UNIVERSITY OF NORTH CAROLINA
De:PBortment of Statistics
Chapel gill, N. C.
,
SOME SCHEFFE-TYPE TESTS FOR SOME BEHRENS-FISHER-TYPE
REGRESSION PROBLEMS
by
Richard F. I'otthofi'
October ·1963
In psychological and other applications, it may be necessary to make
certain comparisons of two regression lines when the variances are
unequal. Such problems arise rather frequently, for example, in
studies comparing two alternative curriculums or two different teaching
metht~. By generalizing an idea which Scheff~ used to obtain a test
for the Behrens-Fisher problem, this paper deVelops some tests for
comparing two regression lines when the two sets of error terms are
nor;ma11y distributed but with two different variances. Scheffe's
test itself is a randomized test, but in this paper we present both
randamized and non-randomized tests. Both simple and multiple re'A gression are considered, but the simple regression tests ape eom'.P~tgt~QP~11y~~sier than the multiple regression tests. The basic
'test statistic which. is used is the ordinary t-s~istic. Essenj tia11y two types of problems are dealt with: {A) determining whether
~the two regression lines are identical when they are kno¥m to be paral-le1; and (B) determining whether the two regression lines are parallel.
~~onfidence bounds as well as tests of hypotheses are a'\71ailable. This
j'paper is on a theoretical level; a more practically-oriented discussion,
With numerical examples, is given in a separate paper.
>
,'>
l
•
/
This research was supported in part by a grant from the NAIDa Research
Grants Programme under a joint arrangement between the Institut de
Statistique of the University of Paris, the Istituto di Ca1co10 delle
Probabilita·.·otc_<~heUniversity of Rome, and the Department of Statistics
of the UniversitY'of North Carolina; it was supported in part by Educational Testing Service; and it was supported in part by the Air Force
Office of Scientific Research.
Institute of Statistics
Mimeo Series No. 374
•
LI UNIVERSITE" DE LA CAROLINE DU NORD
Department de Statistique
Chapel Hill, Caroline du Nord
/
QUELQUES TESTS DU TYPE DE SCHEFFE POUR QUELQUES
,
/
PROBLEMES DE LA REGRESSION DU TYPE DE BEHRENS-FISHER
par
Richard F. Potthoff
octobre 1963
Dans les applications psychologiques et les autres applications,
il peut ~tre necessaire de faire des certaines comparaisons de
guand les variances sont inegales. De
deux lignes, de r~gression
/ tels problemes se presentent assez frequemment, par exemple, dans
les etudes pour comparer deux curriculums alternatifs ou deux
,;
.'"
I
,;",
/
methodes
d~fferentes d enseignement.
En generalisant
une idee
que Scheffe a utilisee pour obtenir un test pour Ie probleme de
B,een,efW"Ftsh.elej ce papier developpe quelques tests pour comparer
deux lignes de regression quand les deux ensembles d1erreurs
residuellessuivent la distribution normale mais avec deux
variances differentes. Le test de Scheffe IUi-m~me est un test
randomis~, mais dans ce papier onpresente a la fois des tests
randomises et des tests non randomises. On considere a. la fois
laregression
simple et la regression multiple,
mais les tests
/
,
de la regression simple sont plus faci1es a calculer que 1es tests
de 1a re'gression multiple. La statistique fondamenta1e des tests
c lest tout simp1ement 1e t de Student. On considere deux sortes de
probl~mes: (A) determiner si deux lignes de regression sont identiques quand on sait qu ' el1es sont paralleles; et (B) determiner si
1es deux 1ignes de r~gression sont para11S1es. Les limites de confiance ainsi que 1es tests des hypotheses sont disponib1es. Ce
papier est p1ut~t theorique; dans un autre papier on presente une
discussion plus pratique, avec des exemp1es numeriques.
", ,;
A
Cette recherche a ete rendue possible partie11ement grace a des
1
fonds du Programme .d I Aide A 1a Recherche de 1 0TAN s ous un arrangement conjoint entre 1 1 Institut de Statistique de l'Universite de
Paris, l ' Istituto di Calco10 delle Probabi1ita de 1 1 Universite de
Rome, et 1e Department de Statistique de l'Universite de 1a Caroline
du Nord; c 'etait soutenue partie11ement par l'Educ~tiona1 Testing
Service; et c1etait soutenue partie11ement par 1 t Air Force Office of
Scientific Research.
~
Institut de Statistique
Serie Mimeographique, Num~ro 374
...
/
SOME SCHEFFE-TYFE TESTS FOR SOME BEHRENS-FISHER-TYFE
REGRESSION PROBLEMS
by
Richard F. Potthoff l
University of Paris and University of North Carolina
= = = = = = = = = = = = = = = == = = = = = = == = = = = = = = = - - =
==
1.
INTRODUCTION
A solution to the Behrens-Fisher problem involving randomized pairing of the
two samples was presented by Scheffe [7J.
The present paper attacks certain
Behrens-Fisher-type regression problems by developing some tests somewhat similar
to the test of [7].
Two basic Behrens-Fisher-type regression problems will be considered in this
paper.
The first, which we will call Problem A, consists of testing whether two
regression lines are identical when they are assumed to be parallel, under the condition that the two variances may be unequal.
we have mutually independent observations
More specifically, we suppose that
(Y , Y2 , ••• ,
l
14)'
(Zl' Z2' ••• , \ )
such that
(i
(la)
= 1,
2, ••• , M)
and
(j = 1, 2, ... , N)
(lb)
where the
eits
are
N(O,
~;), the
are unknovm parameters, the xifts
fj'S
are N(O,
and Wjfts
,
~~), the a's, ~'s, and ~2,s
are (knmm) fixed constants, and
~is resear-'ch 'Vas SUJ?p';ted' in -part by a grant from "the· 'Hmo Res~rch Grants
Programme under a joint arrangement between the Institut de Statistique of the
University of PariSi the istituto di Calcolo delle Probabilita of the University
.of" Rome, and the Department of Statistics of the University of North Carolina; it
'Has supported in part by Ec1ucational Testing Service; and it was supported in part
by the A±~ F~ce ~~fi~ of S~ie~ific R~se~ch.
..
2
M is assumed (without loss of generality) to be < N.
•
(2)
We desire to test
•
The second problem, to be called Problem B, consists of testing whether two
regression lines are parallel when the variances
~y
be unequal.
