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JNSTJ1UJE OF STATlSTtCI
BOX 5457
STATE COLLEGE STATION
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RALEIGH, NORTH CAI'OLfN•
PAIRED COMPARISONS FOR PAIRED CHARACTERISTICS
by
P. K. Sen and H. A. David
University of North Carolina, Chapel Hill
Institute of Statistics Mimeo Series No. 491
August 1966
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Work supported in part by the Army Research Office,
Durham, Grant, DA-3l-l24-G432.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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PAIRED COMPARISONS FOR PAIRED CHARACTERISTICS
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By
2
P. K. Sen and H. A. David
University of North Carolina, Chapel Hill
Summary.
~
The present investigation is concerned with the proposal and study of a
class of nonparametric paired comparison tests for the hypothesis of no differences
among several objects with respect to a pair of characteristics.
1.
~
o b ject h ave n
ij
n. , = n .. for i
~J
J~
t(~2) objects which form (~) pairs, and let the i th
Let there be
(>0)
_
encounters
> j).
. h
w~t
t h e J,th
0
b'Ject, for"~<J= 1, ... ,t ( convent~ona
.
11 y,
We let
t
(1. 1)
_
t
1
N = L:i<j=l n ij -. ~ifj=l n,~J,.
For each of the n ij encounters, it is judged whether the i
th
object is preferred
(or not) to the jth object for each of the two characteristics
of preference is indicated by
J,th
>
(~,~),
and the order
(e.g., ~, ? ~, => the ithobject is preferred to the
~
,.)
0 b'Ject f or t h e c h aracter~st~c
~ .
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Thus, each encounter results in one of the
following four mutually exclusive and exhaustive outcomes (the probability of ties
being neglected):
(, 2,) •
j3 i j3
~i'" ~j' j3 i>- j3 j' A~J
~i >-- ~j'
i -...
j'
(1. 2)
A~~) :
q
~, ~ ~.J ~ 13, >
~
~
j3"
J
A~ ~):
~J
~,
~
-<
~J"
j3
~
-< j3 J"
L
for all ifj=l, •.. , t, and we let
(1. 3)
Tr"
~J.
k =
P(A~~)}
~J
for k=1,2,3,4 and i"!j=l, ... , t.
f
1)
Work supported in part by the Army Research Office, Durham, Grant, DA-3l-l24-G432.
2)
On leave of absence from Calcutta University.
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Thus, the n .. encounters (assumed to be stochastically independent) result in the
1.J
following multinomial distribution
Jrr
4
4
n ij . k
{no
n .. k
Tr1.'J'.k'
1.J. k=l 1.J' ' k=l
·./II
(1. 4)
where n
. is the observed frequency of the event
ij k
A~~),
for k=1,2,3,4 and i<j=l, ... ,t.
It may be noted that by definition
(1. 5)
Tr .. k = Tr .. 5 k and n" k = n .. 5 k for k=1, ... ,4, i<j=l, ... ,t.
1.J'
J1.. 1.J'
J1.'-
The null hypothesis to be tested relates to the equality of perference of all t
objects with respect to both
(1. 6)
'lT
(a,~).
This will imply that
. = Tr for k=1, ... ,4 and all ifj=l, ... ,t,
ij k
k
(1. 7)
Equivalently, (1. 7) can be written as
(1. 8)
-&(1-8) ,
where 8(0 $ 8 $ 1) is a real parameter which we shall term the association paramThus, for convenience of discussion, we shall specify the null hypothesis
by
H:
o
Tr .. k =
1. J'
1
-4(1+8)
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-.
where by virtue of (1.5), we have
eter.
-.
i f k=l,4,
(1. 9)
= -&(1-8), i f k=2,3, for i<j=l, •.. , t.
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The object of the present investigation is to consider some nonparametric tests
for H
o
in (1.9) which may be regarded as natural extensions of some well-known
tests for a single criterion only (cf. David [2, pp. 30-31]).
Various properties
of the proposed tests are also studied.
