One-Way Layouts:
Kruskal-Wallis test for equality of location parameters
  Model:  X(i,j) dist X + theta_i ---  Test H_0: Theta are identical
  
Example 2:
Testing smoothness of paper in four batches
 
Data in 4 treatment groups with sizes 8,8,8,8:
 
  Group #1 (n=8)  387  415  438  445  455  460  477  580
  Group #2 (n=8)  392  393  397  414  418  429  433  458
  Group #3 (n=8)  340  350  390  400  430  430  440  450
  Group #4 (n=8)  340  348  348  354  372  378  412  428
 
In one vertical table for analysis:
(Columns are Ordinal, Value, Treatment group, Midrank, Tiegroups
    1   387   1    9.0   2
    2   415   1   17.0   2
    3   438   1   24.0   2
    4   445   1   26.0   0
    5   455   1   28.0   0
    6   460   1   30.0   0
    7   477   1   31.0   0
    8   580   1   32.0   0
    9   392   2   11.0   0
   10   393   2   12.0   0
   11   397   2   13.0   0
   12   414   2   16.0   0
   13   418   2   18.0   0
   14   429   2   20.0   0
   15   433   2   23.0   0
   16   458   2   29.0   0
   17   340   3    1.5   0
   18   350   3    5.0   0
   19   390   3   10.0   0
   20   400   3   14.0   0
   21   430   3   21.5   0
   22   430   3   21.5   0
   23   440   3   25.0   0
   24   450   3   27.0   0
   25   340   4    1.5   0
   26   348   4    3.5   0
   27   348   4    3.5   0
   28   354   4    6.0   0
   29   372   4    7.0   0
   30   378   4    8.0   0
   31   412   4   15.0   0
   32   428   4   19.0   0
 
Parallel display with midranks with treatment groups as columns:
  387 ( 9.0)    392 (11.0)    340 ( 1.5)    340 ( 1.5)
  415 (17.0)    393 (12.0)    350 ( 5.0)    348 ( 3.5)
  438 (24.0)    397 (13.0)    390 (10.0)    348 ( 3.5)
  445 (26.0)    414 (16.0)    400 (14.0)    354 ( 6.0)
  455 (28.0)    418 (18.0)    430 (21.5)    372 ( 7.0)
  460 (30.0)    429 (20.0)    430 (21.5)    378 ( 8.0)
  477 (31.0)    433 (23.0)    440 (25.0)    412 (15.0)
  580 (32.0)    458 (29.0)    450 (27.0)    428 (19.0)
 
 
By Treatment Group: Number, Rank Sum, Rank Average:
  Group #1  n=8   197.0   24.63
  Group #2  n=8   142.0   17.75
  Group #3  n=8   125.5   15.69
  Group #4  n=8    63.5    7.94

Kruskal-Wallis H = 12.8686 (with no tie correction)
 
Tie-group sizes:
     2     2     2

Tiesum and Tie correction:
 
   18.0000    0.0005

Kruskal-Wallis H = 12.8721 (with tie correction)
 
Large-sample chi-square approximation for H:

  P= 0.0049  (df=3)
 
Doing nsims simulations for the true P-value

nsims =

       10000

Initializing the RNG from the system clock at

startseed =

  2.0860e+005

H=12.8721  and  Hscore=9844.44
 
Number of simulations with values >= Hscore and total number:
 
          18       10000

95% CI for true P-value bracketing estimate of true pvalue:
(Since H and Hscore >= 0, P-values are inherently two-sided.)
 
    0.0010    0.0018    0.0026

 
Test for an alternative of monotonic treatment-group medians:
The Jonckheere-Terpstra test statistic is very similar to
  the Kendall nonparametric correlation statistic between
  values and treatment-group numbers

jj =

   84.5000

The Jonckheere-Terpstra test statistic mean and variance:
(WITHOUT tie correction.)

Jmean_Jvar_Zval_NormPval =

  192.0000  885.3333   -3.6129    0.0002