REMARK ON THE INFLATION FACTOR
The paper includes a method for adjusting the interim sample size
recalculation for the uncertainty in the interim variance estimate,
through the application of an inflation factor (IF). The IF is
computed as described in the right column of page 552 of the paper.
This involves solving a nonlinear equation to find the value of
\omega that fixes the expression "Power" at the desired level.
As the paper was coming out, I heard a simpler way of computing the
IF in a presentation by Steven Julious of Glaxo-Smith-Kline at the
ISCB 2002 meeting in Dijon. Julious's method relies on a result of
Ellison (1964, JASA, Theorem 2) on the expectation of \Phi(Z+cY),
where \Phi is the normal cdf, Z is a normal rv, Y is the square root
of a scaled chi-square rv, the rv's Z and Y are independent, and c is a
constant. In our case, Z is a degenerate rv with constant value z_\alpha.
It comes out that to do the correction we do with the IF, one may just
use the standard sample size formula with the following substitution:
in place of (z_\alpha+z_\beta) one substitutes
TINV(power,df,z_\alpha),
where TINV is the inverse noncentral t-distribution with degrees of
freedom df and noncentrality parameter z_\alpha, and "power" is
the desired power.
In other words,
IF = \omega^2 = [TINV(power,df,z_\alpha)/(za+zb)]^2.
I compared the IF values resulting from this method to those resulting
from the method based on solving the nonlinear equation and they are very
close.
The function TINV is available in SAS. Thus, given the relevant df,
the IF can be computed essentially immediately. For the two-sample
problem with a normally-distributed endpoint, the df is the total
number of interim observations in both groups minus two. For the case
of the mixed linear rmodel, we propose to compute the df using a
Satterthwaite approach, described in detail in the paper.