More specifically,
the set-up for Problem B is the same as for Problem A, except that instead of (1)
our model is
C3a)
Yi
= Ciy
+ I3YIXil + I3Y2Xi2 + ••• + I3YrXir + e i
(1
= 1,
2, ••• , M)
(j
= 1,
2, ••• , N) ,
and
and the hypothesis to be tested is
Scheffe-type tests will be presented for Problems A and B.
All our dis-
cussions, in addition to considering the multiple regression situation with general
r, will also pay particular attention to the special case
tests are simpler.
Section 2
r = 1, for which the
is concerned with randomized tests, while Section 3
develops non-randomized tests closely akin to the randomized tests of Section 2.
For
r
= 1,
the non-randomized tests are considerably easier to calculate than the
corresponding randomized tests and at the same time appear to be almost as powerful, so that probably they would usually be preferred; for
picture is less clear.
r > 1, however, the
Many users of statistics object to randomized tests on
philosophical grounds which have been adequately expounded elsewhere but which have
I--
perhaps failed to convince many mathematical statisticians.
Even if the randomized
tests of Section 2 are not used, however, they still provide both a standard of
comparison for and a basis of constructing the non-randomized tests of Section 3.
~
An entirely different approach to Problems A and
B, based on test statistics
analogous to the Wilcoxon statistic, was presented respectively in [6]
and [5J
3
•
for the case 1'=1; the test for Problem A [6] is rather complicated, but the one
for Problem B [5J
is much simpler.
Welch [8J
has proposed a technique, based
on an approximating t-distribution, which is applicable to Problem B for
Recent results of Hajek concerning a generalized t-distribution [3J
to obtain a conservative test for Problem B when r=L
1'=1.
can be utilized
Thus it seems to be chief-
ly for the case of Problem B that competing tests are available.
No detailed comparison of alternative tests will be attempted here.
we may simply remark that the tests of [6], [5J, [8J, and [3J
However,
are all inexact
(Le., the actual level of significance is only approximately equal to the stated
the
level of significance) but become less inexact with increasing M, whereas alII ttests of the present paper are exact.
Consequently, it appears that the tests of
this paper will be particularly useful vThen M is smalL
The technique of Scheff~ [7J, which provided the inspiration for the tests
4It
developed in the present paper, has also been generalized with respect to an entirely different problem by Mazuy and Connor [4J.
Scheffe-type tests for certain
two-way ANOVA models with unequal variances are considered in [4J.
2.
RAlIDOMIZED TESTS
The tests of this paper, like Scheffe's test [7J, will all be based on the
ordinary t-distribution (except that for Problem B with general r
bility of an F-test will also need to be considered).
the possi-
In this section we will
seek optimal tests which have as many degrees of freedom for
t
as are readily
available and which are based on minimum variance estimators, a method of attack
closely related to that used by Scheffe [7J (which was to obtain a certain confidence interval of minimum expected length).
For Problem A, however, it will turn
out that we will have to seek a minimax estimator instead of a minimum variance
•
estimator • The randomized tests of this section will all be optimal in the sense
4
•
just indicated, whereas the non-randomized tests of Section 3
optimal to a lesser or greater degree.
~ll
all be sub-
(In using the variance of an estimator as
the criterion for judging a test, we are assuming implicitly that the variance of
the estimator is closely related to the power of the test_ an assumption which
seems completely reasonable but which we will make no attempt to prove.)
2.1. Problem A for general r.
Let us first attempt to base a test of H (2)
biased estimator of a
= az
- aye
on the minimum variance un-
OA
"
This estimator, which we ""ill call 0:*, can
easily be found if we apply standard least squares theory to the homoscedastic
- kYl , kY2 , ••• , kYM' Zl' Z2' ••• , ZN' where we define k = af/ae • We
need some more notation. Let y, z, 'if' and Wr(f = 1, 2, ••• , r) denote the means
variables
of the
Yi'S, Zjts, Xir's, and Wjr's
respectively.
vectors containing the 'ir's and Wf's
X(Mxr)
and W(Nxr)
Define two matrices
= X - jMX '
a column vector containing all l'S.
taining the Y 's
i
respectively.
be matrices containing the Xif's
x(Mxr)
and Z 's
j
and w(Nxr)
X
Let
=W-
and W be
rxl
Define U(rxl) = W and Wjf's
X. Let
respectively.
jNWI, where
j
denotes
Let Y(Mxl)
respectively.
and Z(Nxl) be vectors con2
Define K = k • Then it turns out
that
~
=
(z - y) - U'[wtw
+ K xlx]-l
(Wi
Z + K x' Y)
,
and the variance of this minimum variance unbiased estimator is
2
2
1\
af
a
[-1w'w + -1- x'x J- l U
(6) var(Ct*) = N +
+
2
2
at
ae
Unfbrt~na tely, however, (5) is not free of the unknown nuisance parameter K,
t
ut
and so it is not possible to use (5) either as an estimator of a
for constructing a test of H (2).
OA
or as a basis
This suggests that we should try a minimax
5
•
approach.
1\
Consider a completely general linear function a
which is an unbiased
estimator of a, i.e.,
1\
=
a
where
g(Mxl)
,
hi Z - gf Y
and h(Nxl)
must be chosen so as to satisfY
(8)
glX
1\
= a.
in order to have E(a)
var(~) =
(9)
~~
hlh
= h 'V1
Observe that
+ gIg
1\
~~
If we attempt to choose a (7)
so that its variance (9) is minimax, we run
into trouble, because, no matter what selection is made for
variance (9) Will be unbounded with respect to ~; and ~~.
g and h, the
However, in place of
~ (9), let us consider an alternative loss function
1\
1\
(10)
L
= L{a,K)
K h'h + glS
var(a)
=
var(~) = (l/N)K + (11M) +
=
(liN)
which is the ratio of
we select
K UI [wtw + K x'x]-l
U
h'h + (11K) S'g
+ (l/MK) + (11K) U' [(l!K)w'w + xlx]-l
U'
(9) lit (6), 1.e., the ratio of the variance of the estimator
(7) to the variance of the ideal but impossible estimator (5). L (10)
may thus be considered as the ratio of our actual "loss" to our smallest possible
"lOSS", and hence seems to constitute an appropriate 106s function itself.
always be
~
L must
1, but fortunately it is bounded above With respect to K; the closer
L is to 1, the better.
...
