.IN. f,' redr!. 0 P sf..:f, 's {./c.!1
H, ...."• .,gc..,.,C 3 #=-:t 7
.
,.
~,
•
......
, "
.'
'
C• oJ
.
"
,.'.\
........ .:
\,.,0.
•. ,.,,,, -J.Ju_·Lj.t
-- ... ,. "~
a ...:...
' )<.~
........."U..l.L1.
•· .... r··(J,s<';u
_.:. , .~ •
'
,~E .... l1
t"(·;.ti"'tsfcrntatiollS ar'e sC:tlet5,,'7H'=~S i.2.S~O in ;~;~:::
Ct.).'l.;;.:-
I
~'JO'A"~
.j
~"
1. (X)
i'f ".;;e ccni'ine our at.tent.ion to
..... for continuous cnanCH
C1
is
va I>5.a:1(~e :rat:to ~..dst 1 and 0 the?' stIch test s,
co.ntinlHJr;f~
V'~riabl£,'
',Ii tr
u
•
~-rom
sampl:,s Hhether tip "li:<'tributions of
arz the samE! or d·J .. eT'en~"
•
i-lithou'c any
8
o::hance 7ariables ~ &.10 ~:.
1,"'0
J.'rlo~'i kno~lledge
,
Ira) in fests have b8en devised to deal with such
tr,i bution of' either.
C(1.S83; the ';' :1,1,. 1)1' t::lem 'i.s the subjeet of .thls course.
'161oping thera and
cion that
Ii
about the di.s,
I
anp1.~··ing
them
t.-N:
u.sually
Because in
not start from an
atJ
d~~
assu~\p""
chanc<? va:'lable has a distribut:lon funct:lon of' kno,.,ri I'orm
hut involv1rtg cert.ain
un}-;.::I")it:n
pal"f!meters, t:--<2Y
parametric tests; but th.: 'tame 1s not
that someone will soon th ,"k
0;'
11
i'!2.ve '
goer) one snd
a !"uch 'better.
been c311ed non ....
11;
is to be hoped
;,s \.i11 be: seen,
14'9
,
c01'nlng parameters.
•
In the s1.mp1.8st applicc.tion of' the t test we have a set of
..... ,
. n numbers x"
X.
X
r, • v!hlch are observed 'lalues of a c!1!lnCe variable
\le assume that ;..~ 1s ncrmally distribut€:d
'~rit:h unkno~.yn
mean a:ld
·u.n)mown standard deViation, 'or, as \1e often 9xpress it, taat the x are
a random sample from a. normal population of urjknov,'n Int:an and standard
The questi.on \1e wish to decide 1s "Is the mean of the popu"
deviation"
Does the mean of the sample differ significantly
lation zero or not?
from zero'?"
that ·the
mei',!l
the hypoGhesis 1IIh:1.oh we test,
of the
i~OPuJ.qtlon
th,~
null hypothesis, is
is zero (0:,' some otbsr specified valuo),
The t test does this, .and does it more efficiently than '.lny other tpst
v.olJ.ld do
it, but
a1wa~rs ~]ith
t.he assumption that the distribution of
,
X :.5
nQrmal~.
''i!h~~t
cau woe do if we have no
~roun.5s
at all fer assllmi.nt.
that the dist'ributjon of ;.: ts normal, and may even suspect or p."rhaps
•
:.;nln,r that its c1istribution is .far from
~1orEial?
The sign test makes no assumptions about, the furm of the
T.t
teo::::·~
__'"
... ~~
.'
6 I"'·
not
-.YP o....n
v
c~es,
~b0Ut
ttle mean of a
Ma~h.
Stat. R139
.. j"
•
To tast thi:; from the sample
)
'I(
I
J
;; e a -::-'0 assur.1:lng that th·e:te
sirr.plic:lt.y
\7e
are
110
Ti--.e pl'O lx.!. 111 1 i
ZB:Y'D \1 a:!.u8s •
a.ssume this ..
Ii' the null hypothesis. popl,:IDtion medi"l1
positi,,"'e and a negative sign for' any :ti
•
pos1.tive and negatj.ve signs
~'!ould OCCUI'
The chance variables n l
rl", 1:n_..!..
each \-11th mean
J.. n ~L\1d
(,.
and
,
..
Uf.'·:;
0;:
0, is
equa11y .p:t>obabl:s>,-
;,q:!.'ch
atcut
, ..~.
tru~,
Hucl
a
so
€qu.~~l .t!:·e.?nAnt~y.
~ 'u""
- 'fa,;.
.. j .ai)J..
"0
'
1")'.uO~
.c . .!.
..... s,
7'
4
Vc"lufJS of n, (or n,r . ) :1e :;:e. t:i.ng
variance 1;-n.
€,..~
lddely from the expected value..!. n
~
~Ji L,.
be rag-ar3ed
as indicating that the null hypot!wsis is untrue.
'l'1:l8
ty of
3,S
signif learn;,
The p:eobil:·ili·;;y of
normal ""ppl'oximaticil to the binoffi),al ,jj,stl'ibut:i.on \,'ill ;,:ive sur--
fic:ient
ac~uracy
at the oz'dinal':r leve:;.';; (') .. 0:;
Ct'
':;.01)
/ji'
S1"11ific<::110Ei.
This c?.n be cOllvenj,ently' used bJr mean~; O.l til"? prc;batil:i'GY ta':Jle of'
",cem'ac:" •
;;(
r
•
r~ol'
1. e,
2.7
I· "
D.lO
a sample of' 11;·
\'16
0.10
, ,.-
5.8
5. 'i
t .'".,]
0.02.
0.01
haifa the following T'2snlts:
7...'.
.. -."
•0,.
.
•
-I-
L./
/0
S
Ii
.. '
~,"-
4.
-'C.
,j
'I·
"I
==
-
7
o
1,6'
I':) <:
F ""
C ,2. 0
{.
"
::
'7
j'
I ~.
7.
. t:
Z
--~----
-1 ;:; :.:.-(
<'
Or 0·
,
I"
t::
f
I
CJ!.
.. f.
12 ·t
'J
r'c!"an
I '.. "
••• .J •.~. _, JP
< .. ~ .. ..J
~..
•
,(. -=
I
~..J...L[i) -:-
+
Yl
( ..L" ', "
__ "
' ,
..
~
1..
"'y
•
.I.: pO th'si. (-.. ,..0,
t 'ne
"ed.'.~n··
..··'
r.)
'-'I.
__
I
_Vi
I
.?;
'Z:
'I,,'<'l".....
".?,
, ....
}It.i,1..L'_
~
... op ..l.L'-,,\..
Re!l'£l!J1bering t!1at P {"'F.
ill!.
st1:.J use the
0)~..
•
r-' j ,... _.J
'\--·"n
~
.<•.)
_.'
"Z ,:; !1 ..;. k!
;
~lg
z~
met> od p:.'o·i"ided '·,hE.t
'J?
Q
,"
r;' \. 0
50,
>" (1 t el~, .
l: i
'·D,··t
:1,,,
::.lP(in
l'
L
b+
.!'.
c..
)'1".
t":'Tn.-l\
p!";:).;.uC~
.,. I .:..,.
i~l
l'.
V,,-"i·,r-·c:·
~..L.U.C.;l~
1 .,.,
~
\\''.1
(aj
1.
lHIve th·" tqlli,'i.atsj pr(ba~
Canstdf!ri:".t' a6~i.i!1 t.:ne !:iar!l'pl~ 0:- l'+'j tha pl\.o.Jai?i1i,:4 e:: of
0btp.ining HS many or mOl'c posJ.tive sig:c,s
i!otn that
1 ,~
1. ' ..
l"~""''JAV''lU'''S
-...
" ..... (··~a·-·I.P···~'·"~
.:..,,!..r:....
c .....~ ......'\'J• ..:.l;...
{;tLt;>:;IJ..b"·'·~J··,.d'J.l.,.
...J
bil: tlBS.
;',
\.l:,
+
F
.it
10
3
II
D. lEU!;
"
I.!
O,O"os "'" P
th"~se
lr.eci :.an. is·: O.
•
''+,
. '.,.j
r-.
£Inc" 'Gne
0.05
~o
.C;
f>..::
"
"!~j.~:1
tte r,led:i.an i!o really
.< O,! .:
0,0:,
0,0
I
prob>.bilit:tes '.... i1:. be still sma:"1=,' 11' the popuJ.atio[
C8SU:.t 5,:, r.ow
The
rJ3f,l' 'I_:I"
I
00.
I~.,;
I
.'
e"e1
.
