Accelerated failure time model
Encyclopedia
In the statistical
area of survival analysis
, an accelerated failure time model (AFT model) is a parametric
model that provides an alternative to the commonly used proportional hazards models
. Whereas a proportional hazards model assumes that the effect of a covariate
is to multiply the hazard by some constant, an AFT model assumes that the effect of a covariate is to multiply the predicted event time by some constant. AFT models can therefore be framed as linear models for the logarithm of the survival time.
's semi-parametric proportional hazards model is more widely used than parametric models, AFT models are predominately fully parametric i.e. a probability distribution
is specified. (Buckley and James proposed a semi-parametric AFT but its use is relatively uncommon in applied research; in a 1992 paper, Wei pointed out that the Buckley–James model has no theoretical justification and lacks robustness, and reviewed alternatives.)
Unlike proportional hazards models, the regression parameter estimates from AFT models are robust to omitted covariate
s. They are also less affected by the choice of probability distribution.
The results of AFT models are easily interpreted.
For example, the results of a clinical trial
with mortality as the endpoint could be interpreted as a certain percentage increase in future life expectancy
on the new treatment compared to the control. So a patient could be informed that he would be expected to live (say) 15% longer if he took the new treatment. Hazard ratios can prove harder to explain in layman's terms.
More probability distribution
s can be used in AFT models than parametric proportional hazard models, including distributions that have unimodal hazard functions.
. The logarithm of the scale parameter is then modelled as a linear function of the covariates.
The log-logistic distribution
provides the most commonly used AFT model. Unlike the Weibull distribution, it can exhibit a non-monotonic hazard function which increases at early times and decreases at later times. It is similar in shape to the log-normal distribution but its cumulative distribution function
has a simple closed form
, which becomes important computationally when fitting data with censoring
.
The Weibull distribution (including the exponential distribution
as a special case) can be parameterised as either a proportional hazards model or an AFT model, and is the only family of distributions to have this property. The results of fitting a Weibull model can therefore be interpreted in either framework.
Other distributions suitable for AFT models include the log-normal, gamma and inverse Gaussian distribution
s, although they are less popular than the log-logistic, partly as their cumulative distribution functions do not have a closed form. Finally, the generalized gamma distribution
is a three-parameter distribution that includes the Weibull, log-normal and gamma distributions as special cases.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
area of survival analysis
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...
, an accelerated failure time model (AFT model) is a parametric
Parametric statistics
Parametric statistics is a branch of statistics that assumes that the data has come from a type of probability distribution and makes inferences about the parameters of the distribution. Most well-known elementary statistical methods are parametric....
model that provides an alternative to the commonly used proportional hazards models
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...
. Whereas a proportional hazards model assumes that the effect of a covariate
Covariate
In statistics, a covariate is a variable that is possibly predictive of the outcome under study. A covariate may be of direct interest or it may be a confounding or interacting variable....
is to multiply the hazard by some constant, an AFT model assumes that the effect of a covariate is to multiply the predicted event time by some constant. AFT models can therefore be framed as linear models for the logarithm of the survival time.
Comparison with proportional hazard models
Unlike proportional hazards models, in which 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...
's semi-parametric proportional hazards model is more widely used than parametric models, AFT models are predominately fully parametric i.e. a probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
is specified. (Buckley and James proposed a semi-parametric AFT but its use is relatively uncommon in applied research; in a 1992 paper, Wei pointed out that the Buckley–James model has no theoretical justification and lacks robustness, and reviewed alternatives.)
Unlike proportional hazards models, the regression parameter estimates from AFT models are robust to omitted covariate
Covariate
In statistics, a covariate is a variable that is possibly predictive of the outcome under study. A covariate may be of direct interest or it may be a confounding or interacting variable....
s. They are also less affected by the choice of probability distribution.
The results of AFT models are easily interpreted.
For example, the results of a clinical trial
Clinical trial
Clinical trials are a set of procedures in medical research and drug development that are conducted to allow safety and efficacy data to be collected for health interventions...
with mortality as the endpoint could be interpreted as a certain percentage increase in future life expectancy
Life expectancy
Life expectancy is the expected number of years of life remaining at a given age. It is denoted by ex, which means the average number of subsequent years of life for someone now aged x, according to a particular mortality experience...
on the new treatment compared to the control. So a patient could be informed that he would be expected to live (say) 15% longer if he took the new treatment. Hazard ratios can prove harder to explain in layman's terms.
More probability distribution
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s can be used in AFT models than parametric proportional hazard models, including distributions that have unimodal hazard functions.
Distributions used in AFT models
To be used in an AFT model, a distribution must have a parameterisation that includes a scale parameterScale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...
. The logarithm of the scale parameter is then modelled as a linear function of the covariates.
The log-logistic distribution
Log-logistic distribution
In probability and statistics, the log-logistic distribution is a continuous probability distribution for a non-negative random variable. It is used in survival analysis as a parametric model for events whose rate increases initially and decreases later, for example mortality from cancer following...
provides the most commonly used AFT model. Unlike the Weibull distribution, it can exhibit a non-monotonic hazard function which increases at early times and decreases at later times. It is similar in shape to the log-normal distribution but its cumulative distribution function
Cumulative distribution function
In probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
has a simple closed form
Closed-form expression
In mathematics, an expression is said to be a closed-form expression if it can be expressed analytically in terms of a bounded number of certain "well-known" functions...
, which becomes important computationally when fitting data with censoring
Censoring (statistics)
In statistics, engineering, and medical research, censoring occurs when the value of a measurement or observation is only partially known.For example, suppose a study is conducted to measure the impact of a drug on mortality. In such a study, it may be known that an individual's age at death is at...
.
The Weibull distribution (including the exponential distribution
Exponential distribution
In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
as a special case) can be parameterised as either a proportional hazards model or an AFT model, and is the only family of distributions to have this property. The results of fitting a Weibull model can therefore be interpreted in either framework.
Other distributions suitable for AFT models include the log-normal, gamma and inverse Gaussian distribution
Inverse Gaussian distribution
| cdf = \Phi\left +\exp\left \Phi\left...
s, although they are less popular than the log-logistic, partly as their cumulative distribution functions do not have a closed form. Finally, the generalized gamma distribution
Generalized gamma distribution
The generalized gamma distribution is a continuous probability distribution with three parameters. It is a generalization of the two-parameter gamma distribution...
is a three-parameter distribution that includes the Weibull, log-normal and gamma distributions as special cases.
Books
- Martinussen, Torben; Scheike, Thomas (2006), Dynamic Regression Models for Survival Data, Springer, ISBN 0-387-20274-9
- Bagdonavicius, Vilijandas; Nikulin, Mikhail (2002), Accelerated Life Models. Modeling and Statistical Analysis, Chapman&Hall/CRC, ISBN 1-584-88186-0