Accelerated Failure Time Model (II)

Transcription

1 Accelerated Falure Tme Model (II) Purpose:. Introducton to the accelerated falure tme model 2. Estmatng parametrc regresson models wth PROC LIFEREG n SAS Data descrpton: The data conssts of 432 males nmates who were released from Maryland state prsons n the early 970s (Ross et al. 980). These men were followed for one year after ther release, and the dates of any arrests were recorded. We ll only look at the frst arrest here. The WEEK varable contans the week of the frst arrest after release. The varable ARREST has a value of for those arrested durng the one-year follow-up, and t has a value of 0 for those who were not. Only 26% of the men were arrested. The data are rght-censored (type I) so that all the censored cases have a value of 52 for WEEK. The other covarates are FIN: f the nmate receved fnancal ad after release; otherwse, 0. AGE: age n years at the tme of release. RACE: for black; 0 for otherwse. WEXP: f the nmate had full-tme work experence before ncarceraton; 0 otherwse. MAR: f the nmate was marred at the tme of release; 0 otherwse. PARO: f the nmate was released on parole; 0 otherwse. PRIO: number of convctons an nmate had pror to ncarceraton. Let s open the data set n SAS. DATA recd; SET "C:\40.64\recd";

2 One of nterests of survval analyss s to understand the relatonshp between tme to falure and other covarates measured at the studed subjects. Ths can be done by usng regresson models. The regresson model we wll talk about n ths lab s a parametrc model. The LIFEREG procedure produces estmates of parametrc regresson models wth censored survval data usng the method of maxmum lkelhood. In recent years parametrc model has been eclpsed by semparametrc regresson model (Cox s model), whch uses a method known as partal lkelhood. The reasons for sem-parametrc model s popularty wll become apparent n the next several labs. The class of regresson models estmated by PROC LIFEREG s known as the accelerated falure tme (AFT) model. We have ntroduced the most general form of AFT model n the last lab. What PROC LIFEREG actually estmates s a specal case. Let T be a random varable denotng the falure tme for the th subject, and let x, x 2,, x p be the values of p covarates for that same subject. The model s then log T = β + β x + h+ β x + σε () 0 p p where ε s a random dsturbance term, and β 0,, β p, and σ are parameters to be estmated. Note that the only dfferences between the model n () and the usual lnear regresson models are that there s a σ before ε the and that the dependent varable s logged. The σ can be omtted, whch requres that the varance of ε be allowed to be dfferent from. But t s smpler to fx the varance of ε at and let σ change. Ths notatonal strategy could be used for lnear regresson models. As for the log transformaton of T, ts man purpose s to ensure that predcted values of T are postve. If there are no censored data, we can readly estmate ths model by ordnary least squares. Smply generate a new varable, Y = log T, and use the lnear regresson model wth Y as the dependent varable. Ths process yelds the best lnear unbased estmates of coeffcents, wthout dstrbuton assumpton on ε. If ε s normal, the OLS estmates wll also be maxmum lkelhood estmates and wll have mnmum varance among all estmators, both lnear and nonlnear. But survval data typcally have at least some censored observatons, and these are dffcult to handle wth OLS. Alternatvely, we can use MLE wth dfferent dstrbuton assumpton on ε. For each of the dstrbuton of ε, there s a correspondng dstrbuton for T. Dstrbuton of ε Extreme value (2 parameters) Extreme value ( parameter) Log-gamma Logstc Normal Dstrbuton of T Webull Exponental Gamma Log-logstc Log-normal 2

3 Note that all AFT models are named for the dstrbuton of T rather than the dstrbuton of ε or log T. The reason for allowng dfferent dstrbuton assumptons s that they have dfferent mplcatons for the shapes of hazard functon. We wll brefly ntroduce 3 models, log-normal, exponental, and Webull models n ths lab.. The log-normal model To estmate the log-normal model, we specfy PROC LIFEREG data=recd; MODEL week*arrest(0)=fn age race wexp mar paro pro / dst=lnormal; As we mentoned, f there s no censored data ths model wll gve exactly the same estmates as n lnear regresson model. For ths data, the results are as follows: The LIFEREG Procedure Model Informaton Data Set WORK.RECID Dependent Varable Log(week) Censorng Varable arrest Censorng Value(s) 0 Number of Observatons 432 Noncensored Values 4 Rght Censored Values 38 Left Censored Values 0 Interval Censored Values 0 Name of Dstrbuton LNORMAL Log Lkelhood Algorthm converged. Analyss of Parameter Estmates Standard Varable DF Estmate Error Ch-Square Pr > ChSq Label Intercept <.000 Intercept fn age race wexp mar paro pro Scale Normal scale β e has the nterpretaton of the estmated rato of the expected survval tmes for two groups. The scale s an estmate of the σ of equaton (). For some dstrbutons, changes n the value of ths parameter can produce qualtatve dfference n the shape of the hazard functon. For the log-normal model, however, changes n σ merely compress or stretch the hazard functon. 3

