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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

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 \({se}_{0}={se}_{1}=0.04\)\(\kappa ={10}^{-6}\) 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 (\(\kappa =0\))
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