Gehan's procedure, and its comparison against other procedures available for the Phase II trial, are discussed in the Phase II Studies Explained Page . This page provides support for the use of the program in the Sample Size for Phase II Study (Gehan's Procedure) Program Page and the tables in this page.

Gehan's is a preliminary study of a new treatment, to test whether the proportion of success from the treatment satisfies the level required to warrant further detailed study in a Phase II trial.

• Stage 1 :
  • Sample size for stage 1 (s1) is the number of cases to be used in stage 1. This depends on
    • The proportion of success required (p) is defined, a value between 0 (0%) and 1 (100%)
    • The power required, the ability to detect that proportion if it is there. Mathematically power = 1-β, where β is the probability of error in rejecting the new treatment
  • At the end of stage 1, after s1 cases are included, k = the number of successes found
    • If k=0 or k< a critical value where reaching the required proportion of success is unlikely, the treatment is rejected without proceeding to stage 2
    • If k> a critical value where reaching the required proportion of success is likely, the treatment is accepted without proceeding to stage 2
    • Otherwise, stage 2 of the procedure is required for decision making.
• Stage 2 :
  • The sample size in stage 2 is the number of additional cases to be used in stage 2 This depends on
    • The number of cases alr3eady used in stage 1 (s1)
    • The number of successes (k) found in stage 1
    • The acceptable probability of error for a decision to acceot the treatment (ε)
  • The total number of successes is the number of successes in stage 1 (k) + number of successes in stage 2 .
  • The required total number of successes for a decision to accept the treatment is n = p(s1+s2)
  • if the total number of successes reached n, then the trial can be terminated, with a decision to accept the treatment
  • If at the end of stage 2, the total number of successes had not reached n, then a decision to reject the treatment is made.

The program in the Sample Size for Phase II Study (Gehan's Procedure) Program Page produces the results of Gehan's 2 stage procedures.

• The entry data is a single column of 3 rows, where
  • Row 1 = ε, Probability of error for accepting the treatment for further study
  • Row 2 = Power(1-β), where β is the probability of error for rejecting the treatment for further study
  • Row 3 = p, the probability or proportion of successes required for accepting the treatment for further study
• The result are :
  • The sample size for stage 1
  • A table where each row is the parameters related to each value of k
    • k = the number of successes in stage 1
    • s2 = the sample size for stage 2
    • n = the number of successes required (combine stage 1 and 2) for the treatment to be accepted.

The tables on this page provides the results of calculations for Gehan's 2 stage prod=cdures

• Three levels of power (1-β) at 0.8, 0.9, and 0.95
• Two level of error (ε) at 0.05 and 0.1
• Required proportion of successes :
  • From 0.03 (3%) to 0.2 (20%) at 0.01 (1%) intervals
  • From 0.2 (20%) to 0.3 (30%) at 0.02 (2%) intervals
  • From 0.3 (30%) to 0.5 (50%) at 0.05 (5%) intervals
  • From 0.5 (50%) to 0.9 (90%) at 0.1 (10%) intervals
• The cells contains results
  • Blanks for impossible situations, where k>s1
  • R is the combination of s1 and k that allows the treatment to be rejected at the end of stage 1
  • A is the combination of s1 and k that allows the treatment to be accepted at the end of stage 1
  • Cells with numbers are combinations of s1 and k that From 0.3 (30%) to 0.5 (50%) at 0.05 (5%) intervals
  • From 0.5 (50%) to 0.9 (90%) at 0.1 (10%) intervals

References

Gehan E A (1961) The determination of the number of patients required in a preliminary and a follow-up trial of a new chemotherapeutic agent Journal of Chronic Diseases. 13:4, Pages 346-353

Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. Blackwell Science IBSN 0-86542-870-0 p. 255-256

The following is the code for Gehan's two stage procedure for Phase II studies. I have adapted the formulae as described in the text book by Machin et. el.. Those interested can copy, use, and modify these codes

"                      Phase II Gehan
Gehan E A (1961) The determination of the number of patients required in a preliminary and 
    a follow-up trial of a new chemotherapeutic agent  Journal of Chronic Diseases. 13:4, Pages 346-353
Machin D, Campbell M, Fayers, P, Pinol A (1997) Sample Size Tables for Clinical Studies. Second Ed. 
    Blackwell Science IBSN 0-86542-870-0 p. 255-256 
"
# Data
errorI = 0.05   # alpha Type I Error
power = 0.8     # power 1-beta
pa = 0.4        # success rate above which treatment is accepted

# Sample Size for Stage 1
s1 = ceiling(log(1 - power) / log( 1 - pa))

#  Functions for Stage 2
FindGX <- function(n, r, m)
{
  x = 0
  for(s in 0 : r)
  {
     f = choose(n, s)
     p = n - s
     x = x + f * m^s * (1 - m)^p
  }
  x
}

FindGMu <- function(n, r) #iterations to find the correct mu
{
   mL = 0.0001;
   mR = 0.9999;
   m = 0.5;
   xL = FindGX(n, r, mL) - 0.25
   x = FindGX(n, r, m) - 0.25
   xR = FindGX(n, r, mR) - 0.25;
   if((xL * x)>0 & (xR * x)<0)
   {
     mL = m 
   }
   else 
   {
      if((xL * x)<0 & (xR * x)>0) 
      { 
         mR = m; 
      } 
      else 
      { 
         return(0) 
      }
  }        
  while(abs(x)>0.0001)
  {
      m = (mL + mR) / 2
      xL = FindGX(n, r, mL) - 0.25
      x = FindGX(n, r, m) - 0.25
      if((xL * x)>0) 
      { 
        mL = m
      } 
      else 
      { 
        mR = m 
      }
  }
  m
}

# Main Program for Stage 2
# Create vectors to contain results
r1 <- 0 : ssiz1            # number of successes in stage 1
ssiz2 <- 0 : ssiz1         # maximum samples size for stage 2
minSuc <- 0 : ssiz1        # minimum number of successes for tmt to be accepted
ssiz2[1] = "Reject"        # r1=0 treatment rejected no stage 2
minSuc[1] = "Reject"
ssiz2[ssiz1 + 1] = "Accept" # r1=all treatment accepted no stage 2
minSuc[ssiz1 + 1] = "Accept"

for(i in 2 : ssiz1-1)
{
   mu = FindGMu(s1, i)
   s2 = ceiling(mu * (1 - mu) / errorI^2 - s1)
   tot = s1 + s2
   ns = ceiling(pa * tot)
   ssiz2[i+1] = s2
   minSuc[i+1] = ns
}

# Result display

# Sample Size Sate 1
s1
# Stage 2 : r1=number of successes in stage 1
#           ssiz2 = sample size in stage 2
#           minSuc = minimum number of successes required in stage 2 for tmt to be accepted

outDF <- data.frame(r1,ssiz2,minSuc)
outDF