Asymptotic (k-1)-mean significance levels of a nonparametric method for multiple comparisons in the k-sample case

method for multiple comparisons in the k-sample case

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Oude Voshaar, J. H. (1977). Asymptotic (k-1)-mean significance levels of a nonparametric method for multiple comparisons in the k-sample case. (Memorandum COSOR; Vol. 7718). Technische Hogeschool Eindhoven.

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PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 77-18

Asymptotic (k-J)-mean significance levels of a nonparametric method for mUltiple comparisons

in the k-sample case

by

J.H. Oude Voshaar

Eindhoven, October 1977 The Netherlands t

by

J.H. Oude Voshaar

1. Introduction and summary.

It is known that the projection argument, by which the Scheffe simultaneous confidence intervals are derived from the F statistic, can also be applied to the Kruskal - Wallis statistic (see Miller [4J, page 165-172). Here we will consider the case of equal sample sizes, say size n, for n large

(n ~ ~). Then we have under the null hypothesis (F

1

= •.. =

Fk);

for all i,j out of {l, ... ,kU= I - a where r. is the mean of the ranks of the i-th sample and qka is the upper

-1

percentile point of the range of k independent unit normal variables. We will be concerned with the following problem:

if F) - ••• - F_{k- 1}

=

F and Fk

=

G, where F; G, what will be the value (for n ~ 00) of

(1. I) a(F,G) := 1 - P lEi - !'jl s qkY-

_[

-

-

~

Jk

(kn 12 + 1) forall i,j out of {l, •• .,k-I} ] i.e. what is the probability of concluding

among F1, ••• ,F

k_1 are different? a(F,G) is significance level. (cf. Miller [4J).

incorrectly that some of the F.'s

1

called the (k - 1 )-mean

The following statement turns out not to be true in general: a(F,G) S a for all (continuous) F and G.

Even when G is a shift of F, a counter example can be given. However, if G is a shift of F and F is symmetric and unimodal then: a(F,G) S a.

If F is not symmetric, the above statement remains true if we replace unimodal by strongly unimodal (or more general: log F and log (I - F)

both concave). Our main result will be that in the case of shift alternatives the (k - 1 )-mean significance level is depending on the skewness of F (in the sense of the convex order relation introduced by Van Zwet [8]).

2. A general expression for a(F,G)

Let ~11""'~ln;"';~1""'~n be independent random variables (k ~ 3), where x .. has a continuous distribution function F .•

-1J 1

Let r .. denote the rank of x .. among all observations and

-1J -1J

r

} n := -

r

r ..

n j_1-1J

To determine a(F,G) for n ~ 00 t we first must know the asymptotic distribution

of the range of !l""'!k-\ for the case F}

= ... =

_Fk-J and Fk

=

G. If we define

(2.1) p :=

jG

(x)dF (x)

(2.2) q := JG2 (x)dF (x)

(2.3) r :- jF(x)G(X)dF(x)

then for i,j E: {It.o. ,k-l } the following relationship can be shown (see Oude Voshaar [6J) (2.4) (2.5) var<'~i)

=

12

1 k n + (2r - p - 7;)kn + (4p-2p +q-6r+(;)n + 2 1 2 1 1 2 2r 1 +T2 k -p + P - q +

-'6

(2.6) cov(r.,r .) = - -1 kn . + (3p - _P 2

_-

4r + T2)n 1

_-12"

1 -1 -J 12 Remark: Under HO (that is F reduce to: 1

=

G) we have p

= 2

and q

=

r - 3'sO (2.4) to (2.6) I

C,(r.)

==

1..

(kn + 1) -1 2 - I 1 var(!i)

=

12

k(k - l)n +

12

(k - J) cov(r.,r.)

= -

~2(kn

+ 1) -1 -J l '

corresponding to the known formulas (cf. Miller [4J. page 171).