2.
The likelihood ratio test and the spurious degrees of freedom.
By (1.4), the
~1"./'-rv'V'-"-",-rv"'V'.r-r~~-./\./V",""-",-,"v-,."",./'V",",,,,.rV'~./'-/VV
unrestricted maximum of the likelihood function comes out as
(2.1)
L(n) =
n
t
i<j=l
[{n . .
~J
J
4 n ... k' IT4 (n ... kin .. )n ~J.
.. k] ,
1111
k=l ~J
(where we adopt the convention that x
o
k=l
~J
~J
= 1 for x = 0).
Under H
o
in (1.9), the like-
lihood function is
IT
t
(2.2)
[ i<j=l
{
n. .
~J
t
4
I/TIn . .. k'
}]
k=l ~J
where N l = ~i<j=l (n ij . l + n ij . 4 )·
4
-N
(1+8)
N-N
Nl
(1-8)
Thus, the maximum likelihood estimator of 8 is
(2.3)
Hence, the maximum of the likelihood function under H
o
is
A
(2.4)
L(w)
Thus, the likelihood ratio (L.R.) criterion is
(2.5)
_L(W -
1,
N
N-N
1
N l(N_N)
1
1
"'N - L( ) - .....;;-.--~~--
n ..
~J
n ..
t
II
i<j=l
1
4
J
n ~J
.. . k
II n ..
k= 1
~J.
k
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-.
Now, under H in (1.9), the distribution of N is given by
l
o
(2.6)
and hence, under H '
o
"'-
the likelihood function conditional on N or eN is
l
t
II
no .llnn. '.k
i<j=l { 1.J k=l 1.J
(2.7)
I1J .
Since (2. 7) is a completely known probability function, it is always possible to
"'-
find a value of AN' say AE (eN)' such that
(2.8)
E
being the preassigned level of significance of the test.(2.8) implies that
(2.9)
"'-
Hence, the test based on the critical region: AN $ A (8 ) is a similar size E
e N
test for H in (1.9).
o
[More precisely, we require in (2.8) and (2.9) a randomized
test procedure to make the probabilities strictly equal to e; otherwise, we may
take the probabilities as $
EJ.
For large n .. 's, using the results of Wald [5J we
1.J
conclude that under H in (1.9), -2 log AN has asymptotically a X2 distribution
o
with 3(~)-1 degrees of freedom.
(2.10)
Thus, asymptotically
- 2 log A (eN) ~ X32 (t)_1
E
2
,E,
where X2
is the upper 100
r, E
E
% point ofaX 2 distribution with r.d.f.
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t
The L. R. test considered above is really a test for the identity of (2)
multinomial laws in (1.4) under the further specifications in (1.9), and hence
carries 3(~)-1 d.f.
In actual practice, we are often not interested in such a
broad class of alternatives but rather in a comprehensive test for an analysis
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of variance problem posed below.
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Under H
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Let us define
(2.11)
= P{13. >- 13.} = 71".. 1 + 71".. 3' i ~ j= 1, . . ., t.
1
J
1J'
1J'
T
(2. 12)
o
in (1.9), 71" .. (ex.) = 71" .. (13) = ~ for all i'fj=l, ... ,t.
1J
1J
Thus, i f we want to
test for the homogeneity of 71" .. (ex.) , s and of 71" .. (13)' s, the number of d. f. cannot
1J
1J
exceed 2[(~)-lJ.
Further, preference behavior is frequently stochastically tran-
sitive, Le. i f 71"i/ex.)
2:
~ and 71"jk(ex.) 2:~,
then 71"ik(ex.) 2:~.
Indeed, stronger
t
restrictions on the preference probabilities are often reasonable and the (2)
quantities 71" .. (ex.) may be expressible in terms of only t parameters, say
1J
i=l, ... , t.
In a similar manner,. we may proceed with 71" .. (13)'s.
1J
71"~(ex.),
1
Thus, for the
homogeneity of the parameters in (2.11) and in (2.12), we can have alternative
tests carrying only 2(t-l) d.f.
