How to approach Homework 2, problem 1
Let the possible doses be denoted
d
1
, . . . , d
I
. At dose
d
i
, let the probability of a dose
limiting toxicity be
p
i
and write
q
i
= 1

p
i
.
Let’s create some additional notation. Deﬁne
U
i
to be the probability that you escalate
from
d
i
to
d
i
+1
. Note that
U
I
= 0. And let
B
i
be the probability that you go back down
from
d
i
, given that you got to
d
i
somehow (that is, the conditional probability that you
declare
d
i
not safe given that you got there somehow). Note that if you ﬁnd that the lowest
dose is not safe, you’d declare there to be no safe dose.
Note that the ﬁrst time you treat 3 patients at dose
d
i
, there are three possibilities:
•
If you get 2 or 3 toxicities, you go back down right away, and you never do this dose
again. If this was the lowest dose, then there is no safe dose.
•
If you get 0 toxicities, you go up to dose
d
i
+1
unless you are already at dose
d
I
, in
which case you do three more here.
•
If you get 1 toxicity, you give 3 more patients the same dose.
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 Winter '06
 Tsodikov
 Conditional Probability, Probability, Dose, Notation

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