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The log-logistic distribution provides one parametric model for survival analysis. Unlike the more commonly used Weibull distribution , it can have a non- monotonic hazard function : when β > 1 , {\displaystyle \beta >1,} the hazard function is unimodal (when β {\displaystyle \beta } ≤ 1, the hazard decreases monotonically).
Survival models can be viewed as consisting of two parts: the underlying baseline hazard function, often denoted (), describing how the risk of event per time unit changes over time at baseline levels of covariates; and the effect parameters, describing how the hazard varies in response to explanatory covariates. A typical medical example would ...
Survival analysis is a branch of statistics for analyzing the expected duration of time until one event occurs, such as death in biological organisms and failure in mechanical systems. This topic is called reliability theory , reliability analysis or reliability engineering in engineering , duration analysis or duration modelling in economics ...
It is used in survival theory, reliability engineering and life insurance to estimate the cumulative number of expected events. An "event" can be the failure of a non-repairable component, the death of a human being, or any occurrence for which the experimental unit remains in the "failed" state (e.g., death) from the point at which it changed on.
The survival function is also known as the survivor function [2] or reliability function. [3] The term reliability function is common in engineering while the term survival function is used in a broader range of applications, including human mortality. The survival function is the complementary cumulative distribution function of the lifetime ...
In full generality, the accelerated failure time model can be specified as [2] (|) = ()where denotes the joint effect of covariates, typically = ([+ +]). (Specifying the regression coefficients with a negative sign implies that high values of the covariates increase the survival time, but this is merely a sign convention; without a negative sign, they increase the hazard.)
In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions characterised by two distinct levels of a treatment variable of interest. For example, in a clinical study of a drug, the treated population may die at twice the rate of the control population.
This distribution can be used to analyze time-to-event data in biomedical and public health areas and normally called survival analysis. In engineering, the time-to-event analysis is referred to as reliability theory and in business and economics it is called duration analysis. Other fields may use different names for the same analysis.