To sum up, the L. R. test will have a much broader
scope, but, for the specific purpose of testing homogeneity of the parameters in
(2.11) and (2.12), it carries many spurious d.f.
[It is well-known (cf. [3J)
that in X2 tests the effect of increasing the degrees of freedom is to reduce the
power of the test unless the noncentrality parameter increases at a sufficiently
fast rate to compensate; in the present situation, the spurious d.f. may not contribute much to the noncentrality parameter.J
The alternative test to be proposed is a natural generalization of the
paired comparison test for a single characteristic (cf. David [2, pp. 30-31J)
and will be appropriate if the characteristics (ex.,13) can be related to an underlying bivariate trait or random variable (X,Y) in whose locations we are interested.
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Fortunately, this is possible in a wide range of problems involving paired comparisons.
In such a setting, the feasibility and optimality of an appropriate L.R.
test requires knowledge of the underlying bivariate distribution.
This in turn
limits the scope of the inferences and usually renders quite complicated the
solution of the likelihood equations necessary for the evaluation of the L. R.
criterion.
On the other hand, our proposed methods appear to be quite simple and
reasonably efficient.
3.
A
ermutationally distribution free test.
We rewrite (2.7) as
t
,
II
(3.1)
i<j=l
= n
+ n
(2) where n i(l)
.. 3' for i j=l, ••. , t.
j
i j. 1
i j . 4' ni j - n i j . 2 + n 1.).
Thus, the
first factor of (3.1) is a (generalized) hypergeometric distribution, while the
second factor is the product of t(t-l) independent binomial distributions; all
these distributions are simple and well-tabulated.
Let us now define
1
U ••
1.)
= [n .. 1 + n .. 2- n .. 3- n .. 4]/n~., i f n .. > 0,
1.).
1.).
1.).
1.).
1.J
1.J
(3.2)
= 0, otherwise;
1
v .. = [ n.. 1 -n. . 2 +n. . 3 -n.. 4] / n~ . i f n .. > 0,
1.J'
1.J.
1.).
1.J'
1.J
1.J
1.J
(3.3 )
= 0, otherwise, for irj=l, ... , t.
Also let
(3.4)
T(l) =
N, i
t
L: u ..,
. 1
)=
ri
1.J
T(2)
N, i
t
=
L: v ..,
j= 1
ri
1.J
for i= 1, ... , t.
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It may be noted that by definition
"i=t
(3.5)
W
l
T(k)
N ~. = 0 for k=l , 2 •
,
[If n .. = n for all i<j=l, ... , t, we may simplify (3.4) a little further.
On
~J
characteristic (1,«(3), denote the score of the i
. h t h e J. th
counters w~t
n .. 1
~r
+
~J
I
~
~J
(3.7)
Thus,
T(l) =
N, i
=
n .. 1
~J'
+ n .. 2 (b .. =
~J'
~J
Let
t
L: a .. and b.
~
j= 1 ~J
=
=Ii
be the total scores.
~J
~J
t
a. =
(3.6)
object in its n .. (=n) en-
.
b y a.. ( b
), so t h at a ..
0 b Ject
..
n .. 3)' for i~j=l, ... , t.
~J'
th
L: b .. ,
'lq
J=
i= 1, ... , t
=Ii
Then
n-~[2a.-n(t-l)], TN(2~
~
,
~
= n--!-[2b.-n(t-l)], for i=l, ... ,t.
~
the TN .'s are related to standardized deviations of the total scores from
,
~
their expectations].
To formulate the test in a convenient way, we further define
(3.8)
z(l) = -.l(T(l)
2
N, i
N, i
(3.9)
Z(2) = -.l(T(l)
2
N, i
N, i
for i=l, ... ,t.
t
1
T(2~) = L: n~~(n .. 2 - n .. 3)
N, ~
. 1
J=
=Ii
+ T(2~)
N, ~
=
~J
~J'
~J'
t
1
L: n~~(n .. 1 - n .. 4)'
~J'
. 1 ~J ~J'
J=
=Ii
From (3.1), we get after some simple manipulations that
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E{ZN(k~
jH ,
,
0
(3.10)
~
eN}
=
-.
