Biomedical Engineering Reference
In-Depth Information
is often convenient to work with the conditional hazard function of T given
X = x, h(tjx). The Cox or proportional hazards (PH) model
34
is the
most commonly used conditional hazard model. The PH model assumes
that the covariates act multiplicatively on the conditional hazard function
and is expressed as follows:
h(tjx) = h(t) exp(x); (2)
where h(t) is the conditional hazard function of T given X = 0, which is
called the baseline hazard function, and = (
1
;:::;
p
)
T
is a p-vector
of parameters with T denoting transposition. Because no particular func-
tional form of h(t) is assumed, the PH model (2) is referred as a semi-
parametric regression model. The parameter vector can be estimated by
maximizing the partial likelihood
34;35;153
. Kalbeisch and Prentice
79
de-
rived an exact expression for the partial likelihood to accommodate tied
observations. A number of approximation methods have been proposed by
Peto
112
, Breslow
23
, and Efron
42
, among others. The asymptotic properties
of the maximum partial likelihood estimator has been studied by Tsiatis
144
and Andersen and Gill
9
.
The PH model does not propose a direct relationship between fail-
ure time and covariates. In contrast, the accelerated failure time (AFT)
model
79;97;36
has an intuitive physical interpretation in which the eect of
covariate x is assumed to act multiplicatively on the failure time T or addi-
tively on the log failure time, log T, expressed as log T = x + with error
density f
(e). The AFT model with unspecied error distribution is called a
semiparametric accelerated failure time model semiparametric accelerated
failure time model and can be considered a semiparametric alternative to
the PH model. Let h(u) be the hazard function of T
= exp(). The con-
ditional hazard function can be written in terms of this baseline hazard
function as
h(tjx) = exp(x)h(t exp(x)); (3)
where exp(x) is referred to as the accelerated factor. In this model,
the role of covariates is to accelerate or decelerate the time to failure. The
Weibull, log-logistic, the log-normal, gamma, and inverse Gaussian distri-
butions have the AFT property. Among them, the Weibull distribution is
the only one that has both the PH property and the AFT property. See
[114, 118, 145, 150, 87, 88, 159, 55, 78] and [77] for semiparametric inference
procedures for the AFT model.
Hazard functions play a fundamental role in understanding and model-
ing failure time data. See [36] and [21] for a detailed discussion of the role of
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