Hazard ratios (HR) associated with the well-known proportional hazard Cox regression models are routinely used for measuring the impact of one factor of interest on a time-to-event outcome. However, if the underlying true model does not match the theoretical requirements, the interpretation of the HRs is unclear. We consider a new index, gHR, which generalizes the HR beyond the underlying survival model. We consider the case in which the study factor is a binary variable and we are interested in both the unadjusted and adjusted effect of this factor on a time-to-event variable that could be right-censored. We propose non-parametric estimations for unadjusted gHR and semi-parametric regression-induced techniques for the adjusted case. In addition, we estimate the gHR under the so-called marginal Cox regression model that allows population causal effects to be claimed under certain conditions, including on observational data. Sample size requirements are studied using Monte Carlo simulations with practical examples being used for illustrating the utility of the gHR.