0 for k=1,2; i=l, ... , t;
(3.11)
for k,q=1,2; i,j=l, ... , t, where 0uv is the Kronecker delta and N
2
=
N-N .
l
Thus
considering the linearly independent set of random variables {ZN(k?, k=1,2; i=l,
,
~
... ,t-l}, taking the reciprocal of the covariance matrix as a suitable discriminant
of their quadratic form, and finally symmetrizing, we obtain an appropriate test
statistic as
(3. 12)
'" = (N -N )/N.
where eN
l 2
Since ON is a positive semidefinite quadratic form, it will
increase stochastically with increasing heterogeneity among the T(l) and/or
N, i
(i=l, ... ,t).
T(2~
N,
~
Hence, it seems natural to consider the following test procedure:
(3.13)
< ON, € ' accept H0 ,
As in (2.9), the unconditional level of significance will also be equal to E.
to compute the value of ON
, E
THEOREM 3. 1.
Under H
asymptotically a
o
X2
For small values of n. ,'s, we may use (3.1) directly
~J
' while for large n. ,'s, we have the following.
~J
'" is
in (1.9), the conditional distribution of ON given eN
distribution with 2(t-l) d.f.
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PROOF.
By virtue of (3.10) and (3.11), it suffices to show that
(3.14)
has (under H and conditioned on eN) asymptotically a normal distribution with
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zero mean and a finite variance
2
t
t{ ~ (NJN),~ c~k}
k=l
~=l
(3.15)
(3.15) follows from (3.11) and (3.14).
> o.
To prove the asymptotic normality of ZN'
we rewrite it as
2
t
(k)
~
(k)
(k) l
21n , , )/(n" )2.
22:
2: (c'k-c.k)(n,. In, ,)2(n" k-k=l i<j=l ~
J
~J
~J
~J.
~J
~J
(3. 16)
For the asymptotic theory we also assume that
lim n"
(3. 17)
N= 00
~J
IN =
p"
~J
0 <
•
p, .
~J
< 1, for all i < j= 1, ... , t.
Now, from (3.1), it follows that conditioned on
2{n"~J. k -
(-21)n~~)}/{n~~)}t (1
~J
~J
{n~~),
~J
1
~
i < j
~
t, k=1,2}
< i < j < t, k=1,2) are all stochastically indepen-
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dent standardized binomial variables, and hence, asymptotically (under (3.17)),
they have jointly a t(t-l) variable normal distribution with a null mean vector
and a unit dispersion matrix.
Thus, conditioned on
{n~~),
~J
k=1,2, 1
~
i < j
~
t},
ZN in (3.14)( or (3.16)) has asymptotically a normal distribution with zero mean
and variance
(3. 18)
2
t
(k)
~
~ (c'k- c 'k)2 n" In, , .
k=l i<j=l ~
J
~J
~J
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Now, the joint distribution of
n~~)
q
-.
i < j=l, ... ,t, k=1,2 is given by the first
factor of (3.1), and hence by Tchebysheff's inequality it can be shown that
_1
Nk/N + 0 (n.~) for all i < j=l, ... ,t; k=1,2.
(3.19)
P 1.J
-..!.
Thus, the difference between (3.15) and (3.18) is 0 (N 2).
p
The rest of the proof
will follow readily by the method of characteristic functions, and hence is omitted.
~RK.
The condition (3.17) can be replaced by the following.
incidence matrix by «n ..
1. J
for i < j=l, ... ,p.
».1., J-'-1, ••• , P'
We denote the
where n .. = 0 for i=l, ... , p, and n .. = n ..
1.1.
1. J
J 1.