Using theorem 2.1 of Hajek [2J, it is also possible to prove the asymptotic

-

-normality of the vector (!I""'!k-l) when F and G are not identical (see Oude Voshaar [6J).

Hence

Y

_n, defined by:

-! -

-v

=

(v 1"'" v 1-1):

=

n ( r 1 t • • • , r_k 1)

-n -n - n,l!.- -

-is asymptotically normally d-istributed with covariance matrix:

a l a -2 a2 I ai, I I

"

I

"

a 1 a'I

..

a)

.

where (2.7) a 1 2 + (2r - p 1 + 4p - 2p2 + q - 6r I 1 :=

IT

k - -)k 4 +(; and (2.8) a₂ := -) k + 3p -

P

2 - 4r + -1 12 12 where and

J

a_{l -} a₂ y := 1 ±

a

+ {k - 2)a 1 2

then ~n has an asymptotically normal distribution with covariance matrix .(a 1 - a2) Ik_I' (Ik_1 stands for the indentity matrix of size k - ) •.

-!

-1

This implies that the range of (a

l - a2) !nl •••• ,(a1 - a2) ~tk-I has the distribution of a range of k - 1 independent unit normal variables

-l-and so the range of {(n(a

l - a2

»

!i' i

=

I"",k-I} also has this distribution (for n + w ) .

Substituting this in (1.1) and letting Sk-l denote the range of k - ) independent unit normal variables, we now have:

(2.9) for n -+- ClO •

where, a

1 - a2 depends on F and G through (2.1), (2.2), (2.3) and

(2. 10) _a 1 2 1 2 I

1 - a2

=

12

k + (2r - p - 6)(k - 1) + q - p - TZ •

For the right interpretation of (2.9) and (Z.IO) it should be noted that

1 -

-a₁ - a

= --

var(r. - r.)

2 2n -~-J

I Z

and under HO we have a

1 - aZ =

12

k •

3. Maximum value of a(F,G)

(1 S i, j S k - ] and n -+- ClO)

In this section we shall compute the maximum value of a(F,G) and we are anxious to know whether or not a(F,G) is smaller than a for all F and G. From (2.9) we see that a(F,G) will be maximized when a] - a

2 assumes its maximum value.

Writing

(3.1) 2r -p =

J

(2F - I)GdF =

f

(2F - I)GdF +

f

(2F - I)GdF

{xIF(X)<!} {xIF(x»!}

we see that 2r - p is maximized for the pairs (F,G) satisfying the following conditions:

! If F(x)

<!

then' G(x)

=

0 b If F(x)

>!

then G(x) = that is: F

=

~ on the support of G.

Now it happens that q - pZ is maximized by the same pairs (F,G) (satisfying (3.2».

For such a pair both 2r - p and q - p2 are equal to

i,

and (by 2.10)

I 2

a

Hence the maximum value of a(F,G) will be

(3.3)

With the aid of a table for the c.d.f. of the range of unit normal variables, e.g. Harter [3], we find:

Table 2.1.

Maximum values of a(F,G) for a

=

0.01,0.025,0.05 and 0.10

k=3 4 5 6 7 8

I

9 ]0 ]2 15 20

a'" 0.01 .0153 .0181 .0182 .0178 .0172 .0167 .0162 .0]58 .0151 .0]43 .0134 0.025 .0303 .0361 .0386 .0385 .0379 .0372 .0365 .0358 .0347 .0334 .0318 0.05 .0512 .0643 .0682 .0690 .0688 .0682 .0674 .0667 .0652 .0633 .0612 0.10 .0877 .1123 .1208 .1240 .1250 • ]250 .]245 .1238 .1224 .1202 .1172

So it may occur that a(F,G) is larger than a.

1 -

-Intuitively it is also clear, that the value of a₁ - a₂ =invar(!l - !2)

is maximal for the pairs (F,G) satisfying (3.2), since in that case the k-th sample

is

expected to receive the midranks.

