UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
A TEST OF HHETHER TWO REGRESSION LIl'lES ARE PARALLEL
WHEN THE VARIANCES MAY BE UNEQUAL
by
Richard F. Potthoff
July, 1962
Contract No. AF 49(638)-213
The principal topic covered in this paper is the development of
a test of the hypothesis that two regression lines are parallel
under the conditions that the two sets of error terms are normally
distributed but 'With (possibly) different variances. An incidental topic which is covered concerns a test for the slope of a
single regression line; no normality assumption is required for
this second test. ~oth tests are analogous to the Wilcoxon test.
The discussion in this paper is on a rather technical level; for
a less technical discussion of the first test, see Mimeo Series
No. 320.
This research was supported in part by the Mathematics Division of the Air
SCie~tif1c Research and in part by EaQcational Testing Service.
Force Office of
Institute of Statistics
Mimeo Series No. 331
A TEST OF vlHETHER TV10 REGRESSION LINES ARE
1
WHEN THE V.ARI.ANCES MAY BE UNEQUAL
PARALL.t~L
by
Richard F. Potthoff
University of North Carolina
1.
Introduction and
(Y2, X2 ), ••• , (Y ,
M
XM)
SUIlllIlary.
We suppose that
'We
bave M pairs (Y , Xl)'
l
such that
= 1,
(1.1)
(i
(1.2)
(j = 1, 2,
cy,
,
2, ••• , M)
..., N)
•
Z' and t3 z are unknown. The Xi's and
W.'s are (known) fixed constants; to avoid complications in our development, we
The regression parameters
f3 y '
CX
J
'Will assume that the
Xi's are all different fram each other .and that the W.'s
are all different fram each other.
and the
fj
e , e , •••
l
2
'S
J
2
We 'Will assume that the ei's are N(O, O'e)
2
2
2
are N(O, O'f)' where O'e and O'f are unknown and > 0;
f , f , ••• , f
are assumed mutually independent.
l
2
N
222
vIe may define 'T = 0' /(0' + O'f). With no loss of generality, we may assume
e
e
that M < N. (In the proofs, we 'Will also be assuming, 'With no loss of generalitj)
~,
~hiS research was supported in part by Educational Testing Service, and was
supported in part by the Air Force Office of Scientific Research.
2
It is desired to test the hypothesis that the two regression lines (1.1 1.2) are :Parallel; that is,
'We
want to test
(1.3)
against alternatives
t3 y :/: t3 z • The test we are presenting here bears some simi-
larity to the Wilcoxon test [),
of the idea developed in
f!!.7
:27,
and may be considered as a generalization
concerning the use of the Wilcoxon statistic with
respect to the Behrens-Fisher problem.
Thus the approach we are using here is
that of non-parametric statisticsj this in spite of the fact that we assume that
all the error terms are normally distributed.
However, it is possible that our
test may be valid for a wide class of non-normal distributions of the
eirs and
fjts, in addition to being valid for the normal case, but this has yet to be
either proved or dis-proved.
The statistic upon which our test is based is
(1.4)
I:
j<J
where the function
u(V)
,
u (V'Ii4_T)
1. tJU
0 or 1 according to whether V < 0
is
or V > 0
respectively, and where
The summation in (1,4) is over all (i, I, j, J)
and thus embraces
such that
and
(~) (~) terms altogether.
In Section 3 it is shown that, regardless of what
(1.6)
i <I
E(w)
1
= "2
Tis,
if and only if H is true.
o
j
< J,
Sections 4" 5" and 6 are concerned 'With showing that
sup
O<T<l
var
(w)
2 M+
=
5
if H is true.
o
18 M (M - 1)
Section 7 appeals to a theorem of Hoeffding dealing with U-statistics for
independently but non-identically distributed random variables ~l7 to show
2
that" under certain mild restrictions" ~var (wL7- 1/ (w - ~) is asymptotically
1) if Ho (1.3) is true. This result together with (1.6) and (1.7) tells
N(O,
us that a test with critical region
1
(1.8)
where za/2
,
>
w--2
is defined by
zO/2
( 21tf l/2
J
(0/2)
,
-co
'Will be a size -
0
test of Ho (disregarding inaccuracies due to the normal
approximation).
The test (1.8) 'Will of course be a conservative test, since
var (w) generally will be smaller than the value on the right-hand side of (1.7).
Although the problem of finding a Wilcoxon-type test to test the hypothesis
(1.3) under the model (1.1 - 1.2) 'WaS what prinJarily motivated this paper, an
interesting by-product also turned up.
This by-product concerns the one-sample
(rather than two-sample) case of simple regression where it is desired to test
the 13-coefficient.
By 'Way of introduction and for the sake of completeness, we
start out by considering this (simpler) one-sample case in Section 2.
No normality
assumption is required for the one-sample case.
2.
Introductory digression:
where we have M pairs
(Yl , Xl)'
the one-sample case.
t'Y2 "
(i
~)" ••• J (Y ,
M
= 1,
We consider a model
\1)
2, ••• , M)
such that
,
4
where a and
~
are unknown par.ameters, the X.~ I S are fixed constants which for
simplicity we assume to be all different from each other, and the e.~ I S are independent random variables each having the same continuous
distribution function)
c.d,f, (cumulative
F(e) (which is not necessarily normal, or even symmetric),
HI:
~ ;:: ~ , where ~
is a specified
0 0 0
It is desired to test the hypothesis
number, against alternatives
~ ~ ~
o
•
The test statistic we will use is
w'
!:
U
i<I
where the summation is over all
u
,
(ViI - ~o)
(~) pairs (i, I) such that i < I, the function
is as defined in Section 1, and
We now evaluate E(w'),
From (2,3) and (2,1) we get
,
(2,4)
where
vlith no loss of generality, we may assume that
XI
Xl < X2 < ,'. <
~,
> Xi' Using (2,4), we now obtain
;:: P (e
;: J
;::
~
< eI + 6 iI }
i
F(e + 6 iI ) d F(e)
+
J
.LF(e + 6 iI ) - F(e)_7 dF(e)
so that
5
It is not too hard to shOYT (see below) that the integral appearing in the last
line of (2.6) is
hypothesis
H':
o
0
~
= 0, that is, if and only if t.~e null
iI
Furthermore, if this integral is non-zero, its
if and only. if 6
=~ 0
is true.
sign will be the same as the sign Of(t3 - t3 ), regardless of what
o
are. Hence
E(w ' )
=~
if and only if HI
o
i
and I
is true.
