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Table 5 Estimates of vaccine effectiveness (\({\varvec{V}}{\varvec{E}}\)) under non-differentially imperfect sensitivity, small cumulative risk of infection and high rate of false-positive events

From: Mitigation of biases in estimating hazard ratios under non-sensitive and non-specific observation of outcomes–applications to influenza vaccine effectiveness

True Estimation adjusted for \({se}_{0}={se}_{1}=0.04\)\(\kappa ={10}^{-5}\) 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% 9% 0.10 0.11 0.12 − 1 92% 7% 0.09 0.09 0.09 − 3 95%
 30% 29% 0.09 0.10 0.10 − 1 92% 23% 0.08 0.08 0.11 − 7 87%
 50% 49% 0.07 0.08 0.08 − 1 93% 39% 0.07 0.07 0.13 − 11 63%
 70% 68% 0.05 0.06 0.06 − 2 93% 56% 0.05 0.05 0.15 − 14 23%
 90% 86% 0.04 0.04 0.05 − 4 85% 74% 0.04 0.04 0.17 − 16 0%
Cohort of 1,000,000 individuals (50% vaccinated at season onset)
 10% 10% 0.02 0.02 0.02  ± 0 92% 7% 0.02 0.02 0.03 − 3 67%
 30% 30% 0.02 0.02 0.02  ± 0 91% 23% 0.02 0.02 0.07 − 7 0%
 50% 50% 0.01 0.01 0.01  ± 0 92% 39% 0.01 0.01 0.11 − 11 0%
 70% 70% 0.01 0.01 0.01  ± 0 92% 56% 0.01 0.01 0.14 − 14 0%
 90% 90% 0.01 0.01 0.01  ± 0 87% 74% 0.01 0.01 0.16 − 16 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 \(\kappa ={10}^{-5}\) (per person-day) corresponding to a false-positive proportion of 16% among the unvaccinated. Naïve estimation was conducted under the incorrect assumptions of perfect sensitivity (\({se}_{0}={se}_{1}=1\)) and absence of false positives (\(\kappa =0\))
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