t
.
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
A TEST OF WHETHER TWO PARALIEL REGRESSION LINES
ARE THE SAME WEEN THE VARIANCES MAY BE UNEQUAL
by
Richard F. Potthoff
August, 1962
Contract No. AF 49(638)-213
This paper considers a situation where one has two regression lines which are known to be parallel, with the
two sets of error terms assumed to be normally distributed but with (possibly) different variances. The paper
presents a test of the hypothesis that the two regression
lines are identical (i.e., the two ~coefficients are
equal).-Tbetest is analogous to the Wilcoxon test. The
discussion in this paper is on a,rather technical level;
for a less technical discussion of the test, see Mimeo
Series No. 323.
This research was supported in part by Educational Testing Service, and
was supported in part by the Air Force Office of Scientific Research.
Institute of Statistics
Mimeograph . ~. 334
August, 1~2
.. lilt
.'.
...
'.ft. '___
~
"._
A TEST OF WHETHER TWO PARALlEL REGRESSION LINES ARE THE
l
SAME WHEN THE VARIANCES MAY BE UNEQUAL
by
Richard F. Potthoff
University of North Carolina
1.
Introduction and summary.
(Y2 , X2),···,
(YM'~)
We suppose that we have M pairs (Yl' Xl)'
such that
( 1.1)
(i
(1.2)
Zj
=
(j
O:z + f3 Wj + f j
In other words, it is
kr~
that the
ei's
We assume that the
are
N(O,
,
M)
= 1,2, ••• ,
N)
that the two regression lines associated respectively
with (1.1) and (1.2) are parallel.
are unknown.
= 1,2, ••• ,
a;)
The regression parameters O:Y'
Xi'S
and the
o:z'
and f3
and Wj's
are (known) fixed constants;
fj'S
NCO, ~~), with
are
e , e 2 , ••• ,
l
2
eM' f 1 , f 2 , ••• , f N being mutually independent; and that the variances ~e and
2 are unknown. We may define T = ~2/(
~f
e ~2e + ~2f)
It is desired to test the hypothesis
H:QL=O:
o
Y
against alternatives D:y ~ O:Z.
Z
In other words, Ho (1.3) is the hypothesis that
our two (paralleUregression lines are identical.
The test of H to be preo
sented here bears some resemblance to the Wilcoxon test ,["2, §J and is based on
1
This research was supported in part by Educational Testing Service, and was
supported in part:by the Air Force Office of Scientific Research.
2
the same kind. of idea that was used in
L27
and
L!:.7
to develop Wilcoxon-type
tests for a generalized. Behrens-Fisher problemL27 and for testing whether
two regression lines are parallel when tbe two variances are (possibly) unequal
L!:.7.
Thus, even though we are assuming normality (of the
e. I sand
f.
J
J.
IS)
for
our test in this paper, our approach will be that of non-parametric statistics.
The behavior of our test under non-normality of the
ei's
and
fj'S
is an area
which has not been explored.
For our test of Ho (1.3), we will utilize the statistic
(1.4)
where
ViIjJ
(JS: - Wj)(ZJ -
=
the function u(V)
Yi ) - (WJ - Xi)(YI - Zj)
~ + WJ - Xi - Wj
is 0
or 1 according to whether V < 0
or
V>0
res-
pectively, the set 1-1 (over which the sUIllIIla.tion is taken) embraces all quadruples (i, I, j, J) such that
( 1.6a)
and
(1.6b)
and T is the total number of quadruples (i, I, j, J) satisfying (1.6) (i.e.)
belonging to
(1.7)
1-1).
E(w)
In Section 2 we will show that, regardless of what
1
= "2
Section 3 establishes that
if and.
0
nly if Ho is true •
Tis,
3
(1.8)
sup
o < T<
var(w)
=
if H
o
Q
1
where the number Q is an involved function of the
defined by (3.5 - 3.7), (3.9).
I)}
to
is true ,
Xi'S
and Wj's
and is
In Section 4 we utilize a theorem of Hoeffding
show that, under certain mild restrictions, Lvar(wt7 -1/2 (w -
~)
is asymptotically N(O, 1) if H (1.3) is true. This result together with
o
(1.7) and (1.8) tells us that a test with critical region
where
,
will be a size-a test of H (disregarding inaccuracies due to the normal approxo
imation); the test (1.9) will of course be a conservative test, since var(w)
generally will not be as large as Q.
Section 5 is concerned with the consistency
of the test (1.9).
2.
The expectation of w.
