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Water Diffusion across Human Chorioamnion Tissue
  
Data from textbook Table 4.1 p110 (2nd edn)
  X: At term (approx 39weeks, Control)    Y: 12-26 weeks
  
Wilcoxon Rank-Sum tests assumes Var(X)=Var(Y): Is this reasonable?
(See also Section 5.2 p158 in text.)
  
The data:
  
X (m=10):  0.80  0.83  1.89  1.04  1.45  1.38  1.91  1.64  0.73  1.46
  
Y (n= 5):  1.15  0.88  0.90  0.74  1.21
  
Let  Gamma=Var(X)/Var(Y)
  
Observed VarX, VarY, Gamma=VarX/VarY, StdevX, StdevY:
 
    0.1947    0.0389    5.0005    0.4412    0.1973

  
Use Miller Jackknife/Bootstrap procedure to test  H_0:Gamma=1
  (H_0:Var(X)=Var(Y)) and find a 95% Conf.Int. for Gamma
 
BOOTSTRAP TEST OF H_0:Gamma=1 (Var(X)=Var(Y)) and
  BOOTSTRAP 95% Confidence Interval for Gamma=Var(X)/Var(Y):
 
Use a `stratified bootstrap` to generate an array
  of bootstrap values gamma_star = Var(Xstar)/Var(Ystar)
 
Starting random-number seed:
 
     1234567

Three example pairs of Xstar,Ystar bootstrap samples with Gamma=Var(X)/Var(Y):
  1.38  1.89  0.73  1.04  1.46  1.89  1.89  1.04  1.38  1.38
  0.74  0.88  0.74  0.74  0.90
  Gamma=23.4078
  1.45  1.38  0.83  0.73  0.83  1.91  0.73  1.64  0.73  0.80
  0.74  1.15  0.74  0.74  0.90
  Gamma=6.19781
  1.04  1.64  1.04  1.46  0.83  1.38  1.89  1.46  1.91  0.83
  0.88  1.15  0.74  0.74  0.88
  Gamma=5.73939
 
Number of bootstrap replications:
 
       10000

Generate gamma-star bootstrap array:
 
The first 5 sorted gamma-star values are:
  0.5269  0.6337  0.6814  0.7184  0.7263
 
The last 5 sorted gamma-star values are:
  2804.2917  2963.3889  3074.7222  3154.2222  3424.4583
 
Median and Mean of bootstrapped values:
    5.5218   26.5327

Median bias (proportion of gamma-star <= Observed Gamma):
  4193/10000 = 0.4193   ObsGamma=5.00046
 
Bootstrap (two-sided) P-value for H_0:gamma=1:
(Twice the proportion of gamma(star)<=1)
  2*21/10000 = 0.0042 (2-sided)
 
95% bootstrap percentile Conf.Int. bracketing Median
    2.0509    5.5218   52.7316

 
ALTERNATIVE:  JACKKNIFE PROCEDURE (See Section 5.2 p158 in text)
 
Define two sets of jackknife pseudo-values:
Log variance of X sample: -1.6365
Pseudo-values of log variance for X sample:
(These are Ps_i = mm*FullVarX - (mm-1)*Var(X_Delete_Xi) )
  -1.05  -1.26  -0.56  -2.26  -2.59  -2.67  -0.39  -2.06  -0.51  -2.57
 
Log variance of Y sample: -3.2460
Pseudo-values of log variance for Y sample:
  -3.28  -4.09  -4.21  -2.03  -2.08
 
Sample means and sample variances of the two sets
  of pseudo-values (of log variances):

Jxmean_Jymean_Jxvar_Jyvar =

   -1.5930   -3.1373    0.8738    1.1054

Unpooled variance estimator and standard deviation for Jxmean-Jymean:
 
    0.3085    0.5554

Miller 2-sample Z-test for H_0:Var(X)=Var(Y) (two-sided):
 
  Z=2.7805   P=0.0054 (two-sided)
 
Bias-corrected gamma=Var(X)/Var(Y) estimator

Exp_of_Jxmean_Minus_Jymean =

    4.6847

Jackknife 95% Conf.Int. for gamma bracketing Exp(Jxmean-Jymean):
    1.5773    4.6847   13.9135