Download Sen, P.K.; (1966)A class of permutation tests for stochyastic independence."

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A CLASS OF PERMUTATION TESTS FOR
STOCHASTIC INDEPENDENCE, I.
Pranab Kumar Sen
University of North Carolina
Institute of Statistics Mimeo Series No. 458
January 1966
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Work partially supported by Research Grant, GM-12868, from the
National Institutes of Health, Publio Health Service.
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DEPARTMENT OF BIOSTATISTICS
UNIVERSITY O'F NORTH CAROLINA
Chapel Hill, N. C.
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A CLASS OF PERMUTATION TESTS FOR SIDCHASTIC INDEPENDENCE, 1 .. *
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PRANAB KUMAR SEN
Department of Biostatistios
University of North Carolina, Chapel Hill
and University of Caloutta.'
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.§.!a~ In a random sample of n observations drawn from a (bivariate)
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independenoe of the two variates.
population having a unspeoified distribution, a class of non-parametric
permutation tests is oonsidered for testing the hYPothesis of stoohastic
U-statistics and are valid for a wide olass of distributions which are not
neoessarily continuous.
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In this context, a well-known oombinatorial Central
Limit Theorem by Hoeffding (195l)(and extended by Motoo (1957),) is generalized
to a class of U-statistics.
With the aid of this, the asymptotic properties
of the proposed class of tests are studied.
Various applioations of the
findings are made in some speoific problems, and some attention has also been
paid to the test of association in a olass of two way contingency tables.
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These tests are all based on appropriate
1.
Let (X , Y ), a
a
a
= 1, .00,
INTRODUCTION
n be n independent and identioally distributed
(bivariate) random variables (i.i.d.bor.v.) distributed aocording to a bivariate
oumulative distribution function (cdf) F(x,y).
o is
It is assumed that F
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0, where
the set of all bivariate odf t s whioh are nondegenerate in the sense that
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in the two dimensional real space (R ), there is no point (x,y) which has the
probability measure exaotly equal to uni ty.
*Work partially
I t may be noted that we are not
supported by Research Grant, GM-12868, from the National
Institutes of Health, Public Health Service•
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imposing aQY restriction, such as continuity (or absolute continuity), on
F
E
O.
Let us now denote by
F(x~)
=G(x),
F~,y)
= H(y)
for all (x,y)
E
2
R ,
•••
(1.1)
• ••
(1.2)
and let w denote a subset of 0 for which
F(x,y)
=G(x)
2
H(y) for all (x,y) E R •
The null hypothesis of stochastic independence of X and Y relates to
Ho:
FEw.
• ••
In testing this null hypothesis, we are interested in the set of alternatives
2
that at least on a set of points in R of measure non-zero, (1.2) does not
hold.
However, in practical situations, for the convenience of the test
procedure, often we require a more precise and simpler formulation of the
alternative
This may be done very conveniently in terms of some
~potheses.
functionals of the cdf F.
Let Q(F) be a regular functional of the cdf F (also sometimes called
an estimable parameter, cf. Halmas (1946), Hoeffding (1948 a».
Q(F) m~
be termed an estimable a.ssociation parameter of F, if
9(F) = 90 (known) for all FEW,
while for F
¢w
, Q(F) deviates from Qo.
•••
(1.4)
(Particular examples of such
association parameters are (i) product moment covariance, (ii) rank correlation,
(iii) unbiased grade correlation, (iv) difference sign covariance,
(v) probability of concordance, etc.; for some discussion of which the
reader is referred to an expository paper by Kruskal (1958) J Thus, (1.3)
and (1.4) imply that under Ho in (1.3), 9(F)
to Qo' for some speoific F
Wo
= {F:
then we w o.
Q(F)
=9
0
4 w.
}
= Qo'
Q(F) may also be equal
Thus, if w 0 is a subset of 0, suoh that
,
Consequently, 0 - waG 0 - w.
•••
(1.5)
In testing Ho in (1.3), we shall
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be particularly interested in the set of alternatives
H:
F t Q -w
o
c
Q -
• ••
w •
Since Q(F) is estimable, for all F
estimator of it.
& ~~,
there exists an unbiased
The kernel of Q(F) of degree m (~ 1) is denoted by
~([Xl' Yl ], "', [Xm' Ym]),
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where obviously, we can always select ~ in such a manner that it is a
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corresponding to the kernel ~ in (1.7), is given by
symmetric function of its arguments (cf. Hoeffding (1948 a».
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t
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(1.6)
The U-statistic
• ••
where the summation S extends over all possible 1
~
al
< ••• < am ~
(1.8)
n.
It is well-known that as an unbiased estimator of Q(F), Un possesses some
optimum properties (cf. Halmos (1946), Hoeffding (1948 a), Fraser (1957,
pp 140-142),).
Moreover, most of the existing tests in the literature are
based on certain U-statistics.
Thus, we may be interested in developing the
general theory of tests for the null hypothesis (1.3) based on the statistio
Un in (1.8).
If F is continuous, so that the probability of tied observations may be
ignored, and if the kernel ~ is an explicit function of the relative ranks of
X , "', X and Y , ""
ul
am
a1
Y ,then under fairly simple oonditions, the
am
distribution of Un in (1.8), will be essentially non-parametric if Ho in (1.3)
holds. Henoe, strictly non-parametric tests for independence can be constructed
under the above conditions.
The existing tests by Blomqvist (1950), Kendall
(1938), Hoeffding (1948 a, 1948 b) and Bhuchongkul (1964) among others are all
based upon the assumption that F is continuous.
F may not be continuous everywhere.
However, in many situations,
Even if F is actually continuous, the
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intervals which lead to some sort of discontinuity of F,where the possibility
of tied observations may not be ignored, in probability. Moreover, ~ may not
be a simple funotion of the ranks of its arguments.
For such nonsimple ~ts,
the distribution of Un may depend on the unknown F, even when F is oontinuous
and Ho in (1.3) holds.
Due to these reasons, even under H in (1.3), the sampling distribution
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of Un' in general, may depend on F(& w), and as a result, the associated test
may not be striotly distribution-free.
The simplest way of overcoming this
drawback would be to take recourse to the permutation test theory.
The
original idea of permutation tests is due to Fisher (1935), but the essential
rigorous theoretioal developments on this line are due to Scheffe" (1943 a,
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1943 b), Wald and Wolfowitz (1944), Lehmann and Stein (1949), among many others.
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the permutation test based on the sample correlation coefficient.
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For the hypothesis of stochastio independence, Pitman (1937) considered
Wald and
Wolfowitz (1944) looked at the problem from a slightly specialized angle.
They were interested in testing the hypothesis of randomness in a series of
observations, against stoohastic dependence of various types.
This may be
oonsidered to be a particular case of our problem here, if we consider one of
the two variates to be non-stochastic and simply ordered (eg., Xi = i,
i
=1,
2, ••• , n).
In this oontext, they observed that no test would be
admissible against all types of stoohastic dependence, and hence, for specifio
type of alternatives specific test would be found to be suitable or optimum.