We now establish (1.7).
From (1.1-1.2) and
(1. 5) we have
(2.1)
where
Since the e's and f's are normal and independent, and all have zero means, it
follows that ViIjJ is normally distributed with mean (a
z
median (az :- ay), so tha~
.
:c..
.
- ay) •
Hence ViIjJ
~a8
4
aZ
if
C¥z > ay
if a
for all (i, I, j,
J).
= C(y
j"f
Z
< CXy
If we take the expectation of both sides of (1.4) and
apply (2.2), we end up with (1. 7).
Observe that
for all (i, I, j, J) in
1-1.
If a Z >
CXy, it follows from (2.3) that
Hence
E(w)
~~
+ d
An analogous relation bolds when a
< CXy.
Z
We may remark in passing that the relation (1.7) holds even for certain
non-normal distributions of the ei's and fj'S.
Let us use Fy(e) and FZ(f) to
denote tbe cumulative distribution functions of the e's and f's respectively.
Suppose tbat Fy(e) and FZ(f) are both symmetric (about 0) and continuous.
If two
independent random variables are each distributed symmetrically about 0, tben their
sum or difference is also distributed symmetrically about O.
Hence (f - e) is
5
symmetrically distributed about 0, and V
is symmetrically distributed about
iIjJ
(<lZ - d~), so that ViIjJ bas median (az
- Oy),
We thus conclude that (1. 7) holds
whenever Fy(e) and Fz(f) are both symmetric about 0 and continuous ~with certain
trivial exceptions which could occur if there is no density for ViIjJ in the interval between 0 and (az
:;.
- Oy)
_7.
The variance of w.
Next we establish the least upper bound for var(w)
when Ho is true. In the previous section we found that we could relax the normality assumption somewhat and still be able to prove (1.7); in this section, however, we give a proof of (1.8) which is valid only for the case of normally distributed ei's and fj'S.
It is easily shown {refer to ~4, equations (4.2-4.:;}_7} that, if V and
V' follow a bivariate normal distribution with correlation coefficient p(V, V'}
and with E(V}
= E(V') = 0)
then
cov ~u(V}, u(V') _7 = (1/2n) sin-
( :;.1)
l
p(V,v ' )
Now when H 1s true, (2.l) becomes
o
,
where ~IjJ
=1
- PiIjJ'
Thus ViIjJ is normal with mean 0 and variance
22
) (CT 2 + CT 2 ).
(PiIjJ
+ ~IjJ
r
e
var(w)
From (1.4) and (:;.1 } we obtain
= v = VeT}
= (1/ T2 t(1/2n)
Z
Z
sin-
1
P(ViIjJ , ViIIljIJI}
,
1-1 1-1
where the first summation is over (i, I, j, J) and the second summation is over
(1 ' , I I, j I, J I ).We have
6
(1)
= PiIjJ,i'I'j'J,T
(2)
+ PiIjJ,i'I'j'J'
()
1 - T
,
where
2
2)
(PiIjJ + qiIjJ
- -21
_.
..... .....1
,
2
2
\
(PtfIljfd f + ~'I'jfjfJ
2
and
=L'Ojj'(WJ
- Xi)(W J ,- Xi') + 'OjJ,(WJ - Xi)(~' - Wj ,)
+ 'OJ"jl(XI - Wj){WJ1 -
Xi,)
+
"
~\~j:(~I';" IiJ)(X:r' ~ wj ,)_7
- 1
r
1..
The 'O's in
(XI - Wj
)2
+ (WJ - Xi
(3.5) and (3.6) are Kronecker
delt~s;
)2
_7
"2[" (xI' - Wj
1
)2
I
+( WJ' - Xi I
)2 -
_7
"2
.
7
Let us define
~ = (1/21C
(3.7)
2
T ) I:
I:
-1 (k)
sin Pi1jJ ,1'I'j'J'
1-1 1-1
for k = 1, 2.
Then
and
VeT) = ~
lim
T->O
The bound Q in (1.8) we define by
Q = max(Ql' ~)
•
If we can demonstrate that
,
,
O<T<l
then (3.10) together with (3.8) will be sufficient to establish (1.8).
To prove (3.10), observe first that, if we use the simplified notation
,
2
then (for P
(3.12)
I:
1) we have
d2 sin-1 P _ r
dT
2
( 1)
(2 )
-J. P
- P
We now show that (3.12) is? O.
