Biomedical Engineering Reference
In-Depth Information
occurred before time t and is expressed as
Pr(tT < t + tjtT)
t
=
f(t)
h(t) =
lim
t!0
+
S(t)
;
(1)
where f(t) = lim
t!0
+
Pr(tT < t + t)=t =dS(t)=dt is the
probability density function of T. The hazard function is also called the
hazard rate, the age-specied death rate, the conditional failure rate,
the instantaneous death rate, the intensity rate, the mortality intensity,
or the force of mortality. It follows from (1) that the survival func-
tion may be written as S(t) = exp (H(t)), where H(t) =
R
t
0
h(u)du
is referred to as an integrated or cumulative hazard function. There-
fore, the probability density and survival functions can be completely char-
acterized by the hazard function. For a homogeneous population, para-
metric models for failure time include the exponential, Weibull, extreme
value, gamma, log-normal, log-logistic, and generalized gamma distribu-
tions, among others. The exponential, Weibull, and gamma distributions
are special cases of the generalized gamma distribution. For nonparametric
methods, the Kaplan-Meier(product-limit) (K-M) estimator
80
is the most
commonly used nonparametric maximum likelihood (ML) estimator of the
survival function; the Nelson-Aalen (N-A) estimator
103;1
, also called Alt-
shuler's estimator
7
, estimates the cumulative hazard function, which is an
alternative nonparametric ML estimator of the survival function. In the ab-
sence of censoring, the K-M estimator is an empirical survival function. The
asymptotic properties of the K-M estimator have been studied by Breslow
and Crowley
24
, Foldes and Rejto
52
, and Wellner
151
, among others. Padgett
and McNichols
109
reviewed nonparametric density estimation from censored
data.
The failure times of individuals usually depend on characteristics that
are also referred to as explanatory variables, covariates, regressors, or pre-
dictor variables. The explanatory variables may include demographic vari-
ables such as age, sex, and race; physiological variables such as weight,
height, and blood pressure; and behavioral factors such as smoking history
and dietary habits. One can use parametric regression models such as the
exponential, Weibull, and log-normal models to include explanatory vari-
ables. See [79] and [90] for detailed discussions of parametric failure time
models.
Let X = (X
1
;:::;X
p
) be a p-vector of covariates assumed to be time
independent for simplicity of discussion throughout this chapter. To explore
the possible relationship between the censored failure time and covariates, it
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