6
•
A
A
"That we will now do is to find an Ct 'Vlhich minimizes the quantity max L(a,K) •
K
For the time being, let us consider a class of estimators less general than
(7),
of the form
A
Ct
( 11 )
(_
_)
_
c = Z - Y - U'[w'w + C XIX]
-1 (
Note that (11) is formally identical with
(5),
and indeed was suspected as being a
fruitful source of an estimator precisely because of
an unknown nuisance parameter, whereas
quired to choose.
of a
(5); however,
K in (5) is
C in (ll) is a number which we are re-
It is easy to show that
regardless of the choice of C.
(C ~ 0).
w' Z + C x' y)
A
etc
(11)
will be an unbiased estimator
Its variance is
2
O'f
. -1
2
2
2
+ U' [w'w + C x'x]
[w'w O'f + C x'x 0'eJ
If"
. [w'w + C xlx]-l U •
The ratio of (12)
(13)
L{~C,K)
=
=
to
is
l
2
l
(l!N)K + (l/M) + U'[w'w+cx'xJ- [Kw'W+C x t x}[w'vT+Cx'xr U
l
(l/N)K + (lIM) + K Ut[w'w + K xtxr U
(lIN)
2
+ (1!JMK) + U'[w'w + CX'xJ-lrw'w + (C IK)x'x][w 1w + Cx t x]-40
l
(liN) + (1!JMK) + (11K) tit (l/K)wtw + xtxr U
Taking the limits in (13) as
( 14a)
(6)
K -> 0 and as
K ->
0),
we can write
l
+ MC 2 -U' [wtw + C x'x] -1 x'x [ w'w + Cx'xJ-
)
L("
Ctc'O
=1
L(~C'CO)
= 1 + N U [w'w + Cx 1x]-1 w'w ['Wt&t + CX'xr
and
(14b)
l
U
-U
.
7
•
~wrw + K xJx]-l is a
Combining (13) and (14), and then utilizing the fact that
positive definite matrix, we find that
"
L(ac,K)
=
"
"
(11M) L(ac,o) + K(l!N) L(ac'oo)
(11M) + K(l!N) + K Ut[w'w + K xtx]-lu
<
"
<
"
,
max [L(ct,O), L(aC'oo)]
from which it follows that
We
v~nt
to minimize (15) with respect to C.
creasing function of C and
If
(14a)
is a strictly in-
(14b) a strictly decreasing function (which we will
soon prove to be the case), then there will be a unique value of C(which we will
call C*, say) for which
(14a) equals (14b), Le., there will be a unique C
satisfying
and (15) will of course be minimized at the point C
"
= C*.
"
Thus aC* will be the
(11), and we will show later that
minimax estimator in the class of estimatorsa
C
"
a C* is also minimax among all estimators of the more general class aA
(7).
Right now, though, we investigate the two sides of (16) in order to establish
the uniqueness of C* • If we define
Il
=
(M!N)C, G(rxr)
then the two sides of
(16)
= (1;M)
become
x'x, and H(rxr)
= (l!N)wl,w
,
•
8
•
Since G and H are both symmetric positive definite, there exists a non-singular
matrix P(rxr)
such that
=I
and plHP
(19)
plGP
where n(rxr)
= D,
i.e.,
G
z:
(pt )-~-l and H
= (pt r l
D p-l ,
is a diagonal matrix whose diagonal elements (vlhich are the charac-
teristic roots of G-1H) are all > 0 (we are appealing here to a standard matrix
theorem, which is given, e.g., in [1, p. 341]).
two sides of
After application of (19), the
(18) change to
,
(20)
where
= pi U.
Q(rxl)
(16) also]
It is apparent that the left side of
(20) [and hence of
is an increasing function and the right side a decreasing function of
~ (and hence of
C), which is what we were attempting to prove.
Thus (16)
has
the unique solution C*.
We now return to the general class
A
a (7) and show that there is no
A
a for
A
which max
L(a,K)
is smaller than
K
(21)
•
1\
Now clearly max L(a ~ K)
is not less than
K
(22)
A
max [L(a,o)
A
1
L(a~Q.)]
=
max [M gig, N h'h] ,
where (22) was written by taking the limits in
(10) as
K
-> 0 and as K _>
c:;..
Thus it will suffice to show that the minimum of the right side of (22) [with
respect to all
define
(g,h)
satisfying (8)J
is
> (21). When (8) is observed, we may
•
9
a(rx1)
=
Xl g
= W'h
For any fixed value of a, we can minimize Mgf g sUbject to the conditions
and g'X
= a';
g'jM=l
the solution, which can be found rather simply by using Lagrangian
mUltipliers, is
(23a)
min (M g'gla) = 1 + M(a -
X), [x'xr 1 (a - X) •
g
Similarly, for any fiXed a, the minimum of Nh'h subject to the conditions
(23b)
min
(Nh'hla)
=1
+ N(a -
W)'
W) •
[w'w]-l (a -
h
Hence it follows from (23) that the minimum of the right side of
to the conditions
•
(24)
(8)
(22) subject
is
1 + min max [(a a
X), G- 1 (a - X), (a - Wp H- 1 (a - W)J •
vlhat we will now show is that
(24) is equal to (21).
N.I."rtAl first that the minimum in (24) must occur at some value a such that
(a - X), G-1 (a _ X) = (a _ W)' H- 1 (a - w)
Hence we can find the minimum in (24) by using a
mizing
(a - X) f G- 1 (a - X)
( a - -)
X t G-1( a _ -)
X ..
Lagrangian multiplier and mini-
subject to the condition (25).
~[ ( a
Differentiating
. -1( a - -)
.. X)'G
X .. ( a - -)
W 'H-1( a - -)
W]
with respect to a, we easily obtain the relation
after setting the vector derivative equal to zero.
It
Substituting (26) into (25),
we find (after some matrix manipulations) that the tyro sides of (25) can then be
10
•
written respectively as
But (27)
~
is identical with (18), if we set
= ~/(l
+ ~). Thus (25) is equal
to (16) at the point of solution C = C* [i.e., at the point ~ = MC*/(N + MC*)],
from which it follows immediately that
(24)
is equal to (21).
1\
the proof that 0C* is minimax among all estimators of the form
By combining
(16)
This completes
(7).
and (12), we can write the variance of the estimator as
(28)
•
where
s
is the known constant
s = 1 + N -U' [w'w + C*x'x]-1 w'w [w'w + C*X'x]-1. -U
•
A
*'
We now develop a t-test of H (2) based on the optimal estimator aC
Note
OA
first that there exist matrices Ex(M x [M-r-l]) and ~(N x [M-r-l]) such that
(30a)
EW Evl =
I
and
(where
0
denotes a null vector or matrix).
