**Help Aids **Top

**Application: **The main
goal of Phase II clinical trials is to identify the therapeutic efficacy of new
treatments. The formulation and implementation of a Bayesian design is proposed
for planning and monitoring a single-arm Phase II clinical trial.

**Procedure**:

- Enter

a) value of W_{90}, the width of
the 90% probability interval from the 95^{th} to 5^{th}
percentiles

b) value of μ_{S}, the mean
response rate of standard therapy

c) value of d_{0}, the fixed targeted improvement

d) value of c_{E}, the concentration
parameter of experimental treatment

e) value of Nmax, the maximum number of patients to be recruited

f) value of Numstage,
the maximum
number of stages for patient recruitment**.**

**g) the type of **prior distribution used, enthusiastic,
skeptical or flat. Details please click here.

- Click the button “Calculate” to obtain

a) value
of n_{i}, the number of patients to be recruited in stage i

b) value of L_{i}, the lower decision boundary in stage i

c) value of U_{i}, the upper decision boundary in stage i

**Assumption:**

1.
Assume
that a Phase I trial of the experimental treatment regimen has been completed,
from which the proper dosages yielding acceptable toxicity have been
established.

2.
A
maximum of K=1,2 or 3 tests is adopted in the
calculator, since more frequent testing would not seem to be necessary in most
Phase II trials.

3.
Due
to the very limited number of patients who are eligible for a given Phase II
study, and also to the large number of treatment regimens generally awaiting
Phase II trials, the sample size *N*
for a given trial is frequently fixed in advance.

**Theory:**

__Hypotheses__

The null and alternative hypotheses for the Bayesian design are as follows:

H_{0}: p_{E}
£ m_{S}
(experimental treatment does not warrant further investigation) versus

H_{1}: p_{E}
³ m_{S} + d_{0} (experimental treatment warrants
further investigation),

where p_{E} is the true response probabilities of the
experimental treatment, m_{S} is the mean response rate of the standard treatment, and d_{0} is the fixed targeted improvement.

The Bayesian
procedure for designing Phase II clinical trials uses the concept the prior
evidence can be summarized as prior probabilities which are modified by the
experimental data to yield revised beliefs or posterior probabilities. A
clinician can summarize the previous information and knowledge about a
treatment response probability q in order to construct a probability
distribution for q. This distribution is called a prior
distribution because it represents the relative likelihood that the true value
of q lies in various regions of some specified parameter space
prior to the observations of any values of the data.

Let q_{S} and q_{E} represent the patient response rates for the standard and
experimental treatments, respectively. Let X_{n} be the number of
treatment responses after n patients have been recruited. The Bayesian design
recruits n_{min} = 10 subjects and applies the decision rules discussed
below to determine whether the trial should be terminated at the first stage or
if recruitment should continue to stage n + 1 and another patient added to the
study. This procedure is repeated until n_{max} = 65 subjects have been
recruited. The value n_{min} = 10 is chosen because it is the
conventional minimum sample size and the value of n_{max} = 65 is
chosen because it is often impractical to have a larger sample size for Phase
II studies.

__Prior Distributions____Top__

There are several choices of prior distributions to be used
for the response probability q_{E} of the experiment treatment. The prior distributions used
are beta(a, b) distributions with p.d.f. P{q} µ q^{a-1} (1-q)^{b-1}, where 0 < q < 1 and
a, b > 0.

All prior distributions for the standard treatment can be
parameterized in terms of the mean response rate m_{S} = a_{S} / (a_{S}+b_{S})
and W_{90} = the width of the 90% probability interval from the 95^{th}
to 5^{th} percentiles, where S represents the standard therapy. The clinician is asked to specify m_{S} and W_{90} so that a_{S}
and b_{S} may be found.

All prior distributions for the experimental treatment can
be parameterized in terms of the mean response rate m_{E} = a_{E} / (a_{E}+b_{E})
and the concentration parameter c_{E} (= a_{E}+b_{E}),
where E represents the experimental treatment. The clinician must specify m_{E} and c_{E}
so that a_{E} and b_{E} can be found. Note that 2 £ c_{E} £ 10 so that the dispersion of the prior
is no larger than that of the uniform (0, 1) distribution and no smaller than
that of the posterior corresponding to a small pilot study of the experimental
treatment.

The enthusiastic prior distribution
assumes that the mean response rate of the experimental treatment is equal to a improvement over that of the standard treatment, i.e. m_{E} = m_{S} + d_{0}. Enthusiastic
priors are used in situations where the experimental treatment is thought not
to be an improvement over the standard treatment, or when the treatment is
showing negative results. If the
treatment fails to prove promising using an enthusiastic prior, then there is
overwhelming evidence to support the decision that the treatment is not
promising.

