A nonparametric multiple comparison test for differences in scale parameters

Metrika, Volume 36, 1989, page 91-106

A Nonparametric Multiple Comparison Test for Differences in Scale Parameters

By G. Stevens 1

A nonparametric multiple comparison test for differences in scale parameters is sug-

gested The asymptotic distribution of the test statistic is derived. A modification of

the test when the location parameters are unknown and unequal is suggested. This

modified test is not asymptotically distribution free for all underlying location-scale

families; however, we give sufficient conditions on the families under which the test

is asymptotically distribution free.

1 Introduction

Let X I 1 , ...,Xln and X21, ..., X2rn be two independent samples of independent ob-

servations from two populations with continuous cumulative distributions (c.d.f.)

F(u) and G(u) respectively. We shall assume that F and G differ only in the values of

the location and scale parameters, i.e. F(Ou + 7 ) = G ( u ) f o r s o m e - 0 o < 3 , < ~ o and

0 > 0. If we assume that the scale parameter is known (and without loss of generality

can be taken to be 1), then it is desired to t e s t H o : F(u) = G(u)vs. H A :F(u)q=G(u) or equivalently H 0 : 3' = 0 vs. 3' 4= 0. A popular nonparametric test of this hypothesis is

the Mann-Whitney test (Man and Whitney 1947). The procedure here is to joint ly rank

the two samples and sum the ranks of the first sample and reject H 0 if the sum is larger

or smaller than some critical value. Mann and Whitney have tabled these critical values

for small m and n and give a normal approximation when the sample sizes get large.

It is often desired to assume the location parameter is known (or 0) and test the

hypotheses H o : F(u) = G(u) vs. HA : F(u) --/= G(u) or equivalently H o : 0 = 1 vs.

H A : 0 4: 1. There are many nonparametric tests for testing these hypotheses. Three of

the most common were proposed by Mood (1954), Ansari and Bradley (1960), and

1 Prof. Gary Stevens, Dept of Statistics, Oklahoma State University, 301 Math Sciences Bldg., Stillwater, Oklahoma 74078, USA.

0026-1335/89/2/91-106 $2.50 © 1989 Physica-Verlag, Heidelberg

92 G, Stevens

Seigel and Tukey (1960). Since the Seigel-Tukey test is used later we will describe the

ranking procedure in detail. Ranks are assigned so that low ranks get put on the

extremes and high ranks in the middle. We will assign rank 1 to the lowest member of

the sequence and ranks 2 and 3 to the two highest members of the sequence. Then

ranks 4 and 5 are assigned to the next two lowest and ranks 6 and 7 to the next two

highest, etc. The test statistic is the sum of the ranks for the first sample. One would

reject H o if the test statistic is bigger or smaller than some critical value. These critical

values are the same as the ones for the Mann-Whitney test since under Ho, both test

statistics have the same distribution. It should be noted that often times for distribu-

tions with a positive support testing for differences in scale in the same as testing for a

difference in location (i.e. exponential distribution),

2 Steel's Multiple Comparison Test

Steel's test is a rank sum test used for comparing all pairs of treatments in a complete-

ly randomized design. This test is a nonparametric analogue of Tukey's fixed range

test. Just as in Tukey's test one has to compute the distribution of a certain least

significant range; here one has to compute the distribution of a certain minimum rank

sum.

Suppose X 1 . . . . . Xk are continuous random variables that measure a single charac-

teristic on each of k treatments. The c.d.f.'s are F l , . . . , Fk, respectively, and are the

same except for a difference in location parameters. We wish to test H 0 : F l = ... = F k

using ni observations from population i, i = 1 . . . . . k, versus H A : Fi 4~ Fj for at least

one pair. The procedure for the test is as follows.

Step 1: Jointly rank the X x's and X2's, Xx's and X3's . . . . . Y l 'S and Xk's. Then rank

the X2's and X 3,s . . . . , X2 's and Xk's then rank the Xk- l ' s and Xk's. Rank by assigning

rank 1 to the smallest observation up to rank n i + n / to the largest observation in each

ranking.

