COCHRAN'S TEST - Defense Technical Information …s (1941) well-known test for equality of k normal...
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AN EXTENSION OF COCHRAN'S TEST ~ri FOR HO0MOGENEITY OF VARIANCES BY H. SOLOMON and M. A. STEPHENS TECHNICAL REPORT NO. 419 ~\UGUST 8, 1989 Prepared Under Contract N00014-89-i.-1627 (NR-042-267) For the Office of Naval Research Herbert Solomon, Project Director Reproduct ,n in Whole or in Part is Permitted for any purpose of the United States Government Approved [or public release; distribution unlimited. 'EPART>. 'NT OF STATISTICS L)TIC STANFORD UNIVERSITY ELECTE 17 STANFORD, CALIFORNIA AUG 08 3989
Transcript of COCHRAN'S TEST - Defense Technical Information …s (1941) well-known test for equality of k normal...
AN EXTENSION OF COCHRAN'S TEST
~ri FOR HO0MOGENEITY OF VARIANCES
BY
H. SOLOMON and M. A. STEPHENS
TECHNICAL REPORT NO. 419
~\UGUST 8, 1989
Prepared Under Contract
N00014-89-i.-1627 (NR-042-267)
For the Office of Naval Research
Herbert Solomon, Project Director
Reproduct ,n in Whole or in Part is Permittedfor any purpose of the United States Government
Approved [or public release; distribution unlimited.
'EPART>. 'NT OF STATISTICS L)TICSTANFORD UNIVERSITY ELECTE 17STANFORD, CALIFORNIA AUG 08 3989
1. Introduction
Cochran's (1941) well-known test for equality of k normal population variances pro-
ceeds as follows. Let o, i = 1,..., k, be the population variances, and the null hypothesis
is
H0 = 2 = = = , say. (1)
Suppose k sample variances, one from each population. are given, each based on n degrees
of freedom; let the values be s .. k. Let these be ranked, so that, labelled in
ascending order, the values are sf1 ) < 2... < S 2)" To test H 0 , Cochran suggested the
statistic
(2Z = Sh /Y (2)
where Y = E= 1S., Thus the statistic compares the largest sample variance with the
sum of all variances, and is clearly appropriate to test H0 against the alternative that one
population variance is larger than the rest. Solomon and Stephens (1989) have recently
given very complete tables of the distribution of Z and have described, in particular, the
usefulness of the test in process control; for example, to see if the largest variance in a
sample of one week's daily output is excessively large. Solomon and Stephens (1989) also
give references to earlier literature and to other applications, particularly for n = 2, to
testing for white noise in time series analysis.
The question has been raised what to do if it is suspected that more than one vari-
ance is particularly large compared to the others, perhaps, say, in examining 30 daily vari-
ances from one month's output. One might propose to apply Cochran's test successively:
when one variance has been found too high, remove it and repeat the test on k - 1 vari-
ances; continue this procedure, rejecting the largest of the current set of variances, till no
further significance is found for Z. However, for such a technique, it is virtually impossi-
ble to evaluate the overall significance level, that is, to find the probability of rejecting m
variances, if this is what happens, when H0 is true.
In this article extensive tables are given to test the hypothesis of k equal variances,
each with n degrees of freedom, where rejection would indicate that the m-th largest sample
variance is an outlier. This is based on the ratio of the m-th largest sample variance toin.. .i .. Au/or
2 Dist J Special
the sum Y. Thus the test statistic is
s2ZM = S(k+l-).y (3)
Y
Cochran's statistic Z, defined in (2) is identical to Z 1 . The rationale behind this test is as
follows. When the s? are ranked, we can imagine them plotted against the expected values
of k ordered chi-square variables, with degrees of freedom n. If Ho is true, the plot should
appear like a straight line. A commonly occurring alternative to the straight line plot, oc-
curring in data analysis, is a plot in which the smallest s? values are close to a straight
line, but, say, the top m values are all too large to be on the line. Suppose, then, the m-th
largest s?, that is s .. is significantly large. The other larger s?, namely s 2Iag s I
i ,.kl - ) (+2-m),
S2 k+3_,) , etc., will probably be large also, even if not themselves significantly large. be-
cause they must be in order of increasing size. Such an event (s2 significantly large)((k+lm) sgiiatylre
will suggest that m population variances are larger than all the others. When, as usually
happens, the value of a 2 in H0 is not known, it is necessary to divide by Y in order to
obtain a scale-free statistic, and this leads to the statistic Zm. The statistic is not formally
developed as, say, a likelihood-ratio statistic, but will be useful in analyzing real data on
variances, such as arise in quality assurance; in the example quoted above, several vari-
ances during the course of a month may have appeared to be too large, and the test can
be used to examine this possibility.