'j)
~
•
n
;: n
1
+-
n2~>
n"oosf'tl.lvE'')
...... - - - -
O'ht-1'~)'Vf-y'i
,
..'"l.f,
.....
::.
" } n2
_
.•
c:
~,'llul]~,
of Yo. of
~'!}:\~r,
~r'~
i11
~ b~ i.. f'txnd "}ositl-lff1 nU·l!-:-r:r'"
Lf!';j
'-
~'~Jc.;
J." ",
• . ~.p
nf}~Rti'jn
ann
?IAt
-~
!),~t:"I ri;l i Y1'~ ~ }; 0
If "': 't"qkp, v'iluf;2 of n-:>
',';)lich
'..
.
•
/
...
"("'''-'')
;..
~
---,
h
p
--
.\
.....
k
'>,-
j
~rr
T, .
'2
v:,ri'lI1c':'
.
n (l
2i
it! ,
-..
\
l' , n.c
•
Th r ,
pOI'\'(;
r'l
2"
~1
)
"J
--
r
,,
L
'
-I- E:;
~JI"'.
}
of t.he test will b"
G(
)1 )
+ r I.
~
}
.qr~;='
,\
I
0,--_._----~() r~'
,.r~\;:~-(-,---·«'~ )
•
r--
J Yl
e'
f]( ), •
:0
>
•
(I'
~?)
,.,
-->,Cc.
A, :, fixr,c1,
If in
O.
G( •. rn/frl)
-:.K"·r-;::-
k' WI n
_
_.
i.
- '2. 1f
,~
-Kanca the
po~ar
of tha
~
~lgn
test
-)
------ =
r~/~;;;
•
1 l n Drob'l\:
t Ii tv
,
.
•
-> t, .
•• '"
r"\
.'8
•
n\ --to~
~-.
;
\
an n -oJ> e.-o .,
.(~-P1.)
"If! ~ch
,.
"
=:,
m
r-~
~n
E.:.n'i the Ilm'!. tin£!;
~):wr-,r!l
.. U.l OF: the S/l.me 'l.f
2m
•
i'VT
,-f1
~-_.~
\
~
:n: _
n
2.
11
test as
•
,f"t.
r_·
-t-~
n
,-t- ",'1
coropp.r~n
f or.th e
,-.: V
•
.....
,
...
,.., ..
."
-:"!. fl 1i.~i '. 8
........-
""
'~.
..........--l"'~i-- .. -
•
{')l?n~.f:
::tnrluAnc.:;s
the~.·:r
•
\
..
~'7,
-J
5(,
137
~~8
2 1!
8
~11
'~9
•
III
1.4
'-
T .,
6r )
f'i
2~
.• l;/1
.~·r()'7'i~h
D
•
Donot ~
,i,
i
ie
i
i
I
'?t;.
of if
Ii.S
can
l!trge
.:,.
I
I
I.
•
*
l'h',>".Ollcf,hou.··,
t:roesc
n,.',';.ee
C"...
4.'~
...
Q
"S')!:'
~ It' t' tloll'I"tl
l_ ~ .....
_'....
••,
l l•
J
;'
"',",',",'?;
·'z.!~r:·H_"
~'"
.!;n
':''''j;:''')'T.:1~1
··.. -f:;. ....
I.
H
't··'yt'?""'r·
.•. ,. .. c.., •.. 9
l'l·,"",,-"'~oct"
1.t.~·"'4
J,:
•
in
2',
1
."
. _tr::).J
....
.
•
~-"~_'.""
':'
T " 11
_
_.
J.Se-J
....
r_._w·..
'2
-
f
t.I,11."
~-..
V
_.
)
" ':>"j
- :!:-.-.
or
1S.
',"r.~l
_
of ·......
t",.".c{.V
_
H.,
... ·~rl't\.J. n~~
~.
1),,(,~t"'Q
• ,. _, ,-_ .;;;• .
..
.4"Llt
p.!,-""
J. ......
.." ...
....
•
junttce hA
•
\.6
8eB\HF~,i
to 1)·,; gUilty until h? c'.n eGt.r.'bltp;l
I
Jl
•
etc.). Only the p.rt of
th~
tl,~
i5
st;~tisttclc;_n \5 p~r't
Rtrtisticlsn 18
v':1rled
l!U~
v~rledo
Fortun~tp·
{;qn ,'to:r"< out t:,.P.
!,)Ofl$~hle
/
corzSf-:
•
•
ul~r~ces
;)f that 'nt t.. l1
pArH~il
anti
::;gper~
!lfVl nvt ;··)tr.~r
•
I)r ,.. Yr i'. v 1"
•
'
0
•
0
.
. r. J
Put
I,1"-:
1"
,":.OO.lpf) )
,',
•
(
r1
"" _< .0 ,
r
'?J Id.
or
.... '1
in-
•
fin 1 thl!:
oo!'re:"\'jl')nrt1n;~
v,luo13 of tl1'.'!r OUr;'! '7.
flx'3t1 1'nt"'f,0l' lp,AG t,hlln ~n.
SUill by~'..
I' nrri 1 tfl
as Inre;p.
"til
1)enot.l~ any pnrt i-::I~I~1 r 13et by
11' th" nwnber- of ".,ttl '.., t th a sum
ot' lR.f'gp.r than \V' 1s
we ehA-ll cRll I'
i.:t~nlflc··'nt.,
~n-s15nlfic.!:'n!:.
V'3t 0' be !'\
leFHl
than or eflunl t·o F.
Othe.!"w.'lse I' w.111 he CIlIIFlrl
I\. non-!)ign~ficp.nt
Hf,'t
su~h
that the) nu'aDer
,
of "",tB
"~lith 'i
lars!'!'" sum 18 lelSrJ th'1n ;'
'~~ll
•
b," cHlled
-
r",c';)0ct1.vp.l.",.
\',t th til.,
..
J~ ....
'vtth
Since thFl setfl occur I;: Y'iro ,,1' onpoflltell
fH~mo "rtlur.) of
h) •.
we "Jw;:ys take
to to hI'! eV'Jn.
The fllc;nlficp.nt setA are mORt eA81lJ picked out as those
opposites of
the~eo
by ch>.\nce lll.belllng, if lill fletfl ?r" equally Ilrob ',bl 'I 18,
-n
r,~ 2
p. sSY. when there ''U'P. nQ bQr<l"'z'-ltne RetA. :>.ncl
=
•
,
P
~nd ~reHt~r th8~
1 - P
reBo~ct1v~ly,
,
•
~'~l'
d1sproVfHl
n ie
~2 t
th~ ,null
~r . } atf-;r
,',
bor'i~r-11n'l
0
0
so
'vn
8. cr)nv!';~nt~nt
t 11<:l-11 6
o
_3·""',n.,.,.
r.
e; ... vefl
•
hypcthes 18, th" nl'ob.''-hiltty c,r error
Thls 51';!,)fl ',' .." 3·::,-0
./
14
p
=
vA:lue of !.: iF1
8
""
,:.~
n-~
r.'or n
3/fi4 .",-).,')11(-)9.
= -.,:;.,_
_
<)
> fl,
could ftvold
0 .. 0117 •
fl'lt,S by adjustinf!; th" 'TRlue of ;',' t,o the
plJ.rticu18:;'ft Eta set of, obllervm'l differ-encen, but at pt'P.Flent
IV"
•
I
•
"re thin1tlTl[S of det0rmlnln - ;,', Fond P bef,.,l'El the axpFJri-
ment 1t'.
'~ill!llYfled.,
ot:' forfnuIRt.i.n,,: n <1Af'lnite rule which we
automnticnlly follow tn All CAsea.
'1'M.8 tf.·st clJ.n give no fliGnlficllnt result
~t
tl1FJ 5-:'; level ";h"n n 11'1 leSE! th"ln <'.\'h'!l1 n ,>,5 ther8 are
,
,Hiv0.
?hell"l lire?
sie'nHlcant 'v",th P =- 1/3';>= O.031'?? if
,.re e1 vi~n b'Jlo',7,
p ""3/26 '" 0.04(,9
•
,1,..":;
n
2h.
7
3
6
a
';)
10
12'
2 lt
1
2
3
4
r ~ 3/2 8 ", o.nll?'
n
~,.
9
."
-'
:.: .. 1
10
0
1
2
11
12
l~
::!4
3
1;
1',
•
'fhtc 11 ,'1re O:J'lOBttP- in Elli';n to t:"A sum '1, :<rd 10;'(.
j' ;l:mot<~
thp,
\
\ "'n \
suff\clHnt condition for
•
t~8
g1v0n Bet to he
1.1, 1;.6, '1 L 2, 10" 3
F_
A,nr; N:.= 3 -:::-
Ii:':',
,
n.nd r..hr-! RP,t if; 81p;;nl fi":;"\n"t..')
