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ASYMPTOTICALLY. EFFICIENT TESTS
BY THE METHOD OF n RANKINGS
by
Pranab Kumar Sen
University of North Carolina, Chapel Hill
and University of Calcutta
Institute of Statistics Mimeo Series No. 504
January, 1967
Supported by the Army Research Office, Durham,
Grant DA-ARO-(D)-3l-l24-G746.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel
Hi11~
N. C.
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ASYMPTOTICALLY EFFICIENT TESTS BY THE METHOD OF n RANKINGS*
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By
Pranab Kumar Sen
University of North Carolina, Chapel Hill
and University of Calcutta
1.
Summary and introduction.
of ranking after alignment is not practicable, the method of n rankings [cf. Friedman
(1937), Kendall (1955)] is quite useful for the analysis of two way layouts.
of 0l""'Op by the ith observer are denoted by ril, ••• ,r ip ' respectively, for
i=l, ••• ,n.
Well-known tests for the hypothesis of no difference among the p objects
are due to Friedman (1937) and Brown and Mood (1951), and are respectively based
on the test statistics
Je
(1.1)
]
and
(1.2)
1
1
]
Con-
sider n observers, each of which ranks (independently) p objects 0l""'Op; the ranks
~
1
When the block-effects are not additive or the method
x 2 = [12/np(p+l)]
r
n
r (R.-n(p+l)/2f;
. 1 J
R. =
J
J=
j=l, ... ,p;
n
r
M = p(p-l)
(M.-na/p)2;
r
na(p-a) j=l J
where mij is 1 or 0 according as r
is
ij
.L r ij ,
~=l
~
M.=
J
Im .. ,j=l, ... ,p,
i=l ~J
or > a: 1
be the largest integer contained in (p+l)/2.
~
a < p; usually a is taken to
Generalizations of the x2-test to nonr
orthogonal designs are due to Durbin (1951) and Benard and Elteren (1953) and their
asymptotic power-efficiencies are studied by Elteren and Noether (1959).
works on the other test are due to Bhapkar (1961, 1963).
Similar
The present investigation
is concerned with (i) the derivation of a class of asymptotically efficient test for
the same problem and (ii) the characterizations of the optimality of the tests based
on (1.1) and (1.2).
* Supported by the Army Research Office, Durham, Grant DA-ARO-(D)-3l-l24-G746.
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2.
~.
-.
We conceive of some stochastic variables (X.l, •.• ,X. )
lp
l
(mayor may not be observable) underlying the ranks (r.l, ••. ,r. ), for i=l, ..• ,n.
l
lp
It is then assumed that X.l, ... ,X. are independently distributed according to continl
lp
uous cumulative distribution functions (cdf's) F.l(x), ... ,F. (x), respectively,
l
lp
for i=l, •.. ,n.
The null hypothesis states that
(2.1)
= Fin =
H :
o
whatever be Fl, ..• ,F •
n
(2.2)
F., for all i=l, ••• ,n,
l
We are interested in translation alternatives viz.,
Fl.J.(x) = F.(x-t.), j=l, ... ,p, i=l, ... ,n; L.Plt. = 0,
l
J
J= J
t = (tl, ••. ,t ) being a real p-vector.
p
(i)
(ii)
For our study, we shall assume that
n (the number of observers or blocks) is large;
F.(x) is absolutely continuous having a continuous density function
l
el
f. (x) where
l
00
f f7(x)dx
<
l
(2.3)
00
for all i=l, ••• ,n;
_00
(2.4)
(iii) t
=t
~n
8 = (8 , •.• ,8p ) has real and finite elements.
1
~
Also, we shall confine ourselves to the following class of rank tests.
Let
{J(r,p), r=l, ••. ,p} be p real valued functions, where J(r,p) is a function of r
and p, r=l, ••• ,p.
(2.5)
We define
and
=
p
=1
I
r=l
{J(r,p)-J}2
Concerning {J(r,p)} we assume that for any finite p, they are all finite and are not
all equal.
Under these assumptions both J and A2 (J) are finite and A2 (J) is
strictly positive.
where
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We then define a rank statistic (vector) T
= (Tn, 1, .•• ,Tn,p ),
~n
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T
(2.6)
The tests to be considered here are based on the following type of statistics
(2.7)
S
n
. -2
=n
A
(J)~.
1 T .-J-}2 •
n,J
p {
J=
It is easily verified that both (1.1) and (1.2) are particular cases of (2.7).
Our main contention is to select {J(r,p)} in such a manner that the corresponding S
n
leads to asymptotically efficient test.
3.
Asym
totic''"'''--'-''-'''~'''''''''''''';'-'o"""",,-,
distribution of S.
"""'''"''~'''''''''''
n
in the sequel.
