Biomedical Engineering Reference
In-Depth Information
Usually, the Cox proportional hazard regression model is a very use-
ful tool to estimate the coecients in a linear combination of covariates
in survival analysis since both SAS PHREG procedure and SPSS Survival
Package perform regression analysis of the survival data based on the pro-
portional hazards model. However, because of the nature of proportional
hazard regression, neither software packages give an explicit function ex-
pression for the baseline hazard function
h
0
(
t
). In the next section, we will
justify an explicit function of the baseline hazard function
h
0
(
t
)andalso
estimate the parameters in
h
0
(
t
) using non-linear least square technique
based on the result obtained from the Cox regression for the survival func-
tion fitting the data set of lung cancer patients.
2.3.
Baseline Hazard for Lung Cancer Patients
Like any cancer, the exact reason why one particular person is diagnosed
lung cancer and another does not remains unknown. However, certain fac-
tors are strongly correlated with an increase in lung cancer, when groups
of patients are studied. By rank, these factors are listed below [13]:
(i) Tobacco Smoking or exposure to smoke
(ii) Carcinogen Exposures
(iii) Radiation Exposure
(iv) Miscellaneous Risks Factors, including old scars in the lungs.
The first three factors involve an interaction between the individual and
the environment. Presumably an individual is continuously exposed to and
absorbs certain levels of smoke, radiation, or some kind of toxic material
(like carcinogen) which then lead to lung cancer. Though a portion of the
absorbed toxic materials is discharged from the body, the cumulative effect
of retained toxins contributes to the individual's death [6].
For every given
τ
in [0
,t
] and the infinitesimal time element [
τ, τ
+
dτ
],
let the sum
δdτ
+
o
(
dτ
) be the probability that a unit of toxic material
is absorbed during [
τ, τ
+
dτ
] and the sum
νdτ
+
o
(
dτ
) be the probability
that a unit of toxic material in the body is discharged during [
τ, τ
+
dτ
].
Assuming that
δ
and
ν
are independent of time, then the probability that
an individual will absorb a unit of toxic material during [
τ, τ
+
dτ
] and will
retain it in his/her body up to time
t
is given by [6]
δdτ
exp
{−
(
t
−
τ
)
ν
}
.
(2
.
17)
Integrating (2.17) over all possible value of
τ
yields
t
δ
ν
[1
δ
exp
{−
(
t
−
τ
)
ν
}
dτ
=
−
exp
{−
νt
}
]
.
(2
.
18)
0
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