1
(31)
VA
= ;;2 EX
Define an
1
Y
+
2
N- EW z
•
(M - r - 1) x 1 vector
11
•
hom (,oyand (1) we' see that VA (:;1) is always
writing
cr1 = (cr;/M) +
N(O, cr~ I), where we are
(cr~/N). Hence V1VA/cr~ follows the
2
X -distribution with
1\
(M-r-l) d.f.
Furthermore, it follows from (:;Ob) that
are ·ind.~:pet.tA~nt. Since
(~C*
..
ttt:~ 1~
INtO, s
o:C* [see (1;t)J and VA (:;1)
of) [see (28-29)], 14'e conclude that
1
2
follows the t-distribution with
be used to test
H (2).
OA
(M-r-l) d.f. if H (2) is true.
OA
Hence (:;2) can
Although our em:phasis here is formally on the testing of
a hy:pothesis, we can of course also obtain confidence bounds on
0:
if we wish.
1\
The numerator of
(:;2) obViously is unique, since o:C*
(11), taking C = C* to be the solution of (16).
is not unique, inasmuch as the n:a trices
EX
But the denominator of (32)
and ~1 vlhich a:ppear in
be chosen in infinitely many ways and still satisfy (30).
choic~ of
Ex
is calculated from
(:;1)
Furthermore, for any
and ~1' if the columns of either matriX are :permuted, then
will always remain satisfied but VA (31) (and also VlVA)
different value for each permutation.
can
(30)
will generally have a
This suggests that, once EX and
~1
have
been obtained (by some process which itself would necessarily have to be arbitrary),
then the colunms of one of the matrices ought to be :permuted by means of a random
permutation.
vIe will not attem:pt to consider here the question of whether it is :possible
to get a t-test with more degrees of freedom than the
(M-r-l)
d.f. of
(32).
Rough preliminary investigation indicates, however, that this problem may involve
mathematical difficulties whose effect could be such as to render the problem of
little practical importance anyway.
2.2.
Problem
, A for
r=l.
For the special case
r=l, the solution C = C*
of
(16) is simply
•
where we deftJ;te CT~ = w'w!N and CTi = x·x;M.
Hence, upon substitution of (33),
(29) becomes
.
~
and from (11) and (33) we obtain the formula
"cz *
C
- -
Thus, when
tit
by plugging
and
2.3.
Ew
-L
(W'Z/NCTW) + (X,y;MCTX) I
(Z - y) - (W - X ) .
J'
.
.
CTW + CTX
=
r = 1, the t-statistic (32) for testing H (2)
OA
(35), (34), and
in (31)
is calculated
(31) into the formula (32), where the matrices
were selected so as to satisfy (30).
Problem B for
r=l.
In tackling Problem A, we first tried to base a test of H (2)
OA
minimum variance unbiased estimator of a = a Z
the estimator
(5)
r
cry;
contains a nuisance parameter.
such difficulty as this is encountered:
general
EX
and letting t3y (rx1)
on the
but this didn't work, because
For Problem B, however, no
considering for the moment the case of
and t3 (rxl) denote vectors containing the
Z
t3 y ('S and t3 Z!' S respectively, we find easily that the minimum variance unbiased
estimators of the r elements of t3 (rxl) = t3 - t3 y are given by
Z
(36)
e
and
~*
(rxl)
=
[w 1 w]-1 w1z
(36) [unlike (5)]
_
[x'x]-l x'y
is of course free of the unknown nuisance parameter K.
[Note here the obvious point that our hypothesis
HOB (4)
is merely the same thing
•
as
f' = O.J
For the case
HOB (4)
r=l, it is an easy matter to discover a randomized
1ft.
based on the estimator t)* (:36).
First note that
2
=
when
r=l.
(:31)
which we proceed analogously to
(:,8)
=
VB
VB (38)
2
N(O, O"B).
is
N(O,
M-'2
satisfying
and define the
1
(say)
O"B
Evl
EX and
Now we can find matrices
t-test of
(:30), following
(M-2) x 1 vector
1
O"il
EX
I
+
N -'2
EW
0";1
O"~ I) and independent of
1\
Z
(36); also"
t)*
/\
(~*
• f')
is
Hence
1
V'V
w'Z _ X'I) / ( B B)
( w'w
M • 2
XIX
=
follows the t-distribution with
statistic
(M-2)
d.f.
if
'2
HOB (4)
is true.
(39), which like (32) would evidently involve us With some randomiza-
tion in the calculation of its denominator, may be used for testing
the case
Thus the t-
HOB (4)
for
r=l.
2.4. Problem B for general r.
In attempting to find a test of HOB
(4)
for the case of general
natural approach to take might be to try to generalize
r, the
(39) and obtain some sort
I
of
F-test analogous to
(39). This method of attack seems to run into difficulty,
however, because of the way that the nuisance parameters
volved in the inverse of the
~
2
2
O"e and O"f are in-
rxr variance matrix
By using an entirely different approach, though, it is definitely possible
to develop an
F-test [With
rand
(M - 2r - 1) d.f.]
of
HOB (4), by employing
•
a device which will be mentioned briefly in Sub~eeet4Qn ;;~4. But we are deferring
our consideration of euoh F-tests to Section 3 because of the fact that they
generally would be constructed strictly as non-randomized tests rather than as
randomized tests.
What we will propose here, however, is a randomized test based on a group of
r (randomized) t-statistics
element of the vector
each analo8ous to
(;;6).
Let
dx(
and
dw(
(39).
Let
,.
(41)
represents the (-th diagonal element of the matrix (40).
(M x [M-r-l])
[analogously to
e
denote the (-th
be defined by the equation
(41)
Ex(
A
f3t
and Ew(N x [Ml'"r-l])
(;;8)]
the
satisfying
We find matrices
(30), and then define
(M-r-l)xl vectors
(42)
for
(= 1, 2, • u, r.
and
Ew(
for all
(One could of course use the same two matrices for
r values of
(, but we attache~ the
suspect the added flexibility may be advantageous.)
Ex(
(-subscript because we
For each
f,
the (marginal)
distribution of the statistic
(43)
(M-r-l) d.f. if t3 y ( = t3 f.
z
At this point we utilize a simple device similar to one described (e.g.) by
will clearly be a
Dunn in
[2].