The test procedure is set out below, and the t',Ory of the test, together with some
comments, is given in Section 3.
2. Test Procedure
The steps in making the test are:
(a) Rank the k given sample variances (which must be independent and each based
on n degrees of freedom): s0,) <(2) k)< )"
(b) Calculate Zm from (3), where Y = s2 + + + 2
(c) Refer to Table 1 for the appropriate values of k, degrees of freedom n and n..
(d) If Zm is larger than the value given for significance level a. reject H0 at level a.
3
Table 1 contains the percentage points of the null distribution of Z,,. These have
been found as described in the next section.
EXAMPLE. Duncan (1986, p. 388) gives a data set of inside diameters, consisting of
20 samples of size 5; hence k = 20 and n = 4. The values of s? are 16.5, 12.3, 10.3, 15.2,
2.7, 11.3, 7.5, 19.8, 5.8. 5.8, 5.7, 11.3, 12.3, 3.5, 8.2, 12.5, 18.5, 6.5, 4.7, and 6.0; with a
total of i96.4. This example was used in Solomon and Stephens (1989) to show the use of
Cochran's statistic (Z 1 in our present notation), and, with Z1 = .1008, it was shown that
the largest variance was not unusually large. It might be thought that the fourth largest
sample variance 15.2 is an anomalous value, so we test the hypothesis of equal Or2 using
Z 4 , which equals 15.2/196.4 = 0.0774.
Reference to Table 1, with k = 20, n = 4 shows Z4 to be not significant even at the
50% level, so that there is no reason to reject H0 .
3. Theory of the Test and Calculation of
Percentage Points
In Solomon and Stephens (1989) it was shown how the distribution of Z1 could be
calculated exactly for small k, and could be very well approximated., for larger values of
k, by fitting Pearson curves. Again, for very small k, the exact distribution of Z,,, could,
in principle, be found, although with greater difficulty, but here we use only the Pearson
Curve approximation. Z, can be constructed as follows:
(a) Let Y1, Y2,... Yk be i.i.d. random variables, each with the distribution a2,yn, where
a2 is any positive value; let Y(1) < Y(2) < ... < Y(k) be the order statistics of the set yi.
(b) Let Y = Ejyj.
(c) Then Z,, = Y(k+1-,,m)/Y.
It is clear that the distribution of Z.. is independent of a, the scale parameter of y,; also
Y is a completely sufficient statistic for a 2. Thus, by the Ba.u/Hogg/Craig Theorem,
Z, and Y are independently distributed. We can henceforth assume that a = 1. Then
4
ZmIY - Y(k+1-m), and we have
E (Yrk+l-m)) (4)E(M, E(y- ) ,(4
where E(.) denotes expectation. The denominator of (4) is easy to find, since Y is a X2 -
variable with kn degrees of freedom: then
EI r ) = 2r{(kn + 2r)/2}E(Y) r(kn/2) (5)
For the distribution of Y(k+l-m), let G(t) be the distribution, and g(t) the density, of y2;
then E(yr)), for any i (1 < i < k) is
. ,.)) n! 00 trG'-l(t)[1- G(t)]n-ig(t)dt.
(i (n - i)!(i - 1)! (6)
The moments of Y(k+l--m) can be calculated from (6), and hence, using (4) and (5), the
moments of Zm can be found. The first 4 moments have been used to fit Pearson curves to
the distribution of Zm, as described by Solomon and Stephens (1978), and hence to obtain
percentage points of Zm. These can be expected to be very accurate, especially in the long
upper tail which will be used, in general, in the present application.
COMMENTS ON TABLE 1. (1) The case n = 2. For this case, the ' distribution is
essentially the exponential distribution, and the special properties of this distribution can
be used to give an exact answer to the probability (Zm > z). This was done by Fisher
(1929) in connection with testing for white noise in time series, and by Stevens (1939),
in connection with a problem in geometric probability. Fisher (1940) discussed the two
problems, and gave a small table of upper 5% points of Zm, for m = 1 and 2. (Z 1 and Z2
are there called gi and g2). A comparison of our points with Fisher's is given in the small
table following: P.C. refers to the Pearson curve points, F to the exact points as given by
Fisher.
5
k: 5 10 20 30
m = 1 F .6838* .4449 .2704 .1978
m = 1 P.C. .6838* .4437 .2700 .1977
m = 2 F .3670* .2651 .1755 .1336
m = 2 P.C. .3653 .2641 .1753 .1335
*These are exact results, see comment (2) below.