(He now prove formally that the integral in the last line of (2.6) is
if 61: O.
Note that
J
(2.8)
~ 0
LF(e +
~)
- F(e17d F(e)
=
=
=
~
{l -
J
F(e) dF(e + 6»)
J
J
LF(e + 6) -
F(el7
~
d F(e + 6)
["F(e + 6)- F(el7 d
J
-
LF(e + 6)-F(eL72 d
L F(e)~F(e+6)_7
f
Fee)
~
F(e+6g
if 6 > 0 •
But if follows from a result of LehmannL2, pp. 112 - 1127 that this last integral
is strictly > 0 •
In case 6 < 0, the proof is almost the same. )
The variance of our test statistic
where
when
Hence
BiI
6
H
denotes the event that
is true, the event
B
iI
(2.2) is
ViI - ~o
> O. Observing (2.4), we note that,
is eqUivalent to the event that
e
I
> ei
•
6
2M + 5
== 18 M eM - 1)
if 13 == 13 o •
It will be shown in Section 7 that, if
(w'
-~) is asymptotically
N(0 1 1).
H~
is true, then
Thus we can test
L var(w
l
1 2
t7- /
H~ at level ex (approxi-
mately) by using
(2.11)
F 2 M+ 5 7
- IBM (M-l)-
-1/2
I WI
-
1
>
'2
as the critical region.
Once the "only if" part of (2.7) has been established, it is not difficult
to show, by using an argument similar to that given in
test (2.11) is consistent against all alternatives
so that
(2.12)
1
E( w') > 2.
P (rejecting
-
< (~)
-2
zex/2 }
l
2
M- !2 z / 7
ex &-
M sufficiently large'2.that LE(W') - t.7 exceeds
tells us that, nO matter what 13
>
l 2
> M- / zex/2 - LE(W') - t7 }
rE(w') - .!2
)
SUppose that 13 > 13 0 ,
'\Tar (w')
1 _
-
var(w l
13 0 •
2 M+ 5
-1/2
1
H~} > PC £18 M (M_l)_7
(w' - 2)
>
(2.13)
~
pp. 58-527, that the
'
Then we use Tchebycheff's inequality to
obtain
> PC w' - E(W')
for
13
['"3,
1 2
M- / zex/2.
But (2.9)
is,
.L(~) + 2(~) + 2{~) + 2(~t7 < 4 M- l
,
2
The argument given here is slightly over-simplified: in order to ensure that
.LE(w t )-1/27 is bounded below by a number> 0, we should :lJn;pose the assumption (e.g.)
that there-exists a fraction p(O<P<l) and a number e{e > 0) such that, for all M,
the number of pairs (i, I) Lwhere
i < J} satisf'ying JS:-X > e is ~ P (~) •
i
7
from which it follows that the last line of (2.12) approaches
infini~Y. In case ~ < ~o an~ E(w i
)
1 as M approaches
<~, the proof of consistency is similar.
3. The expectation of w. We return now to the main problem, which
introduced in Section 1.
Here in Section 3 we will prove (1.6).
not necessary to assume normality of the
ei's
or fj'S
'WaS
Actually, it is
in order to prove (1.6);
we need to utilize the normality assumption (specified in Section 1) only in
proving
(1.7)
and in Section
The proof of (1.6) Which we are about to give
ei's have a continuous c.~.f.
assumes only that the
have a continuous
7.
c.d.f.
FZ(f), with e , e 2 , ••• ,
l
Fy(e)
iM'
and the
f/S
f l , f , ••• , f
being
2
N
mutually independent.
Combining (1.1 - 1.2) with (1.5), we obtain
where
,
Remember that
i < I .Lin (1.4)_7 implies
It is simple to show that
is
YiI'
about
X:c'
~Z - ~y
and j < J
•
implies Wj < W •
J
ZjJ is symmetrically distributed about 0, as
Now if any two independent variables are each symmetrically distributed
0, then their sum or difference is also synnnetrically distributed about
0 .. Hence
P
Xi <
6 =
(ZjJ + YiI) is synnnetrically distributed about 0.
fZ jJ + YiI
= oj
is
0
Also
since all the random variables involved have continuous
Thus, for the case where Ho (1.3) is true (6 = 0, ~z = ~y)' we write the
result
=
P {ZjJ + YiI >
° J =21
if ~Z
= ~y
•
8
For the case ~z > ~y (~ > 0), we can write
But
,
1
"2
and
p [~~ V
=
iIjJ > 0 }
p [-~
= pC
=
< zOJ
J
-~/2
J
< zOJ
< 0) x
J -
fFZ{f +
xJ
+ YOI
< 0
J.-
~~
fFy(e +
J
p [-~/2
< YOJ.I -< oj
[VlJ - Wj J) - Fz (f).7d Fz(f)
~~
(XI-Xi) ) - Fy (e17 d Fy(e) •
Now note that, since the two integrals in the last line of
(3.2) are both of the
same form as the integral on the left-hand side of the first line of
can conclude that the last line of
proved in Section 2.
Er
-
(2.8), we
(3.2) is strictly> 0 thanks to the result
Hence we conclude that
-7
u(V 0I oJ)
J.
J
>
1
-2
if ~z > ~y
Similarly, it can be show that
The result (1.6) now follows immediately if we apply the formula (1.4) to the
relations
(3.1), (3.3), and (3.4).
9
4. The variance of w. Sections 4, 5, and 6 will cover the proof of (1.7).
As alrea<tr noted, we will utilize the normality ass'L1.1llPtion in proving (1.7).
in Section 4 we will obtain expressions for the variance of w when
ffiote:
Here
H is true.
o
In Sections 4-7, every statement or formula relating to var(w) is to be
interpreted in the context that
H
o
is true, even though this will not always
be explicitly state~7
Note first that, by (1.1 - 1.2) and (1.5), every V
will be normallY
iIjJ
distributed with mean 0 if H (1.3) is true;
o
(4.1)
ViIjJ
=
Furthermore, any two ViIjJ' s will follow a bivariate normal distribution.
it is well known (see, e.g., ~6, p. 620, equation
Now
(1427) that, if V and V'
are two random variables which follow a bivariate normal distribution with
correlation coefficient
(4.2)
p( V ~ 0, V'
p(V, V'), and if E(V)
= E(V') = 0,
then
~ oJ = ~ + (1/2~) sin- l p(V, V')
From (4.2) it immediately follows that
(4.3) cov ~u(V), u(v')_7
= (1/2n)
sin-
l
p(V, V') •
(The value of the arc sine function is assumed to lie between -~/2 and +n/2.)
