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Table 2 Estimates of vaccine effectiveness (\({\varvec{V}}{\varvec{E}}\)) under differential sensitivity and small cumulative risk of infection in absence 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 \({se}_{0}=0.05\)\({se}_{1}=0.03\) 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% 0.10 0.11 0.11  ± 0 91% 45% 0.06 0.06 0.36  + 35 0%
 30% 30% 0.08 0.09 0.09  ± 0 92% 56% 0.05 0.05 0.27  + 26 1%
 50% 50% 0.07 0.08 0.08  ± 0 92% 68% 0.04 0.04 0.19  + 18 4%
 70% 70% 0.05 0.05 0.05  ± 0 93% 80% 0.03 0.03 0.11  + 10 19%
 90% 90% 0.03 0.03 0.03  ± 0 93% 93% 0.02 0.02 0.04  + 3 57%
Cohort of 1,000,000 individuals (50% vaccinated at season onset)
 10% 10% 0.02 0.02 0.02  ± 0 92% 45% 0.01 0.01 0.35  + 35 0%
 30% 30% 0.02 0.02 0.02  ± 0 91% 56% 0.01 0.01 0.26  + 26 0%
 50% 50% 0.01 0.01 0.01  ± 0 93% 68% 0.01 0.01 0.18  + 18 0%
 70% 70% 0.01 0.01 0.01  ± 0 94% 80% 0.01 0.01 0.10  + 10 0%
 90% 90% 0.01 0.01 0.01  ± 0 94% 93% 0.00 0.00 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 under differential sensitivity of 0.05 (\({se}_{0}\)) and 0.03 (\({se}_{1}\)) and a cumulative risk of 0.25 in the unvaccinated in absence of false-positive events. Naïve estimation was conducted under the incorrect assumption of perfect sensitivity (\({se}_{0}={se}_{1}=1\))
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