Biomedical Engineering Reference
In-Depth Information
3.3. Partially Linear PH Models
In practice, if without loss of generality the rst p
1
covariates of x are
assumed to have linear eects and the functional forms of the other covari-
ate eects are unknown and smooth, one can use the model
P
p
1
j=1
j
x
j
+
P
p
j=p
1
+1
g
j
(x
j
), considered by Gray (1992) for (x) in model (15) instead
of the additive model (27), where the g
j
s are unknown smooth functions. To
simplify this discussion, we will assume that the functional form of the pth
covariate eect is unspecied. Gray
64
considered the partially linear
model
(x;
1
;g) = x
1
1
+ g(x
p
)
(29)
for (x), where x
1
= (x
1
;:::;x
p1
), the rst p1 covariates for the linear
terms,
1
is the vector of the associated parameters, and g is an unknown
smooth function that gives the pth covariate eect on the outcome. Engle,
Granger, Rice, and Weiss
43
were the rst to consider the partially linear
model. The conditional hazard model
h(tjx) = h(t) exp( (x;
1
;g))
(30)
is referred to as a partially linear or semiparametric additive PH model.
Let B
1
(x
p
);:::;B
K+4
(x
p
) be the cubic B-spline basis for the space of cubic
splines with the prespecied interior knots
1
;:::;
K
. Gray
64
parameter-
ized g as
K+2
X
g(x
p
) =
0
x
p
+
k
B
k
(x
p
);
(31)
k=1
where
0
;
1
;:::;
K+2
are unknown parameters. Because the space of cu-
bic B-splines includes the constant and linear functions, the constant is
absorbed into h(t), and the linear term is specied separately in (31),
only K + 2 of the B-spline basis functions are used in (31). Therefore,
any two of the K + 4 B-spline basis functions could be dropped, provided
the resulting parameterization is of full rank. Let #
1
= (
1
;:::;
K+2
)
T
and
#
2
= (
0
; #
1
)
T
. To investigate the eects of the covariate x
p
, Gray consid-
ered two hypotheses about g: (i) the hypothesis of no eect, g(x
p
) = 0, i.e.,
#
2
= 0. (ii) the hypothesis of linear eect of x
p
, g(x
p
) =
0
x
p
, i.e., #
1
= 0.
Constructing tests for hypothesis (ii) is exactly the same as that for
hypothesis (i), so we will conne our discussion to testing hypothesis (i).
To estimate
1
and g, Gray
64
maximized the penalized log partial
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