From
(1.4) and (4.3), we obtain the formula
It will be convenient for us to define functions
10
=
peT: a, b: 1
* I/j * J;
l'
* I'I
j'
* J'
)
1
T
2
-
=
a
172
1
'2 (1 - T)
172
'T
+ (1 -
Tt7
and
(4.6)
A(T: a, b: 1
* II j * J;
l'
* I'I
j'
* J') =
(1/2~) s1n- 1 peT: a, b: 1 * I/j
* J;
l'
* I'/j' * J')
'VJe may then define
(4·1)
822 (T; 11 , 12 ; jl' j2) = 822
= A(T:
2, 2: 11
* 121 jl *
j2; 11
* 12/j l *
j2)
=
1
"4 '
11
=
1
1j:
::: ~
A(T: 2, s {(J-j)(J'-j)J
E
E
P:;,:; P:;,2
A(T:
S
1*I/j*J;1*I/j*J') ,
((I-1)(I'-1») ,2: 1*I/j*J;1*I'/j*J) ,
(4.10)
E
A(T:s{ (I-i)(I'-i») ,s (J-j)(J'-j»)
P3,:;
i*I/j*J, l*I'/j*J') ,
(4.11)
(4.12)
= i6
E
E
A(T:O, 2: 1*I/j*J; i' * I'/j * J)
,
P4,4 P4,2
(4.1:;)
A(T: s ((I-1)(r'·i) J, 0: i*I/j*J;1*I'/j'*J') ,
:::
(4.14)
84:;(T: i 1, 12,
::: ~
E
i:;, i
E
P4,4 P4,:;
4; j1' j2' j:;, j4) ::: 84:;(7)
A(T: 0,
S
{(J-j)(J'-j)J : i*I/j*J; i'*I'/j*J') •
12
The notation in the definitions (4.7 - 4.14) requires some explanation.
example, consider (4.13).
and that
S34(T)
It is assumed that
jl' j2' j3' j4 are all different.
As an
i , i , i , i
are all different
l
2 3 4
Whenever the abbreviated notation
is used, the eight arguments in the i's and jls are to be understood.
The first sUIllJllB,tion sign, which has a
P4,3 beneath it, indicates that the
summation with respect to (i, I, I') is across all P4 ,3 == 3!(~) = 24 permutations
of 3 integers chosen from the 4 integers (i , i , i , i ). The second
2 3 4
l
summation sign, which has a P4,4 beneath it, pertains to the summation with
respect to
(j,
J, j', J'), and indicates that this summation is across all
==
24 permutations of 4 integers chosen from the 4 integers.
4
4 == 4! (4)
(.1 1 .) -1 2 1 ~,
P ,4
~4)'
The double sunnnation contains P4 ,3 x P4 ,4 == 576 elements
altogether, but these are actually just 72 distinct elements each repeated 8
TI1 which appears in front of the double sUIIlJlIB,tion
times; hence the fraction
sign.
We use
s (.) to denote a function which is
tive and which is
-1 if its argument is nega-
+1 if its argument is positive.
The explanation we have just,
given for the definition (4.13) also applies, with appropriate modifications, to
the rest of the definitions (4.7 - 4.14).
Now that we have the definitions (4.5
- 4.14), we can re-write (4.4) in the
form
(4.15)
var(w)
= v = VeT:
-2
==
(~) (~)
M, N)
-2
.L E
1
E 822 + 2(M-2(
(~) (~)
E
(~)
+ 2(N_2)-1 E
(M)
3
M 2 -1
S33(T)+ 2( ; )
E S23(T)
(;)
E
(N)
S32(T)
3
N 2 -1
E E S24(T)+ 2( ; )
(~)(~)
E ~ S42(T)
(~)(~)
13
where the first double summation is over i 1 < i
and jl < j2 Land thus em...
2
braces (~) (~) element~7, the next three double summations are over i 1<1 <i)
2
and
jl < j2 < j3' and the last four double summations are o'ver
and jl < j2 < j3 < j4'
The factors (other than the
i 1 < 12 < i < i 4
3
2' s) preceding each double
summation sign are used to get rid of repeated elements.
Now observe that we can use
(4.8.- 4.14) and (4.5 - 4.6) to obtain
1 . -1( 1)
2 s~n
. -1
832 ()
1 = 41 P3,3 P3 ,2 (I
1 2~ ) Lr 3 s~n
- 2 +3
Combining these results
(4.17)
Iim
t
->
var(w)
1 7 = 114
a-
,
(4.16) 'With the formula (4.15), we have
=
v(l: M, N)
1
-2
-2
=
(~) (~) L(~)(~)(1/4) + 2(M-2rl(~)(~)(9/4)
+
2(N-2)"'1(~)(~)(1/4) + 2(~)(~)(1/2)
=
2 M +
5
18 M(M-l)
J
14
In other words, var(w) approaches the bound
(1.7) as ~~ •. > 0, a result which
could (alternatively) have been deduced directly from (2.10).
If we can demonstrate that
(4.18)
then
v ( T: M, N)
~
2 M +
18
5
,
M(M.l)
(4.18) together with (4.17) will be sufficient to establish (1.7). Our
proof of
(4.18) will be in two steps. First we will show that (4.18) holds for
the special case M = N, i.e.,
(4.19)
,
v( T: M, M) <" 2 11 + 5
- 18 M(M • 1)
O<T<l.
Then we will show that (4.18) also holds for the more general case M < N:
will establish that
(4.19) implies (4.18).
Note that" for M = N, we can write
+
we
2 (M-3)
-1
Z
(4.15) in the form
Z
(~)(~)
In Section 6 we will prove that, regardless of what the arguments (the integers
i l , i 2 , i , ill.' jl' j2'
3
(4.21)
(4.22)
(4.23 )
j3' j4) associated with the S functions are,
15
for all
T,
0 < T < 1.
If we substitute (4.21 - 4.24) into (4.20), we will end
up with the inequality (4.19).
