')
A MULTIVARIA'I'E EXTENSION OF' FRIEDMAN IS X"- -'I'EST
r
WITH RA.TlJDOM COVARIATES
by
Tbomas M. Gerig
Institute of Statistics
Mimeograph Series No. 928
Raleigh - May 1974
A MULTIVARIATE EXTENSION OF
FRIEDMJU~'S Xr2 -TEST WITH
RANDOM COVARIATES
Thomas M. Gerig*
* Thomas
M. Gerig is an associate professor, Department of Statistics, North
Carolina State Uni versi ty, Raldgh, North Carolina, 2'(60{.
P..BS'L'AACT
This article investigates a non-parametric procedure for testing for
no treatment differences in multi vaY.'iate two-way layouts wi th random covariates.
The method used is an extension of the Friedman
xI'2 -test.
Permutation and asymptotic permutation distribution.s are given a.l1.d various
asymptotic relative eff'idency results are
vided to illustrate the method.
p!'esentf_~d.
An example is pro-
-31.
INTHODUC1'ION
In situations whf"l1 the asswu-p-tions neCf,SBa:cy i'01: B.l1plying
clas~;ical
statistical techniques are impossible to check or are hlatantly violated,
non-parametric procedures of'fe:r practical alternatives.
multivariate data
In the analysis of
the number and complexity of assumptions makes non-
parametric alternatives all the more attractive.
In an earlier article) Gerig ell considered the problem of testing for
no treatment differences in multivariate data
complete two-way layout.
collect~d
according to a
2
By extending Friedman's x -test) the usual
I'
assumptions of multinormality, additivity of block effects) and block to
block homoscedastici ty necessary for applying classical MANOVA tests were
relaxed.
Frequently, in addition to the dependent variables of interest) a set
of concomitant variables are measured which gauge such things as the
initial level or expected performance for the sampled unit.
This article
deals with an extension of the methods in Gerig [1] to take the concomitant
vari ab les into account.
We shall assume that concomitant variables are random variables which
are jointly distributed with the dependent variables"
We shall further
assume that they are unaffected by the treatments.
Specifically, suppose we are given (p+q)-variate data which form a
comp 1e t ewo-way
t
1ayou.
t
Y~.
......lJ
=
Le t
2
P ) ..
auu~
XI...-- (Xl.. , ·X .. , .•. , x...
IJ
~lJ
IJ
(Y~., Y~., .•. , Y?), and let Z!. =(X!.,
lJ
lJ
plot in the i
(k ;;;: 2).
F .. (x, v),
lJ...... x..
lJ
th
......lJ
"'lJ
lJ
y!.) bt:: the response from the
"'lJ
block that received the jth treatment; 1 ~ i :s; n, i :s; j :s; k)
Assume that the cumulative distribution function of Z.. is
AoIIJ
(x, y)
~
......
E
q
RP + •
-4We wish to test the null hypothesis of no treatment differences,
H
I
F .. (x Y)
J.J "'" -
o
I
F. (x y),
=
1
,..,
for
.1
(1. 1)
s: i s: n,
N
against translation type alternatives of
for
F .. (xly) = F. (x - ex .Iy),
J.J N _
1 -J12
P
where 01 ~ = (01 ., 01., ••• , 01.) • The null
-J
J
J
J
thf~
form
1 :s: j :S: k,
H :
A
(1. 2)
states that the
hypothe.~oiE.
conditional distribution of the dependent
variables, _lJ
X.. , given the con,
comitant variables, Y.. , is the same for all treatments (but not necessarily
-lJ
for blocks).
The alternative states that the conditional distributions
differ by a translation of location.
THE PERMUTATION TEST
2.
s s
Let R .. be the rank of X.. among the observations (X~., l:s: j :S: k} for
1J
lJ
1J
t
s: s s: p and the rank of [\.j'
1
s
T. =
J
s: j s: k} for s = t + p, 1:S:
1
t
(1)
n
s
- L:. 1 R .. , 1 :S: s :S: p and U.
n
1=
lJ
J
Wri te .:E 1 = (Ti, .•. ,
... , U~),
and finally~'
Sections 2 and
W (denoted by
-
s: q.