Some further notable works on this line are due to Noether (1950), Mann (1945 a,
1945 b), Moore and Wallis (1943), Ghosh (1954), Stuart (1954), Stuart and
Foster (1954), among others.
For the hypothesis of stochastic independenoe
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in the bivariate case, Hotel1ing and Pabst (1936) first proposed the use of
ranks.
Further use of ranks are due to Kendall (1938), B10mqvist (1950),
Hoffding (1947), among others.
These relate to the use of rank correlation,
unbiased grade correlation, difference-sign correlation and the probability
of oonoordance in the above problem, and an account of these is given by
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Hoeffding (1948 a).
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been developed by Konijn (1956), who has also studied the power properties of
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the asymptotic power properties of such tests.
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The concept of asymptotic efficiency (in the Pi tman-
sense) of some of these tests against alternatives of linear dependence has
some of the above tests.
Very recently, Bhuchongkul (1964) has oonsidered
the use of rank order statistics in problems of association and has considered
Incidentally, her treatment
is valid only for continuous cdf's and for the study of the asymptotic power
properites she has also to assume the absolute continuity of F.
Now Pitman's test is ev'ident1y a permutationally distribution-free test.
whioh requires the existence of moments of F up to the order 2 + 5 ( 0
whioh is also very sensitive to fluotuations of extreme values.
> 0)
and
The other rank
tests are based on much less restrictive assumptions on F and moreover, if F is
oontinuous, their permutation distributions tally with the corresponding
unconditional null distributions.
But, if F is not continuous, these tests
are (unconditionally) not strictly distribution-free.
The class of tests
oonsidered here contains most of the above tests as special cases and moreover
they remain strictly distribution-free, no matter whether F is continuous or
not.
The tests by Mann, Stuart and others also fall in this class of tests.
In part II of the paper we shall consider some extensions of these results
to the case of more than two variates.
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2.
THE BASIC PERMUTATION PRINCIPLE
Let (Xa , Ya ), a = 1, ••• , n be n i·i.d.b·r·v. distributed according
to the cdf F(x,y)e
'"Xn
= (x..,
--J.
Let us denote by
Q.
••• , Xn ), Yn
= (Y:I'
N.Io
••• , Yn ) and "'!n
Z.
= (X
Nn
; '"Yn)
•••
The null hypothesis (1.3) implies that the joint distribution of (1:~, !~)
remains invariant under the nl
Z{R )
"'" ""'n
= fl [Xa ,
permutations of the type
a =1,
Y ],
R
a
••• ,
n]
•••
(2.2)
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where ~
t
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Now, let Nn
I be the set of nl possible.Bn of the permutations of n
numbers 1, ••• , n, and let T(ZNn ) denote the set of nl points (in an 2n
= (I1.' ••• ,
Rn ) is any permutation of (1, ••• , n).
termed the matching inva,riance.
This may be
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Now, under Ho in (1.3) and conditioned on the given ~ in (2.1), the set
T(Z
) oontains nt possible values 1(Z(R)
f ,which ~e (oonditionally)
",n
'" n
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equally likely.
Thus, oonditioned on the given ,...n
Z in (2.1), the permutational
probability distribution of "'Z(R
ReI"",n would be uniform when H0 in
" "",n ) over ",n
(1.3) holds. Let now 0(Z ) be any test function which with each possible
\ ""n
Z assooiates a probability ~(Z
) of rejecting H.
....n
! ....n
0
Now, if this probability
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function ""{Z
T "",n ) (= n(z
r ....n T(Z",n ») . is based on the permutationally uniform
distribution over the nl
possible realizations in (2.3), then it is easily
shown that 'j'~(Z
) possesses the structure See) of tests [cf. Lehmann and
Nn
Stein (1949),).
£:
0
Thus, corresponding to any preassigned level of significance
< e < 1,
we can seleot ch{z
1 ",n ) suoh that
Z ~(Z{R»
e, for all T(Z ).
R £1 'T ",n
"",n
Nn "",n
N
=
•••
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Henoe Q(Z ) will be a size E similar test for Ho in (1.3).
l""'fl
Thus, our problem reduoes to oonstruoting suoh a test funotion whioh
satisfies (2.4).
Now, as has been discussed in the preceding section, we would
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like to base our test on a statistic whioh is an exp1ioit function of Un' in
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(1.8).
For this purpose, we shall consider first some permutational properties
of U-statistios under matohing invariance.
For some related permutational
properties of U-statistics (under partitioning invariance, app1ioab1e to
multi-sample procedures), the reader is referred to Sen (1965 a, 1965 b).
it
3.
PERMUTATION DISTRIBUTION OF U-STATISTICS UNDER
MATCHING INVARIANCE.
Corresponding to Z(R
,.., .-.n ) in (2.2), let us define
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t
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I'
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•••
where the summation S extends over all possible
~
al
< ••• < am ~
(3.1)
n.
Thus conditioned on the given T(Z ) in (2.3),
"'n
;;;....-.n
U(R ):
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.-.n
.-.n J
•••
forms a set of nl
possible values of Un in (1.8)(not necessarily all distinct)
which are permutationally equally likely. Thus, for small samples, we can
evaluate the permutational distribution function of U(.fin)(given T(Zn),) from
the nl
possible values in (3.2).
However, the labor involved in this
prooedure inoreases prohibitively with the increase of the sample size, n.
To remove this difficulty (whioh is a common feature of almost all permutation
tests available in the literature,) we shall consider first some limit theorems
on the permutation distribution of U-statistics under matching invariance, and
SUbsequently, we shall use these for simplifioation of the large sample
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approach to our proposed class of permutation tests.
In the sequel,
induced by the nl
rn
will stand for the permutational probability measure
equally likely (permuted) values of U(R
) in (3.2), and
....n
the following conditions will be assumed to be satisfied by the kernel in (1.7) •
(i) Let
•••
.
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where the summation P extends over the ml
m
possible permutations of
(~1' "" ~m) over (PI' ... , ~m)· Thus ~* is the completely symmetric
form of~.
Then we assume that
... , Y m) = e
... ,
13
for all (Xa ,
1
(ii)
•••
0
2m
Y~ ) € R
•
m
For some positive k (which will be more precisely defined in some
specifia theorems),
E
il ~([Xal'
Y ], ... , EXam' Y )
R1
Rm
I k } <00,
•••
*... f
Rm = 1, ••• , n. If Ho in (1.3) holds, (3.5) means that
the kernel ~ in (1.7) has a finite k-th absolute moment under Ho ' On the
for all ~
otherhand, if Ho in (1.3) does not hold, it means that there are (rot1)1
different kernels of the type (1.7), (where in mt kernels, ~ is a function of
mtj observations of the original sample, for j
=0,
1, "', m), and for any
suah kernel, (3.5) holds,
(iii) Let
Wo([xl'Y1 )
= E {~([~'Yl]'[X2'Y2]""'[~JYm])IF e
w} - Qo'
•••
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We then assume that for all F £ w,
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where the summation S extends over all possible 1
nl possible
Uo
= e0
R £ ....n
I.