In other words, the establishment of (4.21 - 4.24)
will be sufficient to prove (4.19).
5.
Some inequalities.
Before we can prove (4.21 - 4.24), we will need some
inequalities, which will be presented in this section.
and
for all T(O < T < 1).
The truth of (5.la), 5 .lb ), and (5.2) becomes immediately evident if we
observe the last expression for
creasing function of
Inequality 5B.
(5.3)
A(T: 1,1: i
p in (4.5)
and then note that A is an in-
p.
For all
T
* I/j * J;
(0 < T < 1) ,
i'
*
I'/ j'
* J')
< 1/12
To prove (5.3), we first use (4.5) to write
(5.4)
1 - 4p2(T:l,l: i
=
*
I/j
* J;
~I(XI - Xi)(WJ ,
-
i'
*
I'/j'
* JI)
Wj,)1 - I (WJ
-
Wj)(XI' - Xi,)1_72~;~;
16
P~
From (5.4) we can conclude that
21 '
and so (5.3) then follows at once
from (4.6).
Inequa1ity 5C.
For all
,. (0
A{T:1, 0: i*I/j*J; i'*I'/j'*J') + A(T:O, 1: i*I/j*J; i'*I'/j'*J') ~ 1/12 •
(5.5)
Observe first that, if
o~
< ,. < 1),
P1 ~
(5.6)
1
and 0 ~ P2 ~
'2
sin
-1
and
PI
1
'2'
P2
are any two numbers such that
then
-1
-1
2 1/2
2 1/2
-1
P1 + sin
P = sin
["
P1 (1- P2)
+
P2(1- (1)
_7
~
sin
(P 1 + ( 2 ).
2
-<
Hence ;-by (5.6)7 the left-hand side of (5.5) is
-
-
Inequality 5C
now follows innnediately if we apply (5.3).
Inequality 5D.
P2 ~ = P1 P4;
-1
sin
A(~:l,l: i*I/j*J;1'*I'/j'*J').
Suppose that
and that
+
P2
P4
. -1
s~n
0
~ Pr ~ ~ , for r
=:
P2' P4
~
< sin
=:
-1
P3 •
Then
P1 +
sin
-1
= 1, 2, 3, 4;
that
P4
First note that
cos ( sin
-1
-1
P2 + sin
~
)
= (1
2 1/2
2 1/2
2 2 2 2 1/2
- (2 )
(l-~)
-P2P3=(1-P2-~+P2P3)
- ~
and
(5.9)
1
1
2 1/2
2 1/2
2 2 2 2 1/2
cos(sin- Pl+sin- (4)=(1-P1 )
(1- P4)
-PIP4=(1-PI-P4+P2P3)
- P2 P
Since
2
2
2 2 .
P2+ P3 ~ pl + P4' it follows that (5.8) ~s
3
=:
(5.9).
•
This is sufficient to
establish (5.7), inasmuch as the cosine is a decreasing function when its argument
is between
0 and
Inequality 5E.
n/3.
If
t , t , .•. , tv
l
2
are random variables all having zero
means, then
This well-known relation (5.10) will be utilized several times.
17
Inequality 5F.
Let us define
,
d1/d'0 )
1,1: d/d'
0"
1 = A(T:
A
A =A(T: 1,1: C d + d'}
2
A.;
=A(n 1,,1: ( d + d' }/d~; d/ Cdo + d'0
are all > O.
d" d' , d0' d'0
where
for all
T(O
<
/d0"• d'/ Cd + d f
o
0
T
}
)
,
1)
Then
< 1).
If we can show that A ~ min (A , Aj ), this will be sufficient to prove
1
2
(5.11), thanks to Inequality 5B. To show that A ~min (A , Aj ), it will suffice
1
2
to demonstrate that
(5.l2a)
whenever p > P
-
0
and
where
,
whenever p < P
(5.12b)
-
p
r
is the
where p and p
o
p ;-see (4.6)
-
-
7 associated with
are functions of the d's
A
r
0
for
r
= 1"
and are defined below.
2, j, and
utilizing
(5.4), we find that the relations (5.12) are equivalent respectively to
<
(1 - p o '1)
2
gG
whenever p > P
-
0
and
whenever p < P
<
where p
= died
+ d')" p0
=0
d /(d
+ d'), '1
0
0
G = (d + d,)2 ~~.
-
=1
- p, '10
=1
- P0 , g
Proving (5.1ja) is equivalent to proving that
= (do
0
,
+ do,)2~e2,
18
2
3
4
2
1>oq Lr2p( q0 + p0q0 ) + q( q0 .. p0)-7g
2 + 2J? q(l-pq ) r pq - p 22q 7 gG
0
oL
0
0whenever p ---> P0 •
Now let 1> - P = B, B > O. If we plug this relation (i.e., p - p = 5,
o
0
'10 .. q = 8) into the left-hand s ide of (5.14), it '1uickly becomes evident that
the three expressions inside the three s'1uare brackets will all be
(5.14) holds, and so (5.13a) is proved.
O.
~
Hence
The proof of (5.13b) is exactly analogous.
Thus Inequality 5F is established.
6. Proof of the upper
(4.18).
bOl~d
for var(w).
In this section we will prove
As indicated in Section 4, this proof vdll be in two steps.
First we
will prove (4.21 - 4.24) Lwhich will be sufficient to establish (4.19L7.
Then
we will use (4.19) to deduce (4.18).
In our proofs it will be convenient for us to utilize the abbreviated notation
and also
In all eases, the four kls and the four Kl s will all be integers between 1 and
inclusive.
Let us remember that, because of the conventions we are using,
will always imply Xi.. > Xi.
K
Proof of (4.21).
K
1
and L > K will always imply Wj
From the definition (4.10) we obtain
> W.
L
Jf
K
>k
4
19
+ (1,1.12.12.13.13) + (1,1.23.23.13.13) + (1,1.12.23.13.13)
+ (1,1.23.12.13.13)
If 'We apply the relation
(6.2)
A(na,b:i
*
I/j
* J;
i' * I'/j' * J')
= -A(T:-a,-b:i*I/j*J;i'*I'/jq~J')
and the inequality (5.2) to the pairs 'Within the first four square brackets in
(6.1), 'We find that the sum of each of these four pairs is
:5 o.
If "re apply
(6.2) and Inequality 5F to the triplets within the last two square brackets, 'We
find that the sum of each of these triplets is
:5
1/12.