A sufficient condition for theorem 3.1 to hold is that each
row (or column) of this matrix contains at least one non-zero n .. and only the set
1.J
of non-zero n .. 's satisfies (3.17).
The proof of this follows on the same lines.
1.J
This extension covers also some incomplete block designs where not all possible
pairs are considered.
THEOREM 3.2.
Under H in (1.9), D has also (unconditionally) aSymptotically a
o
N
X2 distribution with 2(t-l) d.f.
PROOF.
(3.20)
As in the proof of theorem 3.1, we express ZN in (3.14) as
L.t
~,
{2
L. (c .. -c ·k)(n .. k- n .. 5 k) In?1
1.J J
1.J'
1.J' 1.J
i<j=l k=l
t
which is a linear function of (2) independent sets of multinomial variates, and
hence has asymptotically a normal distribution with zero mean and variance (under
H )
o
t
(3.21)
(t/2){(1+e) L. c~l
i= 1 1.
t
+ (1-e)
L. c~2}.
i= 1 1.
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Consequently, by routine analysis we find that under H ,
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(3.22)
Now, under H ' 8'" in (2.3)
o
N
has asymptotically a X2 distribution with 2(t-l) d.f.
is the minimum variance unbiased (MVU) estimator of 8, and hence, from (3.12) and
(3.22), it is easily seen that under H ,
o
D
(3.23 )
N
£ D*N
Hence, the theorem.
Let now
eN be
any other consistent estimator of 8, and by substituting eN
for 8 in D& (in (3.22», define a statistic
....
DN.
It follows from theorem 3.2 that
p
under H in (1.9), D - D&, and hence has asymptotically a X2 distribution with
o
N
2(t-l) d.f.
Thus, we may consider a large sample test:
~f "DI
L
N
(3.24)
2
2
•
X2 (t-l),€' reject
H'
0
~n (1. 9)
which will have level of significance e(O < € < 1).
From theorem 3.1 and (3.23)
it follows that DN,e' defined in (3.13), also asymptotically reduces to X~(t_l),€'
Hence under H, the two tests in (3.13) and (3.24) will be equivalent.
o
We shall
see later on that this asymptotic equivalence also holds for some non-null cases.
However, we may add a few remarks here.
First, 8'" is the MVU estimator of 8 (under
N
N
H )' and hence, among the class of all possible D , D in (3.12) is optimal in a
N N
o
certain sense.
Second, the exact null distribution of D given eN (and under H )
o
N
can be obtained from (3.1).
Hence, for small samples, some exact (conditional)
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test can be constructed, a task which may be considerably more difficult for other
'"D
Finally, 8'" is really a very simple estimator of 8.
N
.
N
All these considerations
argue for the use of the test given in (3.12) and (3.13).
4.
Performance characteristics of the tests.
We shall now prove the consistency
of the tests against appropriate alternatives and also study their power properties.
t
When the (2) multinomial laws of the type (1.4) are not all identical, let us define
1
(4.1)
'lT
t
N = 2N 2.: n . .'IT •• k for k=1,2,3,4,
k,
i,,/ j=l ~J ~J.
where'IT .. k's are defined in (1.3) and (1.5).
~J.
(4.2)
i 1, N = iF 4, N' i 2, N = iF3 , N'
whatever be the (~) multinomial laws.
and
(1.5) and (4.1) imply that
Ti\, N + i 2, N = i 1, N + iF3 , N = t,
THEOREM 4.1.
N
also lies in the interval (0,1).
Whatever be the
71" ••
~J.
k's (i < j=l, ... ,t, k=1,2,3,4), '"
8 in (2.3) is
N
the MVU estimator of eN in (4.3).
PROOF.
Writing 8'" in (2.3) equivalently as
N
1
(4.4)
t
2.:
(n .. 1 - n .. 2 - n .. 3 + n .. 4)'
2N i"/j=l
~J.
1.J'
~J..
~J.
the unbiasedness of 8'" as an estimator of 8 , follows readily.
