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The LIFETEST Procedure |
Kernel-smoothed estimators of the hazard function are based on the Nelson-Aalen estimator
and its variance
. Consider the jumps of
and
at the event times
as follows:
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where =0.
The kernel-smoothed estimator of is a weighted average of
over event times that are within a bandwidth distance
of
. The weights are controlled by the choice of kernel function,
, defined on the interval [–1,1]. The choices of
are as follows:
uniform kernel
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Epanechnikov kernel
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biweight kernel
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The kernel-smoothed hazard rate estimator is defined for all time points on . For time points
for which
, the kernel-smoothed estimated of
based on the kernel
is given by
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The variance of is estimated by
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For , the symmetric kernels
are replaced by the corresponding asymmetric kernels of Gasser and Müller (1979). Let
. The modified kernels are as follows:
uniform kernel
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Epanechnikov kernel
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biweight kernel
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For , let
. The asymmetric kernels for
are used with
replaced by
.
Using the log transform on the smoothed hazard rate, the 100(1–)% pointwise confidence interval for the smoothed hazard rate
is given by
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where is the 100(1–
)th percentile of the standard normal distribution.
The following mean integrated squared error (MISE) over the range and
is used as a measure of the global performance of the kernel function estimator
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The last term is independent of the choice of the kernel and bandwidth and can be ignored when you are looking for the best value of . The first integral can be approximated by using the trapezoid rule by evaluating
at a grid of points
. You can specify
, and
by using the options MISEMIN=, MISEMAX=, and MISENUM=, respectively, of the HAZARD plot. The second integral can be estimated by Ramlau-Hansen (1983a, 1983b) cross-validation estimate
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Therefore, for a fixed kernel, the optimal bandwidth is the quantity that minimizes
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The minimization is carried out by the golden section search algorithm.
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Copyright © 2009 by SAS Institute Inc., Cary, NC, USA. All rights reserved.