Let
t-distribution with
e
t-distribution with
t a/
r
denote t~e. point such that
(M - r- 1)
d.f.
lies between -ta /
r
might be the joint distribution of the
thesis
100 [1 - (a/r)]
(4), the probability that the r
r
variables
relations
0/0
of the
and +ta / • 'Whatever
r
tf (43)
under the null hypo-
(44)
(f
-tar
/ -< t(/x -< +tar
/
are s1m.ultaneously satisfied is ~ 1 - a
have a test of size < a
(44):
if HOB (4)
ta/r
is true.
2, ••• , r)
Hence we will
if we take as our critical region the complement of
i.e., our test is to reject HOB (4)
exceeds
= 1,
if one or more of the tf's (43)
in absolute value.
S1m.ultaneous confidence bounds associated with this test (i.e., bounds on the
r
elements of
~)
are easily obtained.
3. NON-RAlIDOMIZED TESTS
All the tests of Section 2 we consider as being randomized tests, because
of the fact that randomization should apparently :playa role in the choice of the
arbitrarily dete:rnli~ed··E-matricesap:pearing in (31), (38), and (42). Although
certain o:ptimal pro:perties are associated with the tests of Section 2, many ex-
tit
:perimenters may nevertheless :prefer not to use these tests, either because of a
general objection to randomized (or otherwise arbitrary) test statistics, or else
because of the computational labor required.
Hence we try to develo:p, here in
Section 3, same tests which can be used in lieu of the randomized tests of Section
2, and which (i) are non-randomized, (ii) require less computational labor, and
(iii) come as close as :possible to the o:ptimal standards of the tests of the
:previous section.
In form, the sub-optimal and non-randomized tests of the present
section resemble strongly the O:Ptimal and randomized test of Scheffe [7J, and in
fact were ins:pired by the latterj but the resemblance is in form only and not in
method of attack. The method of Scheff~ [7] (i.e., his rationale for selecting a
test) influenced the develo:pment of the :previous section, ",hereas the form of
/
Scheffe's test [7J
influenced the present section.
4i'
3·1.
Problem A for
r=l.
The form of SCheffe's test r7J
for the Behrens-Fisher problem is based on a
randomized pairing of the two samples.
In similar fashion, each test here in
Section 3 will be based on a pairing of the two samples, but these pairings will
all be non-randomized rather than randomized.
For Problem A, we will develop a
test of R (2) based on the M variables
OA
where Z(i)
Yi (i = 1,2, ••• ,M).
denotes the Z-observation which is paired with
Observe that
(45)
is fonnally equivalent with Scheffe's (-d.) (see p. 37 of
~
[7J) •
The
~
M
TAi t s (45) will be nonnal, mutually independent, and homoscedastic
with variance
(46 )
)
var (
TAi
=
[Note the point that TPJ.
stant times (28).]
2e +
CT
(IN
M )
CT
f2 =
M
2
A
CT
is such that its variance (46) is equal to a known con-
Considering Problem A for the case of general r
for the time
being, we can write
(i
= 1,2, ••• ,M)
,
where we define
(48)
1 1
Ui ! = -Xi! + (M/N)2 H(i)! .. (1/MN)2
(W(i)l' W(i)2' ••• , W(i)r)
e
being the set of r
N
M
E vl(I)! + (1!N) E WJfI
I=l
J=l
x
,
W-variables associated with Z(i)'
We nOlv conclude that all the conditions for the genera.l linear model (or, more
specifically, for the multiple regression model) are fUlfilled, and so we can obtain
17
eat-teat of HOA (2)
theory to the TAi's
h
Using a'
(Le., of a
= 0)
by applying sta.ndard multiple regression
(45) and Uif's
(48).
to denote the least-squares estimator of a under the model
(47),
we write the formula
1\
(49)
a'
Z - -Y) - -U' [u·uJ-l u' TA
(-
=
[The notation in (49) is analogous to that defined at the start of
TA (Mxl)
r
(48). Note that the mean of the TAi's (45) is TA
means of the Uif's (48) for the
r
values of
f (i.e., the
r,~=l Uif) are given by the previously-defined vector U (rxl)
(49)
variance of the estimator
e
(45); and u(Mxr) = U - jM U', where U(Mxr)
contaj.ns the TAi's
tains the Uif's
the
l
M-
Sub~$c~t~n 2~1:
= Z - Y,
r
=
conand
quantities
W- X.] The
is
,
(50)
is equal to s'~~' where
which [by virtue of (46)J
(51)
s•
=
-11 + M U' [u' i11]
U
Using (49) and (50), we easily find that
1
{(Z - y) - U' ['U·3iJ.]-l
t
=
1J'
1
(11M) +
U·
[!1;J."Ur
is the t-statistic (d.f. = M-r-l)
l
vi),.),
wl1(~}=Wo _ ~
wO '
;}rld 1:t 'We 1.&'0 '1.0. (Mxl}
1
uj2 tTl TA - M T~ - Tl u[iUl'Ur\l' TA}2
for testing H (2).
OA
an Mxr matrix whose i-th row consists of
define
T } (M_r_l)2
A
(where WO '
If we define WO to be
(W(i)l' W(i)2' ••• , W(i)r)
and then
contains ~he colutml'mea'ns Of''bhe' matrix
be a vector containing the M Z(i) 's
and let ZO be
18
their mean, then we can utilize (45) and (48) to write the following formulas
which one might want to employ in .computing t (52):
1
1
0'
0
0
- (M/N)2
0'
,.
(53a)
u'u = x'x
(53b)
o
U'TA = x'Y + (M/N) w ' ZO .. (M/N)2 x' Zo _ (M/N)2 wo Y ,•
w - (M/N)2 x' w
+ (M/N) w
w x
1
1
,
This takes care of everything except for the question of how the pairing is
to be done in the first place.
The t-test using (52), which isJ:tandard applica-
tion of least squares theory based on the model (47), is of course valid for any
In the case of Scheff~'s test [7J for the Behrens-Fisher
pairing whatsoever.
problem, there:is no reason for preferr1ng one pairing to any other, and so a pairing
has to be chosen at random.
But in the present situation, by contrast, the different
pairings will not all be of equal desirability.
choose that pairing which minimizes
For the special case
var
Looking at
(50) and (53a), we see that,
A
r=l, var (a') will be minimized for that _pairing which maximizes
.!
