Biomedical Engineering Reference
In-Depth Information
4.3
Regression Analysis with Proportional Odds Model
This section introduces the regression analysis for current status with propor-
tional odds model. It is well-known that the proportional odds model provides
a good alternative when PH models are not appropriate for use. The propor-
tional odds model is defined as
logit[S(tjZ)] = logit[S
0
(t)] + Z
0
;
(4.2)
where logit(x) = log[x=(1 x)] for 0 < x < 1, and S
0
(t) = S(tjZ = 0) is the
baseline survival function. Let (t) = logit[S
0
(t)]; then
exp( + Z
0
)
1 + exp( + Z
0
)
:
S(tjZ) =
Thus the likelihood is proportional to
Y
[1 S(C
i
)]
i
[S(C
i
)]
1
i
L(;)
=
i=1
i
1
i
Y
exp( + Z
0
)
1 + exp( + Z
0
)
1
1 + exp( + Z
0
)
=
i=1
exp[(1
i
)((C
i
) + Z
0
)]
1 + exp((C
i
) + Z
0
)
Y
=
:
i=1
To maximize the likelihood, it is equivalent to maximize the log-likelihood,
which is given by
X
f[(1
i
)((C
i
) + Z
0
)] log[1 + exp((C
i
) + Z
0
)]g:
l(;) =
i=1
As discussed in Section 4.2, the diculty in estimating eciently is to deal
with the innite-dimensional nuisance parameter (t), the baseline log-odds
function. Here, we can also use the approach in Section 4.2, in which we use
a step function as the approximation of (t). Suppose there are k distinct
observed time points with 0 < t
1
< t
2
< ::: < t
k
; we can define the function
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