4 2. The exponental model Ths model specfes that ε has a standard extreme-value dstrbuton, and constrans σ =. Under exponental assumpton, equaton () s equvalent to * * logλ ( t) = β + β x + + β x (2) 0 * p p where β j * = -β j for all j. The change n sgns makes ntutve sense. If the hazard s hgh, then events occur quckly and survval tmes are short. Let s ft ths model usng PROC LIFEREG PROC LIFEREG data=a; MODEL week*arrest(0)=fn age race wexp mar paro pro / dst=exponental; The LIFEREG Procedure Model Informaton Data Set WORK.RECID Dependent Varable Log(week) Censorng Varable arrest Censorng Value(s) 0 Number of Observatons 432 Noncensored Values 4 Rght Censored Values 38 Left Censored Values 0 Interval Censored Values 0 Name of Dstrbuton EXPONENT Log Lkelhood Algorthm converged. Analyss of Parameter Estmates Standard Varable DF Estmate Error Ch-Square Pr > ChSq Label Intercept <.000 Intercept fn age race wexp mar paro pro Scale Extreme value scale Lagrange Multpler Statstcs Varable Ch-Square Pr > ChSq Scale <.000 4

5 The coeffcent for AGE s twce as large n the exponental model, and ts p-value declnes from.08 to.0. The coeffcent for PRIO ncreases somewhat n magntude, and ts p-value also goes down substantally. The p-value for FIN ncreases to slghtly above the.05 level. The SCALE parameter σ s forced equal to.0. The last lne s a df test for H 0 : σ =. Here the null hypothess s rejected, ndcatng that the hazard functon s not constant over tme. It mples that we mght need some more complex dstrbuton assumpton n terms of the shape of hazard. 3. The Webull model The Webull model s a slght modfcaton of the exponental model. We retan the assumpton that ε has a standard extreme-value dstrbuton, but we relax the assumpton that σ =. When σ >, the hazard decreases wth tme. When 0.5 < σ <, the hazard s ncreasng at a decreasng rate. When 0 < σ < 0.5, the hazard s ncreasng at an ncreasng rate. And when σ = 0.5, the hazard functon s an ncreasng straght lne wth an orgn at 0. Under Webull assumpton, equaton () s equvalent to * * logλ ( t) = α logt + β + β x + + β x (3) 0 * p p where β j * = -β j /σ for all j and α = /σ -. PROC LIFEREG data=recd; MODEL week*arrest(0)=fn age race wexp mar paro pro / dst=webull; The LIFEREG Procedure Model Informaton Data Set WORK.RECID Dependent Varable Log(week) Censorng Varable arrest Censorng Value(s) 0 Number of Observatons 432 Noncensored Values 4 Rght Censored Values 38 Left Censored Values 0 Interval Censored Values 0 Name of Dstrbuton WEIBULL Log Lkelhood Algorthm converged. Analyss of Parameter Estmates Standard Varable DF Estmate Error Ch-Square Pr > ChSq Label Intercept <.000 Intercept fn age race wexp mar

6 paro pro Scale Extreme value scale Compared wth the exponental model, the coeffcents are all somewhat attenuated. But the standard errors are also smaller, so the ch-square statstcs and p-values are hardly affected at all. If we convert the coeffcents to the format n equaton (3) by changng sgn and dvdng by the estmate of σ, we get coeffcents much closer to the coeffcents n exponental model. Snce the SCALE, estmate of σ, s between 0 and, we conclude that the hazard s ncreasng at a decreasng rate. Although we have already dscussed the log-normal model and appled to the recdvsm data, we have not yet consdered the shape of ts hazard functon. Unlke the Webull model, the log-normal model has a nonmonotonc hazard functon. The hazard s 0 when t = 0, and ncrease to a peak and then declnes to 0 as t goes to nfnty. The log-normal model can not be expressed as a proportonal-hazard-lke form as exponental or Webull model. As we wll see later that log-normal model s not nested n exponental or Webull models. 4. Model selecton We have seen that the AFT model encompasses a number of submodels that dffer n the assumed dstrbuton for T. Clearly, we need some way of decdng between the models,.e., the shapes of hazard. Here we wll ntroduce the lkelhood-rato test for comparng nested models. A model s sad to be nested wthn another model f the frst model s a specal case of the second. More precsely, model A s nested wthn model B f A can be obtaned by mposng restrctons on the parameters n B. For example, the exponental model s nested wthn both the Webull and the standard gamma models. Let s calculate the lkelhood rato test for the recdvsm data. H 0 : exponental model H : Webull model The log-lkelhood for exponental model s and the log-lkelhood for Webull model s The lkelhood-rato Ch-square statstc s -2( (-39.38)) = 2.9 wth degree of freedom. Clearly, we can reject the null hypotheses, that s to say Webull model fts the data better than exponental model. As we know, the generalzed gamma model s more general than Webull, so we can further ft generalzed gamma model on our data and test f Webull model s enough. Other technques to dscrmnate between dfferent models nclude the graphc methods, whch can be found n reference books. REFERENCE: P. D. Allson, Survval analyss usng the SAS system: a practce gude, SAS company,