Pi'I'j'J"
J
2 (1
2)-3/ 2
- P
P
Because of (1.6b), we know that PiljJ' ~IjJ'
-) )
~'I'j'J' are all? 0 (for all quadruples 1ni _ •
(1)
Therefore P
8
(3.5) and p(2) (3.6) must each be
~ 0 (for all T~ 0
~ O~
< T < 1). Since p ~
from which it follows that p (3.11) is
O~ the right-hand side of (3.12) is ~ O~
so that
>0
Taking the second derivatives of both sides of (3.3) with respect to T and then
applying (3.l3)~ we conclude that
(0 <
T
< 1)
F1nally~ (3.14) taken together with (3.8) establishes (3.10).
The proof of the bound (1. 8) is thus compJe te.
4. Asymptotic normality.
This section establishes that w (1.4) is
asymptotically normal under H (1.3)~ provided that certain mild restrictions are
o
obeyed. To prove this result~ we will appeal to a theorem of Hoeffding Ll~
Theorem 8.1_7 concerning the asymptotic normality of U-statistics for random
variables independently but not necessarily identically distributed.
A U-statistic must be of the form .Ll~ equation (5.1)_7 ,
(4.1)
U =
(n)
m
-1
E'.2 ("'"1 ' "'"2 • ... ~ ~ )
.
m
where the sUllllllation E' is over all ( : ) sets satisfying 1
~
n, the function
~ is
symmetric in its m (vector)
~ "1 < "2 < ...
arguments~ and
< "m
the x0:1s (0: =
1, 2, ••• , n) are mutually independent (but not necessarily identically distributed)
9
= (xiI),
r-dimensiona1 random variables of the form Xo
= M + N,
suppose we set n
(4.2)
X
0:
m = 4, r
2
r
xi ) , ••• , xi
».
In
(4.1),
= 3,
CI) x(2) x(3 »
=( X0
: ' 0: '
0:
- (e
0'
-
=
X
0:'
1)
for 1 <
(fa.-M' Wa.-M' 2)
<M
0
for M + 1
.:s 0: < M + N
,
and
u
where the summation in
(0:
1
;
0:
2
,
0:
3
,
f. (2) (2)J[ (1) (1)1 [: (2) . (2)J r (1) (ln0"\
LXH -xh '
XHr -~ J- XH' -~ . LXE: "~'.J
,
(
(2) + (2)
(2)
(2)
XII
xH ' - ~
-~,
(4.3)
is over all 4:=
4) (thereby ensuring that
0:
= 0
~
24
permutations (h, H, b l
,
H') of
is symmetric), and where
otherwise
Although the xi2 ),s and xi3),s are fixed constants, we consider them for our present purposes to be random variables for which all the density is concentrated at
a single point;
true, then the statistic U determined by
(1.4)
(4.4)
where
If H (1,.3) is
o
will be related to w
thus the xo:'s are not identically distributed.
(4.1), (4.2-4.3)
by the equation
U =
~, N k"-11, N w
,
10
and
(4.6)
As we already indicated, we will introduce some mild assumptions in order
.r
to prove that, under Ho '
var(wt7 -1/2(w -~) 1s asymptotical1.y N(O,i).
Assumption 4A. We assume that n - > 00 in such a way that N/M approaches
some constant c > O.
Assumption 4B.
We assume that there exists a fraction PI > 0 such that, for
all (M, N), the number of quadruples (i,I,j,J) satisfying (1.6) is ~ Pl(~)(~);
i.e., the proportion of potential quadruples belonging to 1-2 is ~ PI for all n.
We assume that there exists a fraction P2 , 0 < P2 < 1, and
a number e, 0 < e < 1, such that, for all n, the fraction of the T quadruples
Assumption 4c.
(i, I, j, J) belonging to 1-2 for which the relations
and
hold 1s ~ P2 ;
i.e., (4.7) is satisfied by at least P2 T quadruples 1ni-2 •
It 1s not claimed that these three assumptions are necessarily indispensable to
an asymptotic normality proof.
However, none of the three appears to be parti-
cularly restrictive.
From (4.5) and Assumption 4A we obtain
11
lim
~,N
n->oo
( 4.8)
= 6c
.~
;: 4
.1(1 + c)
From (4.6) and Assumption 4B we have
for all n
Looking at (4.4), we see that, if we can prove that
L var(U) _7 - .1/21 "& - E(u2]
is asymptotically N(O, 1), then it will follow immediately that
(w - ~) is likewise asymptotically N(O, 1).
