A Review on Accelerated Failure Time Models

Transcription

1 Iteratioal Joural of Statistics ad Systems ISSN Volume 12, Number 2 (2017), pp Research Idia Publicatios A Review o Accelerated Failure Time Models Riku Saikia 1 ad Maash Pratim Barma 2 1 Research Scholar, Departmet of Statistics, Dibrugarh Uiversity. Dibrugarh , Assam, Idia. 2 Assistat Professor, Departmet of Statistics, Dibrugarh Uiversity, Dibrugarh , Assam, Idia. Abstract Survival aalysis is the aalysis of statistical data i which the outcome variable of iterest is time util a evet occurs. I Statistical literature, it is observed that a good umber of models have bee developed for aalyzig survival data or life time data.the most popular amog them is the Cox Proportioal Hazard (PH) model. Accelerated Failure Time (AFT) model, which is maily used to study the reliability of idustrial products ca also be cosidered as a good alterative of Cox PH model i aalyzig survival data. I this paper, the attempt has bee made to preset a review o Accelerated Failure Time models. Here the historical developmets, techical developmets ad past research o AFT models are discussed. Keywords: Survival data, regressio model, AFT model, Cox PH model, evet 1. BACKGROUND OF THE STUDY: Survival aalysis is the aalysis of survival data i which the outcome variable of iterest is time util some evet occur. Evets are geerally referred as failure because the evet may be death, disease, icidet etc. Due to cesored data, various statistical methods for failure data are developed. They are o-parametric methods (Kapla-Meier method, log-rak test), semi-parametric method (Cox Proportioal Hazard (PH) model), parametric model (Parametric PH model ad Accelerated failure

2 312 Riku Saikia ad Maash Pratim Barma Time (AFT)) model etc. Cox PH model is the most commo approach for modelig survival data. Parametric AFT model provides a alterative to PH model for statistical modelig of survival data (Wei,1992). AFT model is basically used i idustrial fields ad seldom used i the case of survival data. If the appropriate parametric form of AFT model is used the it offers a potetial statistical approach i case of survival data which is based upo the survival curve rather tha the hazard fuctio. It is kow as Accelerated failure time model because the term failure idicates the death, disease etc. ad the term Accelerated idicates the resposible factor for which the rate of failure is icreased. That factor is called Acceleratio factor. The AFT model is also kow as the log-locatio scale model give by Lawless (1982). It is called log-locatio scale because the logarithm of time variable is cosidered. Details of this AFT model or log-locatio scale model will be discussed later. Accordig to the literature foud, Pike (1966) Proposed the AFT model i case of carciogeesis data. He developed the basic statistical methodology ad discussed likelihood estimatio for the Weibull distributio. But i his paper he ever metioed it the ame as Accelerated failure time model. Agai i 1972, Nelso ad Ha, preseted the failure time data i case of idustrial life testig. They cosidered the umber of hours to failure motorettes operatig uder various temperature. The ame Accelerated life test came from this type I cesorig study. The aim of their study was to determie the relatioship betwee failure time data ad temperature. I survival aalysis radom cesorig is cosidered. To modelig failure time data for radom cesorig i D.R. Cox (1972) itroduced a model kow as Cox Proportioal Hazard (PH) model where the effect of covariates act multiplicatively o the hazard fuctio. The Cox PH model is a semi-parametric model i which the baselie hazard fuctio is completely uspecified. Whe the exact form of the parametric model is ot kow the the Cox PH model preferred tha the other parametric models. But whe the correct form of the parametric model is idetified the the parametric models are more suitable tha the semi-parametric or o-parametric models (Kleibaum ad Klei, 2002). The Acceleratio failure time model is a parametric (AFT) model which was itroduced by Cox (1972). Kalbfleisch ad Pretice (2002), itroduced the semi- parametric class of survival model, which was the class of logliear models for time T. I AFT model, the covariate effects act multiplicatively o survival time. Both the PH ad AFT models are regressio models. Though the parametric models are liear regressio model but the differece betwee the liear regressio model ad the survival regressio model is that i case of survival model the cesored observatios are cosidered. The mai objective of this paper is to review o the AFT models ad its techical developmets ad past research i case of survival data.