Finally, the proof of
(4.21) is complete if 'We apply Inequality 5B to the last four elements in the
(6.1).
Proof of (4.22).
(6.3)
Let us define
t1
= T1212
+ T
+ T
,
t 2 = T1212 + T23l3 + T1323 '
t
= T1223
+ T
+T
,
1313
t 4 = T1223 + T23l, + T1312 '
3
2323
23l2
13l3
S1.mJ.
20
where
1
"2
~Te
now apply Inequality 5E to the random variables (6.3) and obtain
by treating the mean of the
equal to
t
f
S
L used
2/6 times the sum of the
on the left-hand side of
(5.1°17
as being
9 distinct TIS. The relation (6.4) is
equivalent to
from which
(4.22) follows at once if we apply (4.21) to the right-hand side of
(6.5).
Proof of
(4.23). We Will again utilize Inequality 5E. This time let
,
where both
(k , k , k , k ) and (f ,
l
l
2 3 4
f 2 , f , ( 4 ) are
3
permutations of
(1,2,3,4),
such that
Note that there are
3x3
~
9 octuples (kl , k2, k3, k 4; f l , K2 , K , f 4 )
3
satisfying (6.6) and the permutation conditions.
t
2, we obtain
Applying (5.10) to
ti and
21
where
NOvT (6.7) reduces to
1
(6.8)
"2
By summing (6.8) over the
9 octuples mentioned above, we obtain the inequality
(4.23) .
Proof of (4.24).
The proof of this fourth relation is a bit lengthier.
First let us define
- (ldC.{L.KK'.{'L')
and
(6·9b)
SOl(k,K,k',K' ,{,L,L')
= (O.kK.{L.k'K'.fL')
+ (O.kK.{L'.k'K'.LL')
-(O.kK.!L.k'K' .LL') •
Expanding (4.13 - 4.14) and using (6.2), we can then write
,
where the two summations are across
(6.1la)
(k,K,K')
= (1,2,3),
(1,2,4), (1,3,4), (2,3,4)
and
(6.11b)
(f,L;!' ,L')
= (1,2;3,4), (3,4;1,2), (1,3;2,4), (2,4;1,3), (1,4;2,3),
(2,3;1,4)
22
respectively, and
,
(6.12)
where the two sUllmJations are across
and
(6.13b)
({,L,L ' )
~
(1,2,3), (1,2,4), (1,3,4), (2,3,4)
The sum (6.10) contains 24(=4x6) 810 elements, and the sum (6.12)
24 SOl elements. NovT note that we can use Inequality 5D to deduce that
respectively.
contains
(6.14)
S10(k,K,K ' ,{,L,t' ,L') ~
(kKl.(L.kK' .('L')
for all values covered by (6.11), and that
(6.15)
for all values covered by (6.13).
P2 P3
.LTo
demonstrate that the required conditions
= P1 P4
and P4:: P2' P4:: P associated with Inequality 5D are satisfied
3
in the first place, we simply appeal to (4.5) and (5.1) respectively (in the
case of both (6.14) and (6.15) } ._7 If we substitute (6.14) into (6.10), we obtain
(6.16) S~4(~) < 2
.;
-
E
E
kK' {Lf'L'
(kK' .(L.kK'.('L ' )
,
where the summations are across
(6.17)
(k,K') = (1,3), (1,4),(1,4),(2,4) and ((,L;f',L')
~
(1,2;3,4),(1,3;2,4),
(1,4;2,3) .
Substitution of (6.15) into (6.12) gives us
CO.kK.(L' .1t'K' .lL')
,
23
where the summations are across
I
(6.19)
(k,K;k' ,K')
~
(1,2j3,4),(l,3j2,4),(1,4;2,3) and ({,L')=(1,3),(1,4),
(1,4),(2,4).
Now we again utilize Inequality 5D:
(6.20)
we obtain the relations
(kK' .12.kK' .,4)+(kIf' •13.• kK t .24) ~ (kK'. 'Y •1rK ' ''Y 2 )+(kK' ·'Y · kK '
3
1
''14)
for all four (k,K' )'s (6.1'), and
for all four ({,L')'s (6.19), where ('1 1 ,'12 ,'1 ,'14) denote the four quantities
3
(Iwj - Wj I, Iw. - Wj I, IWj - Wj f, IWj - Wj I) arranged in order from
. 2
1
J4
3
.3
1
4
2
smalle~t to largest, and (Ql' O , 0.3' Q4) denote the four quantities
2
(Ix. - x. I, IXi - X. I, Ix. - Xi I, IXi - X. I) arranged in order from
~2
~1
4
~.3
~.3
1
4
~2
smallest to largest. If we add (6.16) and (6.18) together, substitute the four
relations (6.20) and the four relations (6.21), and re-arrange the order of the
terms, then we can end up finally with the inequality
(6.22)
-I-
£(14.14.14.23 )+( 0.14.14.23.14]"7+['(14.14.14.2.3 )+( 0.14.14. 2.3 .14_27
+£(Qb·'Y3·0b·'Y4)+(0.03·'Yb·04''Y~7+~(14·'Y3·14·'Y4)+(0·Q3·14.0
4 , 1417
+1(Qb,14.0b·23)+(0.Q.3,14,Q4·1417+~(14·'Y3,14·'Y4)+(0.14.'Yb·23.'Y~7
+~(Qa·r3,Qa·'Y4)+(0,Ql·14,Q2·1417+~(14.rl,14.r2)+(0·Q3'ra.04'Ya17
+~(Qa,14·ga,23)+(O,Ol·14·Q2·1417+~(14.rl·14.r2)+(0.14.'Ya,23.Ya17J
,
Ix.J., -
Xi
1
I
IX
and
i
4
- Xi
2
I,
and where 'Y
24
and 'Y b denote respectively
a
Iwj
the smaller and the larger of the quantities
Note that
apply
Wj rand
rWj
- W
I.
j2
1
4
'Y 1 ~ 'Y 2 ~ 'la' '1 ~ '14 ~ 'lb· Now if we
3
1 ~ Q2 ~ Qa' 0, ~ 04 ~ 0b'
together with Inequality 50 to each of the 12 pairs within the 12
Q
(5.1)
•
,
sets of square brackets in (6.22), then we can coneiliude that the sum of each pair
is ~ 1/12.