N
N
Also, for multi-
nomial distributions of the type (1.4), (n .. l-n .. 2-n .. 3+n .. 4)/n .. is the
~J.
~J'
~J'
1.J'
-.
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Let us therefore write
(4.3)
Like 8 in (1.8), 8
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MVU estimator of
('Tr ••
1J'
1
-1T •.
1J'
2
-1T ••
1J'
estimators are all independent.
3
+1T ..
1J'
4)' for all i
< j=l, ... ,t, and these
The rest of the proof is simple and is omitted.
Hence, the theorem.
Since the multinomial law in (1.4) carries 3 d.f., we may write
= -41[(l +
+
E .. )
+
(eN
+ Tj. .) ] ,
E •• )
-
(eN
+ Tj .. ) ] ,
+ E.1J.)
-
(eN
+ Tj l' J' ) ] ,
+
(eN
+ Tj .. )J, for irj=l, .•. ,t,
(4.5)
1T.. 1
(4.6)
1T.. 1 =
(4.7)
1T.. 3
=
(4.8)
1T •• 4
= ¥(l - .6 .. )(1 1J
1J'
1J'
1J'
1J'
where (6..,
1J
E .. ,
1J
interval (0,1).
1
-4 [
(1
¥ (1
+
J1
= -
(4.9)
E •• ,
1J
1J
6 .. )( 1 -
1J
- 6. .)( 1
1J
1J
1J
E .. )
1J
1J
1J
1J
Tj .. ) are all real parameters which can assume values in the unit
1J
The 6's and E'S account for heterogeneity of locations and the Tj's
for heterogeneity of association.
E ••
6 .. ) (l
Thus, from (1.5) and (4.3), we have 6 .. = - 6 .. ,
J1
1J
Tj .. = Tj .. , so that
1J
J1
1
2N
t
n ..6 .. =
L,
ir j=l 1J 1J
Using (3.17), let us then define
(4.10)
=
t
-.!.
2: p? .~ . .,
. 1 1J 1J
J=
i
r
and let
(4.11)
=
t
-.!.
E p?€ .... ,
j=l 1J 1J
ri
for i=l, ... ,t,
-14-
THEOREM 4.2.
62
+
E
PROOF.
2
The test based on ON is consistent against the set of alternatives
> o.
We rewrite ON in (3.12) in the alternative form
_8
(4.12)
T(q)]tt
f'
N N, i
where (k,q) is any permutation of (1,2).
Now, using (3.2), (3.3), (3.17), (4.5)
through (4.11), it can be shown that
(4.13)
Thus, if 6 2 +
E
.
2
-.
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> 0 it follows from (4.12) and (4.13) that ON can be made stochastically
indefinitely large as
N~.
Hence, the theorem follows from (3.13) and theorem
3.1 (whereby 0N,E: in (3.13) tends to X~(t-l),E:).
In view of the consistency of the test, we shall consider alternatives
infinitely close to the null hypothesis to study its asymptotic power.
By an
adaptation to the categorical situation of Pitman's [4J types of alternatives,
we let in (4.5) through (4.8), eN = 8 and
(4.14)
~:
where J.l .., V .. and
1.J
1.J
6 ..
1.J
s..
1.J
= N-l2
J.l ••,
1.J
(ir j= 1, ••• , t)
-l
€. •
1.J
= N
2
-l
and TJ. • = N 2S .. ,
1.J
1.J
1.J
'V..
are all real and finite.
Further, as in
(4.10), we write
t
(4.15)
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J.l.
1.'
=
L,
l
p?
t
.].1. ",
j= 1 1.J 1.J
'Ii
Then we have the following.
V.1.. =
l
.E p~.v".,
j= 1 1. J 1. J
'Ii
for i=l, •.. , t.
-.
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-15 -
THEOREM 4.3.
Under
(~}
in (4.14), D has aSymptotically a noncentral X2 distriN
bution with 2(t-l) d.f. and the noncentrality parameter
(4.16)
Under {R_} in (4.14), the joint distribution of (u .., v .. ), defined by
-~
1J
1J
PROOF.
