Hazard ratio
Encyclopedia
In survival analysis
, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two sets of explanatory variables. For example, in a drug study, the treated population may die at twice the rate per unit time as the control population. The hazard ratio would be 2, indicating higher hazard of death from the treatment. Or in another study, men receiving the same treatment may suffer a certain complication ten times more frequently per unit time than women, a hazard ratio of 10.
Hazard ratios differ from relative risk ratios in that the latter are cumulative over an entire study, using a defined endpoint, while the former represent instantaneous risk over the study time period, or some subset thereof. Hazard ratios suffer somewhat less from selection bias versus the endpoints chosen, and can indicate risks that happen before the endpoint.
of the number of events per unit time divided by the number at risk as the time interval decreases.
where N(t) is the number at risk at the beginning of an interval.
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
that treat the log of the HR as a function of a baseline hazard
and a linear combination of explanatory variables:
Such models are generally classed proportional hazards regression
models (they differ in their treatment of
, the underlying pattern the HR over time), and include the Cox
semiparametric proportional hazards model, and the exponential, Gompertz and Weibull parametric models.
For two individuals who differ only in the relevant membership (e.g., treatment vs. control), their predicted log-hazard will differ additively by the relevant parameter estimate, which is to say that their predicted HR will differ by
, i.e., multiplicatively by the anti-log of the estimate. Thus the estimate can be considered a hazard ratio, that is, the ratio between the predicted hazard for a member of one group and that for a member of the other group, holding everything else constant.
For a continuous explanatory variable, the same interpretation applies to a unit difference.
Other HR models have different formulations and the interpretation of the parameter estimates differs accordingly.
In simplified terms the hazard ratio is use to describe time-to-event in survival analysis. It is the ratio of the rate at which subjects in two groups are experiencing events where a slower rate suggests a longer time of event-free-survival. This type of analysis is frequently used to evaluate a drug's ability to prevent disease as a function of time.
Survival analysis
Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or sociology...
, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two sets of explanatory variables. For example, in a drug study, the treated population may die at twice the rate per unit time as the control population. The hazard ratio would be 2, indicating higher hazard of death from the treatment. Or in another study, men receiving the same treatment may suffer a certain complication ten times more frequently per unit time than women, a hazard ratio of 10.
Hazard ratios differ from relative risk ratios in that the latter are cumulative over an entire study, using a defined endpoint, while the former represent instantaneous risk over the study time period, or some subset thereof. Hazard ratios suffer somewhat less from selection bias versus the endpoints chosen, and can indicate risks that happen before the endpoint.
Definition
The instantaneous hazard rate is the limitLimit (mathematics)
In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric...
of the number of events per unit time divided by the number at risk as the time interval decreases.
where N(t) is the number at risk at the beginning of an interval.
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
Regression analysis
In statistics, regression analysis includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables...
that treat the log of the HR as a function of a baseline hazard
and a linear combination of explanatory variables:
Such models are generally classed proportional hazards regression
Proportional hazards models
Proportional hazards models are a class of survival models in statistics. Survival models relate the time that passes before some event occurs to one or more covariates that may be associated with that quantity. In a proportional hazards model, the unique effect of a unit increase in a covariate...
models (they differ in their treatment of
, the underlying pattern the HR over time), and include the CoxDavid Cox (statistician)
Sir David Roxbee Cox FRS is a prominent British statistician.-Early years:Cox studied mathematics at St. John's College, Cambridge and obtained his PhD from the University of Leeds in 1949, advised by Henry Daniels and Bernard Welch.-Career:He was employed from 1944 to 1946 at the Royal Aircraft...
semiparametric proportional hazards model, and the exponential, Gompertz and Weibull parametric models.
For two individuals who differ only in the relevant membership (e.g., treatment vs. control), their predicted log-hazard will differ additively by the relevant parameter estimate, which is to say that their predicted HR will differ by
, i.e., multiplicatively by the anti-log of the estimate. Thus the estimate can be considered a hazard ratio, that is, the ratio between the predicted hazard for a member of one group and that for a member of the other group, holding everything else constant.For a continuous explanatory variable, the same interpretation applies to a unit difference.
Other HR models have different formulations and the interpretation of the parameter estimates differs accordingly.
In simplified terms the hazard ratio is use to describe time-to-event in survival analysis. It is the ratio of the rate at which subjects in two groups are experiencing events where a slower rate suggests a longer time of event-free-survival. This type of analysis is frequently used to evaluate a drug's ability to prevent disease as a function of time.
See also
- Survival analysisSurvival analysisSurvival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. This topic is called reliability theory or reliability analysis in engineering, and duration analysis or duration modeling in economics or sociology...
- Failure rateFailure rateFailure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. It is often denoted by the Greek letter λ and is important in reliability engineering....
and Hazard rate - Proportional hazards modelsProportional hazards modelsProportional hazards models are a class of survival models in statistics. Survival models relate the time that passes before some event occurs to one or more covariates that may be associated with that quantity. In a proportional hazards model, the unique effect of a unit increase in a covariate...
- Odds ratioOdds ratioThe odds ratio is a measure of effect size, describing the strength of association or non-independence between two binary data values. It is used as a descriptive statistic, and plays an important role in logistic regression...
- Relative risk ratioRelative riskIn statistics and mathematical epidemiology, relative risk is the risk of an event relative to exposure. Relative risk is a ratio of the probability of the event occurring in the exposed group versus a non-exposed group....