-Ion" J~CJh9 3,,2 .. ::1 2. -4 .. 1& 8 5:> 1 ... 2~ 0 ... 9
0
•
0
8t~nlf1cAnt
\R
V···~
~J.n
lues
r}
.~n'l itr. ·1f:1.:ri~ncp. in w?",2
•
0
ThA C:1!inne "tn~:r't'ibl~ '" iB th!-~
o
1TI-;
x' tn l .tl1 of
".
..,eli
"a
:'In",
2
,.. ,,-
If
~., 1"'
"
•
,
nU""'!Dcx' of theno is' ].1 T
•
1 +- 3~-
- !:FL
/\.180
3n.v
tljr;~~
•
'~I
If
13
•
1;2
-1(.,
20
7~
.11
Ii')
h9
5~
~f
50
54
T:'
8'1
}')'7
17
;>8
@
9'3
41
0'.
;>
"7,77
..1,
•
)
'y,
•• J,;
4?
1';1)
60
fiF,
( 13)
( 1'"")
59
63
77
1J:J
95
1':7
(1
r:'"
, ".
fi9
78
·79
83
6F;
7'.1
71
1-)5
Fl!)
8A
97
"'12
100
(2~)
"',
71.;
::m:i
SQ 00,
uP.
72
713
9~
1')'(
87
,Q3
10~)
lOr.; 10'3
87
91
91
(29)
97
RuhfHiquemt. entrt85..
N .-== 1B2 nnd
80
flf)
thA nurol)f: r" of
Tho.;e lees tlFm 1'n 'ire
th0 true
.
q::;
.
(6)
88
0)
96 ( 10)
(21))
The numbn.rR in ::>n.retl t11i.~f)e~ are
•
'''::::.1
2<;
(;0
78
"lv~r,
f'l_
~.
:.153.
0
8
17
20
41
2,"')"
.~)
no w
171:
to
__
-
V:llur1
vrtlt1·~A
,q
11.
C)f l' is Ifl2!2
10
11~
-==
of V which
ter1. o
0.178.
•
1')
('tnts f."lirly Hccuf'l'lt'lly
12
35
sho":n b.y ti-:e t'l.blp. belo"1.
hi
V
J'~
a~
491,
429
405
283
49
80
If8
92
11:32 ·153
z
p'
2,389
2.088
],,971
0,0168
0.0117
O.01188
1.377
n.lr-:i1F.
0,0469
0.177'(
l'
o,o3f.if3
(),O3'12
P ls, t,hl;l true probp.btl1.ty I'md p' th!'l v"lue given by the
,
of
s~m"lin.3
froID
!l.
norm'll population "'e constder
•
t
~en
~re
n tends to
p.Rymptotlcally
00,
z t'mdr, to
t uni formly 'qi t,11
.or V'luDfi of n not too small
~~utvqlent,
equlv;,lent i f th'! chstribut1. rm ot'
1:"
•
<vere atand1rd norm'lL
the truth when n
>9
'['hill is
an<i t 2
dt ff"rent 1l6aum!1ttons,
difference9
P.
> 1.
Th'! t
fp,ir.l:, good approximation to
'l'hf! teats are
l)!Hl"lr1
test aAAum",s th"t the
on ver'y
obAervc~d
•
:iI'''? p.
5'lm'J18 f)"om a
mer"'ly th'3.t thiJ
!l<)I""'~1
'Che z t:'lst RaRUm<'lS
pOl?tlllat.l.on'.
')f then'" W i're such th"'i:. the <'J.l.strl.bv.-
1I'Rll.1f)f!
-
M.on of ..; is ',pnT'oxinntely normal. IinJ1 Il.B;;U~:P ti.on whi~h
OR?!
In o rrl FH' to lnvAflttsnte. further thFJ rol->tion of t'le
w test nn'J. thA t
tion.
tew~
let UR find thA moments of the w q.lstribu·
)}9note the mom'mts about () of 1,1:A I'. r,'l'mr;e!'s
''1/1 7
...
. i
•
ftl ...
Th" cumuVmt function hI th81'Afore
I
•
f'mc;ion of
~ Y\)I., ~
!.1
'rhn odd moments of 'V !ire A11 zero, "nrl.
3
"l2
::
Put
I, L
/"f ~
1.';1,
--,.,
I
S{V"')
n_-I) t'l t st Y':i. hI! t i on hRS
a
first. t'",o mom"lnts
_\
am
n )
__
w~
n
'i:( IJ )
•
:V is
.l: e, }j
./
4
;
"
w~
,:.:t"«·-•
iJ
\'1n
•
momnjj I~ of
AI
V
I~
/.1.'4 ,.
A.ne)
'"
L
It hi-?8 the:;
....
r
It w11.1. be 1lr.r:pox1mllt '> ly 1
If m e)~' th0 \ '!l 1,\
thrm ':Ii the l'p.st..
hRV8 "bout the 139In," VA.Iue
v:1.11 bf1
which 1:0
I /
I'm "
Hence
t110
)
lJ
•
Bllt ~f '.~ ':re s!'31ipl:1.!13 from 'a normB.l populA.t'ion of Hl~':.'1n O.
£
1:. "-to'"
)
thlfJ
\]
•
1.180
01' t
"0
-I
c1if,t]"'.~;l1ti.on.
'('t:p.l"of·Jrp.
the USE' of this
t test we iH'" not m,;kirv"; fl.ny aesumpti')n of
6.1stribl~tiono
F'1du~Y~l !::~.'nl.ts ( or .C0'l~ldp.n<:.~_Ir!Y!r·::::I17L..!~>r'.::\}.~ Dl}:~p.r:~IY:'"
•
In LocAtion
or
---......__._q
tton thAt tha
y
-
Two
PODul~tlonB
cf the ----S~mp. Form •
--~------_._-_.-.-
b ,-
~IRtribut\ons
On tl1ts
of X Bn1 Y dlffar only In IDcRtion,
"ilS!llP.pttOl1
(1)
>"• I •
that thp.y
•
R~A
not
st~n'ftcnnt
z
is
thp. chAngp. 1n h 19 s.fflclHntly amAll.
1 -
P<:
If it Is
Rl~nificRnt
,nni hAS R neAI"lve aum It will still l)e R15nlf1cnnt with a
Rum of any of the
•
the set (1) 10 s.\.gnlf1cant
w~ th
a
:")r)'ll
t,il/"
sum. and
h
2..
..
•
folIc .... :'! thflt (1) lH sle;nHlc'1nt if
non-·n i!';nlf1cant
if
h
-= h
I
~
h
11,
or
h
> h
<!
1f
h
0)"
=
h,~
h
has
l'iduc~co,l
!)ro1)ilb\-llty
~
J. - P, for' it 1s ertUiV"ll'mt
non-ll~nir1cMnt.
to the etntqrnent th:'Jt the set (1) 18
Th,~
atat~ment
11 _
h
,-
h"'s nduci';l pro];flhUity
•
1 - P.
X Rod Y flT·e
If
boriier11ne Bet '1s 0, :om,i flO the ttouci'll pl"ol:Jrlhlll.ty
If the mSAn
~~x
":(Y)
•
-!f.
and
h
~~(
(3)
-9 4-h r
-27-h,
..
X)
V~lUHR
of X Rnd v both exist
h)
V( 1'1",
"\
':;.:
.I
... h
:r.:(y)
II_h, 'IS·· h., 19- 11, 2(J-h, 23-11. 2<1_1'1
p -
:;
.~
"
-- O,O'l(,'i
J
",',
\ ., I
'
•
II 'H' :3
"'£'!,
Por
-
~
.
3, I.: ,-
(3)
t,l-J0 Sl'lt
2 .'
I
~ (1<,(,- ~ )
-:.==-
J
~ ( oN ..
t,y
1:,h t3ref(1!'s
-
~,
"2
'-
~
(
r-i-
yt~-" 27..
I
•
A9a'..lmin~
F
thryt
z
is SNV':t
f I zI
<:
I, q 87
~ --
o.
C; 5' 3/
This r;ives
i.QS'1/"'(,,,8 ~ - 8.
j
I .
~I??'
,
3'-1. IL1
'flIluen of h OU\.!llde the rAnge -
,
J ..
'to. "1.),
\':111 ()orrp.R~:Jnr1 to lart>;er values of
atgntfic.'lnt.
•
· : h
,
7-
~"I.
I '-f
nnl'l flO '1111 be
the! normal I'lpprOx'dnRtton gives
Ti'lUf!