---
(3.2) and (3.3), can be shown to be asymptotically bivariate normal with means
1
1
(p~
.11 ..,
1J 1J
p~.V
.. ), variances unity, and correlation coefficient equal to 8, for all
1J 1J
i < j=l, ... , t.
(1)
Thus (TN .,
,1
TN(2~)
,1
with i=l, ... ,t-l (defined by (3.4», will have
asymptotically a 2(t-l) variate normal distribution with means (11. , V. ), defined
l'
by (4.15), variances equal to (t-l), covariances between
TN(l~,
,1
l'
(2)
TN, j equal to
(0 .. t-l)8, for i, j=l, •.. , t-l, and finally,. covariances between T(k) T(k)
N, i'
N, j
1J
-1 for all irj=l, •.. , t-l, k=1,2, where 0 .. is the usual Kronecker delta.
1J
sequently, by routine methods we see that
D~,
equal
Con-
defined in (3.22), has asymptotically
a noncentral X2 distribution with 2(t-1) d.f. and the nvncentrality parameter in
(4.16).
Finally, by theorem 4.1, eN converges to 8 (under
{~}), and hence DN £ D~.
Hence the theorem follows.
N
It may be noted that under (4.14), the test based on D in (3.24) will be
N
asymptotically power equivalent to the one based on D in (3.13).
N
Finally, as
compared to the parametrically optimum paired comparison test (for bivariate normal
distribution) based on Hotel1ing's T2 -test, the efficiency of the proposed paired
comparison test will be the same as that of the bivariate median test (cf. Chatterjee
and Sen [lJ) for the c sample problem.
For brevity, these results are not reproduced
again.
5.
A further remark.
~
only.
We have so far considered the case of paired characteristics
The general case of p-tuple characteristics
(~(l), •.. ,~(p»
for some p
~
1,
-16-
In this case, there will be 2 P possible outcomes
can be tackled in a similar way.
(as compared to 4 in (1.2», and there will be (~) two-way marginal tables.
(a
(k)
,
a
(q»
denoted by 8
the same structure holds as in (1.8), (1.9); the corresponding 8 is
"'-
kq
, kfq=l, ... ,p, and the estimators by 8
krq= 1, ... , p.
We write
(5.l)
®
.... = «8 kq » k, q= 1, ... , p,
«8
N, kq
"'-
N,kq
»
~
(defined as in (2.3», for
_
,
k, q-l, ... , p
where conventionally 8 kk = 8 N, kk = 1 for all k=l, ... , p.
matrix of A...
For
Let
A -1
~N
be the reciprocal
and define T(k) as in (3.4), for all k=l, •.. ,p, i=l, .•. ,t.
N,i
Then the
test statistic will be
=
(5.2)
1
t
P
~
P
~
"'-kq
eN
k=l q=l
It can be shown that under H of homogeneity of the t objects, DN(p) will have a
o
known permutation distribution which asymptotically reduces to X2 distribution
with p(t-l) d.f.
Hence, a test procedure essentially similar to (3.13) and (3.24)
can be proposed.
Because of the similarity of approach, the details are omitted.
REFERENCES
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[ 1]
CHATTERJEE, S. K. and SEN, P. K. (1964), Nonparametric tests for the bivariate
two sample location problem, Calcutta Statist. Asso. Bull., 12J 18-58.
[2]
DAVID, H. A. (1963), The method of paired comparisons, Charles Griffin and Co.,
London.
[3J
MANN, H. B. and WALD, A. (1942), On the choice of the number of class intervals
in the application of the chi square test, Ann. Math. Statist., 12J
306-317.
[4J
PITMAN, E. J. G. (1948), Lecture notes on the theory of nonparametric inference, • •
Columbia University, (Unpublished).
..,
[ 5J
WALD, A. (1943), Tests of statistical hypotheses concerning several parameters
when the number of observations is large, Trans. Amer. Math. Soc., ~
426-482.
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