(54)
(a') (50).
r=l, to which we nOVl finally turn our attention, this
optimal pairing is not too hard to find.
when
Therefore it would seem logical to
A
M
o
(M/N)2 E (w(.) .. W )
i=l
~
2
M
O
.. 2 E (Xi' - X) (W(i) _ W )
i=l
.
We pause briefly tor some preliminaries.
assume that Xi < X. 1 for all i
-
1+
No generality will be lost if we
and W < W. 1
j -
J+
than this, however, and assume also that no two Xi's
for all
j.
We will go further
are alike and no two Wj's
Xl < X < ••• < ~ and W < W < ••• < W • [If this assump2
2
N
l
tion of no "ties" is not observed, then some device (preferably something other than
are alike, so that
19
randomization" if possible) may have to be used to break the ties.]
In order to find the pairing which maximizes (54)" we first establish two
initial steps:
(I.)
Of the
N w/s" (N-M) will be unpaired and M will be paired; given
the group of M Wj's
which is to be paired" the best pairing is the one such that
W(l) > W(2) > ... > W(M) (i.e." the W1 s
are arranged in order opposite to the
X's) •
Proof.
It will be sufficient to showthat" if the M Wj f S
are arranged in
any order other than the one just indicated, then it is possible to find a better
pairing.
If the Wts
(i, I) such that
are not in order opposite to the
i < I (i.e."
Xi <
XI)
pairing which results from trading Wei)
e
X's, then there exists
and Wei) < WeI).
'With 'V1(I)
the same, it is not hard to demonstrate that
If we consider now the
and keeping everything else
(54) J;nust be larger for the new
pairing than for the original pairing •. That completes the proof.
(II.)
(N-M) l\l!lpaired Wjts
The
means [thanks to
(I.)]
must be consecutive.
In other words" this
that the optimal pairing is of the form
l<i<\)
,
= WM+l_i
where
\)
\) +1
-< i
-< M
is yet to be determined.
Proof.
two integers
If the
j
<J
(N-M)
unpaired W's
such that
We assume that the paired W' s
W
Furthermore,
Clearly,
consecutive, then there exist
is :paired but W
j
and W + are both unpaired.
J l
are arranged in optimal order [see (I.)]. Then,
J
considering all the W's except W
J
WJ , say cp(WJ ).
are ~
as fixed, we may write (54) as a function of
cP liS quadratic in W , 1.e., cp(W ) = Y2W5+Y1WJ+vO (say).
Y2 = (M_l)/(MN)2.
J
Since 'V2 > 0 and
J
since Wj < W < W + , it
J l
J
:II)
follows that either 9(W ) > cp{vl ) or else 9(H + ) > 9(W ) (or both). Hence
j
J l
J
J
(54) can be improved upon by moving W from the paired to the unpaired group and
J
or Vl + (one if not the other). Thus we conclude
J l
that a non-consecutive choice of the unpaired W's cannot be optimal I and so the
then replacing 'it:-with
W
j
optimal pairing must be of the form
It remains only to determine
for the pairing (55).
(55) •
in (55).
v
Let
ev
denote the value of
(54 )
Then it is easy to show that
1
eV +l
(56)
-
1\ •
2(M-l) (MN) -'2
where
(Vl _ - W ..)
M
N v
N
M-v-l
ov =
VJ
N-v
r.
+ vl
M-v
2
The coefficient of 0v
j=l
,
0v
Vl
j
+
r.
W
j=N-v+l j
(xv+ 1 -;X) •
M - 1
in (56) is > 0 1 and 0v itself (57) i6 clearly a decreasing
function of v which turns from positive to negative somewhere between v
v
=M-
(58)
1.
=0
and
Hence the optimal v to use in (55) is
v (optimal) = the smallest integer v
such that
0v (57)
is
< o.
Although this value of v (58) should not be too hard to find l the experimenter might prefer to settle for a slightly sub-optimal v in order to reduce the
calculations required.
v
= M/2
It would be perfectly legitimate, e.g., to arbitrarily set
if M is even or
= (M-l)/2 (say) if M is odd.
For (52) as well as for the other tests to be given here in Section 3, it is
no problem to obtain confidence bounds associated With the tests, even though
(just as in Section 2) we are putting our main expository emphasis on the tests
themselves.
Our non-randomized t-test of HOA (2) for r=l, which is specified by (52-53)
with the pairing made according to (55, 58), 1-Till of course be somewhat less than
optimal in comparison With the randomized t-test of Section 2.
"
Since var(a l
)
(50)
e
A
and var (ac*) (28)
2
are each equal to erA times a known constant, it is possible
to compare the two tests by comparing these t~'1O known constants
respectively),
For the case
(5,a), we can write
s' = 1 +
s'
r=l,
(for r=l)
s
(,4);
is given by
(s I and
and, by using
s
(51) and
in the form
2
erX +
where er2O = wO , WO /N (note that er2 < er~) and where p is the correlation
H
WO
coefficient between the Xi 's and the Wei) IS. Off~nd, it appears that p should
2
be quite close to -1, and that er2
should not be much less than erW
unless N is
WO
considerably larger than M. Based on this, it looks as t1:\ough s' (59) would
normally be very little greater than
the case
s
(,4). Thus we
~trould
con¢lude that, for
r=l,
the randomized t-test of H
usually has little to recommend it
OA
over the non-randomized t-test (since the latter has the two advantages of bed.ng
non-randomized and easier to calculate).
3.2. Pt'obleJ:n A for general r.
Except for the question of the pairing, the non-randomized t-test for Problem
A for general
r
was already completely determined inSUb_section ,.1 [see (52)].
In choosing a pairing, our objective will be to make
possible.
all:
U'
[ulu]-l ij as small as
From a strictly theoretical standpoint, this presents no problem at
there are only a finite number of poeaible,~afrmggs [NI/(N-M)I, in fact),
and we simply select the one for which Ut[u'u]-l
U
is a minimum (assuming that
the minimum is unique).
From a practical standpoint, however, the computation involved in evaluating
tit
l
UI[Ulur U for
N!/(N-MH
different pairings would generally be prohibitive, ex-
cept that perhaps in some cases it might be feasible with a large computer.
We
have not discovered a direct systematic way of finding the optimal pairing for the
we
case of general r as/did for the case r=l.
Hence it may be necessary to settle for a somewhat sub-optimal pairing in order
to avoid massive calculations.