In the next chapter we shall see that, even when G is a shift of F, var(!] -!2) can assume larger values than under H

O

'

large enough to let

~

2

a 12 k

qk a - a

1 2

in formula (2.9) be smaller than

q:-l .

4. Supremum of a(F,G) for shift alternatives

Now we shall consider only pairs (F,G) for which there exists a real number c such that:

(4. t) G(x)

=

F(x - c) for all x € lR

and ask whether or not a(F,G) ~ a for all such pairs (F,G), that is for all F and c.

our notation:

a(F,c)

:=

a(F,G) where G is defined by (4.1) • Furthermore we define a(F) by the relationship

(4.2) a(F) := sup a(F,c) •

CElR

From this moment also p.q,r and a

1 - a2 should be regarded as functions of

F

and c.

First we try to maximize 2r - p over F and c.

Suppose c > 0 (shift to the right). Then G(x) ~ F(x) for all x E lR and

consequently 2r - p

f

G(2F - l)dF +

J

G(2F - l)dF

~

{xIF(x)<!} {xIF(X»~}

f

2 [2 3 1 2J F= 1 5

o

+ (2F - F)dF

= -

F - - F

= -- •

3 2 F-~ 24 {xlf(x»!} -5

If c < 0, then also holds: 2r - p <

24

(the proof is analogous to the case

of c > 0).

The next example shows that 2: is the lowest upperbound for 2r - p.

Example 4.1. Let F be defined by (4.3) and F(x) := {: :

t ::

m 2 J G(x) := F(x - -) 2 then for m + ~ one will find:

5 1l( 1 2r - p = -₂₄ - V(-) _{m '} 1

- 2

~ x ~ 0 1 O~x~2m

o

Maximizing q - p2 for shift alternatives means maximizing var(F(! - c» over F and c, where ~ has the distribution function F. This is realized by the same F of example 4.1 for m + 00 (see Statistica Neerlandica.

1977, solution of problem nr. 45). but here the shift c should be chosen differently, viz.:

In that case we have

(4.4) q - p 2 -+

24

5 (3 - is) r,:-, ~ .159 for m -+ 00 ,

5 r,:-, 2

so

24

(3 - is) turns out to be the supremum of q - p for shift alternatives.

1

If c

=

2

(then 2r - p was maximized) we have

(4.5) q - p 2 + 192 29 ~ .151 for m + 00 •

Combination of (4.4) and (4.5) gives:

1 2 1 5 1 2 1 5 r,:-,3

IT

(k +

2

k +

16) ::;;

sup (a

1 - a2) ::;;

12

(k +2" k + 2"(3 - is) -

2)

F,c

and after substituting this in (2.9) we obtain the following results:

Table 4. 1 •

Lower and upper bounds for the supremum of a(F) (for shift alternatives). a = .01 a

=

.025 a = .05 a "" • 1 0 k-3 .0079 - .0081 • 0 I 75 - • 0 1 80 .0325 - .0333 .0612 r- .0625 4 .0101 - .0104 .0230 - .0235 .0431 - .0439 .0816 - .0827 5 • 0 1 09 - .01 1 I .0253 - .0256 .0478 - .0483 .0909,... .0918 6 .0113-.0114 .0263 - .0266 .0501 - .0506 .0958 - .0965 7 .0114-.0115 .0268 - .0271 .0514 - .0518 .0987 - .0993 8 .0114-. 115 .0271 - .0273 .0521 - .0525 .1005 - .1010 9 .0114-.0115 .0273 - .0274 .0526 - .0529 . 1019 - • 1022 10 .0114-.0115 .0273 - .0275 .0529 - .0531 • ID25:- .1028 12 .0114 - .OJ 14 .0273 - .0274 .0531 - .0533 .1034 - .1037 15 .0112-.0113 ,0272 - .0273 .0532 - .0533 .1039 - .1041 20 _{.0111-.0111 .0270 - .0270 .0530 - .0531 .1041 - .1042}

In table 4.1 we see that also for shift alternatives a(F) might be larger than a, however the exceedances. if any, are rather small for the usual values of a.