L var( w>-7
-1/2
HoeffdingI s Theorem 8.1 ["1_7, which
is concerned with no~identically distributed xa's, presents a set of conditions
Ll, (8.2 - 8.4)_7 under whichL var(ut7 -1/2 LU - E(ut7
N(O,l).
is asymptotically
InL4, Section 7_7 it was shown that the conditions ["1, (8.2 - 8.4)_7
are satisfied if 0
~
t~
1 and
2
n var(U) ->
( 4.10)
as n - > co
CD
Now clearly 0 ~ ~ ~ 1 ["see (4.3)_7.
Hence all that remains in proving the
asymptotic normality of w is to prove (4.10).
Note that, by virtue of (4.4) and
(4.8 - 4.9), the relation (4.10) is equivalent to
( 4.11)
n
2
var (w)
->
Thus we want to prove (4.11) for all
denote that subset of
1-1
1-.2.
as n-> co
00
T,
0 <
T
< 1.
C.2.
let us use ..
r
to
consisting of all quadruples (i,I,J,J) which belong to
and which satisfy (4.7).
Then, since p (3.4) is always ~ 0, it follows from
(3.3) that
( 4.12)
Since sin-
•
l
p ~ p (0 ~ p ~ 1), we obtain from (4.12) and (3.4) the inequality
12
+ (1 -
T
) ~
l
(2)
PiIjJ,i 'I I j 'JIJ
.:
11' 11'
Thus
-
(4.13) tells us that (4.11) will be established (for all T) if we can show
that
(4.14)
as n->
for k=
k
=2
00
1, 2. It will suffice to prove (4.14) for k = 1, since the proof for
is analogous.
Let
i-)*
-2
denote the set of all octuples (i, I, j, J;
i', I', j', JI )
such that both (i, I, j, J) and (i', I', jl, JI) are ini-l', and such that i
and I
= I'; let i-l~ denote the set of all octuples (i, I,
such that (i, I, j, J) and
8ii , + 8iII + 8Ii , + 8II1
(i', I',
= 1.
j, J;
=i '
i', I', jl, JI)
j', JI) are both ini-l', and such that
Then we can write
•
(3.5)' and (4.7) that any p(l) associated with an octup:}e in
r-)*
1 4
(1) associated with an octuple in 1r-)*
~ -1 will be ? '2 e , and any p
-2 will be ? e4•
Now it follows from
Hence
(4.15) mmplies that
(4.16)
where ~l and ~2 are the numbers of (distinct) octuples belonging to 1-1~ and
1-1; respectively.
13
Let v denote the number of quadruples which belong to
1-1'.
Assumptions
4B and 4c tell us that
,
(4.17)
where P
= P1P2 ( > 0).
= 1,
Let vg (g
2, ••• , M) denote the number of quadruples
(i, I, j, J) in 1-1' such that either i
=g
or I = g.
M
(4.18)
~
v = 2 v
M
~
v
g=l
Then
,
g
and
2
g=l
g
= ~l
The expression on the left-hand side of
to the restriction
(4.18) if
(4.20)
~l +
From
2
v
g
~2 ~
= 2 viM
M
~
g=l
+ 2 ~2
(4.19) attains its minimum value subject
for every g.
2
(2 v1M) = 4
Hence
2
v 1M
(4.16), (4.20), and (4.17) it follows that
(1)
(4.21)
2 -2 2
~ PiI j J , i 'I ' j , J' ~ n T
P (M -
.el 1-1'
I
Since T is ~ (~)(~),
(4.21) implies that
which is sufficient to prove
Lvar(w)_7
-1./2 !\w - ~)
(4.14) (for k = 1). This completes the proof that
is asymptotically N(O, 1) under H '
o
14
5.
Consistency.
In this section we prove that, under Assumptions 4A and
4B, the test (1.9) is consistent against all alternatives
or
~aZ'
Our proof will
use an argument similar :to that given in 1 2, pp. 58-59_7.
As a preliminary, we establish
a bound for var(w).
From (1.4) we have
Hence
(5.1)
var(w)
~ T- 2 E
i
<r:
E
j<J i
I
E
<r: I
E
j I <J I
/cov[" u(ViIJJ), u(Vi'I'j'J,L7 /
<T,2 ~ j~ [G) (:),(:2) (~2)]
[(:2) (~2) J(0)]
i
(1)
{
+
At this point we apply Assumption 4B to (5.1) and obtain
for M> 3, N > 3
This bound (5.2) is of course valid regardless of the values of
other parameters.
ay,
aZ'
T,
or
15
From (5.2) and (1.8) we obtain immediately the inequality
We now proceed with the consistency proof.
same whether O:z >
O:z >
ay,
ay
or O:z <
The proof is virtually the
CXy; we give the proof only for O:z > aye If
then we use (2.5) to write
P {rejecting Ho }
= P{ Q-
i
Iw-
i
I,>
'''/2J
~ P{ Q- i( w - i) > '''/2J
~ P { w - E( w) > Qi '''/2 - dJ
where d is defined by (2.4).