'10.40
0.')5'31
J
•
How
l
/.,' r-( ~,-');i1
'"
r: )
A -h
s
I
Us In'~
thA
(7
distribution
t
F{It I
"-
~,
'iq,9
D"
f
~
we
1;' .. )
hl'!V<')
o. '15" 31
lind t1')"r"fore
f'
t 1/(,-,"1
"'-
"3 '-I . 31, I = 0
J
q5":!1
-'
which e;lves
l."
'"
,
•
1l0':
lnp; theflp.
not th'l t th~
ar"
"6p.umln;~,
thq
r~qutrp.mentB
~"'Jrl)x'l.m" t.}onfl
ri~tnal nu~herfl
for a goon npnroxlmntlon,
w satiify
bu~
r,"sultant nWTlh",rs
vnlup.~
of
v·'llu",s will
b,~
true
'.\'hen
,
h'"
h
9
guidp."
_/1·O, t:,e set
-54, 13, 44,
4~,
m~
1:J~co!r,eA
,
59, 60. 63, 64
and N = 5•
•
I
that the
•
_
L.-~­
"'
h
1'1 " (p
,
'/,
- 38 /2
-
"•
,5
Si '/
,
'f 2 13
and the flAt is
for
bord~rllne,
thp.r~
;'1
7
Rre only 5 setfl 'vlth
-, 38
Thus h.::
,
:0
'Is "
./,
resultqn t , nurn"bera we should ey-pAct thp. norroRl or the t
I
Rpproxlm Q tlon to b" fRirly~oQ~; but tt must be rernembp.:~
ad thnt
h =
~B
•
-
~
-1/(.
on the tails of the distributions.
-,?,I,;
-2"1
--/I) _10J -1I
0)
-'-
fi
=i
20
-1J'i
h
.crktn~
30
- it...,.... -5'1'I
h
are
I
)
- '1'7, -I'~
-1 4' j - I
J
0
,
"3I
N -=,
'I
I~-
2:;'
- '19, -
IS,
-It.
)
- '3,
-z,
I .. Z
This set is "Dordarlin.e for there are only 4 setB wtth
Il
7\ th t'''~ resurt~
or the t apl'r'ox1mr,tlon to be good"
"('he trouP. reflult if!
0, '15 31
'. I~·
•
,
, rJ',·-
4 e~·
t-t:;st
,:r
to yh'.ch
~1.t...her h·'i.~
b("~~n
';'H:t i flr.n
t)I~V
co i
'1C i..}r.:,
t"lr: r3Al11t~~nt vo;..1t.:~3 of ",'; .~ h
(
anDrox\rn~tton
1~
lJnrel\nb],~o
In t:f1n s1mnlp'8t anplic
t~,on
•
w,
r
of
•
'';I.
)
,
w,!,
\Ai,
W
r •
......
)
)
-
VV'l
for
I w,1
.1
>
l"l~l
of
t~·~
t.
t.?Rt,
",I:'>
•
cluRBtfy
'~O~l!)ar-"lnr;
or non-Rt~ntf\cnnt by
r . . . (;;
ttfl '\' v·qluA "lith tho>:e of"othp.r F;p.:r!;.l1p.I1
~ RamDl~
±
4J,
J
RS
-1:
Bl~ntf\cHnt
T,.J~_
nrohnbll-t. ty
•
~ttent\on
WBR rOCURRed on thP.'Rct\vitip.B of t' 0
t· e te t ·... h11"'· ;:;'!') ot,h0r
I,A
tr,q <:lon"<1 form •
•
The ", test is !'.n exact. tf?At.
••
,
flt~t-
It has t"e greRt
•
"1'1
can
of tiF'
~f!tJ
rirl of thlfl r:tsr..rlV'l.nt.fig" by llnin... 1n pI:lOP.
Ob8'H'Vq,~ nU"lbp.rfl thf!1r fllr;nr:lrl rc,n!{fl.
be thf! !'.'lr.k of
I
vJ
~I
Let-I\(I",y !)
-1n the SAt
i vJ" I
t,('
by
post ttve finn by
If
•
no two
w....
-+
.
- I,
~a
~PPly
the :
-R(IW,_\)
w ..
if
h"<vp. thp. fl'lrne rool: ,luf; and tf n"nf) if!
...... ,
-l..
t~Bt
to thlA Ret.
Slnce tllft etlstr\hution
(Btometrtcs.Bnl1et1.n 1 (194-') p.I'l?)
~1.ves Po table
Flhnw-
thll.t COrplfl!lOni ing to v for v.'11uec: of n "from 7 t.o 1"'.
,
•
8,
I,,
,;z,
,
(.0,
7:<-,
78 J
·/07
J
,
\~o
•
repl:'l.ce t':elle numbers by
I
v "'" 17,.
,
,
2
n.
3
-
'-I
I
t.o
-(,,
7
,
8 , <t ,
10,
'7'.lcoxon' Fl tabl<! f5i\>"!ll
11,.
corr~fl"oni.tr:v:;
s
V'
-II
P
'" (V)S3
,,"11 for n = 11
aet 18 not fJi,;n'tftcnnt lit. th'.R 18v"!1.
"Vhnn two 111:'0 !!lore
esten in.
If It apnAllrp.tJ. thflt
abl'l '3ff'ect w~ Rhol.l1o' hRVF! to
•
t'1rw'~
mlp;'lt b",
wo,-k
0",
the
of
0:1. t.hpr . . ;~.'!
conBlstin~
of non- zero VRlues.
arA continuoUR
t~R
Drob~h\llty
If
t~p
chance
of p.lther of
OCC'.l.rr"lrIC"lfl 16 zero.
, I
•
t
con~ide1:'-
v~rl~bl~s
the~e
30
•
Ccnp-lder the open form of FiAher's tent.
5uppone thAt
1s
fl.
W,
,
sample and that '>:e wish to teAt the h:!,'othesis thl1.t
the nonulatlon is fiymm",triclll about 00
z
The ;11ternatl\'p.fJ
<i:' w_
h'lS me:<n 0 and fltRndRrd
d')vi.~t1on
10
If
1s flrna11 then z hflS ap,.,roximAtely N(O.l) <Hstrlbutl0,"
•
Now if
is a flRmple
from a po,.,u1A. t ton of flnitp. vR.ri ance it clln be sho·.m
that
as
o in
n _
QC)
"
Therpfore the d1str1hutl<m of z tendf!
in !'robabiltty to Rt'mdard n0rm'l1 as
n
~-.
trw'! distribution of z given an eXilct t!'1fit.
ex~ct
~
teRt a:· asymntotical1y
Determ\n0 k so that
2
f?F
•
~rohlJ.b11tty
F
'('he
USin,,:
thnn
•
v~luBR
z Z k are
o~
stBniflc~nt.
r:onAirler now thnlintributton of the obA'rye"
of z. (Do <lot confupp. thiR rttstrlbutton '?tth the
,,~lue
distribution of z determine" by the
meiln
obAn~Tert
"qlues),
h:;f= 00
~.
uJ.:--
n
At'! n ->.,.. " th e o.snowlna tor t'mds in ')r'ohl''bl l:l ty to
Th'" distrihution of thp- num",riltor t">n;s
to
nor~al
of mean
r-,n
'1.
z'w",
l'W"'
Iln<i fltandard devta t ton v ..
\
\Z
•
If
al
~nrt
h
~
I -.......
c-""
in fH'ob!Jhili ty
all
1n prohabl11 ty
A.R
Q. t1.is <ilRt r ibutlon of
th", prohAh 1 1ity
P,
->,Co
f
rejectin~
7.
tends
th"> hypothesis
The test is lnp-ffective
a~Rinflt
thts alternative.
'Ve noVl
g,_
In to
rliRCU8S tL :'8 of' the h,Vo-
ot""esis th"t two e;iv'ln a,'?mpl"R c"me f'l"om tt,'! 81me
•
of mp-anA.
f'
Qnd
so
t4~
rro!;Q6;/,T1
of
q
0
32
•
SUPPol>e thp.t lVe have
real number's
mTn
(not nCCASRll.l"11y all ellff8rent). and that their mean
is
zo
of th0
..
m+ tl
numberR ar: paint en on
m
+
n
Con!' Vi8r a. flAp:l.rFition of t:op' n ;;nhers into two r1iffer8nt claflfles of rn and n,
4. , '-',
11,
,
.)
"If" •
.~
.,ith
\~,....,
l~i
rJr-}
merl.J1
U
t th meRn
1>
,.
Thp. 'lUmbAr, N, of aueh flAparatlon8 1il
provVlp.d thllt whe.m
•
m= n
two separations in which thA
tnterc'1finged
;;1.''0
th8
tW0
clRflfles. thone;h
ClA.6SI~fl
H.l'e flimply
regarded as eli ffel'fmt flep:l.rationR.
