Biomedical Engineering Reference
In-Depth Information
The quantity in (2.18) is the expected amount of toxic material absorbed
during the interval [0
,t
] and present in the body at time
t
,whichleadstoa
possible suggestion of a function format for the hazard for cancers caused
through exposure to factors. Suppose the baseline hazard for lung cancer
patients is proportional to the quantity in the following equation:
h
0
(
t
)=
a
−
−
b
(1
exp(
bt
))
.
(2
.
19)
Defining the cumulative baseline hazard function,
H
(
t
), by integrating
0
h
(
t
) and applying boundary condition that
h
(0) = 0 yield:
0
0
H
0
(
t
)=
t
0
a
b
[
x
1
b
(1
h
0
(
x
)
dx
=
−
−
exp(
−
bt
))]
.
(2
.
20)
3. Statistics Methods and Neural Network
3.1.
Maximum Likelihood Estimation
Maximum likelihood estimation begins with writing a mathematical ex-
pression known as the likelihood function of the sample data. Roughly
speaking, the likelihood of a set of data is the probability of obtaining
that particular set of data, given the chosen probability distribution model.
This expression contains the model's unknown parameters. The values of
these parameters that maximize the sample likelihood are known as the
Maximum Likelihood Estimates, or MLE. Maximum likelihood estimation
is a totally analytic maximization procedure. It applies to every form of
censored or multi-censored data, and is even able to be used across several
stress cells and estimate acceleration model parameters at the same time
as life distribution parameters. Moreover, MLE and likelihood functions
generally have very desirable large sample properties because they: (a) be-
come unbiased minimum variance estimators as the sample size increases,
(b) have approximate normal distributions and approximate sample vari-
ances that can be calculated and used to generate confidence bounds, and
(c) likelihood functions can be used to test hypotheses about models and
parameters. Although it has many good attributes, MLE has an important
drawback, that is, with a small number of failures (say, less than 30, and
oftentimes, less than 50), MLE may be heavily biased and the large sample
optimality properties do not apply.
If
X
is a continuous random variable with
pdf
f
(
x, β
1
,β
2
,
···
,β
p
)
,
(3
.
1)
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