Checking Goodness-of-fit of the Accelerated Failure Time Model for Survival Data

The Accelerated Failure Time model presents a way to easily describe and interpret survival regression data. It approaches the data differently than the widely used and well described Cox proportional hazard model, by assuming proportional effect of the covariates on the log-failure time rather than on the hazard function. In order to check if observed data behave according to the model, its assumptions must be verified. In present work we therefore study goodness-of-fit procedures based on both classic regression and martingale theory. On simulated data we try to estimate empirical properties of these tests for various situations. Introduction Let us observe survival data which represent time which passes from beginning of an experiment until some kind of failure, for instance when testing durablity of industrial parts or the life expectation of patients since the first signs of a fatal disease. We suppose that the data may be incomplete in a way that some objects may be removed from the observation prior to reaching the failure, which we call right censoring. We want to model the dependence of the time to failure on available covariates. The most widely used and described is the Cox proportional hazard model, an alternative is presented by the Accelerated Failure Time model (AFT). To check which model suits the data best, goodness-offit tests may be applied. In this work, we focus on studying various testing techniques for the AFT model. A test based on classic regression is introduced, we also study and try to further develop some techniques based on counting process theory. Notation Let T ∗ i , i = 1, ..., n, be the real failure times, Xi = (Xi1, ...,Xip) T covariates, Ci censoring times, Ti = min(T ∗ i , Ci) the times of the end of observation and ∆i = I(T ∗ i ≤ Ci) noncensoring indicator. Suppose T ∗ i and Ci independent for all i. We observe independent data (Ti,∆i,X i), i = 1, ..., n. We assume T ∗ i to be continuous, denote Fi(t) = P (T ∗ i ≤ t) their distribution function, fi(t) density, Si(t) = 1 − Fi(t) the survival function, αi(t) = limhց0 P (t ≤ T ∗ i < t+ h|T ∗ i ≥ t)/h = fi(t)/Si(t) the hazard function and Ai(t) = ∫ t 0 αi(s)ds the cumulative hazard. The data may be represented as counting processes, denote Ni(t) = I(Ti ≤ t,∆i = 1) the noncensored failure indicator, Yi(t) = I(t ≤ Ti) the at-risk indicator, their sums N•(t) = ∑n i=1 Ni(t) and Y•(t) = ∑n i=1 Yi(t), intensities λi(t) = Yi(t)αi(t) and cumulative intensities Λi(t) = ∫ t 0 λi(s)ds. All functions and processes are on an interval t ∈ [0, τ ], where τ < ∞ is some point beyond the last observed survival time. Often we work with the martingale approach. It can be shown, that under the model assumptions, Λi(t) are the compensators of corresponding processes Ni(t) with respect to the filtration Ft = σ {Ni(s), Yi(s),Xi, 0 ≤ s ≤ t, i = 1, ..., n} (Fleming & Harrington, 1992). Therefore Mi(t) := Ni(t)− Λi(t) are Ft-martingales (Doob-Meier decomposition). 189 WDS'10 Proceedings of Contributed Papers, Part I, 189–194, 2010. ISBN 978-80-7378-139-2 © MATFYZPRESS