This is presented briefly, as it will be required
Let us define for each i(=l, ... ,n)
= p{ r .. =r } , p(.j,j') = p{r .. =r,r .. ,=s}, r;'s=l, ••• ,p, j;'j'=l, ••• ,p.
1J
1, rs
1J
1J
(3.1)
Conventionally, we let
p ~j ,j) = p~j) • 0
(3.2)
1, r
1, rs
P ("J,Jof)
i,rr
rs'
where 0 .. , and 0
are the Kronecker deltas.
JJ
rs
1
(3.4)
P
n
l
(s)
o..
,p. ,
JJ 1,r
r,s,j,j'=l, ... ,p,
Let then
(")
I
=J(r,p)p.J , j=l, ... ;p; ~n
n i=l r=l
1,r
(3.3)
I"
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n
. = (l/n)~i=lJ(rij'p), for j=l, •.. ,p.
n,J
o
.. , = 1
n
n,JJ
L.
L.
J(r p)J(s
,
=
~
(3.5)
THEOREM 3.1.
~L.
¥ t
i=l r=l s=l
-n
k
((0
..
n,JJ
,
p)p~j,j')
1 rs
'
-]1
(]1
.]1
n, l' ... ,]1n,p )
."
n J n J
"
,».J,J., =.,
1
••• ,p
;
J·,j'=l, •.. ,p;
.
n 2 (T-n-11J:n ) has asymptotically a normal distribution with a null mean
vector and dispersion matrix
~
-n
•
-4-
OUTLINE OF THE PROOF.
It suffices to show that for any real and non-null a
(al, ••. ,ap ) (other than a(l, .•• ,l»
bution.
Zn = n
e'r~
converges in law to a normal distri-
By using (2.6), we may rewrite Z as
n
Z = n
n
(3.6)
-k n
2
L U.,
i=l
1.
where U. = L:.Pla.J(r.j,p), i=l, ..• ,n.
1.
J= J
1.
As J(r,p), r=l, ••• ,p are all finite, Ul"",U
n
valued random variables.
are all independent and finite
The rest of the proof follows by an application of the
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classical central limit theorem [cf. Loeve (1963, p. 277)] and some routine analysis.
Hence, the theorem.
Under the null hypothesis (2.1), it is easily seen that ~n = J.~,& = (1, ..• ,1),
and L: = A 2 (J)[I - 1 tIt], where I is the identity matrix of order p.
-n
~p
p - -p
using (3.6), the fact that
Ul~""Un
In this case,
are identically distributed and the Berry-Essen
~
theorem [cf. Loeve (1962, p. 288)], it can be shown that the convergence of N
to a multinormal variable is uniform in (Fl, ••• ,F ).
n
k
Further, considering the
(rn-~n)
rando~
-
variables n 2 (T .-J), j=l, ••• ,p-l, taking the reciprocal of their covariance matrix
n,J
as a suitable discriminant of their quadratic form, symmetrizing it and using some
well known results on the asymptotic distribution of quadratic forms associated with
asymptotically multinormal distributions, we arrive at the following.
Corollary 3.1.
Under H in (2.1), S , defined by (2.7). converges in law to a chio
n
sguare distribution with (p-l) degrees of freedom.
We shall now consider the asymptotic distribution of S
alternatives in (2.2) and (2.4).
00
(3.7)
SCi)
s,p-2
n
under the sequence of
For this, we define by
[F.(x)]s[1_F.(x)]p-2-s f~(x)dx, s=0, •.. ,p-2
1.
1.
= S(i)
and conventionally, we let SCi)
p-l,p-2
-1,p-2
1.
0, for all i=l, ..• ,n; (2.3) ensures
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the finiteness of (3.7).
Let then
-en)
ss,p-2
(3.8)
(lIn) ~.n1S(i)
2' for s=-1,0, ••• ,p-1;
~=
s,p-
(i)
-
(n)
A
2 - Sr- 2 ,p- 2' for r=l, ... ,p.
r,n = Sr- 1 ,p-"
(3.9)
THEOREM 3.2.
Under (2.2) through (2.5), Sn , defined
by .(2.7), has asymptotically
.
a non-central chi-square distribution with p-1 degrees of freedom and the noncentrality parameter
~n (J) = p2A-2(J)(~.P18~)(~
P1 J (r,p)A
)2.
J=. J
r= .
r,n
(3.10)
PROOF.
By virtue of the result of theorem 3.1, it is sufficient to show that under
(2.2) through (2.5)
+ 0(1), j=l, ••. ,p;
= - p8 . ~ P1J(r,p)A
r,n
J r=
(3.11)
(3.12)
[~ - A2 (J){I - 1 ~'~}] converges to a null matrix as n~ •
-P
-n
p - -
Now, by definition in (3.1)
00
(3.13)
IF.
_00
~sl
(x) ••• F
i sr_1
(x) [l-F.
~sr+1
(x)] ••• [l-F.
