The '''hazard ratio''' in survival analysis is the effect of an explanatory variable on the hazard or risk of an event.  For a less technical definition than is provided here, consider hazard ratio to be an estimate of relative risk and see the explanation on that page.

The instantaneous limit of the number of events per unit time divided by the number at risk as the time interval decreases.

::<math> h(t) = \lim_{\Delta t\rightarrow 0}\frac{\mathrm{observed \;events}(t)/N(t)}{\Delta t}</math> 

where ''N''(''t'') is the number at risk at the beginning of an interval.

The '''hazard ratio''' is the effect on this hazard rate of a difference, such as group membership (for example, treatment or control, male or female), as estimated by regression models which treat the log of the hazard rate as a function of a baseline hazard <math>h_0(t)</math> and a linear combination of explanatory variables:  

::<math>\log h(t) = f(h_0(t),\alpha + \beta_1 X_1 + \cdots + \beta_k X_k).\,</math>

Such models are generally classed proportional hazards regression models (they differ in their treatment of <math>h_0(t)</math>, the underlying pattern the hazard rate over time), and include the Cox semi-parametric proportional hazards model, and the exponential, Gompertz and Weibull parametric models.

Il '''rapporto di rischio''' nell'analisi di sopravvivenza è l'effetto di una variabile esplicativa a proposito del pericolo o del rischio di un evento. Per una definizione meno tecnica di quella fornita in questa voce, considera il rapporto di rischio essere una stima di rischio relativo e osserva le spiegazioni riportate in questa voce.

Il tasso di rischio istantaneo è il limite del numero di eventi per unità di tempo in rapporto al numero di eventi a rischio diviso per intervalli descrescenti di tempo (fino a tendere a zero).

::<math> h(t) = \lim_{\Delta t\rightarrow 0}\frac{\mathrm{observed \;events}(t)/N(t)}{\Delta t}</math> 

dove ''N''(''t'') è il numero di eventi a rischioo all'inizio di un intervallo.

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The '''hazard ratio''' is the effect on this hazard rate of a difference, such as group membership (for example, treatment or control, male or female), as estimated by regression models which treat the log of the hazard rate as a function of a baseline hazard <math>h_0(t)</math> and a linear combination of explanatory variables:  

::<math>\log h(t) = f(h_0(t),\alpha + \beta_1 X_1 + \cdots + \beta_k X_k).\,</math>

Such models are generally classed proportional hazards regression models (they differ in their treatment of <math>h_0(t)</math>, the underlying pattern the hazard rate over time), and include the Cox semi-parametric proportional hazards model, and the exponential, Gompertz and Weibull parametric models.