3 A Review o Accelerated Failure Time Models INTRODUCTION OF THE MODEL: The Accelerated Failure Time model: The AFT model describes the relatioship betwee the respose variable ad the survival time. I this model, the logarithms of the survival time is cosidered as a respose variable ad icludes a error term which is assumed to follow a specific probability distributio. The assumptio of AFT model is that the effect of covariates act multiplicatively (proportioally) with respect to the survival time. The assumptio of AFT model ca be expressed as s(t/x) = s 0 (exp (β x)t) for t 0. (1) Where s(t/x) is the survival fuctio at the time t ad the s 0 (exp (β x)t) is the baselie survival fuctio at the time t. From this equatio (1), AFT model ca states that the survival fuctio of a idividual with covariate x at the time t is same as the baselie survival fuctio of the at the time (exp (β x)t), where β = (β 1, β p ). The factor exp (β x) is kow as the acceleratio factor. The acceleratio factor is the key measure of associatio obtaied i the AFT model. It is a ratio of survival times correspodig to ay fixed value of survival time. From the acceleratio factor oe may able to kow how a chage i covariate values chage i time scale from the baselie time scale. That is with the acceleratio factor oe ca evaluate the effect of predictor variables o survival time. Suppose cosiderig a compariso of survival fuctios amog the persos takig cacer directed treatmets ad ot takig ay cacer directed treatmets. The survival fuctio of cacer directed treatmets ad ot takig ay cacer directed treatmets are cosidered as s 1 (t) ad s 2 (t) respectively. The AFT assumptio ca be expressed as s 2 (t) = s 1 (exp (β x)t) for t 0 Where exp (β x) is the acceleratio factor which compares the patiets with cacer directed treatmets ad ot takig cacer directed treatmets. If exp(β x) > 1, the effect of covariate is decelerated Ad if exp(β x) < 1, the effect of covariate is accelerated. Cox(1972) itroduced the Cox PH model. The Weibull model ad the Expoetial model ca be derived from this model. But the other parametric models such as logormal, log-logistic etc. are ot derived from this PH model. The other parametric such as log-ormal, log-logistic etc. models ca be derived from the hazard fuctio. (Kalbfleisch ad Pretice, 2002). The AFT model i case of hazard fuctio ca be expressed by λ(t/x) = exp(β x) λ 0 (exp(β x) t) for all t.. (2)

4 314 Riku Saikia ad Maash Pratim Barma for example : I case of Expoetial AFT model, The hazard fuctio of expoetial model is Here the hazard fuctio is costat. λ(t) = λ, λ > 0 Now from the equatio (2), the hazard fuctio for Expoetial AFT model is give by The coditioal desity fuctio is λ(t; x) = λ exp(β x) f(t; x) = λ exp(β x) exp( λt exp(β x)). (3) Let Y=logT ad y = α + ε, ad α = logλ the f(e y ) = e α exp (β x)exp [ e α e y exp(β x)] = e α exp (β x)exp ( e y α exp(β x)) = e α exp (β x)exp [ e ε exp(β x)] Y = α β x + ε. (4) Where ε has a extreme value distributio. From this equatio (3),it is see that the covariates act multiplicatively with respect to the hazard fuctio Cox (1972).It is see that from the equatio (3) that the log-liear form i the equatio(4) ca be obtai ad this model is called the AFT model. The aother represetatio of the relatio betwee failure time ad explaatory / respose variable is the liear relatioship betwee the logarithm of survival time ad icludes a error term which is assumed to follow a specific distributio such as expoetial, weibull, log-ormal, log-logistic etc. The geeral log-liear represetatio of AFT model for ith idividual is give as logt i = μ + β 1 x β p x p + σε i (5) Where logt i represets the log-trasformed survival time, x 1,.. x p are the explaatory variables with the coefficiets β 1,.. β p ; ε i, is the residual or uexplaied variatio i the log- trasformed survival times, that is the deviatio of the values of logt i from the liear part of the model. ε i assumes a specific distributio ad μ is the itercept ad σ is the scale parameters respectively. The iitial step i fittig a AFT model is that for eachε i, there is a correspodig distributio for T i. If the ε i, has a extreme value distributio the T i follows the weibull distributio. Agai if ε i follows ormal distributio the the T i follows logormal distributio etc.

5 A Review o Accelerated Failure Time Models 315 The survival fuctio of T i ca be expressed by the survival fuctio of ε i. S ( t) P( T t) i P(logT log t) P( x... x log t) 1 1 log t x... x 1 1 p p ( i ) log t x... x 1 1 p p i ( ) S ( t) S ( t) i P S i p p i Agai the cumulative hazard fuctio of T i is H i (t) = log (s i (t)) = logs εi (t) = H εi (t) For example : Log-ormal AFT model : If ε i has a stadard ormal distributio the T i is log-ormally distributed. The desity fuctio of ormal distributio is f εi (ε) = 1 2π exp (log t μ β 2 1x 1. β p x p ) /2 σ The survival fuctio of ormal- distributio is S εi (ε) = 1 Φ(ε) The distributio fuctio of ormal distributio is The cumulative hazard fuctio is Ad the hazard fuctio is Φ(ε) = log t μ β 1x 1 β p x p. σ H εi (ε) = log {1 Φ(ε)} h εi (ε) = f εi(ε) S εi (ε) I this way the log-ormal AFT form ca be derived.

