Biomedical Engineering Reference
In-Depth Information
FIGURE 12.3: Illustration of possible shapes of
T
(c;q;Y
E
); considered as
a function q with fixed c = 0.3 and Y
E
= 0.5.
To account for Interval-censoring of Y
E
; given observation interval I
E
= (l; u]
[0; 1], we denote
E;T
(I
E
;y
T
jc;q;) = Pr(l < Y
E
u;Y
T
= y
T
jc;q;):
(12.9)
This is the relevant probability when it is only known that the ecacy event did not
occur by time l and did occur by time u: In this case, infusion is stopped at the end
of the interval, u; so the probability of toxicity is
T
(u;c;q;
T
): It follows that
E;T
(I
E
;y
T
jc;q;) = Pr(l < Y
E
ujc;q;)f
TjE
(y
T
ju;c;q;)
= fF
E
(ujc;q;)F
E
(ljc;q;)g
T
(u;c;q;)
y
T
if
T
(u;c;q;)g
1y
T
;
(12.10)
with F
E
computed from Equations (12:2); (12:3); and (12:5) and
T
specified by
Equations (12:6): The fact that Pr(Y
E
= 0) > 0 and Interval-censoring due to
sequential evaluation of Y
E
give the partition fI
0
;I
1
; ;I
M
g of [0, 1], with I
0
= f0g;
and the likelihood takes the form
L(Yjc;q;) =
p
0
(c;q;)
T
(0;c;q;)
Y
T
if
T
(0;c;q;)g
1Y
T
1(Y
E
=0)
Y
E;T
(I
m
; 1 j c;q;)
Y
T
E;T
(I
m
; 0 j c;q;)
1Y
T
1(Y
E
2I
E;m
)
(12.11)
m=1
f1 F
E
(1jc;q;)g
T
(1;c;q;)
Y
T
if
T
(1;c;q;)g
1Y
T
1(Y
E
>1)
:
For example, if there are M = 8 intervals of 15 minutes each over 120 minutes, then
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