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Application: In phase II clinical trial, randomized design is proposed to establish the sample size for the study to obtain the treatment with greatest response rate for further / phase III study.
Procedure:
- Enter
a) value of p, the lowest response rate among all treatments
b) value of D, the difference between the best treatment and the other treatments
c) value of k, the number of treatment arms
if click the button “Calc B” next step, go 1 (d)
and then step 3, otherwise go step 2 only
d) value of n, the number of patients in each treatment arm
- Click the button “Calc A” to obtain
a) value of n, the number of patients in each treatment arm
b) value of Prob, the probability of correctly selecting
the best treatment.
- Click the button “Calc B” to obtain
a) value of Prob, the probability of correctly selecting
the best treatment.
Theory:
The
treatment selected is that with the largest observed response rate. Foe selecting
among K treatments when the difference in true response rates of the best and
next best treatment is D, the probability of correct selection is smallest when
there is a single best treatment and the other K-1 treatments are of equal but
lower efficacy. If we stipulate that the response rate of the worst treatment
equals a specified value p, then the probability that the best treatment
produces the highest observed response rate is:
(1)
where
is the cumulative distribution function for a binomial
distribution with success parameter p+D and n patients and f(i) is the
probability that the maximum number of success observed among K-1 inferior
treatment is i. Thus
If
there is a tie among the treatments for the largest observed response rate, we
shall assume that one of the tied treatments is randomly selected. Hence, in
calculating the probability of correct selection, we add to expression (1) the
probability that the best treatment was selected after being tied with 1 or
more of the other treatments for the greatest observed response rate. This
probability is:
(2)
where
, b denotes the binomial probability mass function. The
quantity g(i,j) represents the probability that exact j of the inferior
treatments are tied for the largest number of observed responses among the K-1
inferior treatments, and this number of responses is i. The factor 1/(j+1) in
expression (2) is the probability that tie among the best treatment and the j
inferior treatments is randomly broken by selecting the best treatment.
The
probability of correct selection is the sum of expressions (1) and (2). For
specified values of p, D and K, the value of n, number of patients per
treatment, is determined to provide a probability of correct selection equal to
that desired (P).
Discussion:
There are some advantages to the randomized
Phase II design of two or more new agents.
1.
Randomization
helps ensure that patients are centrally registered before treatment starts. Establishment
of a reliable mechanism to ensure patient registration prior to treatment is of
fundamental importance for all clinical trials.
2.
Comparing
to independent phase II studies, the differences in results obtained for the
two agents will more likely represent real differences in toxicity or antitumor
effects rather than differences in patient selection, response evaluation, or
other factors.
Example:
Suppose that three schedules (k=3) of administration are to be studied and the expected baseline response rate is 20% (p=0.2). With 44 (n=44) patients per schedule, we have probability 0.9 (Prob=0.90) of selecting the schedule that has a true response rate of 20%+15%=35% (D=0.15).
Reference: Simon, Wittes and Ellenberg (1985) Cancer Treatment Reports 69: 1375-1381