This completes the proof of (4.24).
Since the relations (4.21 - 4.24) are now proved, this means that (4.19) is
also established, and so the first step of our proof of (4.18) is finished.
The
second step, which consists of deducing (4.18) from (4.19), will be much shorter
than the first.
Proof that (4.19) implies (4.18).
For the general case M ~ N, let us
define
(6.23)
where we assume for definiteness that
jf < jL) embraces all
summation (over
K< L
(i~e., jf
< jL)' Thus
form w (1.4) based on
"'} (Zj
1
M
Wj ).
j1 < j2 <
< jM' and where the second
(~) possible pairs (f, L) such that
o (6.23) is nothing more than a statistic of the
W
(Y1' Xl)' (Y2'
x2 ),···, (YM' XM)' (Zj 1'
W ), ( Zj 2' W ),
j1
j2
Now observe that we can write
M
(6. 24) w - ~
,
==
where thesunnnation is over all
(:) possible sets of M integers chosen from
1}2} .•• , N, and where the w on the left-hand side of (6.24) is of course the
statistic (1.4) based on all M (Yi , Xi)'s and a11N ·(Zjl Wj)'s.
Inequality 5E with
v(T:M,N)
v
= (:)
to
= var(w)=var(w
If we apply
(6.24), we obtain
. 1
N -1
- 2) ~ (M)
L: var
(N)
M
l
.
1
wo(j1,J 2,· ·.,jM) - 2
_7.
25
1
But" by (4.19)" var (wo - -2) = var(w0 ) -< (a.1 + 5)/18M(M - 1) .
Hence
t;
v(nM"N)
(N)
M
2M+ 5
18 M(M-l)
=
2 M+
5
18 M(M -1)
which is (4.18).
Since we have thus proved (4.18)" the proof of (1.7) is also complete.
7.
Asymptotic normality.
The purpose of this section is to establish the
asymptotic normality of w' (2.2) and w (1.4) under the respective null hypotheses
H~
and Ho ' Our proofs will use a theorem of Hoeffding £1" Theorem 8.];7 concerning the asymptotic normality of U-statistics for random variables independent-
ly but not necessarily identically distributed.
A
U-statistic must be of the form 11" equation (5.117"
U = (n)
(7.1)
-1
m
where the summation
~t
< am-< n" the function
xa 's (a
= 1"
sets satisfying 1 ~ a l < a 2 < .••
is symmetric in its m (vector) arguments" and the
is over all (:)
~
2" ••• " n) are mutually independent {but not necessarily identically
distributed) r-dimensional random variables of the form x
a
(He are using x here in place of Hoeffding's
= (x(1)"x(2)"
...,x(r».
a
a
a
X" but except for this we are
trying to maintain notation identical to Hoeffding's throughout this section.)
If H0t
(7.2)
Hence w'
(7.3 )
is true" (2.2) can be re-written in the form
w' = (M)
2
-1
e - et
u (~ .. X )
i<1
1
~
is a U-statistic with n
~ (xi' XI)
=
=M"
e - e
u (.} X1 )
,tI.I - i
m = 2" r
= 2" xa = (ea, Xa ) ,
and
26
Note that, although the expression on the 1'1ght-hand side of
to u(e
I
~ =e ,
a
i < I, we cannot write w'
- e i ) for
= u(eI
@(xi'~)
(7.3) is also equal
as a U-statistic with
r
= 1,
- ei ), due to the fact that u(eI - e i ) is not symmetric
in its arguments.
The
in
XIS
(7.2 - 7.3) are fixed constants, but for our present purposes
we regard them as random variables for which all the density is concentrated at
a single point.
Thus the x
a
f
S
are not identically distributed.
To find a U-statistic related to w
let us set n
(7.4)
= M + N,
m = 4, r
(1.4), we proceed as follows. In (7.1),
= 3,
=
xa ""
(e,
a Xa, 1)
for 1 < a < M
-
for
and
(1)
(C1.'
2 , ~,
ct
(7.5) is over all
Q)
~,
Xfr' -
~,
permutations
(2)
(h, H, h', HI) of
otherwise
(7.5) is clearly symmetric in its arguments. If Ho (1.3) is true,
then the statistic U determined by
by the equation
(7.6)
= 24
,
a4)' and where
=0
Note that
4~
a < M+N
(1)
Xfrl -
~" ~I )u( (2)
where the summation in
M+~
-
U = kH
-~,N
w
,
(7.1), (7.4 - 7.5) will be related to w (1.4)
27
where
We will apply Hoeffding's Theorem 8.1
U ::: ~,N w (7.6).
I)) to u:::
WI
(7.2) and to
The theorem tells us that, if U is of the form indicated by
(7.1) and the accompanying discussion, and if the conditions £1, formulas (8.2 8.417 are satisfied, then
.Lvar(uL7- 1 / 2 .LU - E(uL7
~)
Thus, to prove that 1var(w'17-1/2 (w' -
is asymptotically N(O,l).
is asymptotically N(O,l) under
H~ , all we need to do is to show that the conditions £1, (8.2 - 8.417 are
satisfied.
vle can see at once that
virtue of the fact that
(7.8)
(8.217 and 11, (8'"'17 hold, simply by
(7.3) is bounded in absolute value.
lim
n
o
:::
The condition
,
--> co
is defined by 1 1 , equation (8.117.
where *l( v)
it follows that
Z EI
<P
.Ll,
'ii~( v) I :5
I ~l(V) I ~
Z E
~(v)'
1.
0 ~
Since
It~(v) I ~ ti(v)'
Hence
(p
~ 1 1 see (7.317,
and
Therefore the fraction on the left-hand side of
(7.8) is not greater than
(7.9)
.L
n
Z
E
v=l
2
(tl(
(x») 7 -1/2
v) v ..-
)
m / '-n2 var (W
=
-
where W is defined in fl, bottom of p. 31Q].
2
n var (W)
->
co
as
n
,
Looking at (7.9), we see that
(7.8) will be proved if we can show that
(7.10)
7 -1/2
-
->
00
28
L1,
But it follows from
the condition
Ll,
equation (8.1017
{whose proof is not dependent upon
(8.417 } that the property (7.10) is equivalent to the
property
2
n var (U) --> ~
n -->
as
Looking at (2.10) (and remembering that U = w',
=M),
we see that (7.11) is
Thus (7.8) holds, and our asymptotic normality proof with
clearly satisfied.
respect to
n
~
is finished.
