Biomedical Engineering Reference
In-Depth Information
the hazard function. In addition, Aalen and Gjessing
2
considered the shape
of the hazard function from a process point of view. In Sec. 2, we will dis-
cuss some nonparametric regression techniques used to estimate hazard or
log-hazard functions. In Sec. 3, we will review nonparametric modeling of
covariate eects in the PH model when the linear eects of the covariates
are not appropriate and some semiparametric and nonparametric regres-
sion models for the conditional hazard model (i.e., hazard function with
covariates) as alternatives to the PH model. A discussion is given in Sec. 4.
2. Smooth Estimation of Hazard Function
One can estimate a hazard function in at least two ways. The rst is to
estimate the density and cumulative distribution functions and then use
their estimates to yield an estimate of the hazard function
148;4
. However, the
shape of the estimate of the hazard function can exhibit serious departures
from the functional form of the hazard function. More specically, when
the estimate of the density function is smooth and unimodal and when the
hazard function has a smooth monotone increasing form, as it does with
a gamma distribution, the estimate of the hazard function can still have a
major peak and major valley in the middle of the distribution
20
.
The second way is to estimate the hazard function directly. Once the
hazard function is estimated, one can obtain the estimates of the density
and cumulative distribution functions. The primary advantage of estimat-
ing the hazard function directly is that it can simplify the process when
constraints are placed on the form of the estimate. Therefore, we will re-
view some nonparametric regression techniques used to directly estimate
the hazard or log-hazard functions in this section. A review of estimation
of the hazard function with nonparametric methods was given by Singpur-
walla and Wong
128
and Padgett and McNichols
109
. Wu
154
discussed issues
of smoothing empirical hazard functions.
2.1. Kernel-based Estimation
Let Y = min(T;C), where T denotes the failure time, C denotes the cor-
responding censoring time, and T and C are assumed to be independent.
Let = I(TC) be a censoring indicator for I() being the indica-
tor function. Let (Y
i
;
i
), i = 1;:::;n, be a sample of independent and
identically distributed (i.i.d.) random variables, each having the same dis-
tribution as (Y; ). Let (y
i
;
i
) denote the observed data, where
i
= 1 if
y
i
= min(t
i
;c
i
) = t
i
; 0, otherwise.
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