Multiple Comparison procedures for One-Way layouts
All-pairs comparisons with multiple-comparison correction
 
Numbers of glucocorticoid receptor sites per Leukocyte cell
  for controls and 4 types of leukemia/anemia
Hollander & Wolfe (Table 6.4, p201)
 
Grp #1 (n=14) Normal subjects
Grp #2 (n= 5) Hairy-cell anemia
Grp #3 (n= 6) Chronic lymphatic leukemia
Grp #4 (n= 4) Chronic myelocytic leukemia
Grp #5 (n= 8) Acute leukemia
  
Data:
    Grp#1    Grp#2    Grp#3    Grp#4    Grp#5
     3500     5710     2930     6320     3230
     3500     6110     3330     6860     3880
     3500     8060     3580    11400     7640
     4000     8080     3880    14000     7890
     4000    11400     4280              8280
     4000              5120             16200
     4300                               18250
     4500                               29900
     4500                                    
     4900                                    
     5200                                    
     6000                                    
     6750                                    
     8000                                    
 
5 treatment groups with 37 total observations:
Ord, Values, Group, Midranks, Tiegroups:
   1.   3500   1     5.0    3
   2.   3500   1     5.0    2
   3.   3500   1     5.0    3
   4.   4000   1    11.0    2
   5.   4000   1    11.0    2
   6.   4000   1    11.0    0
   7.   4300   1    14.0    0
   8.   4500   1    15.5    0
   9.   4500   1    15.5    0
  10.   4900   1    17.0    0
  11.   5200   1    19.0    0
  12.   6000   1    21.0    0
  13.   6750   1    24.0    0
  14.   8000   1    28.0    0
  15.   5710   2    20.0    0
  16.   6110   2    22.0    0
  17.   8060   2    29.0    0
  18.   8080   2    30.0    0
  19.  11400   2    32.5    0
  20.   2930   3     1.0    0
  21.   3330   3     3.0    0
  22.   3580   3     7.0    0
  23.   3880   3     8.5    0
  24.   4280   3    13.0    0
  25.   5120   3    18.0    0
  26.   6320   4    23.0    0
  27.   6860   4    25.0    0
  28.  11400   4    32.5    0
  29.  14000   4    34.0    0
  30.   3230   5     2.0    0
  31.   3880   5     8.5    0
  32.   7640   5    26.0    0
  33.   7890   5    27.0    0
  34.   8280   5    31.0    0
  35.  16200   5    35.0    0
  36.  18250   5    36.0    0
  37.  29900   5    37.0    0
 
Sample means and rank means for treatment groups:
  #1 (n=14):  DataAv=  4760.71   RankAv= 14.43
  #2 (n= 5):  DataAv=  7872.00   RankAv= 26.70
  #3 (n= 6):  DataAv=  3853.33   RankAv=  8.42
  #4 (n= 4):  DataAv=  9645.00   RankAv= 28.63
  #5 (n= 8):  DataAv= 11908.75   RankAv= 25.31

Kruskal-Wallis H = 16.6464 (with no tie correction)
  For permutations:  Hscore = 15307.4
  Tiesum=66    TieCorr=0.00130394    (nsum=37)
Kruskal-Wallis HP = 16.6682 (with tie correction)
Large-sample chi-square approximation for H:
    P= 0.00224  (df=4)

Permutation test for large-sample Kruskal-Wallis approximation:
 
Initializing the random-number generator at
    12345678

Number of permutations:
       10000

Number of simulations with values >= Hscore and total number:
    Nge=5    nsims=10000
95% CI for true P-value bracketing estimate of true pvalue:
(Since H and Hscore >= 0, P-values are inherently two-sided.)
    0.00006    0.00050    0.00094
 
ABSOLUTE-VALUE DIFFERENCES BETWEEN ALL PAIRS:
The following are MULTIPLE-COMPARISON-UNCORRECTED P-values:
For each of the  k*(k-1)/2  pairwise comparisons among the k treatment
  groups, consider the large-sample approximation of the
  Wilcoxon rank-sum test, including tie corrections:
Results for k(k-1)/2 = 10 pairs:
  #1 vs. #2:  W=110.0  E(W)=140.0  Z= 2.788   P=0.0053 (2-sided, UNCORR)
  #1 vs. #3:  W=167.0  E(W)=147.0  Z= 1.655   P=0.0979 (2-sided, UNCORR)
  #1 vs. #4:  W=108.0  E(W)=133.0  Z= 2.667   P=0.0076 (2-sided, UNCORR)
  #1 vs. #5:  W=132.0  E(W)=161.0  Z= 1.984   P=0.0472 (2-sided, UNCORR)
  #2 vs. #3:  W= 45.0  E(W)= 30.0  Z= 2.739   P=0.0062 (2-sided, UNCORR)
  #2 vs. #4:  W= 21.5  E(W)= 25.0  Z= 0.861   P=0.3893 (2-sided, UNCORR)
  #2 vs. #5:  W= 32.0  E(W)= 35.0  Z= 0.439   P=0.6605 (2-sided, UNCORR)
  #3 vs. #4:  W= 21.0  E(W)= 33.0  Z= 2.558   P=0.0105 (2-sided, UNCORR)
  #3 vs. #5:  W= 28.5  E(W)= 45.0  Z= 2.132   P=0.0330 (2-sided, UNCORR)
  #4 vs. #5:  W= 24.0  E(W)= 26.0  Z= 0.340   P=0.7341 (2-sided, UNCORR)
Maximum observed absolute Z-value:  2.7885
Note that 3 pair(s) have P<0.01 and 3 more have P<0.05.
 
WHICH OF THESE ARE SIGNIFICANT, allowing for MULTIPLE COMPARISONS?
Using the Steel-Dwass-Crichlow-Fligner multiple-comparison
  procedure described in Section 6.5 of Hollander & Wolfe.
Let  ZObs(i,j)  be the pairwise Wilcoxon rank-sum score between all
  treatment-group pairs, normalized to have mean zero and variance one.
After a permutation of the midranks, let mstar = max_ab |Z(a,b)|
  where Z(a,b) is the corresponding normalized Wilcoxon score between
  pairs (a,b) for the permuted data. Finally, let P(i,j) be the
  proportion of permutations for which  mstar >= |ZObs(i,j)|.
Then P(i,j) is the probability that |Z(a,b)| >= |ZObs(i,j)| for at
  least one pair (a,b), which gives a multiple-comparison-corrected
  two-sided P-value for each ZObs(i,j).
 
Simulating P-values for MULTIPLE-COMPARISON-CORRECTED COMPARISONS
Using 10000 permutations of the data
Pairwise MULTIPLE-COMPARISON-CORRECTED P-values:
  #1 vs #2:  Z= 2.788  P=  128/10000 = 0.01280
  #1 vs #3:  Z= 1.655  P= 4751/10000 = 0.47510
  #1 vs #4:  Z= 2.667  P=  303/10000 = 0.03030
  #1 vs #5:  Z= 1.984  P= 2649/10000 = 0.26490
  #2 vs #3:  Z= 2.739  P=  203/10000 = 0.02030
  #2 vs #4:  Z= 0.861  P= 9281/10000 = 0.92810
  #2 vs #5:  Z= 0.439  P= 9975/10000 = 0.99750
  #3 vs #4:  Z= 2.558  P=  514/10000 = 0.05140
  #3 vs #5:  Z= 2.132  P= 1921/10000 = 0.19210
  #4 vs #5:  Z= 0.340  P= 9994/10000 = 0.99940
Note that none have P<0.01 and 3 have P<0.05.