Biomedical Engineering Reference
In-Depth Information
Sometimes correlations exist within each level of nesting. These could cause
biases and affect variances of parameter estimation [10, 11]. Therefore, tests
need to be done to evaluate the goodness of the estimated survival function.
There are two popular ways to test the model. One is to use 1/2 or 2/3 of
the time scale in the survival data to determine the parameters, and then
use the whole data set to examine the model; another is to use the whole
data set to set up the model, then using resample methods to check the
model. Neural networks are increasingly being seen as an addition to the
statistics toolkit which should be considered alongside both classical and
modern statistical methods. It has been pointed out in [16] many different
ways that classification networks have been used for survival data. In this
study, due to the lack of patient data, we propose a neural network model
to simulate the patients' survival pattern and use the neural network to
generate a long list of “virtual data” to test the survival model.
The remainder of the paper is organized as follows: In Section 2, we
give a description for the survival model. We first introduce the conception
of hazard function and survival function as well as their relationship. We
then outline the method of proportional hazard model and propose and
justify the exponential form for baseline hazard function. In Section 3, we
discuss the parameter estimation by statistical methods including maximum
likelihood estimation (MLE) and non-linear least square estimation (LSE).
We also introduce the idea and conception of the neural network and set up
the proper neural network by MATLAB programs for testing. In Section 4,
we present the computational result with actual patient data. Discussions
and conclusions are given in Section 5.
2. Description of Model
2.1. Survival Function and Hazard Function
Following the notations in Actuarial Mathematics [4], we let T be a nonneg-
ative random variable representing the failure time of an individual in the
population. Assume T is distributed with the probability density function
(pdf) f ( t ), then the cumulative distribution function (cdf) is
t ]= t
0
F ( t )= Pr [ T
f ( z ) dz
(2 . 1)
giving the probability that the event has duration t .The survival
function , S ( t ), is defined as the complement of the c.d.f., that is
F ( t )=
t
S ( t )= Pr [ T<t ]=1
f ( z ) dz.
(2 . 2)
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