The preceding overview of the AFT and TR models contains the connection between the two types of models. The connection lies in the running time transformation r(t|z). We extend the traditional notion of an AFT model by allowing the following more general formulation:
Pr(T >t|z) =S(t|z) =S0[r(t|z)]. (3)
Here again S0(r) is a baseline survival function and r(t|z) is a non-decreasing function of calendar time t that is dependent on the covariate vector z. We require r(0|z) =0 for all z. The transformation r(t|z)encapsulates what we mean by the acceleration and deceleration of time.
The generalized AFT model in (3) is also a TR model if S0(r)is a first-hitting- time distribution for some baseline process{Y0(r)}, baseline boundary setB0and running time r(t|z). The AFT model will not be a TR model if S0(r)is not of the FHT variety. From our experience, it is difficult to conceive of an AFT model that is scientifically meaningful which lies outside the TR family. On the other hand, the AFT model in (3) is a proper subset of the TR family. TR models extend the AFT model (3) whenever the parameters of the baseline survival function S0(t)are made to depend on the covariate vector z. In this extension, we show this dependence by the notation S0(t|z).
A large variety of practical AFT models are created by appropriate choices for the baseline survival function S0(r)and the running time r(t|z). The following examples of TR models that are also AFT models illustrate the range of possibilities:
1. Poisson process. Consider a Poisson process{N0(r)}with a baseline hazard rate λ0>0. The time until occurrence of the first event in the baseline process has survival function S0(r) =exp(−λ0r). Under the simplest acceleration multiplier A=exp(zγ), the running time function becomes r(t|z) =t/A and then the survival function of the AFT model takes the form:
S(t|z) =S0[t/A] =exp
−tλ0exp(−zγ) .
Observe that the baseline hazard rateλ0 may be viewed as the intercept term in the covariate regression function through the correspondenceλ0=exp(γ0).
Conventional methods of statistical inference for survival data can be used to estimate the vector of regression coefficients γ and the baseline hazard rate λ0. It is noteworthy that this AFT model is also a proportional hazards model with a family of constant hazard functions λ0exp(−zγ). If the time to, say, the kth event in the Poisson process is of scientific interest, then the baseline survival function S0(r)is an Erlang distribution of order k with scale parameter λ0(a special gamma distribution). Substituting r(t|z) =t exp(−zγ)for r in the baseline survival function produces an AFT family with a gamma error structure.
2. Wiener process. Consider the FHT for a Wiener diffusion process {Y(r)}
starting at Y(0) =y0>0 and having a boundary at zero. Let the baseline case be defined by the mean parameterμ0<0 and a unit variance parameter. The baseline survival function S0(r)has an inverse Gaussian form that depends on parameters y0andμ0. Again, for simplicity, if the running time function r(t|z)is taken as t exp(−zγ), then the survival function of the corresponding AFT model is given by
S(t) =S0
t exp(−zγ) .
In this scenario, the running time function r(t|z)characterizes the same AFT and TR model. If, however, the boundary of the process were made to depend on the covariate vector z then the TR model becomes broader than an AFT model. Our discussion of practical issues and specific case illustrations later will draw out this important distinction.
The general AFT model in (3) is not new. There is a large literature dealing with survival models having collapsible, composite, and alternative time scales that are essentially of the form shown in (3). See, for example, Oakes [26], Kordonsky and Gertsbakh [16], Duchesne and Lawless [5], and Duchesne and Rosenthal [6]. What is new in our development is the placement of this general class of AFT models within the context of threshold regression and the elucidation of some practical variants of the model that may be valuable in medical applications.
Variants of AFT Model
To give a flavor of the variety of running time transformations that are available for AFT model (3), we present a few illustrations next.
1. Multiplier AFT model. The multiplier version of the AFT model in (2) is a special case of the general formation in (3) as may be seen if we define r(t|z) = t/exp(zγ). The multiplier version is simple in that it postulates a constant rate of progression of illness or disease, with the rate varying with z.
2. Change-point AFT model. An important variant of the preceding model is one in which acceleration engages at a point in time or change-point c. A simple version of this model is:
r(t|z) =
t if t≤c(z),
c(z) + [t−c(z)]exp(zγ)if t>c(z). (4) This version makes the change point c a function of the covariate vector z and is a special case of the exposure AFT model that follows.
3. Exposure AFT model. In many applications, an individual is exposed during different intervals to toxins or other harmful influences, in varying intensities, that can accelerate the onset of a medical endpoint. The following exposure version of AFT model (3) is useful in this context:
r(t|z) =
∑
Jj=1
αj(z)tj, where t=∑Jj=1tj,αj(z)≥0, andα1(z) =1. (5) Here tj is the time an individual is exposed to toxin j during calendar interval (0,t). Toxin 1 is taken as the reference exposure type. The reference type might be, for instance, a non-toxic environment. Coefficientsαj(z)determine the accelerator or decelerator effect associated with exposure to toxin j, relative to
the reference type. Covariates z modify theαjparameters. The equation t=∑jtj
is an accounting equation that ensures that every moment of calendar time t is spent in one of the J exposure types. See Lee et al. [22,23] for an example of (5) in the context of the exposure of railroad workers to diesel exhaust and the onset of lung cancer. This exposure model is an extension of the simple multiplier model (2) becauseαj(z)tj allows for a different multiplier (αj) for each type of exposure (and each vector z).
4. Stochastic AFT model. Running time in some applications will proceed like a stochastic process{R(t)}that has non-decreasing sample paths. The function r(t|z)in (3) would then be such a sample path. Refer to Lawless and Crowder [19]
for an application to crack propagation in which the gamma process serves as a running time (and degradation process).
Further Adaptations of AFT Model
1. Cure rate. A cure rate version of the AFT model in (3) allows for the possibility that an individual will be cured of the disease that would bring on the medical endpoint or be immune to it. A cure rate is accommodated in the AFT model if the baseline survival function is given a probability mass p at infinity, with 0<p<1, as follows:
S0(r) =p+ (1−p)S∗(r). (6) Here S∗(r)denotes the baseline survival function of those individuals that are susceptible to the medical endpoint. A careful look at this AFT formulation, however, shows that it may have limited practical application. As acceleration (or deceleration) of time affects only the running time r, formula (6) shows that all individuals in this formulation must have the same cure rate p. The basic issue is that acceleration, pure and simple, only modifies the time scale and an immune or cured individual would not be influenced by its effect. In contrast, TR models in general do not have this restriction. Thus, the cure-rate case is one type that distinguishes the AFT model from more general TR models.
2. Initial disease progression. In some investigations, individuals do not enter the study at the same stage of disease progression in the sense that each individual has already experienced some “wear and tear” at the outset. This initial progression may be interpreted as an initial running time r0(z), which varies with the covariate vector z. In this case, general AFT model (3) takes the form of the following conditional survival function:
Pr(T>t|z) =S(t|z) =S0[r0(z) +r(t|z)]
S0[r0(z)] . (7) The conditioning in this model is necessary because the individual has experi- enced running time r0(z)without yet experiencing the medical endpoint (i.e., the survival distribution S0(·)is left truncated at r0).