01)'.
s
E. n 1 R..
, s =- t + p, 1 s: t
=
(-
n
... , Ti' ''',
T~,
t
--
= (1'/, U/ ).
1=
lJ
:S:
Let
q.
T~), ;2' '" (u~, ... , u~, ''', ui,
Then, following the arguments of
3 of Gerig [lJ, a permutation distribution may be derived for
P ),
and it follows that
n
s
J
t
= e[[J~lp}
e(T.lp}
n
n
J
k+l
2'
and
Var[wlp}=-VN)t::.=D[(,,)
N
n
_'<;;J _
pt-q k X,( p'"tq ),-,
KJ
(2.1)
<'V
1
where t::. = -(,1 n_k
~
2k)' !k is a. (k X k) identity matrix, l.k, is a (If X k)
~'"
.,.
matrix of uni ties, the symbol (>5)
denotes the K.roc.ec,ktT product de1lned
as
'
_AIC'\B
v.::.J_
=
((a1..JB)) (see, Ander s on [l, p.
v.[s,tJ
=
1
n(k-l)
k
E.
:1=.1 J = l
[E
n
347l ), ..£ =- ((if[ S , tJ ) ), and
R,~lJ.R:.lJ
;J
kqrll-=]oo
(2.2)
... :,-.....
l;)
If we write
- f~ll
V
V
-21
then D. A
-JAJ
- V.£
-J
Q
Vj
~12
""
II
"':"12~2
6., 1
'"
~:]"
D
d.rd
D
j
-l~~
(2,3 )
D ,)
2
""_L
j, 1, s; 2.
~
The statistics of interec;t for L~:"ting H ' giv(-;n by (1. 1), are the
o
vector of rank swns, NT, adju.sted for the concomitant variables, ....,
U.
T
_a
- erTIp
(T
That is,
e(Tlp
_ n })
})
.- (T
',...,
(:?
4)
'"",,,,""
* denotes the genera]j zed invE:isc, of
where £22
~22
v"hi.c!l may be taken as
--1 (X~
n 26 since
V
X
-22 " j
,..,.
(2.5)
Then,
2(~1~;~ @ n~)E21
(21~;~ €) ~)E22(~;~21 (8) n~)
var(!aIPn} = .£11 -
2(21~;~21 ® ~)
= Ell ..
=
(~ll
-
~1~;~21) G0 ~
+
(21~;~21 @ ~)
, : 211. 2
@
As in the MANOVA development (Gerig [ll), we mu~;t asswne that
-
V are non-singular.
,...
For large n ttL:;
i~~
(2.6)
6.
122
true wi th high probabi l i ty.
and
'rhis
4.1 of Gedg [1'1 which shows that
-V converges to a
'"
matrix which under minor conditioDS is itself Don-sinVllar.
A sufficient
follows from Theorem
condi tion for this is the non-existence' of a 1i near relati onship among
th
( )
s )
. .
F[s]i ( Xij , 1 ~ s ~ p+q where }'Cs"j(x j L.: the s-mal'ginal of F i ~n
i
=
1, 2, ... , n.
The high probaLil.i ty for
V,)')
bt'.ing non-singular follO\{s
·"'L.'-
in the same way.
For detail;c~, i',ee ~jection II- of C}erig [1-1.
··6In the event that ~ (or
222)
i,s s.iDgu.lar io prac:tice,it is sufficient
to retain only the highest order non-~:ingu.lar principal minor of
2 (or 222)
and retain only those elern'_onts of W (o.! U) whic:h c')XTt,spond to the remai ning
'"
rows of
2..
,..",
(or 222)'
As a test statistic we shall
takE~
the quadcaUc form
(2.'J)
S,
n
--1
"'. V,), .,'
2.110 2
where
,....... ~~ ..l.• .:..:_
can be interpreted as the generalized ·:U"tance from Teo its permutational
-a
center of gravity or can be seen to be proportional to a Pillai' s trace
[2,
statistic (see Morrison
p.
198J).
