calculateEqualBounds: Computes the exact Pocock bounds
Description
This funtion computes the exact Pocock bounds instead of approximating them by spending function.
The Spending function gives us an approximate Pocock-Design.
To compute the exact Pocock Bounds we will do according to the following pattern:
(1st)we give the bounds with all bounds are equal. As starting value we are using the
mean of the bounds computed by our approximating Pocock spending function. I figured out that
in almost every case this is a quite good approximation so far.
(2nd)we compute the probability according to our equal bounds, as we would do, if we would compute
probability to given bounds.
(3rd)we use Newton Iteration to adjust the bounds in every Iteration until we get the appropriate alpha
Usage
calculateEqualBounds(targetAlpha, provisionallyBounds, n, t2)
Arguments
targetAlpha
is the desired overall size.
provisionallyBounds
is first approximation of bounds by spending function Pocock-Type
n
is the number of interim analyses
t2
is the second or information time scale, for covariances
Value
upperBoundsexact Pocock bounds to given target alpha
Details
We use an algorithm which is called in German "Sekanten-Verfahren".
This one is based on the Newton Iteration but uses secants here
considering we cannot derive our function.
The calculation obeys following pattern whereby $x_{k+1}$ converges against
the value we are searching for.
$x_{k+1} = x_{k}-x_{k-1} {f(x_{k})-f(x_{k-1})}*f(x_{k})$