,
Because of (5.3) and Assumption 4A, Q will become
arbitrarily small as n (= M + N) becomes large.
Hence we can apply Tchebycheff's
inequality to (5.4) to obtain the relation
(5.5)
P {rejecting H }
o
Z1
-
varlw)
2
(d - Q2 zO:/2)
for n sUfficiently large that d exceeds Q1/2 zO:/2.
Finally, from (5.2 ) and
Assumption 4A it follows that the right-hand side of (5.5) approaches 1 as
n
->
00.
6.
~uding
remarks.
We make some final observations:
We might point out that V
(1.5) is just one member of an infinite
iIjJ
class of linear functions of (Y , Y , ZJ' ZJ) haVing medians (means) equal to
i
I
(O:z The original reason why V
was chosen was that ViIjJ is the only
iIjJ
linear function in the class whose variance is some (known) constant times
(i)
ay).
16
(~e
2
2
+ ~f):
this resulted in p (3. 4 , 3.11) being a linear function of
in turn enabled us to prove (3.14) with no difficulty.
T,
which
It appeared that a bound
analogous to (1.8) would not be easy to prove if any function other than ViIjJ
(1.5) had been used.
The choice of the function V
, however, can also be considered in a
iIjJ
O
different light. Let V represent any member of the class (which we will call C)
of linear functions of (Yi , Y , Zj' ZJ) with medians (O:z - ay). Then if the
I
quantity var(vO)/(~e2 + ~f2) is considered as a sort of loss function, it is
easily shown that the minimax value of this "los s function" will be achieved for
Thus, in a certain sense, the choice of the linear function ViIjJ (1.5) constitutes a "minimax variance" choice.
(11)
The original reason for excluding from the summation in (1. 4) those
(i, I, j, J) not belonging to
1-1 was
that, if (1.6b) is not satisfied, then the
coefficients of Yi , Y in (1.5) will not both be negative and the coefficients of
I
Zj' ZJ in (1.5) will not both be positive. On the other hand, for any (i,I,j,J)
in
1-1,
ViIjJ (1.5) will have its coefficients of Y , Y both negative and its
1 I
coefficients of Zj' ZJ both positive, thereby ensuring that pel) (3.5) and p(2)
(3.6), and hence p (3.4, ,.11), are ~ 0;
the fact that p ~ 0 was used 1n proving
(3.14).
Restricting the summation 1n (1.4) only to those (1, I, j, J) which belong
to
1-1
has certain other effects in addition to providing an important link 1n
the proof of (,.14).
It 1s not known whether a relation analogous to (1.8)
17
could be established if the summation in (1.4) were taken over all (i, I, j, J)
instead of Just over those (i, I, j, J)belonging to
1-1,
although such a relation
would probably be difficult to prove even if it were true.
Even if such a rela-
tion could be proved, however, it is not certain that the associated test would
necessarily have better power than our test (1.9).
Although this hypothetical
test based on all ViIjJ'S would obviously have better power in the extreme case
where T is near 0, it would not necessarily also have better power in all other
One effect of taking the summation (1.4) just over 1-1 is that all
2
2
ViljJ's with variance ~ ue + u f are included in the summation while all ViljJ's
2
2
with variance> u + u are excluded from the summation, so that w (1.4) is
e
f
(in a certain sense) based only on those ViljJ's with the best discriminating
cases.
capabilities, so to speak.
For this reason, it seems possible that the test
(1.9) could in some situations be more powerful than the hypothetical test based
on all ViIJJ's.
The test (1.9) obViously is somewhat more advantageous with respect to
calculations than the hypothetical test, since the summations ~
(}.7) contain
fewer terms than would their analogues under the hypothetical test based on all
ViIj / s.
We also note that the (i, I, J,J) in 1-1 constitute all the (i, I, j, J)
such that the intervals (Xi' XI) and (W
W ) have point(s) in common.
J
it does not appear that this effect as such has any consequences.