5. Conditions on F such that a (F) <: a (for shift alternatives)

We now investigate which "type" of distribution functions causes sup a(F) to be larger than a and which conditions on F are required to guarKntee a(F) ~ a. The first theorem we prove is:

Theorem 5. I •

If F is symmetrical and unimodal (and continuous) then

(5.1) 2r - p ~

6'

1

and hence a(F) <: a.

Proof. (due to prof. R. Doornbos)

As the problem is translation invariant, it is no restriction,to let F be symmetrical in x = O. Write:

o

~

2r - p

=

f

F(x - c) (2F(x) - l)dF(x) +

J

F(x - c) (2F(x) - l)dF(x)

o

then substitution in the first term of

and gives: so: (5.2) x

=

-x' F(-x' - c)

=

1 - F(x' + c)

o

m

J

F(x - c) (2F(x) - l)dF(x)

=

J

(F(x' + c) - 1) (2F(x') - l)dF(x')

-=

0 2r - p

=

I

(F(x + c) + F(x - c) - 1) (2F(x) - l)dF(x)

o

Now for x > 0, the relationship

(5.3) F(x + c) + F(x - c) s 2F(x) holds.

If x ~ Icl, this follows from the unimodality of F (then F concave for x > 0). For 0 < x < Icl symmetry and unimodality both are necessary to prove (5.3). Putting (5.3) in (5.2) we find: QO 2r - p s I (2F(x) - 1)2dF (x)

o

=

~

-

1

(2F (x) - 1) 3 F(X)=J == -1 • 6 F(x)= 6

As q - p2 s

'*

(3 - 15) <

!

for all shift

a1~tives

_{we have a) - a2} <

i2

k2 + 1, so the square root in (2.9) is larger

than\l.~

which is not small enough

k +1

a

to compensate the difference between qk-l and q~, so a(F) < a , as shown in the next table.

Table 5.1. Upperbound for a(F), when F is symmetrical and unimodal.

k=3 4 5 6 7 8 9 10 12 15 20

a • .01 .0057 .0071 .0077 .0081 .0083 .0085 .0086 .0087 .0089 .0091 .0093

.025 .0135 .0173 .0190 .0200 .• 0206 .0211 .0214 .0217 .0222 .0227 .0232 .05 .0262 .0339 .0375 .0396 .0410 .0421 .0429 .0435 .0445 .0455 .0465 .10 .0516 .0674 .0749 .0793 .0823 .0845 .0860 .0872 .0893 .0913 .0933

Now we want to relax the conditions on F in theorem 5.1, especially the symmetry, a requirement which is often not fulfilled in practice. But unimodalilty alone is not sufficient to ensure a(F) S a , since F in example 4.1 also is unimodal. As we shall see later, strong unimodality will be sufficient and also a more general theorem will be proved.

As F in example 4.1 is extremely skewed for m 7 QO , one would guess that

a(F) is larger when F is more skewed. We shall see that this guess is the right starting point to come to a more general theory.

To describe the skewness of distribution functions, we shall use the convex order relation, introduced by Van Zwet [8].

During the remaining part of this section we shall restrict ourselves to a class F on which that weak order relation will be defined.

Definition 5.1.

Let

F

be the class of distribution functions F for which there exists an interval IF

=

(xl,x_Z) (xI and X

_z

may be not finite) such that the following

three conditions are satisfied:

(5.4) F(X I) = 0 and lim F(x) = xtx 2 (5.5) _{F is differentiable on IF} (5.6) F' >

o

_{on IF}

Note that F differs somewhat from the class F defined by Van Zwet.

Definition 5.Z.