6 316 Riku Saikia ad Maash Pratim Barma 3. REVIEW ON AFT MODELS : The theory of AFT model has bee a field of active research for last few decades as this model has vast applicability i the reliability theory ad idustrial experimets as well as survivorship data. I the followig sectio, it is tried to have a bird s eye view o a lio s share of such developmets. Pike (1966) i his work suggest two distributioal forms developed the statistical methodology ad discussed the likelihood method of estimatio for the Weibull distributio. Johso ad Kotz (1970) discussed about the estimatio of parametric models ad icludig expoetial, weibull, log-ormal, gamma, log-logistic. Lawless(1982) preseted ad illustrated the statistical methods for modelig ad aalyzig life time data. He used the term log-locatio scale model istead of Accelerated failure time model because the logarithm of the survival time was cosidered. Vaderhoef (1982) i his work applied a parametric method which was preseted for the aalysis of curret status data based o AFT model ad maximum likelihood estimatio. I the paper it seemed that the Weibull distributio model provided a well fittig model. Cox ad Oakes (1984) showed that the Weibull distributio had both proportioal hazards ad accelerated failure time property ad Log-logistic distributio had proportioal odds ad accelerated failure time property.wei (1992) itroduced a o-parametric versio of AFT model, which did ot required the specificatio of a probability distributio for the survival data. Orbe et al. (2002) described that AFT model could be a iterestig alterative to the Cox PH model whe PH assumptio did ot hold. Implemetatio ad iterpretatio of the results of AFT was simple. He applied AFT to two real examples ad carried out a simulatio study ad AFT model lead to more precise results. Nardi ad Scheme (2003) compared Cox PH ad parametric models i three cliical trial studies maily performed at Viea Uiversity Medical School. They used Normal deviate residuals (Nardi,1999) to verify the parametric model assumptios. Their study showed that Weibull model was superior to other parametric models. Pourhoseigholi et al., (2007) compared Cox regressio ad Parametric models i the aalysis of the patiets with gastric carcioma ad foud that logormal model fitted better tha other models. Sayehmir et al.,(2008) studied progostics factors of survival time after hematopoietic stem cell trasplat i acute lymphoblastic leukemia patiets i Shariati Hospital, Tehara ad foud that Weibull AFT model was superior to Cox PH model. Qi (2009) compared PH ad AFT models ad suggested that the Cox PH model may ot be appropriate i some situatios ad that the AFT model could provide a more appropriate descriptio of the data. Ravagard et al.,(2011) compared Cox PH model ad the parametric models i studyig the legth of stay i a Tertiary Teachig Hospital i Tehra ad showed that Gamma AFT model was best fitted for that data. Khael et al., (2012) idetified the importat progostic factors of Acute Liver Failure patiets i Idia by applyig AFT models ad foud that Log-ormal

7 A Review o Accelerated Failure Time Models 317 AFT was well fitted for that data i compariso to log-logistics. Valliayagam et al., (2014) compared parametric models icludig Weibull ad Log-ormal with Cox PH model for boe- marrow trasplatatio data ad Log-ormal model was better fit tha the other models. Nawumbei et al., (2014) compared Cox PH ad AFT models i HIV/TB Co-ifectio survival data ad revealed that Gamma model was well fitted to the Co-ifectio data. 4. ESTIMATION OF ACCELERATED FAILURE TIME MODEL: AFT models are fitted by usig maximum likelihood estimatio (MLE) method. The likelihood of observed survival times t 1,.. t is L = {f i (t i )} δ i{s i (t i )} 1 δ i Where f i (t i ) ad s i (t i ) are the desity fuctio ad the survival fuctio for the ith idividual at the time t i respectively. δ i is the evet idicator for the ith idividual δ i = { 1, if the ith observatio is evet 0, if the ith observatio is cesored Now S i (t i ) = S εi (z i ) Where, z i = logt μ β 1x 1 β p x p σ The log-likelihood fuctio is give by logl = σt i δ i{fεi (z i )} δ i{s εi (z i )} 1 δ i logl = σ i log (σt i ) + δ i log{f εi (z i )} + (1 δ i )log{s εi (z i )} Where z i = logt μ β 1x 1.β p x p ad MLE of (P+2) ukow parameters, μ, σ ad σ β 1,.. β p are foud by maximizig the log-likelihood fuctio usig Newto Raphso procedure.