"e flhall call
!\i -1r1
.thA Anr'ead of
the separRtion, Aincu
·n
Let
If,
be
It
fixed intflger leRA thRn No
particulRr flepfiratlo11 by Ro
aT'll.tione with
•
p..
Denote any
If the number of SAp-
flPT'A!·,d equ'll to or p;reat(H' than t.hat
•
tions ts if",
are mOAt ;QARily plCKP.t1. out as thor.::e ·':'tth the
(~4
Vrllup.8 of
•
proo~b11~ty
of
~
large~T.r
- "., z[
st~ntflc~nt
net lR
1,. P.
Let
.l
A", .. ~,
Jl., ... ~ )
(1)
•
~,
)
)
~n
:t I,
34
•
'\h'm h iB altp-red a sisnificllnt
sign\' f1cant, Ilnn
not
~P.paratton
non-Bir,nificqnt RP-parRtion 'vhich ie
it
bord0r-lln~ ~ill
r~maln
so,
prov1d~d
ch"nge in h lA Ruffict0ntly Hmr<llo
a
separation
Bign~ficant
will rp-ml.lin
If
r0maln
~111
that the
x+
h
"? Y.
Rl~nlficant ~hen
h 1<'1 increased, but a born.;r-line RepaY'ation '1/111
become sifl;n1flcant.
x+
Similarly, if'
h
~
Y. a F!ig-
ntficRnt AefJ/J.rRt'.on wi 11 rel'1flin ste;nlfl.ci'lnt "Ihen h
is dec rp.I'lI"'l Q, but a border-l inc snpi,r;,tlon
'~11l
Der-ome
sle;nificant.
Consider
si~nificAnt.
separation (1) 1s
s~premum
•
h
DAnote by
the
h,
of thORe which RRtisfy the ine1uality
x
a.nd by
valueR of' h for which thA
th~
2
+-
h
Y.
<:
thp- infimum of thoRe
~hich
eatiAfy the
inequality
x+
~rom
the reRults givp.n
h
non-Ri~nific~nt
h,
and
born~r-11ne
h
abov~
it follows that thA
si~ntf1cAnt
separation (1) is
h <:
y.
>
h
or
I
if
>
h
hl
'
if
~
h
:!:
h ...
if
=
h,
or
h
o
•
o
•
)
"
...............
)
into the
19
h .•
rl'
"" "-,
N
"
'-<
.1.,
claHB88
l...,
••
•
t~o
X
"
x.. :-
-,
)
91~n\flc~nt, non-B\~n\f1cant.
Dr bor1!r-l\ne •
•
i3ll.mpla8
""'l
"1''0
~hn
I1: they
at'A
the hypot""'SiS, if non- F; i.:;nlfic··'nt
1'A .lect
1t.
F;1gnlflc'mt or ;-Iot.
probability of an
~rror
1l1"nU'lc"nt
\'Ie
accept
of the flrRt kind.
of
rAj"cttn~,
tlJe hypothAAif1 "'hen it 1[1 rell11y tntn ii; P.
>:~·.'P7-'::
Ar'O th", fo110\'l1ne;
fl"-'!1!?1~6
At",niflcflotly
11fferent7
And
•
aubt1" ct 1.2 fro'll e!J.ch sam!?l'} vAlue flO
t~le
sm,Ql1",flt number to 0"
1-1;; to
reduco
Mid t.hr'n m'":ltipl.f fJRch by
10 to ";8t rid of th'1 d8cl.mril !Joint,F,.
'7e
th'm h'l.ve
0, II, 12, 20
'T'ha meRn vnlua :tfl 17. flO th'lt
The1'~
m Z; ~ 68.
Rre 126 sepllY''lt1.ons of tht'! nn;nhp.1'1'l into clesses
of 4
~lnd
? -=-
1/21.
5,
Her '~A t f
''ie tak" 11'
-=:
f'"
'~e
!lhn 11 have
The groupe of 4 whtch >.;\ve th8 1,Qrge8t
•
v11ues of
It:
1..1
•
• ' I-
1E.'.i
l~'l
')
11
1':;:
.1.F
_:c,
::?l)
0
:1
12
i9
';'.
26
J
n
12
.J')
if ....
25
0
11
12
C- '.:
II ~:;
23
29
24
22
~!0
9~1
27
·?9
Z~ J.+
22
- .'
(j
C);'.
-.
2;~
29
2l~
;;"
_.J
19
C" ,
','.
:?4
29
211
2~~
y;
0
-
""3
•
O":l 'T'
•
,.
L
'.on
,
If
X,
.qnd
~.)~h,
,..,,,
.~
•
•I
•
)
.
,
1
~"
Col "
orr
.1 ... "
.,
r \(
0
-
...- ~1
; .~) ....
,..,
')...
11) J.2,1 20
'16
h'3V~
to c1 !7ijermlne
h,
h
.,,5,
-
h
6.,
Ta!--::e i','
'bcrdo.r-lin" "
~
rj" 15
,
l'
". 1 •
h -to .'.2. t
?-2~
19,
If; ,
:. c ~ . ." '., -
h
•
~
0
15
t
-
h ,-
-
,
1'<
;;>0
":q
fi. :1c1
;;>9
241}
p
-t
1,'21;. th':l1
?"
._ r.
;:>2
=
"rill so
p
•
i.r·;e vnluAfl Of' h u:h ~.~h m'1.kp.
11,
h f-
.~.
('
,.:.
•
(!"':-
"
~,
.J.
i. " :;
'J
..: h
T..
2.~
(
•
-
.t
.,
J ,.-",- ...
39
•
Tre:-ltrrIAnt
A.
1\
'-rrAHtm2nt '3
•
~.
Bum of threp. tr:7''flS,
=
•
)
)
7:.<-
;:jz.
:::t -"
)
)
~"
J
th\.\A
.',
\
~
;;:
x'
,,,
'.
~
... "-
-
( •• 1 .
orfect 07 th8 f0I'ttl1.ty and
P1<I't icul!1!' plot. ,.
ina; R11
t"",
13.11c1
sitl'"'1t~0n""
?te. of t 11P
v'
.......
x ::.n<1 t.hp. y_
rut
t- \,
h,
I
Y'j k.,
•
1. ~
~ J- q
'I
-
I,
)
~f
\" I,
I..,
),(
.~
-h
~
"
- 1:,
,
•
.
r'
'
i)::ln
l,
d' )
j S
--
•
R ~.i"':r~~. f.,. c
'II '"!
tr.
.~.
0;'"
o:..n ~..:.
0',"
--
"
L
no',
~i~O
·.r··..1 0
.":l
b
dc;t~.'-
..
_.' .
'r J.'~ \1(
f'. r.:
Ie.
1'1,
'·'1
.. '-, ,. ton
.'.
..
~
•
J
(
,
J
...........
, \; ,...
•
•
"·r ,1:
")J
"fe
."1~.vp.
q .. " ...... + k,
~.,.
•
-,,
:.
~.
',..
k ~. >
~
ne("~n
;0,
I~'
...
"J
(
=I .' l., ..
' ,.1,
. . '''-;
I8GLn.~
, £..
,
~;.
00)
.......
)
;.,,,\
t)
tt
_"
o !J~;;n
•
E· ...... ;..1
r'j
,
. ~r "
"
,
, .:~ 'L
"
"
"....
~ 1.r..'
i"'t~
'~}':;.'
....,l~-:..··_··~.jl
:},:':
, ... '."
,;
-
'. ",
- ~ ~.~
!:-
:~:: I
) ;:':i'~ T",l 't F- r:
I
) (1;:;," ,
"
;
<,
'
"
..
n
"
1 ,",
"
'j " <
..
-.
ll'-'"
.;-
'1 -",
C'
."'1
::
-i
I'
4
'\' t l't I'
",'"
'
....
0.
'
~.:
r;
1 -1
<::
Z
',.
. ~.
n.," :
o
;: I
l
"
(.,"l'"
C7
lti
~~CCr:
.-
'. !)
"
.-
G '1 ':
,
'j 1":'~,,
'r: c·.. i.
,,
t
)
"',i \' • r)'7
•
: l·,.·
':.~'"
···'r
- ," .(: 'j".
. . . i·....,~ t.l.~·
.....
,.yt:l '"' '~ " S
"
-.- r.
:.~
'I
~l"?
"
.
'~
'c!
. .,'
. ",......
..
"I
in
to
C'
I t.
.~
;
or
"
J.o ::.
"
,...... ,
-
"
/""3
•
u,
..:..