For F,G E

F

we say:

-I

F ~ G if and only if G F is convex on IF

F < G should be interpreted as: G is more skewed to the right then F.

c

If the densities of F and G are called f and g, then we also can say (lemma 4.1.3, Van Zwet [8J):

F < G if and only if c (5.7) f(F-1 (u» -1 g(G (u» is nondecreasing for u E (0,1) •

Furthermore we define for F E F (where x has distribution F)

(5.8) Zr - p(F) := _sup

f

CEm. 0 I

(2u - 1»)(F(F-1(u) -c»du

= Z

sup

cov(F(~),F(~-c»

11

-and

(5.9) q - p (F) 2

:=

sup var F(~ - c)

c€m.

and let F* be defined by

(5. 10) F (x)

*

:= 1 - F(-x)

Note that F* is also an element of F and -~ has the distribution function F*. "F is less skew than G" can now be expressed as:

(5. 11) G* < F < G

c c or G < c F c < G*

2 2

We want to prove that (5. 11) implies 2r - p (F) ~ 2r - p (G) and q - p (F) ~ q- p (G) and hence a(F) ~ a(G), since a(F) is an increasing function of 2r - p(F)

and q - p2(F).

But first we have to state some lemmas.

Lemma 5. I.

F < G if and only if G* < F*

c c

Proof.

-I I

• G F(-x) is convex in x, so F- G(-x) is concave and hence (F*)-IG*(x) = - F-I(G(-x»

~s

conv"ex.

- Note that F**

=

F •

o

Lemma 5.2.

Let fl and f2 be functions on an interval I c:

m.',

where f2 is positive

and f / f 2 is nondecreasing on I.

I f furthermore x

l,x2,x3,x4 € I such that xI ~ x3 and x2 ~ x4 then:

f

This lemma can be proved by elementary calculus.

With the aid of these lemmas we now prove the following theorem:

Theorem 5.2. (5.13) If F,G €

F

and G

*

< F < G then 2r - p(F) S 2r - p(G) c c Proof. First we show: I I

f

-1

f -

I

(5.14) i f F ~ G then sup (2u - I)P(F (u) - c)du S sup (2u -I)G(G (u) - c)du

CE(O.~)O CE(O.~)O

which has been proved, if for any c > 0 there exists a c' > 0 such that:

(5. 15) P(F -1 (u) - c) 2: G(G -1 (u) - c') for u E (0,

n

and

(5. 16) for U E

(l,

1) •

Take c' such that it satisfies:

hence

(5. 17)

c'

=

G -1 (i) - G - 1 - 1 (F(F (i) -

c»

If inf Ip >

-~

and c 2:

F-I(~)

- inf IF' then take

c'

=

G-1 (l) - inf IG •

(If ' _l.n f I _F > -co th en a so 1 ~n . f I G > ~, as G-1F ... ;8 convex).

For u >

i

we use lemma 5.2 with:

f} (G -1) ,

,

f

=

2 (F -1) , Xl = F(F- 1

(D

- c) x₂ =

i

x3 = F(F-I(u) - c) x₄

=

u

Then f)/f2 is nondecreasing because of 5.7, so we can conclude:

and with (5.)7):

hence (as c > 0)

F(F-J (u) - c)

~

G(G-1(u) - c') for u >

I .

The proof of (5.16) is identical. except for the interchangement of Xl and x₂ and also of x3 and x

4•

Thus (5.14) has been proved.

If c < 0 we have to make use of G* < F.

c

By lemma 5.1 this is equivalent to F* ~ G, hence with (5.14) we have:

(5.18) By (5.10): so and hence 1 1

<I

* *-1

I

-I

sup + (2u-l)F(F (u)-c)dussup+ (2u-I)G(G (u)-c)du

C€m 0 C€m 0

= - F-10-u)

*

*-1 -1

F (F (u) - c) = - F(F (1 - u) + c)

1 1

J

(2u - I)F (F

*

*-1 (u) - c)du

=

- J

(2u - l)F(F-1

0-u) + c)du =

o

0

1

=

J

(2u - 1)F(F-1 (u) + c)du •

o

1 (5.19) sup _ C€lR

J

(2u-o

-I

I)F(F (u) -c) s sup +

C€lR I

J

(2u-o

Combining (5.14) and (5.19), the proof is completed.