....n
Using
~ 0.
1
for all ....Zn •
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~([X131'YRf\],···,[X~m'Y~m)
1
that
• ••
(3.l0)
., .
(3.11)
m
2
- go'
where the summation Sc extends over all possible 1
=0,
n and P
(3.4), it is easily shown that
.~ i U{!!n) ~ n} = Uo = eo for all ~.
Let us define
S = n-[n]~ {(n)(m)(n-m)} -1 ~ dl([X 'YR L ••. ,[x 'YR ).
c,n
P
m c m-c
STOo l
a
am a
c
(3.8)
< ••• < am ~
Hence, it can be shown that
1 ~ ~l
...
Let
n-[m] = l/n[m] = I/n(n-l) ... (n-m+l),
over the
(3.7)
S l(G,H) > o.
Also, we require to introduce certain notations.
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•••
•••
~ 0.
1
< ... < am ~
n with a i = Pi i=l, ••• ,a,a"i =} i3 j for any other (i,j),
Using (3.4) it is easily shown .
< ... < ~m ~
1,
~
t •• ,
m.
n,
•
o,n = 0 for all Z
"7l
• ••
It then follows by a routine algebra that
•••
which for large n reduces to
2
2
(m /n) S l,n + [0(n- )] S m,n'
•••
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It may be noted that {t o,n; 1 ~
on ,...n
Z held fixed.
0
~ m] are all random variable as they depend
We shall now oonsider oertain oonvergenoe properties of
them.
Under the null hypothesis.Ho :
F
=. G.H,
let us define the unoonditional
oovarianoe of th([X ,Y ], ' ' ' ' [X ,Y ]) and $([L 'YI~ ], ••• , [X.~ ,Y,~ ])
T a1 a1
am am~l t"1
t"m t"m
(when a i = Pi for i ~
o~
0 ~
m.
and a i =1= Pj for any other (i,j»
I t is then easily seen that
0
by ~ o(G,H) for
• ••
(.3.16)
• ••
(.3.17)
• ••
(.3.18)
whioh for large n reduoes to
2
(m /n)
~l(G,H) + [O(n- 2 )] ~m(G,H).
Further from assumption (iii) we get that
na~,n--i
2
m
S1 (G,H) > 0,
if (.3.5) holds (under Ho ) for k = 2.
If (.3.5) holds for k = 2 then under Ho in (1 • .3), a~(~) is an
2
unbiased estimate of a u,n • Further, if under H0 , (.3.5) holds for k = 4, then
THEOREM 3.1.
P
---1
0;
a~(~) is the minimum v'arianoe unbiased (MVU) estimator of a~,n
(ii)
provided Z is oomp1ete •
.-n
PROOF.
If Ho in (1 •.3) holds, then for any R e I , [Xa'YR ], a
,...n.-.n
a
=1,
••• , n
=
i.i.d·b.r·v,
distributed aooording to a bivariate odf F G.H· Henoe, from
(.3.12), we get that
+ 9 02 is the average over nl terms (i. e., over
o,n
t
~
e
!n),
where eaoh term is a U-statistio in n observations [Xa 'YR ],
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t
a.
= 1, ... , n.
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t
Using this we get by simple algebraic manipulations that
,-~c,n
+
t;i\H
2
• ••
005
for all c = 1, ••• , m.
.
(3.19) together with (3.14) and (3.16) imply that
Hence, the first part of the theorem.
If we now define for any R--n
f'
~ c,n
5' (n) (m) (n-m) ~ -1
(R) ""fl
$([Xi~
,
~ m c
-
1'1
m-c)
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£
--n
~([X Y ] ••• ,[Xa 'Y ] ) •
m Ram
Sc r 0. 1 ' R '
a1
l::
'YR ], ••• ,[X;~ 'YR ]) - g2 ,
i3
I'm i3
0
,.
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(3.20)
• ••
(3.21)
m
1
then from (,3.12), we get that
~ c,n
= n-[n]
l::
p
~ c,n (R).
.....n
Since, (3.20) is a U-statistic in [X
0. 1
,YR ], a = 1, ••• ,n, it can be readily
a
shown using the results of Hoeffding (1948 a) that under Ho and for (3.5)
being true for k = 4,
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• ••
•••
uniformly in c
=1,
••• , m and in Rn £ In"
(3.22)
From (3.21) and (3.22) it is
easily shown that
• ••
for all c
=1,
••• , m.
(3.23)
From (3.14), (3.16) and (3.23), we readily get
through an app1ioation of Tsheby'£heff's inequality that n\a~(Zn)-a~,nl ~
o.
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Finally, using (3.20), (3.21) and a well-known result on U-statistic [cf.
Fraser (1957, p 142)] that under completeness of order statistics, a set of
U-statistic will be the minimum concentration ellipsoid unbiased (MCEU)
estimator of the corresponding set of estimable parameters, it follows that if
r
Z is complete, f".
; 1 ~ c ~ m] will be the MCEU estimator of - t c(G,H);
t ~ c,n
1 ~ c ~ mj. Since (3.14) is a linear fWlction of .i, ~ c,n J) f s whose expectation
is (3.16), the MCEU property of f. s' c,n :): f s implies the MVU property of
"'n
2
(]U(Z
..... n ) •
Hence, the theorem.
THEOREM 3.2.
If (3.5) holds for k = 4, no matter whether Ho in (1.3) is true
2
2
I P O.
or not, n (]U(Z
""n ) - aU ,n )----4
.
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PROOF.
It follows from (3.15) and (3.17) that for our purpose, it is sufficient
to show that
(i) 1
n ~ c,n = opel) for all 0=2, ••• , m, and
(ii)
I
~·l,n - ~ 1 (G,H)
P
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O.
If (3.5) holds for k = 2, then it is easily shown that the expectation
11
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of -n ~ c,n.! tends to zero as n ->00, and hence by Tsheleysheff f s lemma,
we get that 1
S
-!-.;> 0, for all 0 = 1, ••• , m. To prove (ii), let us
n c,n
first define for each 1
v(Xi,Y.) =
J
~
i, j
~
n
(n_l)-[~l](n-ll)-l ~ ~([Xi'Yj]'[X
m-
'YA ], ••• ,[X ,Y~ ]), •••
a 2 1"2
am I"m
C. '
J.
(3.24)
where the summation Ci extends over all possible 1 ~ a 2 < ••• < am ~ m with
Pm ~ n wi th ~k f j, for k = 2,
a k :f i for k = 2, ... , m and 1 ~ 13 2 t .
••• , m; 1 ~ i, j ~ n. Using (3.4) and (3.9), it is easily seen that
Of
1
n
-2 ~
n
~
n i=l j=l
v(X., Y.) =Q.
J.
J
0
:/:
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Also let
*
~ 1
Then,
n
2
2
Z v (X. ,Y.) - Q
n i=l j=l
J.
