Demonstration
October 28, 2000
Logrank statistics
Suppose that X 1 , X 2 ,, X m and Y1 , Y2 ,, Yn are random
variables with distribution functions F (x) and G( y) . To
compare whether the two distribution functions are the
same, we use the following logrank statistic:
Define
R x (t )  no. of X values which are  t
R (t )  no. of X and Y values which are  t
m
 R  X   n  R x Y j  

Logrank   1  x i    
R X i   j 1  R Y ji  
i 1 
m  R  X R  X  
n  R Y R Y  
y
i
x
i
   y j x j 
Variance   
2
2
 j 1 

R X i 
R Y j 
i 1 



Logrank
Statistics 
Variance
The logrank statistic will assume large values if X
sample is smaller than the Y sample.
1. Suppose the F (x) is the exponential distribution with
mean 10 and G( y) is the exponential distribution with
mean 12. m  n  100 and Type-I error is   0.05 . What is
the power of the logrank statistics in this case?
2. Boundary: Haybittle: 3
Number of interim analyses: k=9
Time(s) of the k interim analyses: 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of Statistics: 1
rho(s) of the ih statistic(s): 1
tau(s) of the ih statistic(s): 0
the number of simulations in each iteration: 500
Input the type-I error: .05
Accrual rate: 300
Input 1 or 2 sided test, type 1 or 2: 2
Input the b in the repeated significance test: b should be larger than 0: 3
the number of intervals to specify the baseline survival function: 1
Input the right endpoint(s) of the J interval(s), in increasing order of
magnitude, where J is the number in the last input: 5
Input the J survival probabilities at the points specified in the last input:
The survival probabilities in between those points are interpolated using
the exponential distribution function:.5
Input the alternative survival function. Input the number of intervals
(>=1) to specify the alternative survival function: Up to 20 are allowed:
5
Input the I right endpoint(s) of interval(s) for specifying alternative
survival function, where I is the number in the last input: 1 2 3 4 5
the I survival probabilities: .96 .88 .80 .70 .60
Input the censoring survival function for the baseline (control)
group. the number of intervals :1
Input the right endpoint(s) of interval(s): 5
Input the survival probabilities:.933
Input the censoring survival distribution for alternative (treatment)
group. Input the number of intervals: 1
Input the right endpoint(s) of interval(s): 5
Input the survival probabilities: .937
Input the drop-in (out) rate.
Input the number of times to specify the drop-in rate(s): 5
Input the time point(s) for specifying drop-in rate(s): 0 1 2 3 4
Input the drop-in rate(s): .03 .03 .03 .03 .03
Input the number of times to specify the drop-out rate(s): 5
Input the time point(s) for specifying drop-out rate(s): 0 1 2 3 4
Input the drop-out rate(s): .10 .10 .10 .10 .10