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 1:
Methods for rounding first base in baseball
k=3 methods with average times to second base for n=22 baseball players
   Method 1: Round out near first base
   Method 2: Narrow angle at first base
   Method 3: Wide angle at first base
Table 7.1 (p274) and Figure 7.1 (p275) in text

n=22 blocks (subjects) and k=3 treatment groups
 
Player, Round Out, Narrow Angle, Wide Angle
    1.      5.40      5.50      5.55
    2.      5.85      5.70      5.75
    3.      5.20      5.60      5.50
    4.      5.55      5.50      5.40
    5.      5.90      5.85      5.70
    6.      5.45      5.55      5.60
    7.      5.40      5.40      5.35
    8.      5.45      5.50      5.35
    9.      5.25      5.15      5.00
   10.      5.85      5.80      5.70
   11.      5.25      5.20      5.10
   12.      5.65      5.55      5.45
   13.      5.60      5.35      5.45
   14.      5.05      5.00      4.95
   15.      5.50      5.50      5.40
   16.      5.45      5.55      5.50
   17.      5.55      5.55      5.35
   18.      5.45      5.50      5.55
   19.      5.50      5.45      5.25
   20.      5.65      5.60      5.40
   21.      5.70      5.65      5.55
   22.      6.30      6.30      6.25

FRIEDMAN RANKS are WITHIN-BLOCK ranks, here 1 le R(i,j) le 3
Times for n=22 players with within-subject ranks/midranks
    1.    5.40(1.0)    5.50(2.0)    5.55(3.0)
    2.    5.85(3.0)    5.70(1.0)    5.75(2.0)
    3.    5.20(1.0)    5.60(3.0)    5.50(2.0)
    4.    5.55(3.0)    5.50(2.0)    5.40(1.0)
    5.    5.90(3.0)    5.85(2.0)    5.70(1.0)
    6.    5.45(1.0)    5.55(2.0)    5.60(3.0)
    7.    5.40(2.5)    5.40(2.5)    5.35(1.0)
    8.    5.45(2.0)    5.50(3.0)    5.35(1.0)
    9.    5.25(3.0)    5.15(2.0)    5.00(1.0)
   10.    5.85(3.0)    5.80(2.0)    5.70(1.0)
   11.    5.25(3.0)    5.20(2.0)    5.10(1.0)
   12.    5.65(3.0)    5.55(2.0)    5.45(1.0)
   13.    5.60(3.0)    5.35(1.0)    5.45(2.0)
   14.    5.05(3.0)    5.00(2.0)    4.95(1.0)
   15.    5.50(2.5)    5.50(2.5)    5.40(1.0)
   16.    5.45(1.0)    5.55(3.0)    5.50(2.0)
   17.    5.55(2.5)    5.55(2.5)    5.35(1.0)
   18.    5.45(1.0)    5.50(2.0)    5.55(3.0)
   19.    5.50(3.0)    5.45(2.0)    5.25(1.0)
   20.    5.65(3.0)    5.60(2.0)    5.40(1.0)
   21.    5.70(3.0)    5.65(2.0)    5.55(1.0)
   22.    6.30(2.5)    6.30(2.5)    6.25(1.0)
 
Rank sums and rank averages for each treatment group:
    1.   53.00     2.41
    2.   47.00     2.14
    3.   32.00     1.45
 
With no tie correction:
  S=10.63636   P= 0.0049  (df=2)
 
With tie correction:
  Tiesum=24   Tiecorr=0.0454545
  SP=11.14286   P= 0.0038  (df=2)
 
Simulating P-values for the Friedman test:
Permuting method values independently for each subject
 
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: 6042
Number of simulations with values >= Fscore and total number:
 
          20       10000

95% CI for true P-value bracketing estimate of true pvalue:
(Since H and Fscore >= 0, P-values are inherently two-sided.)
 
    0.0011    0.0020    0.0029