J
0
•••
i,n is also a stochastic variable as it depends on Zn' and using
(3.12), (3.24) and (3.26), it can be shown after a few essentially simple
steps that
It l,n -
t ~,n I = 0p (n- ).
l
•••
(3.27)
We can now classify the (n_l)(m-l] permutations of 13 , ••• , Pm in (3.24)
2
into two sUbsets: the first subset of (n-m) [m-l] permutations over p2' ••• ,
Pm none of which is equal to anyone of a 2 , ••• , am and the second subset
of (n_l)[m-l] - (n_m)[m-l] permutations of 13 , ... , ~m at least one of which
2
is equal to some one of a , ••• , am' Since (n_l)-[m-l]. (n_m)[m-l] ~ 1 and
2
(n_l)-[m-l] [(n_l)[m-1] _ (n_m)[m-l]) = O(n-l)(cf. Sen (1963),), we can
write v(Xi,Y j ) as the average over the first subset of permutations of 13 2 ,
••• , p plus a residual term which is easily shown to be of the order n- l ,
m
in probability.
Then utilizing the property of structural convergence of
U-statistics [cf. Sen (1960, pp. 2-5)] with a more or less straight forward
generalization to vector valued random variables (as here for a
will have the cdf Fo
steps
L2
n
n
n
.
=G.H.),
Z
f
13, [Xa,Yp ]
we get from (3.6) and (3.24) and some simple
2
1
Z E f[v(X.,Y.)-Q -'!' ([Xi,Y.])] IFe~J5 = O(n- ).
i=l J=l
J.
J
0
0
J
• ••
Further, if (3.5) holds for k = 4, it is easy to show (by direct computations,
that
I
~
,n
1
n
= -2 Z
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-14-
I.
I
I
I
1
n 2
'2 1: Z 1VO([X' ,Y.])
n i=l j=l
J.
J
n
for all F
p
~l(G,H)
_.~
(aotually the oonvergenoe will be in mean square).
& Q,
Ie
t
I
I
I
I
I
I
,.
I
From
(3.26), (3.28), (3.29) and lemma of Sen (1960, p. 4), it readily follows that
I
I
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I
•••
•••
P
tl,n --7 ~l (G,H).
Finally, (3.27) and (3.30) imply that
Henoe, the theorem.
The above theorem oan be further relaxed under oertain simplit:ying
oonditions.
In a majority of the oases, we shall observe that 1V o([Xi,Y j ])
satisfies the oondition
1VO([Xi,Yj]) = 1V01(Xi)
1V02(Yj)'
•••
(3.31
2
for all (X.,Y.) & R • In this situation, we have the following.
J
J.
THEOREM 3.3.
If 1Ir0([Xi,Yj]) satisfies (3.31) then theorem 3.2 hold even under
(3.5) being true for k = 2.
PROOF.
It follows from the proof of theorem 3.2 that we are only to show that
for (3.5) being true for k = 2, (3.29) holds under the simplifying oondition
on 1Jro([Xi,Yj]),
Now, if we denote the empirioal marginal odf1s by
'l
1
Gn(x)
=~
Hn(Y) =
~ xl ~
f
(number of Xi
~ (number of Yj ~ y)
,
•••
J
then the left hand side of (3.29) oan be written as
•••
(3.32)
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,.
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follows from Kintchine1s law of large numbers that if (3.5) holds for k = 2,
P
00
S '!t~l(X)dGn(X)
S
2
S 1f01 (x)dG(x)
-00
-00
00
00
----7>
P
2
1f02(y)dHn(y) --7
00
"S
•••
(3.34)
2
1f02(y)dH(y) ,
-00
-00
and hence, the left hand side of (3.29) stochastically converges to
Hence, the theorem.
THEOREM 3.4. If (3.4) and condition (iii) in (3.7) hold, while (3.5) holds
for k = 4(if further (3.31) holds, we may relax k = 2 + $' , $
> 0,)
then the
permutation distribution (i.e., over ""n
R &,'-I' n ) of
nt [U(R ) - Q .?" 1m f t
""n
oj
sl,n
asymptotically, in probability, reduces to a normal one with zero mean and
unit variD.Ilce.
(It may be noted that the permutational probability measure
QDn is
essentially a conditional probability measure, and as such the convergenoe
m~
not hold uniformly for all conditioning factors, but will be shown to hold
for almost all Z .)
n
PROOF.
Let us write
VCR )
,."n
n
=-1n i=l
~ v(X.,Y ), R e I ;
Ri ·.. . n ",n
1.
•••
where v(Xi,Y j ) is defined in (3.24). We shall then prove first that under
the permutational probability measure cPn'
E {n[ (U(R )-Q )-m(V(R )-Q )]2
""n
0
",n
0
I OJ.n "25 = 0p (n-1).
• ••
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(The right hand side of (,3.37) is 0p (n-1), because, the permutational
expectation depends on the variable .....n
Z held fixed and hence, is a stochastio
variable, which will be shown to satisfy (3.37),in probability.)
Now, by
virtue of (,3.4) we have
•••
for all 1
~
i, j
~
n.
(3.37 a)
Henoe, proceeding preoisely on the same line as in the
proof of theorem 2 of Hoeffding (1951, p. 562), we readily get that
E f [VCR )-9 ]2 I~
",n
=
0
n
n
Z Z v2 (X, ,Y j )-92]/(n-l)
'-1'-1
~.
0
~J-
] = [12
n
n
~l* ,nI(n-l)
by theorem 3.2.
=
t l ,n/n +
° (n-2),
•••
p
Similarly, by direct (but somewhat lengthy) computations,
it can be shown that
•••
From (3.15), (3.38) and (3.39), we readily arrive at (3.37).
implies that under the permutational probability measure
1
n2 [U(R )-9]
~n
0
p
r..J
Again (3.37)
r n,
1
n2 [V(R )-9 ].
.
-n
0
• ••
Consequently, it is sufficient to establish the asymptotic normality of the
.1..
permutational distribution of n 2 [V(R
)-9].
Mn
0
To do this, we use the following
(slightly modified) version of Hoeffding-Motoo combinatorial Central Limit
Theorem (cf. Hoeffding (1951), Motoo (1957»:
let bn (i, j) be n2 real numbers defined for each positive integer n,
and let R be a permutation of (1, ••• , n) and
"'n
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,.
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~
1 n n
2 1 n n
2
= --2 ~ ~ b (i,j), a = -~
~ (b*(i,j)] ,
n n2 i=l j=l n
n n i=l j=l n
• ••
where
n
·..
b*(i,j) =b (i,j)_l ~{b (i,g) +b (g,j)} +~,
n
n
n g=l n
n
n
for 1
~
i, j
n.
~
Then, if for some r
>2
·..
n
the statistic
vn- IIn i=l
L: b (i,R.)
n
J.
- 9 .~
o.J
/a
n
has asymptotically a normal
distribution (permutation) with zero mean and unit variance.
Thus, if we let v(X.,Y.) - 9 to correspond with b (i,j), it then follows
J.
J o n
from (3.24), (3.37a) and (3.42) that b~(i,j) = bn(i,j) = v(Xi,yi)-Qo. Further,from
(3.26), (3.27) and theorem 3.2 (or 3.3), we get that
2
an =
ttn
p
-7
~l (G,H).