1~'
". ," ~\)'''''''''lt.;" .
,.
".
•
. .- ,
,
(
[
I:
\
'")',
E
\ '- I
slrc~
(
;
(
cI
( L:
-
.- )
=-
.~
- ,- ,J " (I
-:)
/,!1 a.
---( ...... :-...:-l)~,.
1]
,-
It
_.f.
J'\.1
_ _ •. J.:.
a'
. _/
)
2", ,11- /..., ,' "...... -I) :-: t.~ ' _l. 0
E
•
z
j"
z~
r/~,
\ L.
- ,/t-{ z.
L
r-""111>, .1
from "lhlelJ ths rer-i!lt st",t'1d 1'0110"110_
-~-
•
}
(.~,
,2,_)
J
"
.
}
•
I -.
"~
- :::i)
<.
z
·c
(i:i- ?)~
-_._---
Put.
J?t
}
---- '---'--
•
F.prJroxim·-· V:ly
first mom8nt
Its
as~.
R~cond mornHn~
3
~,
•
if, 1
)
.
)
') v:...
\5
'
,
•
w_
( u·
~
v'/
--'
'!--: r..
•
.....,;.-
~i
w '"
But when the
t"IO
AR.mpleR do come fro<n th,,,
nonnRl
A'ome
pOPlAlation the rUstrlb ;tion of
c
ill eX'lctly iL
2
B(?i.
'~(m ..
r. -
the URe of "hiR approXimAte
e1ulvRlent
':.0
2»
rH8tribution. 'fhqr'')fore
dlstr'h~tion
the us>') of tn') t tellt; but we
mak1.nc; no aSRumr>tionA ?.bout norm" Ii ty.
of u an,l v 'LA '~_PPl'ox'lm?t"ly a
•
tion •
of
B(t.
',e
f,(m ... n
~
R.!'fl
qr8
~xActly
1R
harp.
mp.rely
- 2 »
diatrlbu-
•
3upnose thet
·.IB€
W) (
cr -2")
::=
~ ('Z - if)
=
,'Yl
VI ( ~ • ;;)
)
~ t-:~
a ./4 -z.
-._----
(lOY) , . " )
( 1)
(,.-.: .. .., -.) >"I
•
p
i \u -
r.
vI ". k J
(V'n ;- Vl ) <1 .--to L
---
-
(""'i-I'1-I)""',,!,"
to o. • L
flo
·'Ii.ll cRrtp.lnly
- - - - - -__.
•
,.,
---
'7' r '!.-.
it
(r... .;: + Yl
(v-..,. ,,) l.
47
•
If
n
!11
1n
- r "'"
c nnA t:<.nt
I"l. t i. 0
tend in probqoilit.y to
-
q (
CIt\)
a
Q
~ (h, .
(
C.h l-b'.. ,.
I
'.'
,.-
)t.
G+b
l-.,1'
(Cirb)'-
8.9
m,n ---;
GO
"Y!
\ it lJencl') 11
----'?
•
1
I
h
,7;.
--'f
~
I. Z.
I
I
"
~'hl')
'
as m, n -? """
'rObil:;;\ l i t~r of
il
:0 ~p.:n1fic:'lnt
reflul t
0
"th~ distribution of X if! ~,hl') Sp.llie il.Fl th':! nifltribution of
Y" is conp-tilt3nt ''''ith T'<lfJrwct to 1ho F,ltern"tlve "X,V}\,9 U P.
f1.nite vA.rliln~~6 but <llffAl'p.nt mean
'\'he',
h,
=
V"
ll!es'l,
h~, the ob:;erver'l flr:reM:.
1n 1)ro1)",1)1 Ii t'r A.S m, n -;>- r=
;
but
f\
0 noeR
\ 'it - Y \
-7 0
J<:, I1n'1 thA
'ibove p.rp;ument 1s not delieRte Anough t.o nhO"1 thqt t,hG
U~t
•
us cons \der
(\A-V-)
•
,
i
•
'
,
"
"
:,.- )
:
.'
,"
1
r
,
,--.i
~
,
,
'~.
.
,.
.
..
':.
.
"
;~
-._--
,
<.
/
J.
.~
iI
- ...
...
:
"
~--~_.-
, ~ ~ :. ) ( C: "
\
/'
-
\1
"". . . . J
,-
....
•~
1
,
I l,
,;
,/
d
;1P.... .-. ,': ,'1
';
",
r \ ~.,
,",
'
.. (-;-'
:in
j)l'Oba1:: 1.1:1.ty.
.i,.'-
,
II
:.)
•
..
., , ; "
;
;~l
,
.' . .\
--,-~--
,
, 1' ,n
(
-J .\
~~U'
i
.
J:, II .=.
; '.
r)n
, ,1
-(,1 :nrl·-
n
, '_,r' '. ~-{1'.:~
J'tJ:;t::~
, I;.
I,
lJI)'r'"
~,
")
p" °1(.
,
<I'"':
-,
~-
,.
~
~;~
I
""" ~
r:
,)
....... -t
) (
':1-:
-'.
C
.. ....,
" .
-----~-_
v:. ;' c. -;.. )
~-
t.ends to
.,.
C'
,-I
(
,
". '.
,\)
,
'J
-E
,~
I
I •
•
} f
'\:
•
(1 ll': :--. .... ':1
-"'-_. --
7.1-'
--
.! I : .'
.,
t.·'
t '1~n
'",n
q--r;:
If
I
-
...
~~
~..
\
f\
~..n••."A
a••
o
.·n~.
" "'
In
~ny
CRSll. the prolx'hi11ty of ;; .':lEnific.'lnt I'f-J''lult. ';ill
by 1. ts rank
~ln t,h~
of
whol'?
we
obtalr thp. Rp.I'lil.I'H.1;lol1
.
......
',
"
y-
)
-;.
is the
!D
t·n. t.hll
bp.
'V
~
..
)
)
sO',
J
r~nli::
··1I10p.l1~.ng··
Y""......
of
d:\strH Ition '~.il1 b2 the sp.me
Ol''J::wl out.
Tlo~;ed
by :vtlc())(on (Blomf!trlcs "Mv.llHt'\.n 1945 pi) Bo-83)
•
=.c.
~,'
'.
~. ~ ~
( ',- -1;\
'-'\ J
r (V)
•
v
-:
Y>'I n,M~
--
{"(J)
_
( ...... b '
+.)( ... +" -,)
\ ?
V (V)
•.r )
1:.- ( ;;;;
~
1.
<.
)
v(~)
::
for the test.
•
• 1
nn~
.
m; n
-->c;.oo
if vnlup.A
\"
- l ' .~- IlA
~~
5i
:,o\": let us consider the lJ01JUlution distribution of the ob~.
served value of U/Ell ill the general case \"Ihen the null hypothesis
is'not necessarily true.
Let
rLX, y') = p { x' - y'
p -;
';:>
u::. Z. l
>
0
1
tJ,. ij
\"lhere U~j is a chance voriable \"lith values 0, 1 and correspondine;
probabilities l-p, po , Therefore
];; (Ii:j
)
~
G' ""
E (1):/) ,
E(U) :;1.Z.E(UiJ)=
1::
I"lh("
Fe:!,,)
:'f'
(V") ~ ;[2. F(U,J·....)+ ~2. E(U;j ~~.)T' ~2.. E((.cju.,j,)
•• ~
,*~
jet e
+~~ /?(U'jUiL)
j l'
-:. rn"f'
e
+ m('Il1-') n(t\-·)r..... -t- Yl'l(M_,)nr'-+
-7' 0 as n, n
ihus
Hen~e
17
tTl"
---?
I >"
V(f'lY} I'
"
(l?-')f'~
<>l>,
P in j)l'obubility as B, n
the one-nay test
t"Jo-'."my tes'!;
--7
V'l"\(\
;In ~ K
__
is consistent if p ,.
is consi3tent if p
/
Q,) ,
:F 1/2 ,
-:t)
and the
•
.--::1 01'
,
~,,~
,'<n - <V,., (eo)
rt", (Bo)
G- (>.... )
---7
Eenee
•
I
~
as n - 7
uhen
k ... -
\'Ihere
G-C\)
=-
00
,
0(,
'f,.,(8u-t- ~)
er.. . (8. +1!.)
If(i
1<" - ,r",(8.).
!it "P'n (80 t cit)
J
<SO ((). ~ .Ii )
'"
Thus 110'.. cr i'or
linit fort;ivon
&" 9.
T!.-
..rn
--7
G- (,,\ -
K
c)
, and this
).. k <ieT)cnds only on c.