~sp
(x)]dF .. (x),
~J
where the summation Sj extends over all possible choice of (sl, ••• ,sr_1) from
(1, ••. ,j-1,j+1, ••• ,p), (sr+1' ••• 'sp) being the complementary set.
Now, by (2.2)
and (2.4), we have
(3.14)
Using (3.13), (3.14) and some routine analysis, it follows that under (2.3),
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(3.15)
(l/p) + n
for all r t j=l t , .• t pt i=l t •.• t n.
-~
[p8.(S
J
(i)
S(i))]
2
2 1
2
+ o(n -~ )t
r- tPr- tP-
(3.3) and (3.15) ascert (3.11).
Proceeding in a
similar manner t it can be shown that under (2.2) through (2.4)
(3.16)
p~jtj')
rs
= 1/p(p-1) +
].t
O(n-~)
for all
j~j' tr~s=l"",Pt
i=lt •.. tn.
Hence t from (3.2)t (3.4)t (3.5)t (3.15) and (3.l6)t we obtain (3.12).
Hence the
theorem.
It may be noted that if F.]. possesses a finite variance a~]. for all i=l, •.• ,n,
then (p-l)F(p-l)t(n-l)(p-l) (where F(p-l),(n-1)(p-l) is the classical analysis of
variance test statistic,) has asymptotically (under (2.2) and (2.4)) a non-central
chi-square distribution with (p-l) degrees of freedom and the non-centrality parameter
~* = (~.p 8~)/02.
(3.17)
n
J=l J
n'
Intrinsic
asyrn totic
-..,.............
......... efficiency and the efficient..........S -test.
""oJY'~......,.'"'""'
~....,...-.".'"'"".......,.....;...........,,_'"'-'"'~"'-"_.....,..,.-"J'
n~
From theorem 3.2
and (3.17) the asymptotic relative efficiency (A.R.E.) of the test based on S
n
with respect to the classical analysis of variance test is deduced to be equal to
(4.1)
As from (3.7), (3.8) and (3.9), we have ~ p A
= 0, (4.1) may also be written as
r=l r,n
(4.2)
p
p
p
p
[o2p 2(p_1)
A2 ]{(
[J(rtp) - J]A
)2/(
A2 )( L [J(r,p) - J]2)} ,
n
r=l r,n
r=l
r,n
r=l r,n r=l
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where the second factor of (4.2) is uniformly bounded by 1, and the first factor is
independent of {J(rtp)t r=lt"'tp},
We now define
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0n2 = (l/n) ~]..n=la].~
(3.10) and (3.17) will be used in the derivation of our main results.
4.
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(4.3)
e*
n
as the intrinsic A.R.E. of the method of n rankings as it is the maximum possible
value of e (J).
n
Further. a particular S -test is said to be asymptotically efficient
n
for (Fl, ... ,Fn ) if en(J) =
e~
for that (Fl, ... ,Fn ).
It follows from (4.1), (4.2)
and (4.3) that e (J) = e* if and only if,
n
n'
(4.4)
J(r,p) - J
For p=2, we have Al
J].
[J(2,2) -
,n
k A
for all r=l, ..• ,p, k being a constant.
r,n
= -A
2,n
and also whatever be {J(r,p), r=1,2}, J(1,2) - J
Hence, (4.4) holds.
So any S -test will be asymptotically efficient
n
and will attain the intrinsic A.R.E.
(i)
(i)
If p=3 and Fl, •.. ,F are all symmetric about
n
. '
zero (so that 60 ,1 = 61 ,1 for all l=l, ••• ,n), we have from (3.9), Al,n
A
=
2,n
0.
As such, the optimum Sn reduces to
Xr2 , defined by (1.1).
= -A 3 ,n and
However, if
Fl, .•• ,Fn are not all symmetric, this property of X~ will not hold, in general. (It
may be noted that S
n
corresponding to {J(r,p), r=l, •.• ,p} in (4.4) also corresponds
to the likelihood ratio criterion (for the model (2.2) and (2.4) with any given
~),
and the intrinsic A.R.E. e* is the A.R.E. of the likelihood ratio test for the same
n
problem.
For intended brevity the proof of this statement is omitted.)
For p
~
4, the choice of the optimum Sn (in accordance with (4.4)) in general,
depends on (Fl, ..• ,F ) through A
, r=l, •.• ,n.
n
r,n
If, however, F
l
5
•••
= Fn = F,
then in most of the cases, A
, r=l, ..• ,p, are quite simple and the corresponding
r,n
S - statistics are not difficult to compute.
We shall consider the following
n
illustrations.
(I)
e
=
(4.5)
Uniform distribution:
f(x)
=
Ar,n = J(r,p)
Fl(x)
=
=
f'
0,
-1,
° < x < 1.
dx:
r=l
2
~
r
r=p •
~
p-l
Here, we have
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Thus, for p ~ 4, the optimum Sn is different from X~ in (1.1).