• ••
Hence, from (3.43) and (3.44) it follows that we are only to show that for
some r
>2
• ••
(or (3.45) holds, in probability).
letting 2
<r ~
Now, if (3.5) holds for some k
k, it readily follows that n-(r-2)/2
n i=l j=l
has an expectation which converges to zero as n
holds, in probability.
{L2 ~ ~
"""'"700.
> 2, then
(v·(X., Y. )-g I rj
J.
J
0
Consequently, (J.45)
This, in turn implies that (J.45) holds, in probability.
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,.
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1
Hence, the asymptotic normality of the permutation distribution of n~[V~n)-Qo]
holds, in probability.
The rest of the proof then follows from (3.40).
Hence, the theorem.
Before we conclude this section, we would like to consider a simple
estimate of
h,n or
~l (G,H),
which will make the large sample approach of
the tests (to be proposed) more convenient from practical stand point.
estimate will be usable when
%([X,Y)) satisfies (3.31).
If
This
%([X,Y]) =
o/Ol(x)o/ 02(y), it follows from (3.27) and (3.30) that
1 n n 2
2
1 n 2
1 n 2
P
.!:
!: ~Ol(Xi) ~02(Yj) = - !: *Ol(X,). - !: ~02(Yj) /'--" S:l
•••
n J.=l j=l
n i=l
J. n j=l
,n
"2
(3.46)
A
As we shall see later on that in many cases, we can have estimates
'!tO (X) and
l
1\
0/02 (Y) of ~r 10 (X) and '!t 02 (Y) such that
•••
From (3.30)4 (3.46), (3.47) and lemma of Sen (1960, p. 4), it readily follows
that
2
1
n
A
2
1
n "'2
s = - !: W,Ol(X,)- !: W,02(Y j )
J. n j=l
n i=l
P
rV
~ 1
P
,n
-7
~ 1 (G,H).
• ••
Hence, we arrive at the following
THIDREM 3.5.
If '!to ([X, Y)) in (3.6) satisfies (3.31) and if there exists any
estimate of '!tOl and 0/02 satisfying the oondition (3.47), then s2 defined in
(3.48) is stochastioally equivalent to ~ l,n and converges in probability
to t 1 (G,H).
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4. PERMUTATION
TESTS BASED ON U(R ) AND THEIR PROPERTIES.
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We shall now enter into the formulation of the actual permutation tests
for the null hypothesis
(1.3)
and consider their various properties.
Now, corresponding to aQY observed .......Zn (defined in (2.1),) we have a set
of n!
values ",~n
Z(R ), defined in (2.2).
These n!
(oonditioned on the given . .Z. n ) equally likely.
funotion '\'~(Z
",n ) whioh satisfies (2.4).
possible values (,...
r Z(R
): "'n
REI
}
~n
~n
a manner that (2.4) holds.
values are permutationally
We require to oonstruct a test
In small samples, to eaoh of the n!
we may attaoh probability masses in suoh
However, as has been stressed in seotion 2, we
would mostly prefer to use a single test statistic and to base a rejeotion
rule on the values of this statistio.
properties of
t U("?n) :Sn E ~n 1.
In seotion 3, we p'.V'e studied some
Here, we shall oonsider Un' defined in
(1.8), as a test statistic and base the permutation test on it.
let us divide the set of values
Ho
Yo 1(Yn f
Hl and !!2' where
p
f U(P-n )
€
of our test;
lh
J
into three subsets
1uQtn) ] ,
.)
suoh that
f U(Rn )} ,
such that
en ~ e ~ en + '( , and U2 is the comptementary set
n,~
It may be noted that all these subsets )}o'
and assumed fixed struoture for fixed ,....n
Z.
91
and !!2 are stoohastio
For convenience in notation, we
= <,po' El' .Y2 )·
•••
Then, we formulate the following test function ~(Z
",n ):
~(~n)
r<
1, if Un E 'Yo'
I
=
,.go'
= En ~ E, where E is the preassigned level of significance
(may be empty) is a subset of points
ru
write ~*
!'n E .fn
contains a subset of values of
p ~. U(R
""'n ) E Ull @n ~ = ":
n
of ,PoU!!l.
f U(~n):
For this,
I
[
Il-
(E-E n )/ Y n' if UnE£l'
0, if UneU,... •
2
•••
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II
:1
:1
tl
II
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It is easily seen that ~~n) satisfies (2.4) and hence is a distribution-free
similar test of size e:
fl
,e
'I
I
lr..•...
Now (4.2) simplies to a great extent for
the conventional one (or two) sided tests whereA~o is the set of lower (or
21
both) extreme v'alues of U'»n) and
large samples, we can use theorem
of
Consistency of the test.
Q(F)
f gO,
may be a single (or two) point (s).
3.4 and rewrite
then,U in
n
consequently I' Yn:
f"n
U
It readily follows that if F
¢ w,
and for F ~ w,
(1.8) will converge stochastically to Q(F)
- Q f
0
I
For
.h(Z
'II ,...-n ) in terms of the values
- g 0 1 /m~ *1 ,n •
Vii- f. Vn
will be stochastically large.
+QO,
Hence, if we
J
base our permutation test and utilize the asymptotic normality in theorem
3.4,
we readily arrive at the following.
THEOREM 4.1.
The test (4.2) will be consistent against the set of alternatives:
F e wo ; Wo = f F: Q(F) f Qo} ,
provided the conditions of theorem 3.4 hold.
(For one sided alternative the formulation will be very similar).
In view of the consistenoy of the test
4(3n ),
for any given Q(!)
the power of the test would be asymptotically equal to unity.
f
Qo'
Henoe to study
the asymptotic power properties, we consider certain sequences of alternatives
chosen in such a way that the power asymptotically lies in the open interval
I
!I
< e < 1.
0
Let
l Fn f
be a sequence of distribution functions chosen in such a
manner that
(i)
Fn(X,y) ._-> F(x,y) as n -'7 00, at all points of continuity of F, and
Few;
(ii) . Q(F)
= Q0 + an
(' , where n1: S-n'
-~
n
S (:1:0)
I
as n "--700 •
We then frame our class of alternative hypotheses [H ~ by
n
•••
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,.
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Q(F) = Q(F ) = Q + [ •
non
H:
n
THEDREM 4.2.
•••
Under the sequence of alternative hypotheses:" H )\ in (4.4)
..!.
\.
n
1,..
and under the conditions of theorem 3.4, the statistic n 2 tUn- o
Q )~ /m~ 1 ,n 2
..!.
r
has asymptotically a normal distribution with mean
J /m[ ~ 1 (G,H)]2 and unit
varianoe.
(The parameter
as ~ (F).
S
may depend on F(e w ) and should be properly defined
However, for notational simplicity, we would prefer to use
.l.
The asymptotic normality of n 2 {Un - E(U
n
PROOF:
I Hn ) 1 follows
g.)
readily from
1
Hoeffding1s (1948 a) results.. Now, under 5( n
H ?l , n2 1.'E UnnS
IH {' - Q0)? =
1
n2 {Q(F ) - Qo~ --;; 5. So the only thing remains to be proved is that under
n
f Hn·1 ·
P
(i)
I!'