Various artificial techniques for obtaining such a
pairing could undoubtedly be devised.
We mention here one possibility which ap-
pears intuitively to be a reasonable technique.
(60)
l;(i,j)
Let s(il,jl)
~
Define
g(i,j)'s
denote the smallest of the MN g(i,j)l s • Now discard all of the
for which
the remaining
i
= il
(M-l)(N-l)
all of the s(i,j)t s
or
j = jl'
and let s(i ,j2) denote the smallest of
e
s(i,j)l s • We continue in similar fashion, discarding
for which i
=i2
denote the smallest of the remaining
or
j
= j2
(M-2)(N-2)
and then letting g(i , j3)
3
g(i,j)l s • Keeping on like this,
(i l , jl)' (i 2 ,j2)' ••• , (~, jM)' and we
(I = 1,2, ... ,M) •
then pair the two samples by setting Wei ) = Wj
II
The technique just described is certainly feasible with a computer, and in
we end up finally with the M couples
some cases the MN g(i,j}ls (60) could be calculated by hand.
It would appear
that the technique should generally lead us to a fairly low value of
s' (51), but
in order to find out how effective it (or any other technigue) really is, we would
of course have to compare
From
e
s'
(5i)
wIth s (29).
(51) and (29) it appears that, in general, the problem of pairing will
be less crucial (and also the values of s'
and s
smaller the elements of U are in absolute value.
will be closer to 1) the
Thus the experimenter should
normally try to ~e the elements of U as small as possible in absolute value if
he has any control over the matter.
3.3 Problem B for r=l.
Since what we do in this sub-section will resemble in many ways what we did
in Sub-section 3.1, we will omit some of the details.
in trying to obtain a non-randomized t-test of HOB (4)
A natural mode
for
r=l
of attack
is to utilize
the M independent variables
1
(61)
TBi = +
(11M2 ~X)Yi
1
+
(11m2 ~w)
[Z(i)
for these T ' s (61) have a formula analogous to (45) and have variance equal to
Bi
~~ (37). The (+) in (61) means that either sign may be chosen; the choice of
sign would generally be made in line with the aim of getting the lowest possible
variance of the estimator of (3 = (3Z - (3y.
The expectation of each TBi (61) can
be written as a linear function of (3y' (3Z' and a third parameter.
The t-test of HOB (4) which results from this model has (M-3) d.f. and uses
the statistic
(62)
=
t
where
A
fa
(64a)
(64b)
1\
1\
,
= (3Z - (3y
2 -1
[ (11 •Y T ).:!:
(3y = (1 - p)
B
A
p R
(uz TB )
J
1
,
,
(65a)
1
(65b)
u Iy T
B
:;: (l/M O"i)
Xl
y
'+ (1/MN)2
(l/O"X O"W)
Xl
ZO
J
,
(65c)
(66)
~(R,p) ==
l+R2-+2pR
1 ... I'
=
2 -
(R _ 1)2 +
(R + 1)2
2(1 .:t p)
2(1 :; p)
A
:;:
var(f3 )
2
,
O"B
and
•
The optimal choice of pairing and of
~
2Nl/(N-M)1
(+)
sign for this t-test [there are
possible choices altogether] would be the one for which V(R,p) (66)
is a minimum.
This minimum ~
could be located by
(e.g.) evaluating all
2N1/(N-M)!
possibilities, but there appears to be no simple systematic way of locating it.
Nor
does there seem to be available a simple and completely-defined method for arriving
even at an approximate solution.
* (R,
1') (66) is minimal at the
2
p:;:.:!: l/R [at which point we have $(R,.:!: l/R) :;: R ]; in the short interval
nature of the function *(R,p).
point
The difficulty stems partly from the tricky
on the p-axis from.:!: l/R to .:!: 1,
For fixed R,
~(R,P)
thereby creating something of a trap.
increases rapidly and without bound,
If it is not feasible to locate the exact minimum of
clined to use the following
~airing
Choose the grou~ of
(i)
Wj's
one might be in-
a~~roximate
solution:
(N-M) unpaired Wj's in such a way as to minimize
0
R (67) ( i.e., maximize w0' w).
~aired
method to obtain an
~,
This minimum R is attained by taking the un-
(W _ + ' W _ + ' ••• , W _ ), where
Mv l
Mv 2
Nv
v is the smallest integer
is along the same lines as with (II.)
and (56-58) of Sub-section
to be
such that
is < 0
[~roof
3.1] •
(ii)
'pl.
Pair the M X' s
and
that for
(I.)
WIS
in such a way as to maximize
(which we will designate by Ipo J)
This maximum 'pl
the X's
and the M chosen v1' s
in either the same or else the
o~~osite
is achieved by arranging
order
[~roof
is like
of Sub-section 3.1J, de~ending on which order gives the larger 'pl;
then the u~~er +
or
..
sign (for same order) or lower sign (for op~osite order)
is selected everywhere in
(61, 64.. 66).
This pairing method would a~pear to be qUite satisfactory so long as Ip
out to be ~ 1/R.
But if
11'0 1 exceeds
much more than just a theoretical
1/R
~ossibility),
o
I
turns
(an event which mayor may not be
then
~
could became enormous.
Of course a ~airing could be found which would bring p c~ose to 1/R in absolute value, but it would be necessary to devise a
com~letely-defined
technique for
arriving at such a pairing.
In order to provide a simple and well-defined device for circumventing the
~roblem
just raised, we propose the following addendum to the pairing method
(i-ii) which will now make the method completely clean:
(iii)
In case
fp
I
o
turns out to be > l/R, substitute
wherever the latter appears in
appears in
(62, 64, 66).
(61, 65), and replace
R by
~WO fp 0 f- l
~W
for
'll,-l wherever R
·0
[Everything else in (61-66) remains the same, and the
validity of the resulting new t-test is easily verified.]
Under this change (iii), the variance of the new estimator of 13
is
2
[+
x x
~
*( fp ,-1, p) var (new T ) = p-2
o
Bi
0
~
Therefore, the danger of
+
exploding no longer eXists; however, (66) is not pre-
cisely equal to a known constant times
e
0
2
~B'
which may be a bit of a drawback.
Note that everything simplifies considerably in the special case M=N.
M=N, R (67)
is automatically equal to
Minimizing
~(l,p) = 2/(1
[see (ii)]
whether the
1, so that
(i)
When
and (iii) become vacuous.
