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Fig. 2 | Emerging Themes in Epidemiology

Fig. 2

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

Fig. 2

Estimates of influenza vaccine effectiveness in the Finnish elderly (N = 1,160,986) in 2016/17. A: Kaplan–Meier estimates of the observed survival functions in the unvaccinated (\({\widehat{\tilde{S}}}_{0}(t)\)) and vaccinated (\({\widehat{\tilde{S}}}_{1}(t)\)) and the corresponding estimates of the true survival functions (\({\widehat{S}}_{0}(t)\), \({\widehat{S}}_{1}(t)\)) based on (1) assuming non-differential sensitivity (\({se}_{0}\), \({se}_{1}\)) of 0.04 and absence of false-positive events. The estimated cumulative risks (\(1-{\widehat{S}}_{0}(t)\), \(1-{\widehat{S}}_{1}(t)\)) at \(t=196\) (days) were 0.20 and 0.16. B: The linear relation between the log–log transformed survival functions \({\widehat{S}}_{0}(t)\) and \({\widehat{S}}_{1}(t)\) supports the proportional hazards assumption (cf. Additional file 2: Web Appendix). C: Time evolution of vaccine effectiveness estimates (solid line) and pointwise 95% confidence intervals (dashed lines) based on (4). D: Dependence of vaccine effectiveness estimates at \(t=196\) (days) on the assumed values of \({se}_{0}\) (symbols) and ratio \({se}_{1}/{se}_{0}\) (horizontal axis) based on (4). The plot area has been restricted showing only non-negative vaccine effectiveness estimates. For the full range see Additional file 2: Figure S1 (see Additional file 2: Web Appendix)

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