Non-linear Relationships between COVID-19 and Non-COVID-19 Mortality by Vaccination Status within Age Groups

Background: Observational studies on COVID-19 vaccine effectiveness (VE) are prone to biases such as healthy vaccinee effect and misclassification, potentially distorting mortality patterns by vaccination status. The proportionality hypothesis posits that COVID-19 mortality is proportional to all-cause mortality within subgroups, but selection bias and frailty may introduce non-linearity. Objective: To investigate non-linear relationships between COVID-19 and non-COVID-19 mortality by vaccination status using UK data, modeling with a power function to correct for biases. Methods: Monthly age-standardized mortality rates from the Office for National Statistics (ONS) database (January 2021-May 2023) were analyzed for five vaccination statuses and six age groups. Relative risks (RR) were calculated, and a power function (RRcov ~ (RRnoncov)^a) was used to test non-linearity. Optimal a-values were validated via visual analysis of VE curves, focusing on stability and logical patterns. Results: In older age groups (<70 years), the relationship was non-linear (a = 1.5-2), with VE stabilizing at 60-90% after correction. No significant improvement from additional vaccination doses was observed. Younger groups showed near-linearity (a = 1-1.5). Concentration effects explained higher mortality in groups not receiving additional doses. Conclusions: The non-linear model corrects for biases, suggesting initial vaccination suffices for protection and multiple boosters may be unnecessary. Frailty likely drives this non-linearity in older groups. However, visual analysis limits robustness; future validation with individual-level data and statistical tests is needed.