(iii)
j
,
However,
It was pointed out that (1.7) holds even for certain non-normal
distributions, whereas (1.8) was proved only under the assumption of normality.
It is not known to what extent (}.10) is satisfied for non-normal distributions,
but we might conjecture that (3.10) is satisfied for some non-normal distributions
but not others.
18
If this is the case, it might be possible to obtain some number > Q which
would be an upper bound on var(w) for a large class of distributions.
This pro-
videa an area for further investigation.
(iv)
It appears that the test (1.9) is not unbiased:
ability of rejection when H is true (although always
o
be constant, but rather will vary with T.
(v)
S
0:
note that the prob-
approximately) will not
Only the two-tailed test was discussed in this paper.
However, the
extension to one-tailed tests is immediate.
(vi)
(1.9).
ay)
We can obtain confidence bounds on (O:z -
associated with the test
The technique for getting the bounds is similar to the one often used with
the ordinary Wilcoxon statistic:
subtracted from every V
in
iIjJ
threshhold of significance.
we find that value of 6. = (O:z -
(1.4),
ay)
Which, when
will cause the resulting new w to be on the
Note that, if it is only desired to test the hypothesis
(1.3) and no confi-
dence bounds on 6. are needed, then it is only necessary to compute the numerator
of each ViIjJ
(1.5)
rather than the entire fraction
(1.5),
inasmuch as
However, if confidence bounds on 6. are to be obtained, then the full fractions
ViIjJ
(1.5)
(vii)
all need to be calculated.
The performance of the numerical calculations for the test (1.9) is
covered in a different report
example,
15_7
15_7
in more detail than has been given here.
For
points out that, if the formula
2 sin-1 p
= cos-l( 1
- 2 p2)
is utilized in calCUlating the terms in the summations
need for obtaining any square roots.
(3.7),
this obviates the
19
The calculation of Q rather than.of w of course presents the principal
computational problem associated with the test (1.9).
The calculation of w is
simple compared with the calculation of Q.
Since Q is a function only of the Xi's and WJ'S and not of the Yi's
Zj'S, Q can be calculated before the Yi's and Zj'S are ever even available.
and
This
fact may be of advantage in experimental situations where the Yi's and ZJ'S
represent the outcome of the experiment and the Xi'S and WJ's are available before
the experiment begins.
(viii)
Calculating Q by hand is virtually out of the question:
speed computer is required.
a high-
Moreover, it appears that even a high-speed computer
is not fast enough to calculate Q economically except when M and N are rather
small, since the sums Ql' ~
(;.7)
each involve huge n~bers of terms even for
moderate M, N.
Fortunately, however, there is a way to circumvent this problem of evaluating Q.
Instead of determining Q ~iactly, we can calculate an adequate estimate
of Q by taking a sample of the elements which make up the sums
(;.7).
It appears
that a close estimate of Q can be obtained in this manner without costing an excessive amount of computer time.
elUding stratified sampling.
techniques is given in
Different kinds of sampling are possible, in-
A more detailed discussion of possible sampling
L5_7.
A different approach to circumventing the evaluation of Q might be to try to
obtain a general method for finding some number which is slightly greater than Q
but which is much easier to calculate than Q.
lieu of Q.
Such a number could then be used in
To implement this approach would probably require considerable investi-
gation, and it is hard to tell whether the approach could work at all well.
20
7. Acknowledgments. 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 parallel regression lines are identical when the variances
are not necessarily equal.
·.
21
REFERENCES
£1_7
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A class of statistics with asympto-
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... _ .... _ z:: '¥
~_
£2_7
MANN, H. B., and WHITNEY, D. R. (1947).
On a test of whether one
of two random variables is stochastically larger than the other.
Ann. Math.
Statist., vol. 18, pp. 50-60.
POTTHOFF, RiCHARD F. (1962).
generalized Behrens-Fisher problem.
Use of the Wilcoxon statistic for a
Institute of Statistics Mimeo Series No.
315, Department of Statistics, University of North Carolina.
POTTHOFF, RICHARD F. (1962).
A test of whether two regression
lines are parallel when the variances may be unequal.
Institute of Statistics
Mimeo Series No. 331, Department of Statistics, University of North Carolina.
POTTHOFF, RICHARD F. (1962).
Illustration of a test which
compares two parallel. regression lines when the variances are unequal.
Institute
of Statistics Mimeo Series No. 323, Department of Statistics, University of North
Carolina.
WILCOXON, FRANK (1945).
methods.
Biometrics, vol. 1, pp. 80-83.
Individual comparison by ranking