~
l,n --7 ~ 1 (G,H), and (ii) if ~ 1 , Hn denote the covariance of
O([JS.,Yl ], •• " [Xm,Ym]) and
~1 H
, n
---,?
S 1 (G,H) as n
.-?
¢ ([Xm,Ym],
••• , [X2m- l ,Y2m- l ]) under Hn , then
00-
The first part follows readily from theorem 3.2, while the second part can
be proved by a more or less routine method.
Hence, the theorem.
By virtue of theorem 4.2, we have the asymptotic relative efficiency
of
9(~n) proportional to S2/m2 ~ 1 (G,H), which may be oompared for various
possible
O(Z
)'s.
, ""n
5.
APPLICATIONS AND ILLUSTRATIONS
Before we proceed to oonsider some specific tests, we would like to
disouss the relationship of this type of permutation tests based on U and
n
possible large sample tests based on similar U.
..!.
2
1
n
It follows from the a.symptotio
normality of n fUn - Qo } /m[ ~ 1 (G,H)]2 ~'under Ho ) that if
A
tl
is any oonsistent
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estimate of ~l(G,H), then an asymptotioally unconditional test for the
l
hypothesis (1..3) may be based on the test statistio n 2 {Un - go f
Al
1m ~i
·
A
Further, if ~ 1 has the property that for all Fn -? F
il
.__~~
t
& w ,
(satisfying (4.4),)
1 (G,H), then it is easily seen that the permutation test based on
U and the asymptotic test based on the same U are asymptotioally power
n
n
equivalent for the sequence of alternatives r
" HnJ? in (4.4).
(The proof of
the statement follows by simple arguments and hence is omitted.)
Thus,
permutation test also links the gap between an exact similar test for the null
Qypothesis (1 •.3) and an asymptotic test for the same.
Let us now consider different kernels and study how they correspond to
some
knO\oll1
tests.
Let us first consider the kernel
It. is easily seen that E t ~ IHo $ =
° and for any linear dependence, E f $) H1 f 0.
Now, it is easily seen that
~*(Xl,X2'Yl'Y2) =t[9([Xl ,Yl ],[X2 ,Y2 ]) + ~([Xl'Y2],[X2'Yl])]
·..
So (.3.4) holds.
If Ho holds, then (.3.5) will hold if E/X!k
<00
and ElY Jk
<00.
Further
1Jr o ([xl'Yl])
= E f(~-x2)(Yl-Y2)IF &
w}
·..
=E(~-X).E(Yl-Y) = (xl-~1)'(Yl-~2)'
where
~l
E[
and
1Jr
~2
are the means of X and Y respectively.
~(X,Y) IF
&
w ]
= E(X -
~1)2
E(Y -
~2)2
= a; a; > 0,
if the distributions of X and Y are non degenerate.
condition (.3 •.31).
Thus,
·..
~ also satisfies the
Thus, the asymptotic normality of the permutation distribution
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of the oorresponding Un (whioh is nothing but twioe the product moment
oovariance of X,Y) oan be deduced with the help of theorem .3.4 and suitable
tests might be constructed.
This test is valid for all odf's for which the
marginal distributions do not degenerate to point distributions.
Pitman's (19.37) test for the sample correlation coefficient.
This is
This test is not
a very robust one in the sense that the rate of convergence to the asymptotic
normality of the oorres,onding U-statistic is slow as oompared to some other
rank tests and further this convergenoe depends to some extent on the
fluotuations of the sample extreme values.
We shall now cons,"der the rank tests for independence. Out of the four
measures of assooiatioa considered just after (1.4), we shall specifically take
up the difference sign covariance, while the treatment of any other one will
follow precisely on t:ae same line.
For rank tests, we should differentiate
the two situations; the first one relates to da,ta where there are no tied
observations while in the second case,
some of the observations may be tied
and as a result, som. sort of artificiality has to be introduced for the
definition of ranks.
If the parent cdf F is continuous, the possibility of
tied observations mGlY be ignored, in probability.
Further, if in such a case,
the kernel in (1.7) ~s an exp1ioit function of the ranks of the observations,
then it can be shm. that the permutation distribution of U (in 1.8),) agrees
n
with its (uncondit~nal) null distribution. Since, the unconditional theory
follows readily fr\ltll Hoeffdingts (1948 a) results and many such actual tests
have already been tonsidered by him, we shall not enter into the discussion of
these.
On the otl:j,erhand, if F is not continuous everywhere, so that tied
observations may ~se with a positive probability ~d/or if the kernel is not
an explicit funct~n of ranks, Hoeffdingts (1948 a) theory will fail to provide
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-24us with the permutation distribution theory of U.
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ourselves to this situation.
We shall therefore oonfine
5.1 Tests for association on a countable or finite (bivariate) d;screte sample
space.
Let (X, Y) be a random variable where X assumes v'alues in a spaoe A =
[ ~, a , ••• } having either a finite or a countable number of disorete mass
2
points and similarly Y assumes values in a spaoe B = bl' b , ... } having a
f
fini te or countable number of elements.
single point sets.
2
We only assume that A and B are not
The probability law is defined by
...,
•
• ••
where obviously
Z 1T. j = 1T (l), ~i 'IT
= 'lT (2), ~ ~ 1T = ~ 1T (1) = ~ 1T (2) = 1
j
ij
i
j 1
i j ij
i i
j j
• ••
The null hypothesis of no association implies that
Ho :
1T ij = 1Ti l ) 1T32) for all i = 1, ••• ; j = 1, ••• ;
•••
=
Let then (X ,Y ), a
1, ••• , n be independent observations on (X,Y) and based
a a
on this sample of size n we want to test the null hypothesis (5.7). We now
define a kernel
9([Xa'Ya]'[X~,y~]) = lif(Xa-X~)(Ya-Y~)
= 0 if
>0
(Xa-X~)(Ya-Y~)
=-1 if
It is easily seen that Q(F)
(X -X )(Y -Y )
a ~
a ~
=0
• ••
< O.
= E i~IF} = 0 for
all F e; w, while Q(F) 'Will be
positive or negative acoording as (X,Y) are positively or negatively associated.
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-25The U-statistios oorresponding to (5.8) is given by
·or
n-l
U = (2)
n
~
l~a~~
~([X ,Y ],[XA'Y/:l.))'
a a
t"
t"
• ••
U is nothing but the difference sign oovariance, whioh has been studied in
n
detail by Hoeffding (1948 a) only for oontinuous odfls.
Now, we oan easily
show that (5.8) satisfies (3.4), (3.5)(for any finite k
and B are not single point sets).
satisfies (3.31).
> 0)
and (3.6)(if A
Further, it is easy to show that
palso
In faot,
• ••
(5.10)
for all i = 1, ••• ; j = 1, .,..
