ABSTRACT
Let x(·) = (x0(·), ..., xm(·))T ,
be a possibly time-varying and multidimensional explanatory variable; here x0(t) ≡ 1 and x1(·), ..., xm(·) are univariate explanatory variables. Under the AFT model the survival function under x(·) is
Sx(·)(t) = S0
(∫ t 0
r(τ)dτ ) . (5.1)
If the explanatory variables are constant over time then the model (5.1) is written as
Sx(·)(t) = S0 (r(x)t) . (5.2) The function r is parametrized in the following form:
r(x) = e−β T z, (5.3)
where β = (β0, · · · , βm)T is a vector of unknown parameters and z = (z0, · · · , zm)T = (ϕ0(x), · · · , ϕm(x))T
is a vector of specified functions ϕi, with ϕ0(t) ≡ 1. Under the parametrized AFT model the survival function under x(·) is
Sx(·)(t) = S0
(∫ t 0
e−β T x(τ)dτ
) , (5.4)
and xj(·) (j = 1, . . . ,m) are not necessarily the observed explanatory variables. They may be some specified functions ϕj(x). Nevertheless, we use the same notation xj for ϕj(x). If the explanatory variables are constant over time then the model (5.4) is
written as Sx(t) = S0
( e−β
T x t ) , (5.5)
and the logarithm of the failure time Tx under x may be written as
ln{Tx} = βTx+ ε, where the survival function of the random variable ε does not depend on x and is S(t) = S0(ln t). Note that in the case of lognormal failure-time distribution the distribution of ε is normal and we have the standard multiple linear
of the random able
R = ∫ T(x·) 0
e−β T x(τ)dτ
is parameter-free with the survival function S0(t) Let us discuss the choice of the functions ϕi.
