Testing Parallelism for Several Lines
(See Section 9.5 in text (attributed to Sen, Adichie))
See the comments in ParallelSlopes.m and ParallelSlopes.txt
  (``Approach I``) for a discussion of the method
 
EXAMPLE:
Test of cloud seeding of cyclones
  kk=2 samples: Seeded and not seeded (in that order)
  Y_ij = measure of precipitation
  X_ij = meridional circulation index for cyclones
  Sample sizes n1=11 (seeded) and n2=10 (unseeded)
See Table 9.5 page 436 in text (Problem 28)

K=2 groups with sizes N1=11 and N2=10
 
Data:
 Seeded (n=11):
   Y: 0.180 0.175 0.178 0.021 0.260 0.715 0.441 0.205 0.417 0.498 0.603
   X:  24    28    30    37    43    47    52    57    71    87   115  
 Unseeded (n=10):
   Y: 0.138 0.081 0.072 0.188 0.075 0.435 0.423 0.339 0.519 0.738
   X:  -7    10    17    25    44    51    53    63    75    90  
 
The H_0 model is  Y_ij = beta X_ij + mu_i + error_ij.
  As in ParallelSlopes.m, minimizing over the mu_i for fixed
  leads to the system
 
     Y_ij - Ybar_i = beta (X_ij - Xbar_i) + error_ij       (4)
 
  Let beta be the overall least-squares estimate of beta in (4)
  for all (i,j) pairs (1 le i le kk, 1 le j le nn_i)

Overall beta (beta_bar) = 0.005735
Beta values within groups:  0.004999  0.006413
 
Aligned observations  Ystar_ij = Y_ij - betabar X_ij:
 Seeded (n=11):
    0.042 0.014 0.006 -0.191 0.013 0.445 0.143 -0.122 0.010 -0.001 -0.057
 Unseeded (n=10):
    0.178 0.024 -0.025 0.045 -0.177 0.143 0.119 -0.022 0.089 0.222
 
Within-group ranks:
 Seeded:     9.0  8.0  5.0  1.0  7.0 11.0 10.0  2.0  6.0  4.0  3.0
 Unseeded:  10.0  5.0  3.0  6.0  1.0  9.0  8.0  4.0  7.0 11.0
 
Within-group Ti=Sum_j Xc(i,j)*Rank(i,j)/(n_i+1)  and
   Ci2=Sum(j=1,nn_i) (X_ij - Xbar_i)^2:
 Seeded:     T1=-29.00   C1^2=7722.182
 Unseeded:   T2= 16.42   C2^2=8378.900

Sen-Adichie test:  X=1.6929   P=0.1932  (df=1)
 
Compare with a permutation test of within-group ranks:
(Friedman-like permutations.)
 
Initializing the random-number generator at
    12345678

Carrying out ns=10000 sets of within-group permutations:
Observed Vscore = 0.141078
Number of simulations with values >= Tscore and total number:
 
        4491       10000

95% CI for true P-value bracketing estimate of true pvalue:
(Since H and Fscore >= 0, P-values are inherently two-sided.)
 
    0.4394    0.4491    0.4588