If we let
p
(Y)
n _
= nL:
aL
p l:: P v[ s,
s=l t=l 11
tJ l::
k
(u~ _~ k+ 1) (u t _ k+1)
j=l
J
2
j
2'
(2.8)
and
= nl::p +q
£n (X,Y)
..... _
where ((v[s, t
11
s=l
J ))
=
~-l
-11
l::p+q V[s,tJl::.k (W~ _ k+l) (Wt _ k+l)
2
j
2'
t=l
J=l
J
and ((v[s,t J )) =
V- l ,
'"
then the following theorem
gives an identity which simplifies the computation of £ (X/Y), given by
J1N"""
Theorem 2.1:
£ (xIY) '" £ (X, Y) - £ ('1'),
n,..."",
n"'......
n_
(2.10)
where So (XIY), So (X,Y), and So (Y) are given by (2·7),
n
n
f"<eJ""';
f"'ttJ
f"'tJ
n
~
,
(:2.8L and (2·9)·
Proof:
The theorem may be seen to be tr'J.e by wri ting £n (X
IY'} as a quadratic
'A"J ~I
form in T and U and applying the following lemmas to the matrix of the
......
,..."
quadratic form.
~
,".
'7
Lemma 2.1:
If
if
211 and 222
22 2.1 is
are non-singuhn an'I if V
.....11. 2
13
dpfiw::d
EU3
in (2.7) and
correspondingly defined, then
--,1
V
",,22.1
Proof:
This follows from Raa
-
]
:=
221' and,£
Lemma
:=
[3], page 29, number 2·9
by setting
!:;
222'
--1
211'
2.2:
Under the condi ti ow, of Lerrnro, 2. J.,
f~ll
··1
212
l221 222
Proof:
See for example, Morrison [2J, page
66.
Thus, the procedure for computing £ (XIY) is to compute £
n""""
n
as given by
(3.2) of Gerig [lJ, first for all p+q variables, then for the q concomitant
variables and take their difference.
Using this method and the permutation
argument in Section 2 of Gerig [lJ an exact permutation distribution of
£ (xIY) can be calculated.
n""""
Based on this, a randomized tec;t of H
a
can be
constructed which is distribution free under H.
a
When "amples are large,
an approximation to this distribution is needed.
T'his approximation is the
subject of the next section.
3.
Theorem
I
ASYMP'I'OTIC PERMUTATION DISTRIBUTION OF' £ n"",,,,,
(X Y)
3. 1:
Under the assumptions of Theo!'t~m 4.) of Gerig ell} the permutation
distributions of £ (X, Y) and £. (Y) aTe a:3ymptotically dli-bquare wi th
n ,..,. '"
~l "'"'
(p+q) (k-l) degrees of freedom and q (,k"1) degl'f:et; cf fTeedom respectively.
Proof:
This theorem is the same as '.T.t,eo..':'em
of' Ge!."':ig [1J.
}+.)-t
Lemma 3.1:
Let Q
=c
Q + Q J . .[ here Q and Q are distributed as chi ·-square
l
2
l
variables with a and b degrees of freedom respectively.
Then Q non·-negati ve
2
implies Q is distributed as a chi··square variable with (a-b) degrees of
2
freedom.
Proof:
This is an extension of CoclL'an's theorem (see Hew DJ p. 151J).
Theorem 3.2:
Under the assumptions of Theorem 4.3 of Gerig [llJ the permutation
distribution of ~ (XIY) is asymptotically chi-square >"lith p(k-l) degrees of'
n ..... ,..,.
freedom.
Proof:
Using Theorems 2.1 and
3.1
with Lemma 3.1 and Slutsky's Theorem (see
)\- .,
-1
Sverdrup [4J)J it is sufficient to show that £(x y):= rr'(L.
2
rl ,...,.. '"
"..;a,..,..; ll
~,··l "
is non-negative J where Ell. 2 := .Ell -. Eli:2?.E21 and where
00
Eo
L.
1
L.
~ll
'co
.....12
[ L.
, ..... 21"
,...2;2')
<-'J
is given by (4.1) and condition (ii) of Ge:cig
form of
.....V).
" ow t\--...'c<t
(,-",1
Thus J we mus t ' s.n
~
,~
j
1
But if'we wri teL:- ')
f"IV ll
.t-
=
BE' and
~
st(xIY)
= ,...,a
T'(B ~ yn
n
,..." ~
I"'W
f"'!IJ
which completes the proof.