The skeptical prior distribution
assumes that the mean response rate of the experimental treatment is equal to
that of the standard treatment, i.e. m_{E} = m_{S}. Skeptical priors
are used in situations where the experimental treatment is believed to be
promising, or when the treatment is showing positive results. If the results prove the experimental
treatment is promising, then this would be extremely convincing evidence to support
the decision that the treatment is promising. Hence, the Bayesian design can
incorporate the skeptical opinions about a treatment to yield overwhelming
evidence about the treatment.

The flat prior distribution
assumes m_{E} = 0.50 and c_{E} = 2. The flat prior is used in situations
where little is known about the efficacy of the experimental treatment. The rationale for using uninformative
priors is that inferences are dominated by the observed data rather than
depending on a clinicians knowledge about the
experimental treatment. The flat
prior has a uniform (0, 1) distribution, so it treats all true treatment
response probabilities equally. The
weakly uninformative prior distribution assumes m_{E} = 0.50 and c_{E} =
10. This may also be used in
situations where little is known about the experimental treatment. The weakly uninformative prior has
dispersion like that of the posterior corresponding to a small pilot study of
the experimental treatment.

The likelihood is from a
binomial (n, q_{E})
distribution with p.d.f.

P{X_{n} | q_{E}} µ q_{E}^{Xn} (1-q_{E}) ^{n - Xn}.

The posterior p.d.f. for q_{E} is a beta (q_{E}; a_{E} + X_{n}, b_{E}
+ n - X_{n}) distribution with p.d.f.

P{q_{E} | X_{n}} µ P{q_{E}} P{X_{n} | q_{E}}= q_{E}^{Xn + aE - 1} (1 - q_{E})^{n
- Xn + bE – 1}.

__Decision Boundaries__

The stopping rules are based
on probabilities from the posterior distributions of treatment differences. Let
p_{U} and p_{L} be defined as
predetermined probabilities corresponding to the upper and lower decision
boundaries respectively. For this, p_{L} =
0.05 and p_{U} = 0.95.

U_{n} = smallest
integer x such that P{q_{E} > q_{S} | X_{n} = x} ³ p_{U} (experiment treatment declared promising), and

L_{n} = largest
integer x such that P{q_{E} > q_{S} + d_{0} | X_{n} = x} £ p_{L} (experiment treatment declared not promising).

The value of U_{n} is the smallest value of x at which the probability
that the response probability of the experimental treatment is greater than
that of the standard treatment given the observed number of treatment successes
is at least p_{U}. Any value greater than U_{n}
would give a probability greater than p_{U}, which implies that the
treatment is promising. The value of L_{n} is the largest value of x at
which the probability that the response probability of the experimental
treatment is greater than that of the standard treatment plus some targeted
improvement given the observed number of treatment successes is at most p_{L}. Any value less than L_{n} would
give a probability less than p_{L}, which
implies that the treatment is not promising. These criteria are used to generate
decision boundaries and stopping criteria.

Note that P{q_{E} > q_{S} + d_{0} | X_{n} = x}, like the
probability of any random variable with a continuous distribution, can be
expressed in terms of an integral.

P{q_{E} > q_{S} + d_{0} | X_{n} = x out of n
patients} = 1 - P{q_{E} £ q_{S} + d_{0} | X_{n} = x out of n
patients}

= E[1 - *ò*_{0}^{q}^{s + }^{d}^{o} ¦(z;a_{E} + x, b_{E} + n - x)dz]

= E1 - F(z;a_{E} + x, b_{E} + n - x) |_{0}^{q}^{s + }^{d}^{o}]

= E[1 - {F(q_{S} + d_{0};a_{E} + x, b_{E} + n - x) -
0}]

= *ò*_{0}^{1 - }^{do} [1 - F(z + d_{0};a_{E} + x, b_{E} + n
- x)] ¦(z; a_{S}, b_{S})dz,
where

E[·] is the expected value with respect to the distribution of q_{S},

¦(·; a_{S}, b_{S})is the
p.d.f. of the prior distribution for the standard treatment, and

F(·;a_{E} + x, b_{E} + n - x)is the c.d.f. of the
posterior distribution for the experimental treatment.

__Stopping Criteria__

At stage n, the decision rules are:

·
If X_{n} ³ U_{n}, then terminate the trial and reject H_{0},
i.e. declare treatment promising

·
If _{n} £ L_{n}_{A}, i.e. declare treatment not
promising

·
If L_{n} < X_{n}
< U_{n} and n < n_{max}, then continue to stage n + 1

The trial is
declared inconclusive if X_{n} has not hit a decision boundary by n = n_{max}. This procedure also provides for
continual sequential monitoring.

**Example:**

Since Phase I
studies indicate that the experimental regimen is somewhat more toxic than the
standard, in phase II clinical trial, it is decided to set *W _{90}*=0.2,

**Reference**: Simon, Thall (1994) *Biometrics
*50: 337-349