(II)
A
r,n
•
The corresponding S
-l/P(P-~'
J(r,p) =
r>l
x < 00.
{I'
0,
Here
r=l
if
if r=2, •.• ,po
is a particular case of (1.2) (with a=l).
n
(III)
{
o :s
r=l
lip
(4.6)
f(x) = e -x
Exponential distribution:
Double exponential distribution:
f(x)
e-1xl,_00 < x < 00.
By (3.7),
we have
Ss-1,p-2 =
(4.7)
-p~(P=~""":'"l7"") {p-x
I
r=p-s
(~)2-P
- (p-s)
I (~)2-P}
, s=l, ••. ,p-l.
r=s
Hence, it can be shown that
r-l
J(r,p) = p(p-l)A
(4.8)
(IV)
r,n
1 - 2
I
s=O
(~)2-P, 1
:s
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r < p.
_00 < x < 00.
Logistic distribution:
Here, we have
f(x) = F(x) [l-F(x)], and hence, it can be shown that
(4.9)
A
r,n
- 2(r - p;1)/p(p2_l)~ J(r,p)
r for 1
:s
r < p.
Thus the optimum S reduces to Friedman's X2 •
n
r
(V)
Normal distribution:
f(x) = (2rr)
-!,,;
2
exp{-~x2},
_00 < x < 00.
We denote by
X
the sth smallest observation in a sample of size p-l from the standard
s,p-l
normal distribution, s=l, .•. ,p-l.
(4.10)
Then from (3.7), we have
(p-l)S s- 1 ,p- 2 = E{f(Xs,p- l)}'
s=l, .•. ,p-l.
It may be difficult to evaluate (4.10) for various values of pans s.
2
and 0 s,p-l
be respectively the mean and variance of Xs,p-l·
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Let lJs,p-l
Then, for p
~
4, (4.10)
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may be approximated (crudely) as
(4.11)
(p-l)Ss_1,p_2
2
= f(~ s,p- 1){1+~(~2s,p- 1-1 )0 s,pI}'
s=!, ... ,p-!.
2
As tables for the values of ~ s,p- 1 and 0 s,p1 are available (cf [6]), (4.11) can
be computed for various values of sand p.
It may be noted that for logistic cdf, the rank-sum test is asymptotically
optimum (in one way layouts) and the same property holds for the method of n rankings.
However, for double exponential cdf, in one way layout, the median test
(cf. [4]) is asymptotically optimum, but the statistic (1.2) (with a
[p/2+l]) is not so for the method of n rankings.
=
[p/2] or
There is also difference in the
scores for the normal distribution for the two cases.
We have so far considered the case of complete layouts where each observer
ranks all the p objects.
Certain balanced incomplete layouts are considered by
Durbin (1951) and Bhapkar (1961); the test statistics are straight forward generalizations of (1.1) and (1.2).
If F
1
~
~
F
n
~
F then all what has been
discussed earlier this section also holds for such (balanced) incomplete layouts.
For brevity, the details are omitted.
If, however, Fl, ••• ,F are not all identical,
n
it is quite difficult to simplify the expression for the power-efficiency of such
a scheme and to consider results analogous to (4.1) to (4.11).
REFERENCES
[1]
BENARD, A., and ELTEREN, PH. VAN (1953). A generalization of the method
of m rankings. Indag. Math. 15, 358-369.
[2]
BHAPKAR, V. P. (1961). Some nonparametric median procedures.
Statist. ]1, 846-863.
[3]
BHAPKAR, V. P. (1963). The asymptotic power and efficiency of Mood's
test for two way classification. Jour. Indian Statist. Assoc.
~, 24-31.
Ann. Math.
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[4]
BROWN, G. W. and MOOD, A. M. (1951). Some median tests for linear
hypotheses. Proc. Second Berkeley Symp. Math. Statist. Probe
1, 159-166.
[5]
DURBIN, J. (1951). Incomplete blocks in ranking experiments.
Jour. Psychol. i, 85-90.
[6]
GREENBERG, B. G., and SARHAN, A.E. (1962). Contributions to Order
Statistics. (Editors). John Wiley: New York.
[7]
ELTEREN, PH. VAN, and NOETHER, G. E. (1959). The asymptotic efficiency
of the x2-test for a balanced incomplete block design. Biometrika.
46, 475-~77.
[8]
FRIEDMAN, M. (1937). The use of ranks to avoid the assumption of
normality implicit in the analysis of variance. Jour. Amer.
Statist. Asso. 32, 675-699.
[9]
KENDALL, M. G. (1955). Rank correlation methods. New York: Hafner
(Second edition).
[10]
LOEVE, M. (1962).
-,
British
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Probability theory. D. Van Nostrand Co.
el
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