If \'Ie havc tuo tests of
pOller \Iith respect to the
SOLle
3tl.L1G
hypothesis, and for tho some
alternative one requires a sample
of n and the other a sample of n' > n. the relative efficiency
•
of the second tost -,lith reapoct to the first is n/rf. •
If,..' (fl.)
•
---
H,("Yl) .
\f~ u?o)
Denote the corres"")oJ:'JirIG function for ",;hc second tost by
Juppooe tho second test s:.ltisfies tho
Sf:!".iC
H:>. ("11 )
conc\itions but th"t
c'
1st test
for
Baile
r),
For the· Uniting pO\7or'S to be equal
1<c
•
= ~C'.
TherE-foro
-
..rYi
W
c'
c
and
'.L'hc function
nny be caneu tilc eff'icucy of the; stctistic Try.
•
The Gsymptotic
re13tivc efficiency is the limit of the r::ltio of the e:i'ficacios.
Q
and
L(~) - L( y) ~
e ,
X-Y
has
l'1COn
e
and v'. riallce
•
Ii'
\'10
UGe the rank spread test to test the hynothesis e", 0 • it ccm
be shoun that the statistic::'"
ditions.
E
(.m~)
-
f -
sat.isfies the rel:uired con-
p(X-y>o)
\'Jhich is
\"I11en
nf.JL)-=
v \"In '1l
+:\'1( + I
I 2. 'l't111
(.):0.
-yf\
e::: 0
nhen
The first term is asymptotically standard normnl. TUlile the denominator of second tern tends in probability to
~
0
lienee tIle efficacy of' the t - test
•
and
GO
relative esyn.ntotic efficiency of' the rank 3!Jrcad test is
Let Y.
3
-::
O,<1S.
If"
be j,ndependent continuous cl18nce
Z;
frequency functions in 1C J
i
-p
yJ
ad
2
~
{}
'Ii'~':riQbles
',lith
./'-(x) , Cj(1C) ,
J=f:;.equenc y function of Y-z.
. -; J~(.) ':i ( ~ e) d.x .
,
l p j '(- z ~ eJI :: J +-r~) ~()() oi,)'
aD l
--
d
t
Y-z
~ iJ
l(
<X>
II.
v-o
-06
ouppose 'nOll that X.Y are continuous chonce variables ,nth
distributions ,nich differ only in location and that their fre-
•
quency functions aile
+-u- 8)
1l-f x)
I
B-=
rank spread test to test
0
If' ,'Te a!J"ly the
0 •
f""(f'~,J' f"= pIX~Y1-= p{ z~Y-B-~
vhere Z =x Hence
~ J.
e
e·.
-=
p{y-z!:O}
hao the some distribution as Yo
=
)-1: (~)
1-
_
J JC.
0<'
and the offioacy =
/2.
m" S
hl ... I1+·
f
~_...,
t(?\) 1. oil! }
2..
For the t teat applied to same populations of finite variance
•
ei'f'icacy
r-J
, >r;-
>'Ylll
<,'1»+<1) <r'"
1'b:us the asymptotic relative efficienoy of the rank spread test
compared to
t~e t
test is
12 {
J~;(}li''d'}t } ~
(j
..,tis indopende* of scnle.
Note that
•
For
population
Q rcctongul~r
_~)(
-J.;l-
)
-~
J1..-
)
and the o3yu.!?totic l'c1ativo efficiency io 1.
For a f(J)
f)
population
I
-p (x)""" (0)
e.-X ,X ':l
o
~.x
the oSyI11ptotic l'cla-c1 VIS cfficiGncy is
~
Vb '10
Here the re:nk
spread test is nore efficient them the sproou test ,lh1ch is
:~sycJptotico11y
ctluivo1ent to the t test.
It should be noto'! that the rank spread teBt cun be
•
applicti to populations like the Cauchy population \"Ihicn hove
infinite variance.
It rlas only uhile comparinG it to the t
teot and the spread test that ue assuoed finite variance.
19
Tllli lIrlALD
ilim iIOLFOi,'I'I'Z TEST
(,mna1s of i.Iath • .Jtotistics
1940 PI' 1'17-162)
,,'e have a pair of samples
) ( ,• • QO • • • • • • • • • • •
~J. ;,
,x....,..,.....
J
!:I""""
and rle wish to test the hypothesis that the distribution of X
is "the S:1I:1e as the distribution of Y.
ArrcImc the
r:lt" n nuobcrs
in or:lcr of limc;nitude
•
z"
Z 2...' ••••
0 •••
p
~..,...., t......"..,
then count the number of runs, a run being defined as a succession
of one or more X'll!!- or of one or more Yo
'Je assume that no t\'10 Z
ure e'lucl, \Ihieh \Iil1 be true rJith probcbi1ity 1 \:hen
O(,::,'llinf', fron continouo pOlmluti:);10.
•
only neet! to
1~0\J
0,0,
X
1.e.
In countinG the runo
17110ther any z 10 an x or a y. and
rCl1lncc it by x or y
:::s~ho
l.~
m'o
,fG
caoe :loy be.
DO \'10
\'/0
nay
For tIle pair of om:mles
1.1, 1.G
x y x x Y Y'.[Xyy
\'Ihich hall 0 runs.
If the null hypothcsio is true all
r.J
x's and n y's nrc equally probable.
mentD 1n..wl -+'''''
.~~
•
<:;,..-,
= ~!"!j t
•
distribution of U, the
hypotheDis is true.
pen~tQtions
of tho
·Tho nu..r:Jbcr of such arrange-
,,'e can c1etCr.i·line the exact
nm~bur
of runs,
\nlCn
the null
If the diotrlbutions of A and Yore different
l>o
ne should ex')cct bunching of .( valuGs Gnu Y valueD. and fe':1Cr runs.
"
'i'll,. lJlu'l'h1uiJi'lOIi 01' U ,]iJ.,n T1D, l;T;LL lIYl'OTH.i:.,j'i"'; 10 'i.'HUI..
20.
Tho nUJ'ilbcr of noys in 'Jhieh
~, lil~e
objects
K different Dots, each set non-cLlpty, is ,,77-/
be s00urcted into
C::lll
C;r
-/.
,
:,:8lre Q ro17 of N
crossoe to rcprcmmt t~ ;.~~eJ).o~~~2_'__ ~h~~ the first object
in
(J(icll
nct by puttinc a
ti~ck llUst
croe:;. anu the:'cforc the plocen for the ticks
.-rI-/
C./1-1
If
rum;,
i"/C~'S
HC
Call ',CO
above the first
x chooen in
•
Cl .. ,.. Q l'\8Q
I'/C
GO
arc to ""&I....
ill
X'S cnd n y··s in a
rO\'1
:!nY l'h'st separate the x into k different
so co to ha:ll:o 2k
f;
non-co!,~y
sets and
tho y 0100 'into k different non-el.mty sote. tind then interlace the
t",:o
•
~
;::otso
Tho lntter operation IJDy be done in 2
rJO:is-'J!tjlrt:!rFr;r
oither otol"'t \lil;l1 on x or ot~rt ,.fit!~ y •
Henco
-I
~-/
•
,Ji!.lil:-:r1y
p.{v"= ~.t.-t}= /W1-/(e-").~.[':i-/
--?'YI
sinGe
;.'t:
•
/Y>1-/V-1 :-n-/ Q-:l..
1-/l"IC/YYL.
I
!!lUst h"vc either k-,l( :;cts of x and. k [K'~O of y or
k seta of x
•
T
(,Wl
1:-1
3(,tO
of' y .
•
"
.~
;.
"J
('l·.,
..... . . ·
L13. :.:
~.-
r;
or
,"' \ I. 'f
_.;.::
." .. t
(:,,-1,
r~.'
~
'."-:.".1.
.,
v·T.
If' . ,
1" C;'!
"
~..i. -=: ;',
'f1j
I··~"~
.t.:~;....~
i;
,,-
_.....
""
~-
.~
,
y or b.r ;y, };.
~.. to ~
_-~.
(I~'" ) (.,-. -til'" - i.l
r=
,
:I. I
(lA " JJ
.....
7
~
r > i
"~l .... '1'",
= I \"
,.. , f
,-.... ,,1)
:
.)e.'J'
..
-'
~ "'" '"
.'" .. Yl •
../.. I ,
I __
;;;.
i t: . ,":2 l L-t ..,//
j.)-~ 't-'~··1
t~ Lll~:
"r
i.e be
1
L-'r
,.,f
I. ;,;:
E. i '.Ai' U,_ •.J
,- .,:.1
l-
•
U J '·;·C
, -I
',- -~
~'l'j
oJ}"
.0'::".;.:l.t;
V>lk,.I!lii
......, I
'r l
)
f ,-;11 0 ~.! l. n.';~
'rt ~
r
J,i'
t!l.tR 11=
~., {',-I/WI
~~,-::j'
.::~~.(
~.. '1'71-1 1
/';{ , '"J"
"
<;
•
\)
•
M'H\'~
?:
1
fol:,.o
~
...
