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Cochran`s Test:
Two-Way Layout with 0,1 data:
Data  X(i,j)=0,1  where
   1 le i le nn   Subjects or Blocks
   1 le j le kk   Treatments or Treatment Groups
Same model and H_0 as in Friedman`s test
 
Example: Success for four methods of soothing newborn babies
From: Lehmann ``Nonparametrics: Statistical methods based on ranks``
  pages 267-270
See Lehmann p283 for the full data set.
 
k=4 treatments and n=12 newborns
Treatments:
  1. Warm Water
  2. Rocking
  3. Pacifier
  4. Sound

n=12 blocks (subjects) and k=4 treatment groups
B_i are row sums and L_j are column sums
Row number, warm water, rocking, pacifier, sound:
   1.    0    0    0    0    Sum: 0  (B( 1))
   2.    0    1    0    0    Sum: 1  (B( 2))
   3.    0    0    1    0    Sum: 1  (B( 3))
   4.    0    0    0    0    Sum: 0  (B( 4))
   5.    1    1    1    1    Sum: 4  (B( 5))
   6.    0    1    1    1    Sum: 3  (B( 6))
   7.    1    1    0    0    Sum: 2  (B( 7))
   8.    0    1    0    1    Sum: 2  (B( 8))
   9.    0    1    1    1    Sum: 3  (B( 9))
  10.    0    0    1    0    Sum: 1  (B(10))
  11.    1    1    0    1    Sum: 3  (B(11))
  12.    1    1    1    1    Sum: 4  (B(12))
 
Treatment-group (column) sums (L(j)):
         4    8    6    6    Sum: 24
 
Row and column-sum statistics:
(L1,B1 are sums, L2,B2 are sums of squares)
 
   L1=24   L2=152   B1=24   B2=70

Cochran-test statistics:

   L2=152   Q=3.692   Pval=0.2967  (df=3)
 
Simulating P-values for the Cochran test:
Permuting data values (=0,1) independently for each block
 
Initializing the random-number generator at
    12345678

Carrying out Nsims=10000 Friedman permutations:
Number of simulations with values >= L2 and total number:
 
        3349       10000

95% CI for true P-value bracketing estimate of true pvalue:
(Since H and Fscore >= 0, P-values are inherently two-sided.)
 
    0.3256    0.3349    0.3442

 
 
Applying the Friedman test procedure:
Finding within-block midranks for each block:
  1.  0(2.5)  0(2.5)  0(2.5)  0(2.5)
  2.  0(2.0)  1(4.0)  0(2.0)  0(2.0)
  3.  0(2.0)  0(2.0)  1(4.0)  0(2.0)
  4.  0(2.5)  0(2.5)  0(2.5)  0(2.5)
  5.  1(2.5)  1(2.5)  1(2.5)  1(2.5)
  6.  0(1.0)  1(3.0)  1(3.0)  1(3.0)
  7.  1(3.5)  1(3.5)  0(1.5)  0(1.5)
  8.  0(1.5)  1(3.5)  0(1.5)  1(3.5)
  9.  0(1.0)  1(3.0)  1(3.0)  1(3.0)
 10.  0(2.0)  0(2.0)  1(4.0)  0(2.0)
 11.  1(3.0)  1(3.0)  0(1.0)  1(3.0)
 12.  1(2.5)  1(2.5)  1(2.5)  1(2.5)
 
Rank sums and rank averages by treatment group:
    1.   26.00     2.17
    2.   34.00     2.83
    3.   30.00     2.50
    4.   30.00     2.50
 
With no tie correction:
  S=1.600   P= 0.6594  (df=3)
 
With tie correction:
  Tiesum=408   Tiecorr=0.566667
  SP=3.692   P= 0.2967  (df=3)
 
Note that the second set of values are exactly
  the same as in Cochran`s test