Two-Way Layout:
Data  X(i,j)  where
   1 le i le nn  Subjects or Blocks
   1 le j le kk  Treatments or Treatment Groups
Model:  X(i,j) = mu(i,j) plus error  or mu + theta_i + tau_j plus error
Test H_0:  mu(i,j) = theta_i  and there is no effect of Treatment
 
Example 2:
Strength of cotton fibers as a function of levels of potash (K_2O) in field
k=5 potash fields with n=3 randomized blocks
Potash levels in pounds per acre:
  144(k=1)   108(k=2)   72(k=3)   54(k=4)  36(k=5)
Table 7.5 (p286) in text

n=3 blocks (subjects) and k=5 treatment groups
 
Cotton fiber strengths at 144, 108, 72, 54, 36 pounds per acre:
  1.   7.46    7.17    7.76    8.14    7.63
  2.   7.68    7.57    7.73    8.15    8.00
  3.   7.31    7.80    7.74    7.87    7.93

FRIEDMAN RANKS are WITHIN-BLOCK ranks, here 1 le R(i,j) le 5
Cotton strengths for n=3 replications with within-replication ranks
  1.   7.46(2.0)   7.17(1.0)   7.76(4.0)   8.14(5.0)   7.63(3.0)
  2.   7.68(2.0)   7.57(1.0)   7.73(3.0)   8.15(5.0)   8.00(4.0)
  3.   7.31(1.0)   7.80(3.0)   7.74(2.0)   7.87(4.0)   7.93(5.0)
 
Rank sums and rank averages by treatment group:
    1.    5.00     1.67
    2.    5.00     1.67
    3.    9.00     3.00
    4.   14.00     4.67
    5.   12.00     4.00
 
With no tie correction:
  S=8.80000   P= 0.0663  (df=4)
 
With tie correction:
  Tiesum=0   Tiecorr=0
  SP=8.80000   P= 0.0663  (df=4)
 
Simulating P-values for the Friedman test:
Permuting rank values independently for each block
 
Initializing the random-number generator at
    12345678

Carrying out Nsims=10000 Friedman permutations:
Fscore = Sum of Squares over Treatment Groups
  of Treatment-Group Rank Sums: 471
Number of simulations with values >= Fscore and total number:
 
         347       10000

95% CI for true P-value bracketing estimate of true pvalue:
(Since H and Fscore >= 0, P-values are inherently two-sided.)
 
    0.0311    0.0347    0.0383

 
 
PAGE TEST for monotonic alternatives
L is essentially the sum over blocks
  of the Spearman correlation coefficient of
  within-block midranks and the treatment-group number
Specifically, L = sum(i=1,n) sum(j=1,k) j*R(i,j)
  = sum(j=1,k) j*sum(i=1,n) R(i,j)
 
Observed score and large-sample approximation:
  L=158   Lmean=135   Lstdev=8.66025
  Z=2.65581   P=0.0040 (one-sided)

Carrying out Nsims=10000 Friedman permutations for Page`s test:
Observed L = 158
Number of simulations with values >= Lobserv and total number:
 
          25       10000

95% CI for true one-sided P-value bracketing estimate of true pvalue:
 
    0.0015    0.0025    0.0035