1.2 Problem Statement
2.4.3 Accelerated Failure Time Model
The Accelerated Failure Time Model (AFTM) is a parametric regression anal- ysis which provides an alternative to the popular PHM (Qi, 2009). The two models differ by how they assume the covariates affect the system/component hazard. Komárek et al. (2005) explain that the AFTM establishes a direct relationship between the time to failure and the covariates, and not a rela- tionship between the covariates and the hazard of a system/component like the PHM. Thus, any AFTM assumes that the covariates considered have the effect of accelerating or decelerating the survival time of equipment.
The survival function of a certain piece of equipment is represented by R0(x),
which can be seen as the control group. This survival function is related to the survival function of the same equipment under the accelerated life conditions by R(x) = R0 x θ , (2.4.18)
where θ is a function that incorporates the explanatory variables (covariates). This function then denotes the total effect of the covariates; it is known as the acceleration factor Nachlas (2005). The acceleration factor generally has the form θ(z) = exp(β0z) where z denotes a vector of covariates. The hazard
function of the control group h0 is related to the equipment being considered
by h(x) = 1 θh0 x θ , (2.4.19)
Anderson and Keiding (2006) then provide the survival time on a logarithmic scale as log T = µ + log θ + σ. This equation can then be rewritten in a log-linear form as Equation 2.4.20 and is the most general form of the AFTM; log T = µ + β0z + σ. (2.4.20)
Martinussen and Scheike (2006) and Komárek et al. (2005) explain that β0 is a
set of regression parameters and is a residual error term with an unspecified distribution in the semi-parametric model. The exponentiated value of the parameter coefficients (exp(β)) are known as the time ratio and has a physical meaning. Carstens (2012) state that the time ratio can be used to explain the failure time characteristic of an item, where a time ratio larger than one implies that an earlier failure is more likely while a time ratio less than one implies the opposite, as illustrated in Figure 2.9. In an AFTM, the residual error () is a random variable and its distribution must be assumed for a parametric model, where the intercept is denoted by µ and σ is the scale parameter.
0 50 100 150 200 250 300 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Probability of failure TR < 1 TR = 1 TR > 1
Figure 2.9: Effect of time ratio value.
The survival and hazard functions corresponding to the log-linear regression become conditional functions because the covariate vector z has a definite effect. They are presented as given by Zelterman and Lin (2004);
Ri(xi|zi) = R0 xi exp(β0zi) , (2.4.21) and hi(xi|zi) = 1 exp(β0zi) h0 xi exp(β0zi) (i = 1, ..., n). (2.4.22) Parametric AFTM
The semi-parametric extensions of the AFTM would estimate the regression parameters but will then not be able to predict the survival times and are, therefore, not considered here. In the non-parametric case, the regression
parameters cannot be estimated and is also not covered. The survival function of T (the survival time) can be expressed by the survival function of the residual error (). Qi (2009) provides this relation as
R(x) = R log x − µ − β0z i σ . (2.4.23)
Therefore, the distribution of has a corresponding distribution of T , corre- sponding distributions are provided in Table 2.2. The Weibull distribution is Table 2.2: Corresponding distributions of T and , adapted from Qi (2009).
T distribution family distribution family Exponential 1 Parameter, extreme value
Weibull 2 Parameter, extreme value
Log-logistic Logistic
Log-normal Normal
Gamma Log-Gamma
used for the survival time T and is seen to yield an extreme value distribution for the random variable . This extreme value distribution is used to represent the limiting distributions for the minimum of a large collection of random ob- servations; in this case, it is used to represent the random error . The survival function for can thus be given by:
R = exp(− exp()). (2.4.24)
Therefore Equation 2.4.23 becomes:
R(x) = exp − exp −µ − β 0z i σ x1σ . (2.4.25)
The hazard function for the Weibull AFTM can be written as
hi(xi|zi) = 1 σx 1 σ−1exp −µ − β 0z i σ . (2.4.26)
This model is generally not presented in this form since the model assumes a direct relation between the covariates and the failure time. The AFTM written in its general form is as Equation 2.4.20 and when expanded can be written as log T = µ + β1z1+ ... + βjzj + σ. (2.4.27)
By exponentiating the coefficients the term exp(βj) is known as the time ra-
tio. The parametric AFTM can be fitted by using the maximum likelihood method. For a component with n observed data points, the likelihood function is presented by L(β, µ, σ) = n Y i=1 [fi(xi)]δi[Ri(xi)]1−δi. (2.4.28)
Qi (2009) explains the likelihood function and that it is for the ith subject at time xi and δi is an event indicator, where it is one at a failure and zero
otherwise. Taking the log of Equation 2.4.28 yields
log L(β, µ, σ) =
n
X
i=1
{−δilog[σxi+ δilog(fi(yi)) + (1 − δi) log(Ri(yi))]},
(2.4.29) where yi = (log(xi) − µ − β1z1i− ... − βjzji)/σ and j is the number of param-
eters. This function then estimates the values for j + 2 unknown parameters, µ, σ, β1, ..., βj.
A disadvantage of the AFTM is that although it has been known to model electronic components and is widely utilized in the manufacturing industry, it is unfamiliar to many, especially to the medical industry. This is where most survival analysis research has traditionally been done. Another disadvantage is that even though familiar methods are used to do the parameter estimation, some complex calculations are required to derive the formulae.
One of the advantages of the AFTM is that it can be used with any of the distribution families that are appropriate for survival analysis. AFTM also relates the influence of the covariates to the failure time directly, thus, making it easier to interpret the results obtained. Another important advantage of this model is that packages in software programs such as MATLAB, R and SAS are easily accessible and very easy to implement. The next model to be reviewed is the Additive Hazard Model.