I:::.
,
'*
'~11.
nI:o.
,..,.',-..,;
1)(3
~
~
€)
2
ell
Q
eVn
~"
o
Li the lirni ti ng
1:::.'*) 1~~ llon··negative definite.
,."
I)
""
(L
CVn
\[n 1)/'1',
~!"Va.
I) tr-len
""
Z'Z
~
P'tJ
~ 0,
-9For large values of n, p, and .k the computation.:: [I.eeded to calculate the
permutation distribution become too lengthy to be practical.
Theorem). 2
provides a large sample approximation to the permut.ation distribution.
J':n (~,l,~) ~ X~,p(k_l)'
is, for large n, reject Ho if
where
P(X~ ~ X~'Q1}
That
= l-Ct,
0<0I<1.
For assumptions sufficient for positive definite
V),
Eo
(limiting form of
see Section 4 of Gerig [ll.
""'
UNDER A SEQUENCE 0[' THAW; rATION ALTERNATIVES
Consider the contiguous sequence of trarlslation alternatives, [H
n-1,
given by
1
F .. (x
lJ
where x
s
nj
x
s
x
+ n
2
, ... ,
xl'·, y)
l
""'
-1/2 s
Ct. for 1
J
~
l
nJ
:F'.(x ., x
j
~
k, 1
=~kfails to hold, where
= ,e2 :::
~l
,
~
~
0".
8
=
"'"'J
... ,
2
nj'
xl' . ,. y),
n,] ,...
(4.1)
P and assuming
~
1
J
2
(Ct.,
0'.,
J
'00' Q1~).
J
Theorem 4.1:
Under the assumptions of Theorem
6.1 of
Gerig ell and under
(Hn },
the*
asymptotic distribution of
(n
n
1/2 (T sj
1/2
-s
- ,u Ij ),
S
-8
(U.-,u2')'
J
J
1 ~ j
s:
1 ~
k,
l::;;ss:q})
lS:js:k,
is (p+q)k-variate normal of ra.nk (p+q) (k-l) wi th
persion
iL~j
-s
,u2j
=
=
Eo 0
s s: p;
mE;an
'fector zero and dis-
~, where
*2:i~l
[1 +
2:1,~l
e(Fi1,[sl
(X~l,)}]'
1.
;0;
S
s: p,
1. s: j s; k (4.2)
:j;j
~ 2: n [
k
(
( s )}n i=l 1 + 2:1,=1 e Fi1,[tl Yil, J,
:j:j
t
s +- p,
.1 s; s s; q,
(4.3)
-10s
Fij[sJ is the marginal distr:ibutioD. of Xij , 1 5: s
marginal distribution of Y~
s: p" and F'ij[tJ is the
t := :p + ,:i, 1 ~ s ~ po
0'
lJ
Proof:
This theorem is identical to Theorem 6.1 of Gerig [lJ,
Theorem 4.2:
Under the conditions of Theorem 6.1 of Gerig [(1 and .if
lim
0.-+0:::"
(~)
0.
J0000 f~[
L:. n
1= l
-
1
S
l
(x )dx
=
a
.
s.,.
the asymptotic distributi on of
~
exists for 1
S
~
s
n+q, then under (H }
11 '
.l:'
(X) y) 1,3 non·<:entJ>l.l c:bi .. quare, ,vi ttl
fl. ,... ""'
(p+q) (k-l) degr'ees of' freeJ.om anJ llon-cc-ct<traLi.
~ p ~ p
:= L..s =1 L..t:=l
\2
f\
=
where 01'
"""j
[s,tJ_ _ ~ k
s t
11. 2 O-s O-t L.. j =1 01 jQ'j '"
1
2
p
..
(01., 01 0' ••• , 01 0) and A
dl ag (a 1
(J
'C'
Lo •
k
J
J
-
~
(4.4)
""""""'J
... , a ).
j
J
A_.
"" ....1
0I .. ~11 .. )l""IIoI'.
•
J=.L -J-.
p
.L
Proof:
As in Theorem 6.2 of Gerig [lJ,
S
n 1/2,,-.