Thus, if we define Hn and Gn as in (3.32),
then as in theorem. 3.5 it oan be shown that
n
s2=* ~ [G (X.-O)+G (X +0)-1]2. 1 L [H (Yj-O)+H (Y +0)_1]2
i=l n J.
n i
n j=l n
n j
is stoohastioally equivalent to
permutation test on the nt
~l,n'
• ••
(5.11)
Thus, in small samples, we oan base a
possible realizations of U in'(5.9)(under
n
permutational set up) and prooeed as in (4.2), while for large samples, it
1
will be oonvenient to use the asymptotio normality of n2UJ2s for the formulation
of the test statistio.
The power of this test for sequenoes of alternatives
as in (4.4), oan be readily traoed (asymptotioally) with the aid of theorem 4.2.
I t may be noted that in the above formulation we have oonsidered a.J. and
bj to be discrete mass points.
This oan be also extended readily to the
following situation, where Ai stands for a typioal value of the oell Xi _l
X ~ Xi' i = 1, ••• ; and bj for Yi - l < Y ~ Yj for j = 1, •••• With this
<
simple modification the test is thus also applioable to grouped data, no
matter whether the parent cdf is continuous or not.
Also, Un in (5.9) does
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not depend on the values of
relative magnitudes.
f ~, ... IS
and {bl' •••}
but only on their
Thus, the test is not sensitive to the choice of the
values of {.~, ••• 1 and ~. bl , ••• } • This property is particularly useful
when the extreme cells ma:y be half open and as such, a typical v'alue for such
cells may be indefinitely large.
5.2.
Tests for association for graded qualitative traits or categor.ical data.
Let us now consider a two way contingency table where the A classification
has r l categories~, ••• , A and the B classification has r 2 categories
rl
Bl , ••• , B , where r l and r 2 are two positive integers which are greater than
r2
one. Any observation (X,Y) belong to some cell (Ai' Bj ), and the number of
observations belonging to the cell (A., B.) is denoted by n .. , 1
~
1
~
j
~
~J
~
i
~
rl ,
n2 and let
r
== n.~o ,
r
J
r
l
l
~
i==l
== n .,
oJ
n ..
~J
• ••
2
~
n. == ~ n
== n.
i==l ~o
j==l oj
Here also, we denote the probability 1 aw by
p
·r-
(X,Y)
P
f
X & A.? ==
&
~
}
(A. ,B.)
~
J
?
-
==
TT..
~J
1T~1)
and P
~
;1 ~ i ~ r , 1 < J.
l
-
f Y E B.J,'('
==
1T~2),
J
r
~
r ,
2
·..
and we are interested in the null hypothesis
·..
If nio noj/n is large for all 1 ~ i ~ r l and 1 ~ j ~ r 2 , then under the null
~othesis (5.14), the statistic
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•••
has approximately a ohi-square distribution with (r -l)(r -l) degrees of
2
l
freedom, and the same is mostly used to list for no assoaiation between A and
B.
Mitra and Roy (1956) have studied this statistic from the stand point of
MANOVA for categorioal data and also considered its power properties.
However, this statistic seems to have three drawbacks.
based on the assumption that n. n ./n is large (at least 5) for all (i, j),
1.0
which may not be always true.
oJ
Secondly, we may have rl' r 2 quite large and
this in turn will make the degrees of freedom of Xl
"large.
in (5.14) also appreciably
This has some serious effect on the power properties of the test.
It
is well known that as the degree of freedom of a chi-square variate increases,
the power of the associated test will decrease unless the noncentrality
parameter increases at a much faster rate to counter balance this loss (cf.
Mann and Wald (1942),). Thirdly, in the extreme case, conoeptually r
r
2
l
and/or
may be countably infinite and in this case, the test will be applicable
only after proper amalgamation of cells.
This in turn introduces oertain
arbitrariness in the sense that different statisticians may prefer different
types of amalgamation of cells.
These drawbacks can be removed if we have the reasons to believe that
the caregorical set is an ordered set.
We denote Ai
< Ai'
if the categorical
element Ai' is preferable to Ai and a similar case follows with B and Bj '.
j
Thus, if
A= (~ <A < ••• < A )
2
r
l
I"-'
~ = (Bl < B2 < ••• < Br )
2
I
Firstly, it is
•••
(5.16)
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be the two ordered set, no matter whether r 1 , r 2 are finite or countab1y
infinite, we m83' use the following test. Let
~([Xa'Ya]'[X~,y~]) =1 if Xa < Ya ' X~ < Y~
or Xa
> Ya '
X~
> Yp;
•••
= 0,
(5.17)
otherwise;
It is then easily seen that ~ in (5.17)
and we define Un as in (5.9).
satisfies (3.4), (3.5), (3.6) as well as (3.31).
Thus, we may prooeed
precisely on the same line as in the case of discrete finite or countable
sample spaoe.
Let us first simplify the expression of Un in (5.9) in this
case. Let
Ni
,
j
= k~i
~
for 1
~
i
~
I.~j
~
n. j , N.
J.
J.,o
= ~i
~
~ j ~
r l and 1
nk , N j
0
0,
=I.~j
~
22
+ Z n2
io + Z n j - 3 Z Z nij - 2
j
0
•••
0
(5.18)
r 2 • Then, using (5.17) and (5.18), it can be shown
following a tew simple steps that
2
n(n-l)U = n +4 ~ Z ni . Ni J' - 2 ~ ni Ni 0
n
i j
J,
i
0,
i
n /.,
i j
-
2 ~ n j N .
j
~ ~
0
~
i j I.~j-l
o,J
n. j ni f
J.
A
•••
For actual oomputation of (5.19) it is most convenient to have two two-way
tables, one with the entries {n , nio and noj
iJ
f
and the other with the
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cumulative entries [N. ., N.
~,J
~,o
)~~.
and N j
0,
From these two tables, the
computation of (5.19) will not appear to be difficult.
Similarly, s2 defined
in (5.11) reduces to
s2
= '1..
6
n
[Z
i
[2N.
~,o
- n.
~o
- n] 2"{
~ Z [2N . - n j - n] 2');1
)
j
O,J
0
,
•••
and for large samples, the test is based on
t =
vn
fUn 1 /2 s,
•••
(5.21)
which has (under H ) a normal distribution with zero mean and unit variance.
o
Further, when the null hypothesis is not true, the asymptotic normality of
Vii: fUn - Q(F)} will follow from Hoeffding1s (1948 a) results and hence,
from (5.21), we get that the power of the test based on t can be computed
for any sequence of alternative hypotheses.
The advantages of the proposed test over the classical contingenoy t
test are
(i)
this is valid even when some of the oells are empty or have very
small frequenoies, (as the asymptotic normality of the permutation distribution
of Un does not require the expected frequenoies to be all large);
(ii)
the asymptotio normality of \/n Un makes it insensitive to the
values of r l and r 2 and also avoids the problem of amalgamation of oells
when the expeoted frequenoies are not all adequately large;
(iii)
there is no diffioulty in tracing the power of the test against
suitable families of alternative hypotheses, and finally
(iv)
the test remains valid when r
l
and/or r
2
may not be finite but be
oountably infinite•
Finally, we will show how the proposed test reduoes to a well known test
when r
=2.
