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| In the [[statistics|statistical]] area of [[survival analysis]], an '''accelerated failure time model ''' ('''AFT model''') is a [[parametric statistics|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 rate|hazard]] by some constant, an AFT model assumes that the effect of a covariate is to accelerate or decelerate the life course of a disease by some constant. This is especially appealing in a technical context where the 'disease' is a result of some mechanical process with a known sequence of intermediary stages.
| | I'm Rae (21) from Suderbrarup, Germany. <br>I'm learning Bengali literature at a local high school and I'm just about to graduate.<br>I have a part time job in a backery.<br><br>My weblog - [http://blog.iheardyousinging.com/ wordpress dropbox backup] |
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| ==Model specification==
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| In full generality, the accelerated failure time model can be specified as<ref>{{cite book
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| |last=Kalbfleisch & Prentice
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| |year=2002
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| |title=The Statistical Analysis of Failure Time Data (2nd ed.)
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| |publisher=Wiley Series in Probability and Statistics
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| |location=Hoboken, NJ }}</ref>
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| ::<math>
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| \lambda(t|\theta)=\theta\lambda_0(\theta t)
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| </math>
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| where <math>\theta</math> denotes the joint effect of covariates, typically <math>\theta=\exp(-[\beta_1X_1 + \cdots + \beta_pX_p])</math>. (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.)
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| This is satisfied, if the [[probability density function]] of the event is taken to be <math>f(t|\theta)=\theta f_0(\theta t)</math>, from which is follows for the [[survival function]] that <math>S(t|\theta)=S(\theta t)</math>. From this it is easy to see that the moderated life time <math>T</math> is distributed such that <math>T\theta</math> and the unmoderated life time <math>T_0</math> have the same distribution. Consequently, <math>log(T)</math> can be written as
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| ::<math>
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| log(T)=-log(\theta)+log(T\theta):=-log(\theta)+\epsilon
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| </math>
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| where the last term is distributed as <math>log(T_0)</math>, i.e. independently of <math>\theta</math>. This reduces the accelerated failure time model into [[regression analysis]] (typically a [[linear model]]) where <math>-log(\theta)</math> represents the fixed effects, and <math>\epsilon</math> represents the noise. Different distributional forms of <math>\epsilon</math> imply different distributional forms of <math>T_0</math>, i.e. different baseline distributions of the survival time. It is typical of survival-analytic contexts, that many of the observations are censored, i.e. we only know that <math>T_i>t_i</math>, not <math>T_i=t_i</math>. In fact, the former case represents survival, while the later case represents an event/death/censoring during the follow-up. These right-censored observations can pose technical challenges for estimating the model, if the distribution of <math>T_0</math> is unusual.
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| The interpretation of <math>\theta</math> in accelerated failure time models is straight forward: E.g. <math>\theta=2</math> means that everything in the relevant life history of an individual happens twice as fast. For example, if the model concerns the development of a tumor, it means that all of the pre-stages progress twice as fast as for the unexposed individual, implying that the expected time until a clinical disease is 0.5 of the baseline time. However, this does not mean that the hazard function <math>\lambda(t|\theta)</math> is always twice as high - that would be the [[proportional hazards model]].
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| ==Statistical issues==
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| Unlike proportional hazards models, in which [[David Cox (statistician)|Cox]]'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 for <math>log(T_0)</math>. (Buckley and James<ref>{{Citation | doi=10.1093/biomet/66.3.429 | journal=Biometrika| year=1979 |volume=66 |pages=429–436 | title=Linear regression with censored data| first1=Jonathan |last1=Buckley |first2= Ian |last2=James| issue=3|jstor=2335161}}</ref> proposed a semi-parametric AFT but its use is relatively uncommon in applied research; in a 1992 paper, Wei<ref>{{cite doi|10.1002/sim.4780111409}}</ref> pointed out that the Buckley–James model has no theoretical justification and lacks robustness, and reviewed alternatives.) This can be a problem, if a degree of realistic detail is required for modelling the distribution of a baseline lifetime. Hence, technical developments in this direction would be highly desirable.
