Balanced Incomplete Block Designs
Data  X(i,j)  J(i,j)=0,1  where
   Observation X(i,j) is present if J(i,j)=1, otherwise missing
   1 le i le nn   Subjects or Blocks
   1 le j le kk   Treatments or Treatment Groups
   ss observations in each block (Sum(j=1,k) J(i,j)=ss, all i)
   pp observations for each treatment (Sum(i=1,n) J(i,j)=pp, all j)
   lam observations common between every pair of treatments
 
Example:
Percent conversion of Methyl Glucoside to Monovinyl Isomers
  in presence of acetylene and a strong base under high pressure
See Table 7.14 (p316) in text
 
Data:
Run    250psi   325psi   400psi   475psi   550psi
  1.     16       18                32            
  2.     19                         46       45   
  3.              26       39                61   
  4.                       21       35       55   
  5.              19                47       48   
  6.     20                33       31            
  7.     13       13       34                     
  8.     21                30                52   
  9.     24       10                         50   
 10.              24       31       37            

k=5 treatments and n=10 blocks (runs)
 
ss=3 observations in each block
pp=6 observations in each treatment group
lam=3 observations in each pair of treatment groups
Total number of observations: 10x3 = 5*6 = 30
These two numbers should be the same:
  pp*(ss-1)=12  lam*(kk-1)=12
 
Within-block ranks at observed values:
Run    250psi   325psi   400psi   475psi   550psi
  1.    1.0      2.0               3.0            
  2.    1.0                        3.0      2.0   
  3.             1.0      2.0               3.0   
  4.                      1.0      2.0      3.0   
  5.             1.0               2.0      3.0   
  6.    1.0               3.0      2.0            
  7.    1.5      1.5      3.0                     
  8.    1.0               2.0               3.0   
  9.    2.0      1.0                        3.0   
 10.             1.0      2.0      3.0            
Observed treatment-group rank sums:
        7.5      7.5     13.0     15.0     17.0   

Observed rank-sum score for permutations:  Lscore=795.5
Large-sample approximation:
Durbin etal test:  D=15.1   P=0.00450  (df=4)

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=795.5
Number of simulations with values >= Lscore and total number:
 
          22      100000

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

    (0.00013   0.00022   0.00031)