Step 2: Add the ranks of the X l 's when they are ranked with the succeeding Xi's to

give the rank sums T12 , Tl3 , ..., Tlk. Then add the ranks of the Xz's when they are

ranked with succeeding Xi's to obtain the rank sums T23 . . . . , T2k . . . . do until Tk_l , k

is obtained. Then compute the conjugate of Tq, T~j = ni(n i + n~ + 1 ) - Tq. The smaller

of Tq, T;/will be used as the test statistic.

A Nonparametric Multiple Comparison Test for Differences in Scale Parameters 93

Step 3. Compare the tabulated minimum rank sum value with the computed rank

sums. If a computed rank sum is less than the tabulated value, there is said to be a

significant difference in the location parameters of the two corresponding populations.

2.1 Rational for Using the Minimum Rank Sum

I

Let Sij = ra in ( T i p Ti] ). We now have the statistics $12, $I3 . . . . . S i k , $23 . . . . . S2k , . . . ,

Sk l,k and we reject H o if any of these are less than a critical value c. We have

PHo (of any of the Sij's < c) = c~ (1)

1 - PH o (none of the Sq's < c) :

1 -PHo (min ( S q ) > c ) = a

PHo (min (Sq) < c) = a.

By knowing the distribution of the min (Sij), we can find the c such that statement (1)

holds. From this we see if Sij < c then F i ¢ Fj. If we do this for all Sq, then we get all

pairwise comparisons under the same c~-level which is described for multiple compari-

son tests.

2.2 Distribution of the Min (Ti])

Let X l , . . . , X k be continuous random variables measuring a single characteristic on k

(:C n,.)! populations on which n l , ..., n k observations are made. There are - - possible

~(ni!) permutations of such observations. Under H 0 all permutations are equally likely. One

can generate the distribution of (TI2 . . . . . Tlk, T23, ---, T2k . . . . . T k - l ' k ) and from this

joint distribution one can obtain the distribution of the rain (Tij) by simple enumera-

tion.

94 G. Stevens

3 P r o p o s e d M u l t i p l e C o m p a r i s o n T e s t f o r D i f f e r e n c e s

in D i s p e r s i o n o f K - P o p u l a t i o n s

The proposed test is a modification of Steel's multiple comparison test. The first im-

por tant thing to notice is the ranking procedure used in Steel's test. One can see that

each pair of populations is ranked like the Mann-Whitney test. Seigel and Tukey, by

changing the ranking procedure, were able to obtain a test for variances while under

H 0 they were able to maintain the same distribution as the Mann-Whitney test. This

same idea is used here. Instead of ranking each pair of samples in the Mann-Whi tney

manner, we rank each pair of samples in the Seigel-Tukey fashion.

3.1 Procedure for Procedure Test

Let XI . . . . . Xk be continuous random variables that measure a single characteristic on

each of k populations. Let F 1 . . . . . Fk be the respective c.d.f. 's; also assume the c.d.f. 's

are identical except for a difference in scale parameters. We wish to test H o : F 1 =

... = Fk vs. HA :Fi =/:F/for at least one (i, ])pair. This is the same as testing H o : 01 =

... = O k vs. H A : Oi 4: 0 / fo r some (i,/) pair, where 0 i is the scale parameter o f F i.

Step 1: Jointly rank the X l ' s and X2's and Xa's , . . . , X l ' S and Xk's. Then rank the

X 2,s and Xa's, ..., X2's and Xk'S . . . . . X k - l ' s and Xk's. Rank each pair in the Seigel- Tukey manner.

Step 2." Same as Steel's test.

Step 3: Same as Steel's test except if we reject H 0 we conclude there is a significant

difference in the scale parameters of the two corresponding populations.

The distribution of this test statistic is obtained in the same way as the distribu-

tion of Steel's test statistic. Note that under H0: these two statistics have the same

distribution; therefore, existing tables can be used.