It is clear that the Pearson curve fits give excellent results; calculations show that the
errors in a given by using the P.C. points at the 5% level will be less than 0.25%.
(2) For m = 1 an exact formila is available for P(Zm > z) when z > 0.5. This was
used in Solomon and Stephens (1989) and is again used in the present Table 1, which
repeats results for m = 1 for the sake of completeness.
(3) As a check, we calculated the moments of Y(k+l-m) using several available algo-
rithms for the incomplete Gamma distribution. Table 1 as given uses the most recent,
by Shea (1988). Other algorithms gave essentially the same values in the upper tail, but
slightly different values in the lower tail, especially for small k and n. The lower tail is, of
course, much less likely to be used in applications, and, to make the Table of manageable
size, we give only the upper tail points. A much more extensive table including the lower
tail, is available from the second author.
This work was partially supported by the Natural Sciences and Engineering Research
Council of Canada, and by the U. S. Office of Naval Research, and the authors express
thanks to both of these agencies.
References
Cochran, W. G. (1941). The distribution of the largest of a set of estimated variances as
a fraction of their total, Annals of Eugenics 11, 47-52.
Duncan, A. J. (1986). Quality Control and Industrial Statistics. Irwin: Holmwood. Illinois.
Fisher, R. A. (1929). Tests of significance in harmonic analysis. Proceedings of the Royal
Society, A 125, 54-9.
Fisher, R. A. (1940). On the similarity of the distributions found for the test of significance
6
in harmonic analysis, and in Stevens's problem in geometric probability, Annals of
Eugenics 10, 14-17.
Shea, B. L. (1988). Chi-squared and Incomplete Gamma Integral. Algorithm AS 239,
Applied Statistics 37, 466-473.
Solomon, H. and Stephens, M. A. (1978). Approximations to density functions using
Pearson curves, Journal of the American Statistical Association 73, 153-160.
Solomon, H. and Stephens, M. A. (1989). Percentage points for Cochran's test for equality
of variances, Journal of Quality Technology, to appear.
Stevens, W. L. (1939). Solution to a geometrical problem in probability, Annals of Eugenics
9, 315-20.
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UNCLASS I FIEDSECURITY CLASSIFICATION Of THIS PAGE (When D"e Entered)
REPORT DOCUMENTATION PAGE BEFOREsCrPuN FORM
REPORT MUMmeR 3, GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMMER
419
4. TITLE (and Subtitle) S. TYPE OF REPORT a PERIOD COVEREO
An Extension Of Cochran's Test For TECHNICAL REPORTHomogeneity Of Variances G. PERFORMING 0RG. REPORT NUMUEX
7. AUTHOR(e) S. CONTRACT OR GRANT NUMIER(e)
H. Solomon and M. A. Stephens N00014-89-J-1627
S. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT, TASK
Department of Statistics ARIEA A WORK UNIT NUMCERS
Stanford University NR-042-267Stanford, CA 94305
i. CONTROLLING OFFICZ NAME ANO ADDRESS 13. REPORT DATE
Office of Naval Research August 8, 1989
Statistics & Probability Program Code 1111 13. NUMIEROF PAGES33
t4. MONITORING AGENCY NAME & AOORESS(I dif terent Ina Contrellng 0ffcs) Is. SECURITY CLASS. (o fh #a..)
UNCLASSIFIED
IS.. Oi CL.ASSIFIrICATI ON/OOWMNGRAOI NGSCM IOULI
r
It. O1STRIUUTIOM STATEMENT (of ld Reporf)
APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED
17. DISRIUTION STATEMENT (01 the ebettect entered In lo, k0. II difnemrn fbom Reped,)
I4. SUPPLEMENTARY NOTES
It. KEY WORDS (Contlaue on reowea ide It necoeev and Idaemilt by ledia mmbr)
Control charts; process control; quality control, stability of variance.
20. ASSTRACT (Cenihme an revern alde IS nacoesery sind Identitby b. mumbo)
Cochran's test for equality of k normal population variances consists ofcomparing the largest of the set of k sample variances, all based on the samenumber of degrees of freedom n, with the sum of the sample variances. Moregenerally, it may sometimes be advantageous to compare the m-th largest samplevariance with the total. The distribution theory of the resulting statistic Zmis discussed, and a table is given of percentage points of Zm , for a widerange of k, n, and m, with which to make the test. The test is illustratedwith an example taken from Duncan (1986).
DD I P1OH 1473 EoITION OF I NOV 45 Is o,1SOLTELIELS/N 0102-014-601 1 UNCLASSIFID33 99CUIRITY CL.ASSIICATION OF
rT141S PAQ-9- (Wht Date anforew)
