Two-Way Layout with Multiple Replications per Cell
Data  X(i,q,j)  where
   1 le i le nn   Subjects or Blocks
   1 le q le cc   Replications for (i,j) combination
   1 le j le kk   Treatments or Treatment Groups
 
Example:
Test for consistency across 4 laboratories for amounts
  of niacin detected in enriched bran flakes.
Each laboratory tests 3 samples of bran flakes for each
  of 3 levels of known additional enrichment (0mg, 4mg, 8mg per 100g)
Data source:  Campbell and Pelletier (1962)
See Table 7.20 in text (p331)
 
Data by blocks:
            Lab1              Lab2              Lab3              Lab4      
0mg:  7.58  7.87  7.71| 8.00  8.27  8.00| 7.60  7.30  7.82| 8.03  7.35  7.66
4mg: 11.63 11.87 11.40|12.20 11.70 11.80|11.04 11.50 11.49|11.50 10.10 11.70
8mg: 15.00 15.92 15.58|16.60 16.40 15.90|15.87 15.91 16.28|15.10 14.80 15.70
 
Within-block ranks by blocks:
           Lab1              Lab2              Lab3              Lab4      
0mg:  3.0   8.0   6.0 | 9.5  12.0   9.5 | 4.0   1.0   7.0 |11.0   2.0   5.0 
4mg:  7.0  11.0   3.0 |12.0   8.5  10.0 | 2.0   5.5   4.0 | 5.5   1.0   8.5 
8mg:  2.0   9.0   4.0 |12.0  11.0   7.0 | 6.0   8.0  10.0 | 3.0   1.0   5.0 
 
Summaries by treatment group:
         Xavg Ranksum  Rankavg
 Lab1:  11.62   53.0    5.889
 Lab2:  12.10   91.5   10.167
 Lab3:  11.65   47.5    5.278
 Lab4:  11.33   42.0    4.667

Lscore = Sum ranksum^2 for permutation test: 15201.5
 
Large-sample approximation:
MS=12.9274  Pval=0.00480  (df=3)

Carrying out Nsims=100000 Friedman-like permutations:
 
Initializing the random-number generator at
    12345678

Lscore = Sum of Squares over Treatment Groups
  of Treatment-Group Rank Sums:  Observed Lscore=15201.5
Number of simulations with values >= Lscore and total number:
 
         226      100000

95% CI for true P-value bracketing estimate of true pvalue:
(Since Lscore >= 0, P-values are inherently two-sided.)

    (0.00197   0.00226   0.00255)
 
Rank-sum (and rank-average) differences between pairs of treatments
  and pairwise P-values multiple-comparison-corrected for all pairs:
  Based on maximum pairwise rank-sum differences
    in N=100000 simulations
  Lab1 v Lab2:   38.50   4.278    5148/100000 = 0.0515
  Lab1 v Lab3:    5.50   0.611   98663/100000 = 0.9866
  Lab1 v Lab4:   11.00   1.222   90079/100000 = 0.9008
  Lab2 v Lab3:   44.00   4.889    1568/100000 = 0.0157
  Lab2 v Lab4:   49.50   5.500     373/100000 = 0.0037
  Lab3 v Lab4:    5.50   0.611   98663/100000 = 0.9866

Labs 1,3,4 appear similar.  Lab2 differs from Labs 1,3,4
  but the difference is MCC significant only for Lab2 v Labs 3,4