3.2 Asymptot ic Distribution of the Ti/'s

If we assume that the location parameters 7i are equal or known then the means,

variances, and covariances of the Tij's are (Steel 1962)


E(Tq) : ni(n i + nj + 1 )/2

Vat (Tii) = ninj(n i + n] + 1)/12

Cov (Thj, ~1) =Cov (Thi , Thj ) = nhnin]/12

Cov (Thi , Tij ) = -nhn in] /12

Coy (Tgh, Ti]) = 0

where (i 4=] C g 4 = h). The jo in t asymptotic distr ibution of the Tij's is found by writing

the Tq's as a U-statistic and using the asymptotic theory of U-statistics.

Defini t ion 1: Let H ( X 11 . . . . . Xlrn I " " " ; x k l . . . . . X k m k ) be a funct ion symmetric in

the X~'s . . . . ,Xk 'S . Let X l l , . . . , X l n ~ ; . . . ; x k ~ . . . . . Xkn~c be samples of size n l =pi N from distr ibut ions F1, .. . , Fk. Then a statistic of the form

( ) l ( ) , U= nl nk ~

. . , . , . m I m k c 1 ~ C 1 c k ~ C k

h ( X l C l l . . . X l c l r n l •

X k c l k "'" X k c k m k )

where C i = (cil . . . . . cim l ) and Ci is the collection of all subsets of the m i integers chosen

wi thout replacement from the integers (1 . . . . . mi) i = 1 , . . . , k is called a k sample

U-statistic.

Under H o the distr ibution of the Tq's for the proposed test is the same as the

dis tr ibut ion of the Ti/'s for Steel's test. Since Steel's test statistic is easier to write as a

U-statistic, we use it realizing that under H o the results are directly applicable to the

proposed test. We can write the Tij's as a U-statistic as follows: Let m I = ... = rn k = 1.

Also let

n 1 n k

Uq = (1/n 1 . . . n k ) ~, ... ~, h q ( X t g l . . . ; X k g k ) g i = l g k = l

where

h q ( X l g l ; . . . ;Xkgk) = 1 if Xigi<~Xjgj , 0 otherwise.

96 G. Stevens

We see in this case that Uq reduces to

n i n/ Ui/=(1/ninj) E E hq(Xig;X/k )

g=l k=l

where

hq(Xig;Xjk) = I if Xig < X i k , 0 otherwise.

Now the Tii's can be written as

Tq= (n i +n/+ 1)/2 + nin/U q.

The major theorem on the multivariate asymptotic distribution of r k-sample U-statistics

is due to Lehman (1951). Using the reduction of h(.) as above and the fact that all the

m/'s = 1 we can simplify the theorem as follows.

Theorem 1." Let Uij(i = 1 . . . . . k - 1, j > i ) be k(k - 1)/2 generalized U-statistics defined

in terms of hq and let E[Uq] = Oq. The joint limiting distribution of the variables

Z12 = ~ (UI2 - 012) . . . . . Z k - l , k = V ~ - ( U k - 1 , k - O k - l , k )

is a k(k - 1)/2 variate normal with mean zero and covariance matrix

= (oO.b)),

where

, t . t a = (i, j ) , b = ( t , 1 )

and

k a(a,b)= ~ (I/pq)6 (a'*')

q=l


where the quan t i t i e s 6 (a' o) are given by

6(qa, b) (H(qal), , (hi =E(H(q~),H(a2) ) OaOb = COY /7q2 )

where

H(q] ) = hq(Xigi; Xjg/)

H(a) q 2 = 17 i'j" (Xi'gi; Xj ,g j, )

and the sets (gi, gj) and (gf , gj,) have no integers in c o m m o n if g i = gi or gj = g],.

