Use the Miller method to test whether or not the VARIANCES of two samples are the same (see Section 5.2 in text). Miller's idea is to apply a `stratified' jackknife to the log sample variance. Compare Miller's P-value and 95% confidence interval with a stratified bootstrap P-value and 95% confidence interval. Diffusion of tritiated water (that is, water containing H^3) across human chorioamnion as a test of placental permeability The first sample is at term (approximately 39 weeks) The second sample is at 12-26 weeks. Table 4.1 (p110) in Hollander & Wolfe, 2nd 3dn At term (approx 39 weeks) (Xs, m=10) (Xbar=1.313, SampStdev=0.441, Sampvar=0.195) 0.80 0.83 1.89 1.04 1.45 1.38 1.91 1.64 0.73 1.46 At 12-26 weeks (Ys, n=5) (Ybar=0.976, SampStdev=0.197, Sampvar=0.039) 1.15 0.88 0.90 0.74 1.21 Set gamma = SampVar(X)/SampVar(Y) Observed gamma = 0.194668/0.03893 = 5.00046 JACKKNIFE PROCEDURE: First, get the jackknife pseudovalues for the log sample variance for each sample: LogSampVar of Sample 1 (n=10) for full sample: -1.63646 Pseudovalues of Sample 1 (LogSampVar, n=10): -1.0531 -1.2555 -0.5620 -2.2607 -2.5887 -2.6708 -0.3905 -2.0645 -0.5115 -2.5723 LogSampVar of Sample 2 (n=5) for full sample: -3.24599 Pseudovalues of Sample 2 (LogSampVar, n=5): -3.2830 -4.0893 -4.2068 -2.0265 -2.0807 JACKKNIFE TEST of H_0:gamma=1 for gamma=Var(X)/Var(Y): Sample means and sample variances of the two samples of pseudovalues of log sample variances: Abar=-1.593 Bbar=-3.137 S2Avar=0.874 S2Bvar=1.105 Unpooled variance of Abar-Bbar=0.308 Unpooled StDev=0.555 V1=S2Avar/m (etc) and the Miller two-sample Z-statistic: V1=0.087 V2=0.221 Z=2.781 P=0.0054 (2-sided) For gamma(X,Y)=s2(X)/s2(Y): Bias-corrected gamma estimator and symmetric 95% CI for gamma: 4.6847 (1.5773, 13.9135) Observed gamma=5.0005 Starting random-number seed: 123456 BOOTSTRAP PROCEDURE (n=10,000 bootstrap replications): Use a `stratified bootstrap' to generate an array of gamma(star) bootstrap resampled values for gamma(star) = s(X)^2(star)/s(Y)^2(star) Filling the bootstrap ``gamma(star)'' array: Sorting the bootstrap ``gamma(star)'' array: The first 10 gamma(star) values are: 0.052 0.351 0.418 0.441 0.544 0.551 0.572 0.587 0.617 0.645 The last 10 gamma(star) values are: 2518.736 2546.194 2575.056 2627.792 2680.569 2708.833 2818.069 2822.903 2831.125 3008.403 Bootstrap median bias (Obs_Gamma=5.00046): 4069/10000 = 0.407 For gamma(X,Y)=s2(X)/s2(Y): Bootstrap median and bootstrap 95% CI for gamma: 5.6289 (2.0966, 48.7321) Observed gamma=5.0005 Bootstrap P-value for H_0:gamma=gamma0=1 (gamma(observed)=5.00046): (Twice the proportion of resampled gammas <= 1.) P = 2*21/10,000 = 0.0042 (2-sided) Random numbers used: 150,105 Starting random-number seed: 123456