\J.1
1
o
.J
k+l)
2
-
oS
- -
.....
01. a
J
for
S
1
~
s
~
p,
however,
iL~j
and, hence, n
~ L:i~l (1
1/2 -s
k+l
(J.1
- 2')
2j
k+l
2
+ k;l}
Thus,
O.
* *L: -1A*01.*
A2 = L:. k 1 OIoA
O
J=
""'J- '""
'"" ""'J
L:'~l (OI~,O/) [~
J-
_
k
'A~ll
- L:. 1 01 o~ Aot
J=. ""J"""" - J
0
We obtain
Theorem
0
""J ""
(4.3)
2J
0
""
r
L:
"'"
I:
11
L: l'-='
~
~
1-
"'"
-
2l
L: 2°
~
L '""
I
ol
L
A.
(l
'"
""
0
0
'"
'"
rOil
I -j i
1I,...0 ;I
I-
...
•
by observing ttl.at 2.;1l
""
=
1
2.:. (Lemma 2.2),
",,11. 2
4.3:
Under the conditions of Theorem 6.1 of Geri e,".. ~Ll-J· awl under rH 'J S, (y)
1, n '
n '....;.'
is asymptotically chi-squared wi th q(k-l) degrec:s of fr<~edom,
-11..
Proof:
This fol.lows from the proof of Ttleorem
4. ;?
Theorem 4.4:
Under the conditions of 'rheorem 6.1 of GE,rig [1] and under {H}, £ (XIY)
n
is asymptotically non-central chi-squarE;
n -,...
wi th p(k-l) degrees of freedom
"
1
(4)'
and non-cen t ra 1l" t y parame t
er'~ 2glven)y
.4}.
Proof:
from Theorems
4.2 and ).... 3 using
thf: same iugumE:,nt:::: as those of Theorem
3.2.
Some conditions of practics.i intered. are glvc,nln Sections 4 and 6 of
Gerig ell which are sufficient for the conditiorw required in Theorems
4.1
through 4.4.
5.
ASYMP'I'OTIC RELATIVE EFFICIENCY
It is not surprising to discov'er that the limiting Pitman efficiency
n__
of £ (xly) with respect to £11_
(X) satisfieccl
This is true since the left tand
~)ide
equals
.. 1
,. .k
fA " ..IIA'
--.1 01." L. 11 ') nw.
,J-, ""J- ~_,_'"_.=_~.
'" k
fA ",-1 11_.
L."
L."1
01.
Loll n.r.
J =- .. -J"'" "" ,,- ...,.,J
It is easily seen that thi", can be wri tten a:: orw pIlle. a
nGn~negati Vt'
quanti ty.
It is of interest to compa['e tr,is test with the likelihood ratio test
of H
o
derived under the assumption. of
'~lOrma.lity, a;;~um.jn.g
linearity of
-12regression of Y.. or X..
-lJ
1:s; i :s;
-lJ
dispersion matrices are equal..
1:s; j :s; k, and
rl,
a~)surning
tbat block
Then, the likf';lihood ratio te,;t, U
(XIY),
n""J~
under (H }, defined by (4,1), "rill ba,"r} asymptoti cally a Eon-central chin
square distribution wi th p(k-l) degrees of f,reedom and non-centrali ty
parameter
_ E k
1.- 1
+ E k
01. 1\11 ,~OI.
J. = 1 J - , . a...J
J. = 1.
.11
01. 1\
,.....J~
01.
-J
where
1.611
,...A
I
!
L~21
1:':;0
r "".1J.
,
1\1 ,.
""'.L.(~
t_C..
11.- 1
1\
J
I
"'"
I
01
A·'~
L
.J
l')1,
A--~
-,y)
AU:'
I
1
""
and
Therefore, we have that
""k 1 Ci.'A L,
-.1.1 nw
!\A'
.
J= ""J- "'"
---J
L,.
.... _
e(sn (XIY),
U(XIY))
[1
__
k
Ct. \
J=- ....J- -
If we let p
"--
~11.2
1
=
,.,....,J
1, then (5.1) becomes
2
al
where a
(5.1)
-;A-- 11
A
Aot.
I:. 1
k
oo
J
_rv-.