To apply this t h e o r e m we n e e d the fo l l owing e x p e c t e d values

E[h ii (Xie ; X]a ) = P(Xic <~ X/d) = 1/2

E[hq(Xic ; X j d ) h h ' f ( X f c ; Xj'd')]

= 1/3 i f i = i ' , j - - j ' and c 4=c' ,d =d ' or c = c',d--t=d'

= 1/4 i f i = i ' , j = j ' and c -- c ' , d = d '

= 1/3 i f i = i ' , / 4 = / a n d c = c ' , d = d ' o r c = c ' , d ~ d '

= l / 3 i f i - ~ i ' , j = j ' a n d c = c ' , d = d ' o r c 4 : c ' , d = d '

= 1 / 4 i f i = i ' , j 4 = j ' a n d c - - / = c ' , d = d ' o r c 4 = c ' , d ~ d '

= 1/4 i f i 4= i ' , j = j ' and c = c ' , d 4= d ' or c 4 = c ' , c 4= d '

= 1/6 if]" --- i ' , i 4=/.' and d = c ' , c = d ' or d = c ' , c 4= d '

= 1/4 if]" = i ' , i 4:/. and d = c ' , c = d ' or d 4= c ' , c ~ d '

Using these e x p e c t e d values we can n o w c o m p u t e the ~(a, a),s.

d ~.°~ = (l&~ + 1/pj)(1/12)

98 G. Stevens

o (a'b) = (1/12Pi) i f i = i ' , j =j'

= (1 /12pj ) i f i =/=i',j =]'

= - ( 1 / 1 2 p f ) i f i' =j, i:/=j '

=Oi f i 4=i',j ~ j '

From these we get the following lemma.

Lemma 1: Zl2, . . . , Z k - l ,k have a joint limiting multivariate normal distribution with

mean zero and variances and covariances given by :

Var (Zij) = (1/17 i + 1/pj)(1/12)

Coy (Zhi, Zhj) = (1/12Ph)

Coy (Zhj, Zq) = (1/12pi )

Coy (Z h i, Zq ) = ( - I / 12Pi)

Cov (Zgh, Zii) = O,

for (i 4=j =/:g :¢= h).

To perform the test we declare any Tii < c to be significant and conclude Oi :¢:Oj. The critical value c is determined from the distribution of the minimum of a multivari-

ate normal with mean and covariance structure given in lemma 1.

4 Unknown Location Parameters

Consider tile situation where the X's are shifted by an unknown constant ~i so that

X i ~ F(x - ~i)- Since the new test is not robust in this situation it is desirable to elimi-

nate the effect of the shift. To do this make the pseudo-observations Xi~ = Xii - }i

where ~i is a consistant estimator for the location parameter ~; such that P(l~t - ~il >

b i / x /N )-+0 for some finite fixed number bi. Now perform the test on the XiT's and

obtain the rank sums Ti~'s which can be written as:


Ti~ = ni(ni + I) /2 + ninjUi~

where

n i nj Ui~ (l/n~nj) ~, ~, *" * = h (X~ , X/,,)

g=l /~=l

and

h(Xig, Xjk ) = 1 if Xig <~ X ~ , 0 otherwise,

Tos study the limiting distribution of the Ti~'s we can take ~i = 0 (i = 1 ... k) without

loss of generality. By the use of a multivariate extension of Slutsky's theorem it is

shown that the jo int limiting distribution of the

z,7 = (V~/,,O(T,7- ,,,jA(~,, O)

is the same as that of the Zq's, where A (t x, t2) = E((hXig - t 1 ; Xik - t2 )).

Theorem 2." If Xn is a vector which converges in distribution to a multivariate normal

random vector with mean ~ and covariance matrix X and Yn is a vector which con-

verges in probabil i ty to 0, the Zn = Xn - Yn converges to a multivariate normal distri-

bution with mean ~ and covariance matrix 2;.