Remat:k. -I l}G(G (u) - c)du •

*

Also (5.11) implies 2r - p(F) S 2r - p(G}, as 2r - p(G )

=

2r - p(G) • 2

To prove the analogon for q - p • we will need the following lemma:

Lemma 5.3.

If fl and f2 are functions on (0,1), such that

(i) 1

f

fl (x)dx

=

o

1

J

f 2(x)dx

o

< eo

(ii) there exists a Xo € (0,1) such that fl (x) S f

2(x) for x € (o,x

o)

and fl(x) ~ f₂(x) for x € (xO,I) then 1

J

xf_l (x)dx ;::

o

I

J

_{xf 2(x)dx •}

o

This lemma is a special case of a theorem due to J.F. Steffenson (see Mitrinovic [5J, page 114, theorem 13).

Theorem 5.3.

*

If F,G € F and G < F < G then q - p (F) 2 c c 2 S q - p (G).

Proof. Let 2 and ~ have distributionfunction F and G. As ~, defined by

y :-

F(~

- c), has the distribution function H(u)

=

F(F-1(u) + c), we

have:

(5.20) and (5.21) First we (5.22) We take (5.23) 1 1

eu

=

1

-J

H(u)du - 1 -

f

F(F-I(U) + c)du

0 0

1 1

£:l

= ] -

2

J

uH(u)du .. ]

-

2

I

uF(F -1 (u) + c)du •

0 0

prove that for any c > 0 there exists a c' E: lR such that

c'

var F(! - c) :::; var G(l: - c') • such that eF(! - c)

₌

CG(l: c I) ,

1 1

f

F(F-1 (u)

o

+ c)du

=

I

G(G-1 (u)

o

that is + c ')du •

There exists such a cl

, since F and G are continuous and hence the integrals are continuous functions of c and cr.

By (5.20), (5.21) and (5.23) we have that (5.22) is fulfilled if

1 (5.24)

_f

uF(F

-}

(u)

o

1 + c)du

~

f

uG(G-J(u)

o

This follows from lemma 5.3, if we substitute

+ C I )du

-1

f) (u) ~ F(F (u) + c) and f -]

2(u)

=

G(G (u) + c') •

Condition (i) of lemma 5.3 is satisfied by (5.23) and condition (ii) is satisfied because:

-1 -)

1. By (5.23) there exists U

o E: (0,1) such that F(F (uO) + c)

=

G(G (uO) + c')

-1 -1

(F, F , G and G are continuous).

2. As F < G we can use lemma 5.2 in the same way as in the proof of theorem 5.2

c

with! replaced by u

o• This gives F(F-I(u) + c) :::; G(G-1

(u) + c') for u E: (o,u

o)

and the reverse inequality for u E: (uO,I). Hence now we have:

(5.25) F ~ G .. sup + var F(! - c)

c~lR

~ sup var G(X - c) • cdR

*

*

I f c < 0 we use G < F, which 'is equivalent to F < G.

c

*

c

As -! has distribution function F , (5,25) gives:

*

sup + var F (-! - c) ~ sup var G(X - c)

celR ce:lR and because

*

var F (-! - c)

=

var F(! + c) we find

*

G ~ F .. sup _ var F(! - c) ~ sup var G(X - c) •

ce: lR ce:lR

Together with (5.25) this completes the proof.