~
~,k
L.
j=l
01
2/--~
1
'" k
L.
j ~ll.2 j=l
Ct
2
j
2
1.
[f(un du and f(u) L~ the p.d,L of Y."
lJ
1:s; i :s;
fl.,
1. :s; j :s; k under H •
o
Now remembering that
tl:F.:'
Ewymptotic n,laU,v<,; .:fi'iciency of Priedman I s
2
X -test wi th respect to the clasr:.i cal. two-!,{ay ANO'/A tes t ,1 s
r
e(X2 ,F)
r
so we have
=
.
d h ence, Slnce
an,
o'llcr 11
<!:
1. an d '1\1.",11.
l' L1-
~
1,
lr
t ·11a
t
J
e(x2 F)~-- :;;; e (('I
\,1., (xlV')
.•...•.
,
U,..._
r'
11
Al1 A
By definition All
=
[A
elements of Y, ,
11
-lJ
are linearly unrelated to
From thi s i t
follows that
To illustrate the tecrJ.Liqu(; pl'E'sented in th.i:: art.i cle, the data in
table
wi 11 be analyzed.
tl'1'2
The data :eepresent tht: results of an experiment
to determine the effect of stac-:k posi tion of tol'acco leaves on the
chemical makeup of tobacco.
There are three treatments representing the
three positions on the stalk (high, middle, and low) and six blocks
corresponding to six different locations (.::'arm';).
TiJree dependent variables
were measured for each treatment-·lJlock combination:
soluable sugar, and total ash.
of leaf.
per'_~ent
nicotine, percent
Orle concomitant variable was
measun~d;
color
The data and associatE.,d within block ra.nk::; appear in the table.
Using a computer program, the permutation distribution for s.(xly)
was
tl
~......,
obtained using the procedure
out in Sec:tir)[i ;? of Gerig [lJ and
Thi s entai 1;,; ,;ovaluati.ng £ (X, Y) an:i £, (Y) (and, hence,
equation (2.10).
r n"",,,,,,
(X II))
~::pF~lled
n
A""
~
~l
,..."
for each possi bh~ reB.:rrar:gtc:ment of the, multi'!.9,ri ate observati on"
Since, for ttiL; exarnpl(:~, p
within blocks.
there are (3!)
distribution.
6
=
46,656 evaluat.icns
c=
~).
w;~ce:;Ea.Y'Y
q
=
1,k
:=
),
and n.
:=
6,
tcJ IJroduc'F: the permutation
Adding a.nother block would result in ?I9, 9Y· c~·\fa1.uation8,
another treatment in 11.9,976 E:valu8.tiot1E'.
It L; ot,yiou:; that for moderately
large values of nand k, the pEC:rmutatioD di stributi on cannot
b~
obtained.
To illustrate that the chi··.:;quare dL;tributicn \ori til p (k··l) degrees of
freedom gives a reasonably good approximation to
bution, their
figure.
c·'J.mulativf~
tr~e
permutation distri-
distrily,xt.ion. functions ",.roe plotted together in tLe
Also included in the) f1 e.uc: .is the
c'~mulative
distribution
function of c times a beta randem vo.riaJ)le wi th par8.meten~ 0./2 and b/2,
where
a
p(k-I),
(k - l)(ck - 1-' 1)
<' .- J .
t' .
r
.:;;
fey.
P
:2: k
I
..
1
and
p(nk
c .-
1
-
(k
-
k
-f
1) (nk
1)
-k
+ 1)
J'(ll'
.f' S k
for
p
~
k
1
..
1
.1.
This is the approximation USed in conjunction wi til the Pi11ai I s trace and,
in practice, seems to be better than the chi-square approximation.
In
fact, the pattern shown in th- flg:J.l\: is typical of all (,xamples ever run by
the author.
That is, both the chi-square and bf,ta approximEttions lead to
conservative tests with the latter coming rWJcb clost'r to the desired level
of significance.
will be identicaL
Of
cour~~e,
as n tends to inf':ir,i. ty these
lviO
distributions
It is wortb pointing out that the permutation
distribution is a step function despite its smooth appearance in the figure .
.
Also, it should be made clear that this raI·ticular peTmutation. distribution
is for this data only.