To see the application of this theorem in our situation let Xn be the vector of Zij's and let Yn be the vector made up of components of the f o r m ( % ~ - / n i n j ) [ T i j - p q ) ] -

(Ti~ - A ( ~ i , ~1)), where laij = g(Tij ). Then Z . is the vector of Zi~'s. To see the Zi~'s are

asymptotically distributed the same as the Zii's, it suffices to show that Yn ~ 0. It

further suffices to show that each component of Y, converges to zero in probabili ty,

i.e.,

(v~/ , ,~ , j ( [ (T , ; - m ) - (T~ - A(~,, 0 ) 1 ~ 0

Theorem 3: If the densities of X and Y exist and are bounded then

100 G. Stevens

This theorem, which is analogous to Theorem 3.1 of Sukhatme (1958), is a two sample

extension of his Theorem 3.1. The proof of Sukhatme's Theorem 3.1 follows directly

from his Lemma 3.2. Similarly, the proof of the above Theorem 3 follows from Leam-

ma 2 (below), which is the two sample extension of Sukhatme's Lemma 3.2. Without

loss of generality we consider Z~z, assume n 1 = nz and N = 2n.

L e m m a 2." If the densities of X and Y exist and are bounded then define

H q r z N ( X , y , ta, t2) = sup I h ( x - z l ; y - z 2 ) rid/x/N~zi<ti

- Eh(x - z 1 ;Y - z2 ) - [h(x - r l d / x / ~ ; y - r 2 d / x ~ )

- Eh(x - r l d / ~ ; y - r2d /x /N)] l

and

S r l r 2 N ( t l , t 2 ) = ( , ~ / N / n l n 2 ) X ~ {h(x i - t l / V ~ - ; y i - t2 /~c /N) i i

- E h ( x l - t l / x / N ; y j - t z / x / N ) - [ h ( x - r l d / x / N ; y - r 2 d / v / N )

- Eh(x - r l d / ~ / A ; y - r2d /x /N)]}

then

(i) E H r l r 2 N ( x , y , (r l + 1)d/vCN -, (r2 + 1 ) d / , v / N ) < ~ A l d / ~ / ~

where A 1 is some constant, and

(ii) E([Hr l r2N(X, y , q l , q2) -EHr l r2N(X' . Y', q l, q2)]} ~ 0

where

qi = (ri + 1 )d / v /~ , and

(iii) E [ S r l r 2 N ( t l , t2)[ 2 <~ C[(t 1 - r i d ) + (t2 -r2d)]/~c/N.


Using the lemma the proof of Theorem 3 follows in the same manner that Sukhatme's

Theorem 3.1 follows from his lemma 3.2. Thus we get the following corollary.

Corollary 1: The joint limiting distribution of the Zi~'s is a multivariate normal with

mean zero and variances and covariances given in Section 3~1.

We can see this modified procedure is not asymptotically distribution free. We can

also see that when ~i 4 :0 then Zij depends on ~i- The quantity A(~i ~, ~j) becomes

A(~i - ~i,~ i - ~j) and thus depends on knowledge of the parameters of ~i and ~/. We

would like to have

(x/~lnini)(Ti ~ - tJq) ~ (x/Nlnin/)(Tq - I~ii),

A necessary and sufficient condition for this is

OA(tlt2) tl Otl =t2=o

_ OA(tl , t2) q

at2 =t2= 0 = 0

This is clearly seen in the one sample case.

For

( x / ~ - / - , , O ( T , ~ - u , j ) = ( x / - f /n i , O ( T,~ - A (~ - ~ ) )

+ ( x / f - / , ~ , j ) ( A ( ~ - ~) - U , j )

But

A ( ~ - ~) ± -- A ( O ) - (~ - O A ' ( h )

SO

( ' , / f f / , , , ,O(Y~7 - U~j) = (x /N/ , , i , ,p(T,"] - A(~ - O)

+ ( x / ~ / n ~ , O ( ~ - ~) × A ' ( h )

Since x / f f (~ i - ~i) is bounded in probability and i fA ' (O) = 0 we get the desired result.

