科学文献

The Interquartile Test of Dispersion
INTERQ.SAS is a SAS ® Subroutine written by Paul Johnson that calculates the Interquartile Test of Dispersion (see Shoemaker, 1999). The test is based upon the interquartile range and is valid for a variety of both skewed and symmetric distributions when location is known or unknown. The subroutine is written using SAS (see SAS/STAT, 1993).
FORMULAE
Without loss of generality the program calculates the p-value for a one-tailed test. For a two-tailed test, double the value obtained for the p-value. The data example used is the times to breakdown (in minutes) of an insulating fluid under elevated voltage stresses
(see Shoemaker, 1999). Population 1 are insulating fluids under 32 KV elevated voltage stress.Population 2 are insulating fluids under 36 KV elevated voltage stress.
H 0: Variability is the same for the two populations of interest
Ha: Variability of Population 1 is significantly greater than that of Population 2.
Let p) (X , p)1 (X − and p) (Y ,p)1 (Y − be the pth and (1-p)th percentiles of the X sample and the Y sample.()()
2
2y 12x p p n n Y X T ωωγγ+−=where:
()p) ()p (1p X X X −=−γ and ()p) ()p (1p Y Y Y −=−γ;
2x 2x 2x x 2
x 2x 2x Q /P /))Q p(P Q p(P +−+=ω and 2y 2y 2y y 2y 2y 2y Q /P /))Q p(P Q p(P +−+=ω;
()()p)1(F f Q ,p)(F f P 1x 1x −==−− and ()()p)1(G g Q ,p)(G g P 1y 1y −==−−.
x P is estimated by (x
P ˆ):[The number of observations falling in () h X ,h X 11n p) (n p) (+−] / 21
n 1h n ; and x Q is estimated by (x
Q ˆ):[The number of observations falling in () h X ,h X 11n p) (1n p) (1+−−−] / 21
n 1h n ,where 5/111n n s 3.1h 1
−= with 1s being the standard deviation of sample X.
If 1n 1s 68.0h then 25n 1
=≤; likewise for 2n of sample Y.P and Q are estimated similarly for sample Y.
Under conditions outlined by Shoemaker, the author points out that T is then asymptotically distributed N(0,1).
RESULTS
Summary Statistics for the Two Treatment Populations
Univariates for the Two Treatment Populations
Treatment Population = 1
Variable=X
Moments
Quantiles(Def=5)
100% Max 215.5 99% 215.5
75% Q3 82.85 95% 215.5
50% Med 13.95 90% 100.58
25% Q1 0.79 10% 0.4
0% Min 0.27 5% 0.27
1% 0.27
Range 215.23
Q3-Q1 82.06
Mode 0.27
Treatment Population = 2
Variable=X
Moments
Quantiles(Def=5)
100% Max 25.5 99% 25.5
75% Q3 3.99 95% 25.5
50% Med 2.58 90% 13.77
25% Q1 0.99 10% 0.59
0% Min 0.35 5% 0.35
1% 0.35
Range 25.15
Q3-Q1 3
Mode 0.35
H Values for Treatment Population 1 and 2
X(p), X(1-p) and diff = X(1-p)-X(p) for the Two Treatment Populations
Interquartile test Statistic (T) with p-value (one-tailed result): A test to compare the variability of two populations. Do the two distributions show the same spread?
T P-VALUE ()X p γ Y p γ STD
Here we reject H 0 and conclude that the variability of the times to breakdown of an insulating fluid for population 1 (32KV elevated voltage stress) is significantly higher than the variability of times for insulating fluid under 36KV elevated voltage stress. A two-tailed test would result in a p-value = 0.0056, showing strong evidence that the two populations do not have the same variability. In this example p was set equal to .1875, since then (n+1)*p is a whole number and the estimates of the quantiles are whole number order statistics. The user needs to input the value of p into the logic of the SAS macro . For this example the code is: %let p = 0.1875. Shoemaker indicates that it may be better to use interquartile tests and estimates for 0.25p 1.0<<. The author comments that in general that his simulations indicate that the interquartile test is a valid test over a variety of distributions.
REFERENCES
SAS Institute. Inc. (1993). SAS/STAT User's Guide, Version 6, 4th ed., Volume 1 and Volume 2,Cary, NC: SAS Institute Inc.
Shoemaker, L. H. (1999). "Interquartile Tests for Dispersion in Skewed Distributions," Commun.Statist.-Simula ., 28(1): 189-205.
ACKNOWLEDGMENTS
SAS, SAS/STAT and SAS/GRAPH are registered trademarks or trademarks of SAS Institute Inc.in the USA and other countries.
® indicates USA registration.。