:t p) then becomes a question merely of determining
XIS
and
WI S
shoUJ.d be arranged for pairing in the same
order or the opposite order to obtain the maximum
fp I.
The type of non-randomized t-test proposed in this sub-section rregardless of
whether it is based on the scheme (i - iii), or on some technique to locate the
pairing which gives the exact minimum of
~(R,p)J
the randomized t-test of Sub-section 2.3
in two different respects:
one less degree of freedom
mator of 13
[(M-3)
will be slightly inferior to
there will be
versus (M-2)], and the variance of the esti-
will generally be slightly larger [but not much larger, since
should (for the case
Jpol ~ l/R) normally be close to
IJ.
~(R,p)
However, it would
appear that these two slight disadvantages would usually be much more than offset
by the two advantages of the test of the present sub-section:
it is easier to
27
~
calculate [if the scheme (i - iii) is used] and is non-randomized.
3.4. Problem B for general r.
In principle, it is easy enough to generalize the non-randomized t-statistic
(62) and obtain a set of r
non-randomized t-statistics akin to the set (43) of
randomized tIs, in order to test
section 2.4)
of
r
(4) for general r by using (as in Sub-
HOB
a critical region of size
critical regions each of size
critical region (f
= 1,
2, ••• , r)
aIr.
~
a
which is constructed as the union
More specifically, the f-th such
may be based on the non-randomized t-test
(d.f. = M - 2r - 1) of ~Y~ = ~z~ that is obtained by applying standard regression
theory to the M independent variables
which are so defined as to be homoscedastic with variance equal to
~
subscript
is included in Z(i~)
[The
in (69) in order to allow a different pairing
values of ~.J Thus each one of the r groups of
one
(69) is used to compute f t-statistic (the ~-th group of M TBi(s
to be used for each of the
M TBi~'s
(41).
r
being tailored specifically for the t-test of ~Y~
= ~Z~)'
in a way similar to that
in which the group of M T t s (61) is used to compute the t-statistic (62). We
Bi
will not spell out here the detailed fOT-mula for the r t-statistics, but this
fOT-mula is obtained [just as (62-67) was obtained] by a straightforward application
of ordinary multiple regression methods.
Once the set of r
non-randomized t-
statistics has been calculated, the remainder of the procedure for testing HOB (4)
is of course exactly tue same as in Sub-section 2.4.
There remains, however, the problem of how to choose (for each f) the pairing
e
and the
(+)
sign.
As before, this is no problem at all in a strictly theoretical
sense, because we can simply select (for each
f)
that particular one of the
e
2N! I(N-M) !
possibilities which minimizes the variance of the estimator of
(/3zf - /3Yf) (assuming that the minimum is unique).
But from a pa:a.ctieal standpoint,
such a lengthy technique as this would usually be out of the question.
alternative method we have to offer is to apply (for each
(i - ii) of Sub-section ;.;:
for
f
The best
separately) steps
f=2, for example, we choose the pairing and
(+)
c>f ;lu&tleect1\'lm. ;.; to the M X 's and N Wj2 's. H~~;
i2
it is not clear whether this method will generally give a test with satisfactory
sign by applying
(i-~1)
power; we might anticipate possible trouble on the basis of the potential explosive-
W(R,
ness of the function
p) in Sub-section ;.;, but as yet we have found no
v~y
to generalize (iii) of SubO!"section ;.; in order to provide a means for averting such
trouble.
Using a set of
r
non-randomized t-statistics is not the only way of getting
a non-randomized test of HOB (4)
for general r.
which was already alluded to briefly in Section 2:
F-test, with d.f.
rand
variables of the form
(M - 2r -
There exists a second way,
we can develop a non-randomized
1). Such an F-test could be based on M
(69) [but now we would drop the subscript
f
in
(69)
everywhere that it appears]; the F-statistic would be calculated via standard
regression formulas.
But the major problem would be to figure out a satisfactory
way of choosing the pairing, the (+) sign, and the value of
dw/dx.
Thus we are
faced again with the same type of problem which we have encountered many times
throughout this paper:
we have available to us a huge (or infinite) number of
possible tests, all of which are valid; but in order to select from among these
a single test, we must have a selection method which will be well-defined, which
will not be too cumbersome to use in practice, and which will produce a test for
which the power (or some other related criterion) is close to the best possible.
We will not attempt to tackle here the unresolved quest.ions pertaining to the
matter of how an F-test should be selected for testing HOB
(4). This looks like
29
e
a difficult problem, and, in fact, even the choice of a criterion (by which to
judge different possible F-tests) may present difficulties; furthermore, it should
be remarked that the F-test approach mayor may not lead ultimately to a better
test than the approach based on r
t-statistics.
[Incidentally, one might wish to
consider a type of F-test which is more general than the type based on the M
variables of the form
(69) :
linear functions of the Yl s
it obviously is possible to construct a set of M
and Z's
which are homoscedastic and mutually in-
dependent, but which are not of the form
( 69); and F-tests of HOB (4) can be
based on this generalized kind of linear set.]
We should note that, under either of the non-randomized approaches suggested
in this section (F-statistic or r t-statistics), there are only (M - 2r - 1)
d.f.
for error, as compared with
section 2.4.
(M-r-l)
under the randomized approach of Sub-
This may be a serious difference if
r
is relatively large and/or
if M is relatively small.
For the case r=l, we feel confident that, for both Problems A and B, our
non-randomized tests of Section 3 will almost always be preferable to our randomized
tests of Section 2.
For the case of Problem A for general r, the situation re-
garding the non-randomized test of Sub-section 3.2 based on the S(i,j)l S
(60)
cert.ainly requires further investigation (perhaps of a practical rather than of a
theoretical nature), but it should not be too surprising if this non-randomized
test turns out in most cases to be entirely satisfactory (in comparison with the
randomized test of Section 2.1).
For the case of Problem B for general
r,
however; the randomized test of Section 2.4 may often have some pronounced advantages
over any non-randomized test which is presently available, since we were
not able to arrive at any very decisive results perte1ning to non-randomized tests
~
h~re
in this final sub-section; in fact 1 the questions raised in this final
sub-section seem to provide the greatest opportunity for further study.
4. ACKNOWLEDGMENr
The author wishes to thank Dr. Frederic M. Lord of Educational Testing
Service for suggesting the problems concerning comparisons of two regression lines
when the variances may be unequal.
31
e
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