But in our situation we can see that

102 G. Stevens

aA(t , t2) ' , _ aA(t , , t2) I = ? f2(x)dx Ot] =t2= o bt2 tl=r2=o -**

and this is never zero. Sukhatme shows a sufficient condition for (x/~-/ninj)(Ti~ -laii ) to have a limiting normal distribution is that Tij, ~i and ~j have a limiting joint normal distribution.

Under the stronger condition that x/~-(~i - ~i) ~ 0 we get the fotlowing theorem.

Theorem 4: A sufficient condition that

is that the densities of X and Y are bounded and

v ~ ( ~ - ~) ~ o.

This is useful for densities with supports on the half real line that are shifted so that

the first order statistic can be used as an estimate of the shift. An example of such a distribution is the Weibull with shape parameters between I and 2.

5 C o m p a r i s o n to O t h e r Tests o f D i spe r s ion

The next question to be answered is whether the test has good power against alternative

hypotheses and how it compares to other nonparametric multiple comparison tests for

difference in scale? The second part of the question is easy to answer. According to

Miller (1966) the are no other multiple comparison tests for scale (parametric or non-

parametric); therefore, an "unfair" comparison between the proposed test and other

tests for differences in dispersion is made. This comparison is "unfair" because we

know that multiple comparison tests inherently have less power than the "all or none"

type tests, that is, procedures that test the hypothesis HA ai 4= oj for at least one (i, j) pair.

The proposed test will be compared to the nonparametric multi-sample analogues of the two sample Mood statistic and the two sample Ansari-Bradley statistic. It will

also be compared to Bartlett's test (1937), the Box test (1953), and a jackknife test


(Miller 1968). In these comparisons the distribution shall be symmetrical and the loca-

tion parameters assumed equal. The case of asymmetrical distributions must be handled differently because of the

distributions involved (exponential and gamma). The mean of the exponential is a 1/0

and the variance is 1/02. Both depend directly on 0 so if the variances are unequal the

means are also unequal, which violates one of our assumptions. We can, in this case, indirectly test for the equality of variances by testing the equality of means. (This is

the same procedure used by Tsai, Duran, and Lewis 1975.) To do this Steel's multiple

comparison test for means is used and compared to the multisample analogues of the

sum of squared ranks test (Layard 1968) and the Kruskal-Wallis test (1952), and again

the Box test and the jackknife test. The reason for comparing with the above tests is that they are the "best" and

power studies on them have been done (Tsai, Duran, and Lewis 1975). The symmetric

distributions used were normal, double, exponential, and Cauchy. The asymmetric

distributions were exponential and gamma (shape parameter 2 and shape parameter 5).

The distributions were generated by the IMSL routines. Three equal sized samples (n l , n2, n3) = (5, 5, 5) were generated 1,000 times for each of the underlying distribu-

tions under each hypothesis. To obtain samples under the alternative hypotheses the

observations in the second and third samples were rescaled to produce unequal scale

parameters. The ratio of the scale parameters used in this study were 1:1:1, 1:2:4, 1 :4:6 , and 1:5:25. The proportion of rejections under the various hypotheses were

recorded. This format was used to facilitate comparisons to the Tsai, Duran, and Lewis.

We restrict the following analysis to the proposed test (see Table 1). The true a-level as given by Steel (1960) is a = 0.0413. The proposed test was

robust in the a-level, i.e., it did not fluctuate from distribution to distribution. This

was expected becuase is a nonparametric test and therefore the distribution of the

sample should not affect the test. The simulated size of the test is within the simula-

tion error bound. We can see in the symmetric distributions the power of the test was less than the

Bartlett, the Mood, the Ansari-Bradly, and the jackknife test. This was again expected

because of the nature of multiple comparison tests. The proposed test did have more

power against the alternatives than the Box test. In the asymmetric distribution the power of the test was less than the Kruskal-

Watlis test and the sum of squared ranks test. The proposed test did have more power

than the Box test and the jackknife test.