Wi
We turn now to w (1.4).
As indicated in Section 1, we will introduce some
mild assumptions in order to prove that, under Ho (1.3), Lvar (wt7 -1/2
(w -
1
2)
is asymptotically N(O, 1):
Assumption 7A.
some constant
We assume that
~
c(c
Assumption 7B.
number
e,
°< e
1, since N ~
n --> ~
in such a way that
N/M
approaches
M).
We assume that there exists a fraction P,
° < P < 1, and a
< 1, such that, for all n, the fraction of the (~)(~) possi-
ble octuples (il , i 2 , i , i 4; jl' j2' j3' j4) Ll ~ i l < i 2 < i < i 4 ~ M,
3
3
1 ~ .11 <: j2 < j3 < j4 ~ N_7 for which the properties
for all k
#K
t
~ L
and
(7 .12b)
hold is
e <
-
? P;
I W.J
L
.. Wj
I <
t--
1/e
for all
i.e., the number of octuples satisfying (7.12) is
?
P (~)(~) •
It is not claimed that these two assumptions are necessarily indispensable for a
proof of asymptotic normality of the test statistic under the null hypothesis.
However, neither Assumption 7A nor Assumption 7B is particularly restrictive, and
it appears that neither assumption would create any difficulty with respect to
29
practical applications.
Observing (7.7), we see that Assumption '7A implies that
2
4
6c /(l+c)
lim
n
-->
which is strictly > O.
,
00
We conclude from (7.6) that, once we prove that
£var (UL7 -1/2 lU - E(uL7
is asymptotically N(O,l), it will follow innne-
diate1y that Lvar (wL7 -1/2 (w -
~)
is likewise asymptotically N(O,l).
To prove that Lvar (UL7 -1/2 £U - E(UL7 is
will again apply
L1,
Theorem
the same argument that
viaS
8.1_7.
Since
asymptotically N(O,l), 11e
0 _~ ~ ~ l£see (7.527, we can use
just given in the discussion about
if (7.11) holds, then the conditions
L1,
(8.2 - 8.417 for
L1,
w'
to show that,
Theorem 8,'!:.7 are
satisfied.
Hence all that re:mains for us to do is to verify (7.11).
From (7.6) and
(7.13) it follows that the property
(7.14)
n
2
var (w) -->
is equivalent to (7.11).
as
00
We will prove that
n
-->
(7.14) holds for all
From (4.15) we obtain
At this point we must pause to prove some
Inequalities 7A.
(7.100)
For all
T(O <
T
QIJ
ineqt~lities.
< 1) ,
S10(k, K, K', {, L, (', L') > 0
T
(0 < T < 1).
30
and
(7.l6b)
where
f,
SOl(k, K, k', K',
SlO and SOl are defined by (6.9).
in (7.l6a) and
f < L < L'
,
L, L') ~ 0
~It is understood that
k < K < K'
in (7.l6b)._7
\V'e need prove only (7.100), since the proof of (7 .16b) will be exactly
analogous.
n =d
Let us write
=I
+ d'
X.
-
~I
x.
d
= lX~ - X~ I,
1,
J. k
d
o
d' = IX~,
=
d'
o
- Xix I ,
= IWj
-
L'
w.
Jf' I·
Vie then define
i\l
=
or +
d2
d2
(1 - or)
,
i\2
= or
0
i\3 =
2
or + n
T
+
(d' )2
(d' )2
0
(1 -or )
0
,
i\4
= or
n2
+ --2
(1 - or )
(d'0 )
and
This enables us to express the left-hand side of (7.l6a) in the form
We may also w..:ite (7.17) as
,
where
,
(1 - or)
,
31
2 1/2
2 1/2
1
1
P4 = P3( - P1)
- P1( - ~)
.
=
,. .L(4"'1"'4-
2 1/2 .
2 1/2
,. )
- (4"'1"'2 - ,.) .]
4"'1 ("'2(1,4)1/2
=
•
We have
= ,. "';~~o,.3 + ~1,.2(1-")+~2T(1-,.)2+ ~3(1-,.)3+
2"-4("'1"'4 where
and
1 2 1/2
1j. T)
("'1"'2 -
1 2 1/2
1j. T)
_.7)
32
1.1
o
=.2,
2
1J.1
=
~
(d' )
n2
5
+
'2
(d' )2
0
d2
2 -2
d
+
0
°
(7.21)
1J.2
=
3d2 n2 + d2(d,)2
d2 {d S )2
o 0
(d' )2 f )n 2 _ (d,)2
+
7
(d' )4
0
2n3d2d'
' 1J.3='" d2 (d S )4
o 0
Since
it follows that
~
o' 1J.1 , 1J.2 (7.21) are all > 0, we con0 for all T. The fact that
~
0,
Using (7.22) along with the fact that
clude that the expression (7.19) is
P2 - P4
~
0 implies that (7.18) is
lJ.
-1
thanks to the property that sin
is an increasing function of ita argument.
Inequalities 18.
p
Thus (7.16a) is proved.
If the relations (7.12) are satisfied, then there exists a
number e > 0 such that
o
(7.238)
1
whenever '2 <
T
whenever 0 <
T
<1
and
< 1:
-
2
It will suffice to prove (7.23a.), since the proof of (7.23b) will be analogous • If (7.12) holds, then
2
Hence A2, A , A4' (A1A4 - ~ T )1/2
3
From (7.20) it then follows that
33
( 7·2 4) i\5
:5
2
-6 r -4
L €
€
+
€
7 ( € .. 2_r € . 4 +
.. 4.
_
€
.. 4
_
7+ € -2
E
- 4)
__
substituting (7.24) and (7.22) into (7.19), and remembering that ~1' ~2 (7.21)
are
~
0, we find that
€
whenever
T
1
> 2'
-1
d sin
16
Since
2- 1 /2
e
:=
(1 - p )
>
,
1
d p
it follows from (7.25) that
(7.26)
~Thenever
sin
-1
~ ~
. -1
-7
P2 - Sin P4?- 1 x 2
€
16
Comparing (7.26) with (7.18), we see that
satisfied if we set € := ( 2~ )-1 2-7 E16 •
T
•
(7.23a) will be
o
Now that we have proved Inequalities 'fA and 7B, we can obtain some further
relationships.
