UMBRELLA TESTS for one-way layouts
Test whether or not medians of treatment groups increase to that of
  a particular treatment group and then decrease afterwards
Two versions: Given peak and unknown peak
 
Initializing random-number generator at
    12345678

Number of permutations:
      100000

SECOND DATA SET:
Fasting metabolic rates in deer by month pairs (Jan-Feb, Mar-Apr, etc)
Table 6.8 p215 in Hollander & Wolfe 2nd edn
6 treatment groups with 26 total observations:
  #1 (n=7):  36.0  33.6  26.9  35.8  30.1  31.2  35.3
  #2 (n=3):  39.9  29.1  43.4
  #3 (n=5):  44.6  54.4  48.2  55.7  50.0
  #4 (n=4):  53.8  53.9  62.5  46.6
  #5 (n=4):  44.3  34.1  35.7  35.6
  #6 (n=3):  31.7  22.1  30.7
 
Ord, Values, Group, Midranks, Tiegroups:
   1.  36.0   1    14.0    0
   2.  33.6   1     8.0    0
   3.  26.9   1     2.0    0
   4.  35.8   1    13.0    0
   5.  30.1   1     4.0    0
   6.  31.2   1     6.0    0
   7.  35.3   1    10.0    0
   8.  39.9   2    15.0    0
   9.  29.1   2     3.0    0
  10.  43.4   2    16.0    0
  11.  44.6   3    18.0    0
  12.  54.4   3    24.0    0
  13.  48.2   3    20.0    0
  14.  55.7   3    25.0    0
  15.  50.0   3    21.0    0
  16.  53.8   4    22.0    0
  17.  53.9   4    23.0    0
  18.  62.5   4    26.0    0
  19.  46.6   4    19.0    0
  20.  44.3   5    17.0    0
  21.  34.1   5     9.0    0
  22.  35.7   5    12.0    0
  23.  35.6   5    11.0    0
  24.  31.7   6     7.0    0
  25.  22.1   6     1.0    0
  26.  30.7   6     5.0    0
 
Sample means and rank means for treatment groups:
  #1 (n=7):  DataAv= 32.70  RankAv=  8.14
  #2 (n=3):  DataAv= 37.47  RankAv= 11.33
  #3 (n=5):  DataAv= 50.58  RankAv= 21.60
  #4 (n=4):  DataAv= 54.20  RankAv= 22.50
  #5 (n=4):  DataAv= 37.43  RankAv= 12.25
  #6 (n=3):  DataAv= 28.17  RankAv=  4.33
 
THE UMBRELLA TEST for KNOWN p=4 (Mack-Wolfe):
Single-umbrella test:
Is the data significant for a `peak` in treatment-group medians at p=4?
 
First Step:
Matrix of Mann-Whitney differences between treatment groups:
  1.   -1.0   15.0   35.0   28.0   21.0    5.0
  2.    6.0   -1.0   15.0   12.0    6.0    2.0
  3.    0.0    0.0   -1.0   12.0    0.0    0.0
  4.    0.0    0.0    8.0   -1.0    0.0    0.0
  5.    7.0    6.0   20.0   16.0   -1.0    0.0
  6.   16.0    7.0   15.0   12.0   12.0   -1.0
 
Example of an Ap statistic:
  Ap(4) = U(1,2)+U(1,3)+U(1,4)+U(2,3)+U(2,4)+U(3,4) + U(5,4)
 
The umbrella statistics Ap with normal approximations Astar:
(P-values are one-sided normal.)
 p=1  Ap=125.0   Mean=138.0  Var= 493.17   Astar=-0.585   P=0.72086
 p=2  Ap=111.0   Mean= 82.0  Var= 269.17   Astar= 1.768   P=0.03856
 p=3  Ap=148.0   Mean= 83.0  Var= 291.50   Astar= 3.807   P=0.00007
 p=4  Ap=157.0   Mean= 85.5  Var= 291.92   Astar= 4.185   P=0.00001
 p=5  Ap=156.0   Mean=109.5  Var= 383.92   Astar= 2.373   P=0.00882
 p=6  Ap=151.0   Mean=138.0  Var= 493.17   Astar= 0.585   P=0.27914
 
Simulating the P-value of Ap=157 given p=4 using 100000 permutations:
Number greater-or-equal to old Ap(p) and number of permuations:
    Nge=0    nsims=100000
95% Conf.Int. for one-sided P-value bracketing estimate:
     0     0     0

 
THE UMBRELLA TEST with UNKNOWN p (Chen-Wolfe):
 
Multiple-umbrella test:
Is there a `peak` in treatment-group medians at SOME p, and if so,
  at which treatment group?
 
Note that the one-sided P-values in the table above are NOT
  multiple-comparison corrected.
 
Max_Astar=4.18482  is attained at  p=4
Simulating the P-value of Max_Astar=4.18482 using 100000 permutations:
 
Number greater-or-equal to Max_Astar and number of permuations:
           4      100000

95% Conf.Int. for one-sided P-value bracketing estimate:
    0.00000   0.00004   0.00008