Because Puri's extension of the Mood test and the Ansari-Bradley test are "all or

none" type tests we expected them to have more power against the alternatives than

the proposed multiple comparison test. Their power was not that much greater than

the multiple comparison test for any distribution or alternative. From this we can

conclude that the proposed test is a fairly good multiple comparison test for differ-

ences in dispersion.

104 G. Stevens

Table 1. Empirical Power Funct ions for Tests of Scale (Three Samples of Size 5 at a = 0.05)

Scale Rat lo

T e s t 1 : 1 : 1 1:2:4 1 : 4 : 1 6 1 : 5 : 2 5

~ o r m a l D i s t r i b u t i o n B a r t l e t t . 0 4 5 . 5 8 8 . 9 8 9 1 . 0 0 0 Mood . 0 3 5 . 3 1 9 . 7 5 5 . 7 9 2 Ansar] .031 .269 .684 .734 Box .O46 .167 .462 . 5 9 7 Jackknlfe ,O38 .318 .836 .930

Mood (X 2 ) .024 .269 .714 .746

Ansarl (X 2) .028 .260 .669 . 7 2 2 New MCT . 0 3 8 . 2 6 3 .566 .643

Double Exponential Distribution Bartlett .137 .599 .980 .995 Mood .043 .220 .654 .689 Ansarl .042 .200 .601 .639 Box .046 .136 .412 .490 Jackknife .056 .296 .744 .827

Mood (X 2) .036 .190 .601 .638

Ansari (X 2 ) .040 .192 .584 .620 New MCT .037 .163 .474 .538

Cauchv bistrlbutlon Bartlett .597 .746 .924 .962 Mood .047 .168 .456 .532 Ansarl ,041 .161 .434 .520 Box .041 . 0 9 0 . 245 . 2 7 8 JackknSfe .156 .283 ,534 .610

Mood (X 2) .036 .135 .403 .481

Ansari (X 2) .O38 .150 .421 .500 New MCT .048 .140 ,370 .447

(Table 1 continues)

References

Ansari AR, Bradley RA (1960) Rank sum tests for dispersion. Annals of Mathematical Statistics

3 1 : 1 1 7 4 - 1 1 8 9

Bartlett MS (1937) Properties o f sufficiency and statistical tests. Proceedings from the Royal

Statistical Society, Set. A, 160:268 282

Box GEP (1953) Normali ty and tests o f variance. Biometrika 40: 318 - 335

Conover WJ, Johnson ME, Johnson MM (1981) A comparative s tudy of tests for homogenei ty o f

variances, with applications to the outer cont inental shelf binding data. Technometr ics 23:

351 362


(Table 1 continued)

Scale Ratio

Test I:I:i 1:2:4 1:4:16 1:5:25

Exponent 5a] Distribution K-Wa/lis .O51 .302 .802 .851 Sq. ranks .O35 .333 .822 .887 Box .O31 .135 .347 .401 Jackknife .046 .300 .647 .749

K-W (X 2 ) .038 .291 .784 ,846

Sq. rank (X 2 ) .024 .284 .781 .856 New MCT .038 .222 .692 .795

Gamma Distribution [shape par. 2) K-Wallls .047 .571 .978 .995 Sq. ranks .044 .591 .987 .997 Box .941 ,151 .410 .517 Jackknife .056 .313 .787 .844

K-W (X 2) .O41 .552 ,973 .992

Sq. ranks (X 2 ) .O31 .536 .980 .995 New MCT .046 .467 .948 .983

Gamma DistrSbutlon (shape par. 5) K-Wallis .048 .944 1.000 1.000 Sq. ranks .O52 .947 1.000 I.OO0 Box .050 .154 .436 .543 Jackknife .O41 .319 .827 .875

K-W (X 2) .043 .936 1 .000 1.000

Sq. r a n k (X 2) .039 .927 1 .000 ] . 0 0 0 New MCT .O61 .908 1.000 1.000

Source: Portions of Table 1 are from Tsai, Duran, and Lewis (1975)

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Received January 1988 Revised version May 1988

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A nonparametric multiple comparison test for differences in scale parameters