It mUc:t
bt·~
recomputed for "':l.cb. ne" problem.
The computed value of £ (Xl'{) for t.bf; data
D",,,,,
[U;
from the :figpre carl be seen to be highly ;.:;.ignificant..
ob;oerve'l i.,: 18.931 which
STALK POSITION .EXPBR.IMEN'I':
Po~;;ition
Location
Variable
1
2
3
4
5
6
on tbt:
StCi..lk
'---~--~---~---"---------~-~------'
--_._"--'._-~-_._~'''--------
2, 28
n.60
25~
43
28,6
Upper
--._--..
Lm-rer
Observation
Nic.
Sugar
Ash
Color
RAW DA'rA AND RANKS
Middh~
._.-._~._,-~_.
Rank
Ob~;E'rvati on
~-~.
Har'.,k:
Observation
_.~-_._--~_._----_.,_.-"--,_---~,,,~~-~
3
,.'
2~ 4C)
:,'1. 90
12, 40
.1.
94,)
1
1
'.z;
:;
2
~~o
3
2
2
15· 80
10. 15
.l00. 0
3
,')
,-
')
~J
•
1
~)
:). 16
2
/'
.~)
':)(1
AsL
3, 6:~
:3, I
:5,56
Color
54, 0
5
Nic.
Sugar
3,66
;',1
~) ,
Nic.
Sugar
Ash
Color
2. 20
4. :20
22.'(3
25, 0
Nic.
Sugar
Ash
Color
L
74
,-
L
1
1
ll,
S'i
>:
5, c·",)
';;I"
"l
4. 87
1;::. '(1
:?
ll. S,O
')0. 4
:.~
28.6
1
1
;-::
)
4.
,::.j
.3
12. 80
"-
10. 10
68. 1
2
·z
.J
~
1
,)
L.o
S'{
3
15· 80
14. 2j
1
79· 0
1.
.1
;~?
0"5
~3
1
~)9· -'
09
1
26.35
O. 00
l lL:5
,
1..
ll. 90
1;), 48
3. 20
,,,.\...
1
1
j
3· 20
25.68
16, 0
l-
.
J .
04
4. 01
18. 15
51.6
:?"
i
~
Nic.
Sugar
Ash
Color
Nic,
Sugar
Ash
Color
, 1
::)0 .1-.,_
Rank
r;
'J
~.
7)
f"j'Z
,)
L
3, 79
,)
12, 20
(-
:?
~?
,-
1.5'7
2
1
~?L30
;z,
5
j
10,78
1
1
1':),
r)
'7
I
5· 81
L3. 8L~
6:.-!. 9
,)
.3
2
')
1.
3
5
1
:;
--'
'39
;)0
1J.. 8.3
3
,y{, 2
3
f._,'
2
:2
PERfVlUTATION DISTRIBUTiON OF
.In (XIY)
AND APPROXIM~TIONS
PROBABI LITY
1.0
0.8 .-
~
-·BETA
'CHI SQUARE
0.6 .-
O.4r
il
i
O.2r
I
n
/
I
//
A
t"'h'~_. J~,~"~,~,,~ .. ~~nJ"_ . ~ " ~;'\,lh~., , ~, ~,l"~<" ... n..~..-.. ,.~l,-,·,~.,_~_,_ .. N_l,~~_ .
\J ()
.•.
:3
6
u .•"
9
12
15
VALUE OF .In(XIY)
nL
......--.1
18
21
_ 1 ("7_
,'-I
EB:;<'ERENCE2
[lJ
:?
Geri g, Thomas M., "A mu1t..i'va:::·iate extensi un of' F'l'ieQlllan I s X- -Test ",
l'
1595-1608.
[2J
Morrison, Donald F.
Multiv8.riate Statistical Methods~
New- York:
McGraw-Hill Book Company, 1967.
[3J
Rao, C. R. Linear Statistical Inference.
Ne\V' York:
John vIi ley and
Sons, Iuc., 196';).
[4 J Sverdrup, Erling, "The limi tiistl'ibiJtion of a cuntinuom:; function of
random variables", Skandinav-isk Aktuarietidskrift,
(1952) 1-10.