$) of 0.04 and a cumulative risk of 0.25 in the unvaccinated. The false-positive events occurred at rate $$\kappa ={10}^{-6}$$ (per person-day) corresponding to a false-positive proportion of 2% among the unvaccinated. Naïve estimation was conducted under the incorrect assumptions of perfect sensitivity ($${se}_{0}={se}_{1}=1$$) and absence of false positives ($$\kappa =0$$)