In this case, it follows from (5.17) that
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• ••
which corresponds to the usual statistic used for testing association in
2 x 2 tables.
In this case, the test based on Un and the ohi-square test
in (5.15) can be shown to be equivalent, and we have the same underlying
assumptions.
On the otherhand, if r
> 2,
1
in order that t (G,H) in (,3.7)
is positive, we require only the assumption that the marginal distributions
of the two way table do not degenerate to point ones, while the use of chisquare test in (5.15) requires that the expected frequencies of each cell
must be at least equal to
5 or
more (cf. Cochran
(1952),).
Let us then consider a related problem of independence which often
arises in some psyohometric analysis of traits.
Here we assume that underlying
the categories A , •• ', \: (whioh we assume to be ordered) there is a trait
l
(t) whioh is not observable.
Such a trait oan often be graded into a k(Ll)
point soale (eg., intelligence of students, attitude towards a political or
social event etc.), and the oategories A
l
r
l
< •• ' < Ar
may be regarded as the
l
graded levels of the underlying trait t ; t may have a oontinuous
distribution.
as the r
2
Similarly, the ordered oategories B
l
graded levels of another trait s.
< ••• < Br
may be regarded
2
The joint cumulative distribution
of (t,s) is denoted by F(t,s) and their marginal cdfts by G(t) and H(s)
respectively.
It is desired to test the null hypothesis that t and s are not
assooiated with each other.
in
(5.21).
In such a oase, we may of oourse use the test
But in psychometrio analysis, often we prefer to use certain scores
to the grades of the traits t and s.
A, we attach the soores a
n, 1
l
levels of
< ••• < an,r and similarly for the r 2 levels
l
< ••• < bn,r • We want use some measure of
n, 1
B we have the soores b
This corresponding to the r
of
2
association whioh utilizes the information on G(t) and H(s) oontained in these
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soores.
There are various ways of attaohing suoh scores and the oomparative
performanoes of some of these have been studied in detail by the present author
[Sen (1966)].
For our purpose, we consider the following method.
=1,
We define {Ni,o' i
=1,
... , r 2 } as in
Also for an unessential simplicity, we assume that G(t) and H(s) are
(5.18).
... , rll and {No,J' j
suoh that
StdG(t)
= 0,
St2dG(t)
=1
and SsdH(s)
= 0,
Ss2dH(s) = 1.
•••
(5.2.3)
(In faot, if G and H possess finite seoond moment, we oan always have (5.2.3)
by simple change of origin and scale.)
=
a solution t = t n, i' for i
1, ""
Then, we formulate the soores
t
a
n,i
Then by equating G(t) = Ni,o/n we get
r l and we define toby
G(tn, 0) = 0.
n,
t n,l..
tdG(t)1 S
dG(t),
n,i = S
t n,i-l
t n,i-l
r
l
By definition, we have 1: a i ni In =
i=l n,
0
~
•••
° (by (5.2.3),)
r
and also
l
(ni In)a2 .
i=l 0
n,l.
1:
2
. St dG(t) = 1.
On the otherhand, 'fa
- n,l.., i = 1, "" rIll is a stochastic
vector as it depends on the vector £. nio ' i = 1, "" rll' This will be
termed the observed score vector, while the true soore vector {ai' i = 1,
••• , r lS may be defined in the same manner (as in (5.24), ) with the only
ohange that t n, its are replaced by ~ils, where G(~.)
= E(Ni ,0In), i
0, ""
1.
=
Similarly, let fbn, j' j = 1, "', r 2 ?
) be the observed score vector of the
levels of B, whioh is defined precisely in the same manner as in (5.24) and
let
rP
j' j = 1, ... , r
f
2 be the true soore veotor. Now, in partioular, if
G(t) is assumed to be normal, the scores will be termed the normal scores
[often called the Z-score, of. Garrett (19.37, pp• .307-.310)].
Similarly, if
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G(t) is assumed to be uniform, the soores will be termed the peroentile
soares.
In aotual praotioe, mostly either of these two soores are used.
However, there is no logioal diffioulty in extending this definition to any
cdf not necessarily normal or uniform.
Once we define the soores in any suitable manner, the r r 2 cells are
l
defined in terms of a set of paired soores, and thus conditioned on the
given {nio ' i = 1, .. "
i
rlf and {n oj ' j = 1, ... , r 2
~,i' i = 1, "', r l } and [bn,j' j
=1,
j (whioh in turn fix
... , r2~)' we have a (bivariate)
discrete sample space and we can use our results in section 5.1.
rank measures of association, they will be independent of the partioular set
of values of fan,i' i = 1, ... , r l } and
r bnj ,
j = 1, ... , r2'~'
so that
the tests oonsidered earlier this section will also apply to this situation.
However, these will fail to utilize the information oontained in the soores.
To do this, we may oonsider the produot moment covarianoe of these soores and
this oomes out to be
rl
r2
Tn = 1.,2 1: n j a . b j'
n i=l j=l i
n,J. n,
•••
and there is no diffioulty in applying our permutation test theory as based on
Tn' and the same will follow essentially on the same line as in the oase of
produot moment oovarianoe oonsidered earlier.
r
s2
n
'Zl a2 n. ?
n i=l n,i 1.0)
= [1.
[1.
(
n
~
j=l
For large n
l
if we define
b2
n '2 ,
n,j oj J
•••
.1.
then it follows from our theorem .3.4 that n 2 T / s
n
n
will have asymptotioally
a normal distribution (under H ) with zero mean and unit varianoe.
o
if we oonsider the sequenoe of alternati ves
I
If we use the
Further,
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Hn :
•••
for i = 1, ••• , r , j = 1, ••• , r 2 (so that we are oonsidering a sequence
1
of joint distributions F (t,s) which asymptotically converges to G(t).H(s)
n
in the usual Pitman-sense), where
Jij r s
are all real and finite quantities,
then it can be shown following a routine algebraic manipulations that under
(' H ) ,nt Tis
1. n J
r
l
.I:
n
r
I:
i=l j=l
2
S ij
n
C1i~j
will have asymptotically a normal distribution with mean
and variance unity.
With this, the comparison of the
asymptotio power properties of this test with that of the others does not
appear to be difficult at all.
In fact, for the type of dependence (linear)
of t and s as considered Konijn (1956), we can also find out the asymptotic
(extended-) Pitman-efficiency of this test with respect to any other test.
But, the same will be a funotion of the density functions of G(t) and H(s)
and for allpraotioal purposes, may not be a very suitable thing to study.
The non-parametrio tests for randomness against various types of
stochastic dependence considered by Mann (1945, 1946), Moore and Wallis
(1943), Stuart(1954)and Stuart and Foster(1954)are really based on U-statistics
and the theory of permutation distribution studied here also enables us to
use these tests for disorete processes.
Since, these will be a partioular
case of similar tests for independence where one of the variates is nonstochastic, the results follow precisely on the same line as in the tests
considered here.
Hence, the details are omitted.
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~,
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