Substitution of (7.16a) into (6.10) and of (7.16b) into (6.12)
gives us the results
(7.27a)
(0 < T < 1)
and
(0 <
T
< 1)
Similarly, if we substitute Inequalities 7B into (6.10), (6.12), we find that
34
(!::2 < ,. < 1)
and
> 24 e 0
if
:5 ~)
(0 < ,.
-
(7.12) holds. Finally, we can combine (7.27) and (7.28) to determine that
(0 < ,. < 1)
whenever
(7.12) holds.
Returning now to
(7.15), we re-write (7.15) in the form
+
where
r
denotes the set of octuples satisfying (7.12) and
complementary set.
square brackets in
r'
0(1) ,
denotes the
The first two of the three terms which are summed inside the
(7.30) will be
~
0 by virtue of
be proved if we can show that, for all ,. (0 <
T
(7.27). Hence (7.14) will
< 1),
(7.31)
approaches infinity as
n -->
co
•
Applying
(7.29) and recalling that
Assumption 7B) the set r
contains at least P (~)(~)
we see that the expression
(7.31) is
(by
elements (octuples),
~
,
35
which clearly increases lo11thOtlt bound as
~)
proof that £var(w17 .. 1/2 (w -
8.
Concluding remarks.
(i)
n
becomes large.
is asymptotically
N(O,
This completes the
1)
under H '
o
vTe make a few final observations:
Under Assumptiot's7A ~1d 7B, the test (1.8) will be consistent against all
13 y
alternatives
~
13 '
A basic step in the consistency proof is the establishZ
ment of the "only if" part of (1.6), and this 'WaS already taken care of in
Section 3.
Once we have the
1I
0nl y if" part of (1.6), the remainder of the con-
sistency proof proceeds similarly to
(2.13)
we can write the inequality
(8.1)
var(w)
(2.12)
and
(2.13),
except that instead of
I:
i'<1'
for M > 3.
other
Observe that (8.1) holds regardless of the values of 13y ' 13 ' and
Z
pa~eters.
(ii)
It appears that the test
bility of rejection when H
o
(1.$)
is not unbiased:
is true (although alwys
not be constant, but rather will vary with
note that the proba-
< a approximately) will
or.
There seems to be no such a priori reason for believing that the test
(2.11)
is not unbiased, however.
(iii)
We can obtain confidence bounds on
test (1.8), and confidence bounds on 13
(f3z" 13y ) associated with the
associated with the test (2.11).
The
technique for getting the bounds is similar to the one often used with the ordinary
Wilcoxon statistic:
6,. -= (~z
(~z
for the case of
- ~y)' e.g., we find that value of
- ~y) which, when subtracted from every ViIjJ
in (l. 4), will cause the
resulting new w to be on the threshhold of significance.
(iv) Only two-tailed tests were discussed in this paper.
However, the
extension to one-tailed tests is immediate.
(v)
In practical applications of the test (1.8), some of the
equal and/or some of the lV ' s
j
in Section l.
iIjJ
)
may be
may be equal, contrary to the assumptions made
If such equalities occur, perhaps the most natural way of handling
the situation would be to count a tally of ~
u(V
Xi's
in the sum (l.4) for each
whose argument would be undefined by virtue of its two
equal and/or its two W's
X's
being
being et.lU&l.
Similar remarks would apply to the test (2 .11) if the
Xi r s
are not all
different.
(Vi)
In this paper we have proved the validity of the test (l.8) only under
the assumption that the
e
i
f
s
and the
f j ,s
are normal.
The normality
assunption was utilized in proving (1.7) and in the proof in Section 7;
not
utilized in prOVing (1.6).
it was
Although the test (l.8) may very well not be
valid for all non-normal continuous distributions
Fy(e)
and FZ{f), it is con-
jectured that the test will be valid at least for some non-trivial class of such
distributions.
This problem provides an area for further research.
OUr proof of (1.7) consisted of proving (4.17) and then prOVing (4.18).
Although the normality assumption was of course used in proving (4.18), it was
essentially not needed in establishing (4.17).
referring to (2.10) that, if Fy(e)
tribution, then it seems that var(w)
value
More explicitly, we can see by
is any continuous (normal or non-normal) diswill (under H) approach the limiting
o
(2M + ')/l8M(M-l) Lindicated by (4.1717 as
Fz(f)
approaches a distri-
37
bution whose density is all concentrated at a single point.
This relation is
not inconsistent with the conjecture made in the previous paragraph.
(vii)
The literature contains several different proofs which establish that
the Wilcoxon statistic [)" 27 is asymptotically normal under different conditions.
It is interesting to note that" if an assumption like Assumption 7A is
made" a rather short proof of the asymptotic normality of the Wilcoxon statistic
can be obtained by utilizing Ll" Theorem 8.};7 and employing an argument analogous to the one in Section 7 which established the asymptotic normality of w
under Ho • For the Wilcoxon statistic, it is a simple matter to verify that
(7.11) holds under either the null or the non-null hypothesis.
(Viii) We should mention that certain similarities can be noted between the
statistic w'
(2.2, 7.2) and the well-known statistic t
f see, e.g.,
equation (9.917J which 'Was investigated by M. G. Kendall and others.
Ll"
For
example, observe that, under certain conditions" (2w') has the same variance as
t
(com,pare (2.10) with
9.
Acknowledgments.
Ll,
equation (9.l8t7J
The author wishes to thank Dr. Frederic M. Lord of
Educational Testing Service for suggesting the problem of developing a test of
the hypothesis that two regression lines are parallel when the variances are not
necessarily equal.
The author is also indebted to Professor Vlsssily Hoeffaing, who made a suggestion which was responsible for Inequality 5E being utilized in the proof of
(4.18).
Inequality 5E is appealed to at three separate places in Section 6.
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I)}
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£'5,7
L-"2.7
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A C'laSB of statistics with asymptotically
Consistency and unbiasedness of certain non-
Ann. Math. Stati~, vol. 22, 1'1'. 165-179.
MA1'JN, H. B.) and I'lHITNEY, D. R. (1947).
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Ann. Math. Statist.,
vol. 18, pp. 50-60.
£!!7 POTTHOFF,
RICHARD F.
(1962).
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£27 WILCOXON,
FR.I\M{
(1945).
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f§}
ZELEN, MARVIN, and SEVERO, NORMAN
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(1960).
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