Biomedical Engineering Reference
In-Depth Information
R
p
for x
i
ranging over the subsetX
i
of R, i = 1;:::;p. Let 1q <1
and G a q-dimensional linear space of functions on [0;1)Xsuch that
g(jx) is bounded on [0;1) for g2G. LetB
1
;:::;B
q
be a basis of this space.
They used the following linear combination of linear splines and their tensor
products to develop the following HARE model:
q
X
(tjx; ) =
j
B
j
(tjx);
(38)
j=1
for (tjx), where = (
1
;:::;
q
)
T
. The method is similar to the MARS
technique
53
. It can be seen from (38) that the approach to modeling (tjx)
does not depend on the validity of the basic assumption of the PH model
that the conditional log-hazard function is an additive function of time
and the vector of covariates. One can obtain the ML estimate
^
of by
maximizing the log-likelihood corresponding to the HARE model (38) for
(tjx)
Z
X
n
y
i
`() =
i
(y
i
jx
i
; )
exp[(ujx
i
; )]du
;
(39)
0
i=1
which is a concave function on R
q
. Consequently, the corresponding ML
estimates of the conditional log-hazard function, hazard function, sur-
vival function, and density function are given by ^ (tjx) = (tjx;
^
),
h(tjx) = h(tjx;
^
) = exp((tjx;
^
)), S(tjx) = exp(
R
t
0
h(ujx)du), and
f(tjx) = h(tjx) S(tjx), respectively.
To resolve the problem of choosing the linear space G (i.e., the selection
of the nal model), Kooperberg, Stone, and Truong
82
proposed an auto-
matic procedure involving the ML method, stepwise addition using Rao
statistic, stepwise deletion using Wald statistic, and the Bayes information
criterion
126
. If the selected space is the space of constant functions, then
the HARE model (38) has q = 1,B
1
(tjx) = 1, and (tjx; ) =
1
, which
means that the conditional distribution of T given X = x is exponential
with mean exp(
1
) independent of x. If none of the basis functions of the
selected space depends on both t and x, then the HARE model (38) is a PH
model, hence the HARE models include PH models as a subclass. However,
if any of the basis functions in the nal model depends on both time and
a covariate, a PH model might not be appropriate. Therefore, the presence
or absence of interaction terms between time and covariates in the nal
model can be regarded as a check on the proportionality of the underlying
conditional hazard model.
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