0

Application:

Now we take G defined by (the negative exponential distribution):

(5.26) G(x)

=

1

-*

-x

e (0,00) •

Then G < F < G is equivalent to:

c c

(5.27) log F and log(1 - F) both concave •

Since for G defi~ed by (5.26) the following relation holds:

3 2 J

2r - peG)

= --

and q - p (G)

=

-16 9

we may conclude that if F satisfies (5.27), then by (2.10)

k=3 4 5 6 7 8 9 10 12 15 20

a-.Ol

.0053 .0073 .0083 .0088 .0092 .0094 .0095 .0097 .0098 .0099 .0100 .025 .0127 .0176 .0200 .0214 .0223 .0229 .0234 .0237 .0241 .0245 .0248 .05 .0249 .0345 .0393 .0422 .0440 .0453 .0462 .0468 .0478 .0486 .0493 .10 .0496 .0682 .0777 .0834 .0870 .0895 .0914 .0928 .0947 .0965 .0979 Corollary.

Table 5.2 is also valid if F is strongly unimodal.

Proof.

If F is strongly unimodal, that is log f is concave. then log F and 10g(1 - F)

both concave.

If F is twice differentiable, this can be proved with the aid of lemma 5.2, but here a more general proof (due to F.W. Steutel) is given:

(5.28)

(5.29)

log f concave .. log f (u + a) - log f (u) ;,: log f (x + a) - log f (x)

.. f (u + a) ;,: f (u) f (x + a) f (x) f (x + a) < f (x) log F concave .. F (x + a) - F (x) .. F (x + a) ;,: F (x) f (x + a) f (x)

for all u < x and a > 0

for all x € lR. and a > 0 •

Now (5.28) implies (5.29) because

F(x + a) f (x + a) x

=

f

x f (u + a) du;,:

f

f (u) d == F (x) f (x + a) f (x) u f (x:)

6. Sgme final remarks

Of cgurse a(F.G) (or a(F) for shift alternatives) can only exceed a if the distribution of !I""'!k-I depends on the distribution of the k-th sample.

This is not the case with the mUltiple comparisons procedures for normal models (e.g. the methods of Tukey and Scheffe). Consequently, if not all samples come from the same distribution , the probability of concluding thgse samples to be different, which in fact are identical, is smaller than a fgr these methods.

The same holds true for the nonparametric method proposed by Steel [7], since here the rank means of the i-th and the j-th sample are computed from thgse two samples only and not from all k samples together. Besides the fact. that for Steel's procedure the outcome of the comparison of two

samples is not influenced by the other observations, it also has in general a larger power than the method based on the Kruskal-Wallis test (see:

de Boo [I]).

So we may conclude that Steel's method is more suited for nonparametric simultaneous inference in the k-sample case.

Finally it should be remarked that the method derived from Friedman's test (see Miller 4, page 172-178) suffers from the same defect and

conse-quently the (k-l)-mean significance level may also be larger than a. But this subject will be treated in a forthcoming paper.

References

[I] de Boo, Th.M. (1973): Een vergelijkend onderzoek naar vier para-metervrije k-steekproeventoetsen, Memorandum CaSaR 73-11,

Eindhoven Technological University.

[2J Hajek, J. (1968): Asymptotic normality of stmple linear rank statistics under alt~rnatives, Annals of Mathematical Statistics, vol. 39,

[3] Harter, H.L. (1969): Order statistics and their use in testing and estimation, vol I, Aerospace Research Laboratories,

Government Printing Office, Washington.

[4] Miller, R.G. (1966): Simultaneous statistical inference, McGraw-Hill, New York.

[5] Mitrinovic, D.S. (1970): Analytic inequalities, Springer-Verlag, Berlin-New York.

[6] Oude Voshaar, J .H. (I 976): "(k-l )-mean significance levels" van de asymptotische versie van de methode voor simul tane ui tspraken voor het k-steekproevenprobleem gebaseerd op de toets van Kruskal

en Wallis, Memorandum COSOR 76-27, Eindhoven Technological University. [7] Steel, R.G.D. (1960): A rank sum test for comparing all pairs of

treatments, Technometrics, vol. 2, page 197-207.

[8] Van Zwet, W. R. (1964): Convex transforma'ions of random variables (dissertation), Mathematical Centre Tracts, Amsterdam.