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| 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.<ref>{{Citation | journal=Statistics in Medicine| year=2004 |volume=23 |pages=3177–3192 |doi=10.1002/sim.1876| title=Parametric accelerated failure time models with random effects and an application to kidney transplant survival| first1=Philippe |last1=Lambert |first2= Dave |last2=Collett| first3=Alan| last3= Kimber |first4=Rachel |last4=Johnson | pmid=15449337 | issue=20}}</ref><ref>{{Cite doi/10.1002.2F.28SICI.291097-0258.2819970130.2916:2.3C215::AID-SIM481.3E3.0.CO.3B2-J}}</ref>
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| The results of AFT models are easily interpreted.<ref>{{Citation| title=On the use of the accelerated failure time model as an alternative to the proportional hazards model in the treatment of time to event data: A case study in influenza | journal=Drug Information Journal | year= 2002 | last1=Kay |first1= Richard| last2= Kinnersley|first2= Nelson| volume=36| pages=571–579| url=http://cat.inist.fr/?aModele=afficheN&cpsidt=15244883| issue=3}}</ref> 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.
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| ===Distributions used in AFT models===
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| 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 expression|closed form]], which becomes important computationally when fitting data with [[censoring (statistics)|censoring]]. For the censored observations one needs the survival function, which is the complement of the cumulative distribution function, i.e. one needs to be able to evaluate <math>S(t|\theta)=1-F(t|\theta)</math>.
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| 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. However, the biological applicability of this model may be limited by the fact that the hazard function is monotonous, i.e. either decreasing of increasing.
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| Other distributions suitable for AFT models include the [[log-normal distribution|log-normal]], [[gamma distribution|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 distribution|Weibull]], [[log-normal distribution|log-normal]] and [[gamma distribution|gamma]] distributions as special cases.
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *{{Citation| journal=British Journal of Cancer |year=2003 |pages=431–436 |title=Survival Analysis Part II: Multivariate data analysis - an introduction to concepts and methods |first1=MJ| last1=Bradburn| first2= TG |last2=Clark |first3=SB | last3=Love | first4=DG |last4=Altman| volume=89|doi=10.1038/sj.bjc.6601119 |issue=89| pmid=12888808| pmc=2394368}}
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| *{{Citation |title=Fundamentals of Survival Data| first=Philip |last=Hougaard| journal=Biometrics| volume=55| year=1999| pages= 13–22|doi=10.1111/j.0006-341X.1999.00013.x |issue=1 |pmid=11318147}}
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| *{{Citation | title =Modelling Survival Data in Medical Research|first=D. |last=Collett | year=2003 | edition=2nd | publisher=CRC press| isbn=1-58488-325-1}}
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| *{{Citation | title=Analysis of Survival Data | first1= David Roxbee | last1=Cox | author1-link=David Cox (statistician) | first2= D.|last2= Oakes | publisher=CRC Press | year=1984 | isbn= 0-412-24490-X }}
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| *{{Citation| title=Analysing Survival Data from Clinical Trials and Observational Studies| first1=Ettore| last1= Marubini| first2= Maria Grazia| last2=Valsecchi | year=1995| publisher=Wiley |isbn=0-470-09341-2}}
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| * Martinussen, Torben; Scheike, Thomas (2006), Dynamic Regression Models for Survival Data, Springer, ISBN 0-387-20274-9
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| * Bagdonavicius, Vilijandas; Nikulin, Mikhail (2002), Accelerated Life Models. Modeling and Statistical Analysis, Chapman&Hall/CRC, ISBN 1-58488-186-0
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| {{Statistics|analysis}}
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| [[Category:Survival analysis]]
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I'm Rae (21) from Suderbrarup, Germany.
I'm learning Bengali literature at a local high school and I'm just about to graduate.
I have a part time job in a backery.
My weblog - wordpress dropbox backup