8 318 Riku Saikia ad Maash Pratim Barma Estimatio of parameters for the Weibull AFT model without covariates. The desity fuctio of Weibull AFT model is Where < y <. f(y; μ, σ) = 1 exp(y μ) /σ exp ( exp (y μ)/σ) σ If the ε i follows the extreme value distributio the T i follows the Weibull distributio. The survival fuctio ad the desity fuctio of the extreme value distributio are respectively The likelihood fuctio is The log-likelihood fuctio is s(z i ) = exp ( exp(z i )) log(s(z i )) = exp (z i ) log(s(z i )) z i = exp (z i ) f(z i ) = exp(z i ) exp ( exp(z i )) logf(z i ) = z i exp (z i ) log(f(z i )) z i = 1 exp (z i ) L = 1 σ [ {f i(t i )} δ i{s i (t i )} 1 δ i ] logl = δ i log ( 1 σ ) + δ i log {f εi (z i )} + (1 δ i ) log{s εi (z i )} logl = rlogσ + δ i log {f εi (z i )} + (1 δ i ) log{s εi (z i )} = rlogσ + δ i log { exp(z i ) exp ( exp(z i ))} + (1 δ i ) log {exp ( exp(z i ))}

9 A Review o Accelerated Failure Time Models 319 = rlogσ + ( δ i z i exp (z i )). (6) Differetiatig this equatio (6) with respect to parameters μ ad σ oe ca get the parameters values of Weibull AFT model. I this way the parameters of the AFT models ca be estimated. 5. MODEL CHECKING: To check the appropriate distributio of the AFT model various methods have bee used. They are such as Akaike Iformatio Criterio ( AIC), Baysia Iformatio Criteria( BIC), etc. (i) AIC: To compare various semi-parametric ad parametric models Akaike Iformatio Criterio (AIC) is used. The AIC is proposed by Akaike (Akaike, 1974). It is a measure of goodess of fit of a estimated statistical model. For the model i this study, AIC is computed as follows AIC = 2(log likelihhod) + 2(P + K) Where P is the umber of parameters ad K is the umber of coefficiets (excludig costat) i the model. For P=1, for the expoetial, P=2, for Weibull, Log-logistic, Logormal etc. The model which as smallest AIC value is cosidered as best fitted model. (ii) BIC: The Baysia Iformatio Criteria (BIC) is give by Schwarz (Schwarz, 1978). It is computed as follows BIC = 2(log likelihood) + (P + K) log () Where P is the umber of parameters i the distributio, K is the umber of coefficiets ad log() is the umber of observatios. The distributio which has the lowest BIC value is cosidered as best fitted model. (iii) Cox Sell-Residual: the Cox Sell Residuals ca by used to check the goodess of fit of the model which was give by Cox ad Sell ( Cox ad Sell, 1968). The Cox- Sell residual for the ith idividual with observed time t is defied as r ci = S i (t) = S εi ( logt μ β 1x 1.β p x p ) σ where the parameters are already defied above. The Cox- Sell residual ca applied i ay of the parametric AFT model.

10 320 Riku Saikia ad Maash Pratim Barma 6. CONCLUSION: From the above discussio, it is observed that the AFT models which is widely used i aalyzig idustrial data ca also provide a good alterative of the Cox PH model. If the correct form of the parametric model is kow, the the AFT model has a vast scope for the future researchers. This model is comparatively easy to iterpret. A umber of research studies have already bee coducted by usig the AFT model which provides fruitful results. There is a scope for further developmet of the AFT model. Till ow, AFT model has bee fitted for distributios like expoetial, weibull, log-ormal, log-logistic, gamma etc. Oe ca use other distributios such as skew-ormal, geeralized expoetial etc. to model survival data. I AFT model, the depedet variable is log of the survival time T. The depedet variable ca also be used by other strictly icreasig fuctio which is also the further scope of the study. REFERENCES: [1] Akaike H(1974).: A New Look at the Statistical Model Idetificatio. IEEE. Trasactio ad Automatic Cotrol AC-19, pp [2] Cox D.R., ad Sell E.J(1968).: A Geeral Defiitio of Residuals (with discussio), Joural of the Royal Statistical Society, A.. [3] Cox, D.R(1972). : Regressio Models ad Life Tables, Joural of the Royal Statistical Society, Series B, (Methodological),,Vol 34, No. 2, [4] Cox ad Oakes(1984): Aalysis of Survival Data, Chapma ad Hall, Lodo New York. [5] Kleibaum,G.David ad Klei, Mitchel(1996) : Survival Aalysis : A Self- Learig Text, Secod Editio,, Spriger, New York. [6] Khaal Prasad, Shakhar, Sreeivas,V. ad Acharya,K. Subrat (2014) : Accelerated Failure Time Models: A Applicatio i the Survival of Acute Liver Failure Patiets i Idia, Iteratioal Joural of Sciece of Research (IJSR), ISSN: ,Impact Factor(2012):3.358,Volume 3, Issue 6,Jue 2014, pp [7] Kalbfleisch, Joh D. ad Pretice, Ross L.(2002) : The Statistical Aalysis of Failure Time Data, Joh Wiley & Sos, Ic. [8] Lawless, J. F.(1982) : Statistical Models ad Methods for Lifetime Data Aalysis, Wiley, New York.

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