~f.(~,.lA;,"'I}
.
.~.,.:~:
"",-'
".
..
,
r ...··~·
. r.
(]
<J
~~
g.
r7.'
)1
;i/..
(1'
)t
'&-
J!
(5
I .
•
;'~O"
l' .
'1\
. J
(r'
c I
('?)'I'
~
.--
,·c· ~ .. !
c
'Ie
.'0·...
, . .oJ........
·-<;·U
t:t.\
.&~
J ---
1'<;"':-f
\ (U)~ E:{(U-!(~' ~.{ f(~_);'J. -:.
2.n\'<l (2~,,,,·d
.0--- _ _ ,.. ...
~
~_
(tn·{·',)(V.'l ·m-,)
"2. It>' ;,[''2. w.!o - m _. VI)
~--=:-""~-c.~_~_.::z.=:;-
( V'I\ 'i' VI
'f( b''') -l" Vi ~ ,)
•
. 1
.-:..
".
~',
~
.,
." ;.,~-
,
•
:.
I
•
Trnn
,
T;le distributior. of U .ihen the Eull ilYpotl1eois 10
21.
:.o:~
T2U8
o
n valu 38 of y, and tLn.t the number of puns 12 \).
'i;? C1~all C.SSU!f:"3 th8t t.he nu:nb'?r of' };oint.c of (~lf;!coIi::::..n~.t\ty ('1'
't, ',.:;>
' •• ""
•.•••.
L.~19 i.1"•.Hl.[}·..;r
at'
;/r.lell f(x)Ja 0 if f( x I if, conti:luous at
•
fini tf.?
1'::
and
nur::lv~r
,)'X (-J~ }
r~
r:.
~
'-f.)
'P{Y c
oU'iJ~ide .i.(l
E(I)..' f,. E (m'*,,!'J =- -pm
Thus
F (_u,,) <. E-,
L')t I r
";
in I:.
value~,
-
~
J;} L
'1"\J
~
0
()
I f Ug
(1.i3ncL'.)~:
::'\.',.~L..
Th::r,"i'ore
'ttl
L
~('m'H,L
be an;,' in t,fJrva1 of length
C'
;.)encte by Of'"
tr
the
V"low.-in,: to I
numb~r
p
the
of r'...lns :-.t:.:.rt,lnr,,:
in I r' .. an". , b";; U',("
not C0unt a run \ihien
~tarts
I
Ur'" U..
•
_ll'~~
• ...
Morn
Imx onrnplenent c:f' 10
Cbvio'lr, ly
j
of int ~rvals Guch tt18.t
are bounded
I/n1ue' ana y
':1Irj
0\"
at the first (x or ::) vali.
U; tl,
'?
in I .....
"
•
r.lm~bc'!''''
m" n' of x values and y values in Il-' 1)11 p~rn;ut:.:.tionf of'
•
t:le x ,;,.nd y v::.tlues ars equally
"
,
d..-nn
,
,
~)robable,
I
2 !J!... , !L
m
n
I
!L
rn
-m
ant. so fo!" flxed
0
m .... n
.•
( ~ + E.:.
n.&)
0..
("'1'1
T~e
chnnce
v~rlab1e
;..,1
Yn'to
•
r.n
m» n -.., "'" 1 t tends in
Y1
~;robCo.bl1ity
to
H,.. !;! hr ~ F~ ~..-
H,Ju-IjI".b
cJ t-b!j
:J JnCe its rT'ean valu<, must t.end to the name limit.
2
f
I; ... t{~)
YY1 ....
In th·3
~~ame
J1.!
'" lim
C
','lay Ne
[{(~rJ
and
•
t''1~t
tIle limit of this is
q-G(. L
J"
..,
V~ I -:.
~ ~
c
a.: . ~L
("i'"
C(t+hj
Thus
he.
f.j t r
:A ~
/
"
•
r, c
Gon"lder no':l the ['"oner8.1 case ·.-I:18re
CCJTl"tant over Ie
are not neces£.arlly
If the value of fIx) 13 increased thrcul'hout
0
Ir--w~th a coopansatlnr decrease elsewhere-··b~th ~(U.) ard ~(u:)
w111 be increased, for this Is equivalent to throwinc in
n·..lmbe:" of runso
"eimilar stat"",ont applies to /"0
of l!
"
Clore I'D})lac:ed over I.,..
interval"
and lJ r
...
bJ thqir suprsreum
valu8~
Th,H'efoI'e
I in>
•
Subdhridin'- I into intervals,
,,/8
cot
"lmilarly
:n+
J,.,.,
'::here f
I
P
E j~) ~ 2-d 2" ~. 'j. 1,.
lm+o)
o..+~
"f,+~9.
Ere tha infimum values of f(X'I,
['.' ,
1 akinr each h. -7' 0,
C
I j.,
~
':12
5 v- tJ.. l
l "" ,.."..\
h'1.ve
~
2"-
b
"'... b
f
j:(.,) 5(') -
"f(.) .. "'S(,)
T
i'J 0\1 makinG f:: -7 " ,
,vEl obtain
2."-
b
c....... b
f
o<J
_C>~
•
+(7')
~(~) J)l
o.H..) .. I,'3(~)
d,r
.
~ora
x values
In particul:lr,
in thlz
•
.
(
....
LU.+-b) J:'3 + a.b F -9) '"
" b +- .. -+ l."- ~+ (,') ~ 5 + a b 'j 2... f + 65
~f +- b~
I
ali
-bF "t"j
-- ""
~+b)"
- co
Henr~e
E[J!...
1 ~ 2".1,_._.1
rn"''' 5 ("...by ~<-
Q....
•
Ii m
<.
•
unless f(x)= (,(x) •
.Put
b"
f'Ct: -9 )"'-th
... b)'
"f+b'j
-00
2a.b
<S-+-b)~
H(x) "- ~ +.(?<) ~(?')
"-... b
,..J(~)t b 5C7.) )
then "'(x) is continuous at all pOints ':Ihare both f. care continuo'Js,.
Correspondinc to any positive
i HM d~
such that
€:
there exists a set I"
< t
I.
and H(x) is continuous and bounded outside Ie
tlilf:tlllBDl infimum.
Denote by U"
1m
the numb<lr of runs starting in 1 0
o
•
~
(Jb.. ?
~ l.~.,.n
j
__
)
(Ll(?,)
r.
I.
5ince
<.
with a positive
Let K. ~ be its supremum and inrli'um values outs ide
the variance of U" /(m..-n). then
I
of intervals
U.
:!:. I
ch
,
and by
•
Let I r
be; any lnt"rval of length h r
thG numoer' of run" 2,tart1nc in I,",
in I, t:-t8 cOI:lplement of
Di ro
G";.
>-
the mean and
U~/(m+n).
variance of
I, '"
ir. IH~>
rn_ '"
~
Clnd also
,1"
/.J.
If
.:::
I'r
1,,..
'Hhero H, 15 t:le :blxli!l infimum of H(x) irl
,I,,"
~H.1.h'"
-
r
1tt t:1e supremUir} of' :f(x) in It-~
':"here :-1 '1.
I;~
for all ,-
•
e-{(~:~Y-J
svr
"'~ ~
sOr
l"r"~
/1:-- H," -"-
continuity of H(x) in I.
lihen the h
/<. OL - H) <. t'.: ..
~1,.
the h . . . Gore su.fficiently small
'.\'~len
I, m
2.
*~
5
"p
i ~~f'
bec~l,uGG
Thus
~r ~
c=
<.
::Ire f'uffic13ntly sI:lall.
-r
!I;,)!1ce
nu t
So<J;>
m. +
anrJ so
If
cr
,1'l ...
I;", '~f'
,-
-r . ..
:>,
T Q .. T' "
1'1'),+
1"
<.. (:
rfh.+··~
the -.D. of U/(m~n),
G"
~
•
trc..,. 0, +- er,.,'.,. .. \
Therefore
Hence
and BO r,'-?
•
co
as m. n ---1''''''.
I ;""
f
-';e therefore have that
'"
.2.
1,
c..
b
.,..
tt,) ~(~"..
o.f{l)+bj(<t)
- ""
of
t~1a
uuiforr;
•
in probability as ro,
and t\1i8 is less than
n-?~,
,'1hen f(x)$ r(x).
U
If the null hypothesis is true
in T)rcbability.
eLmce if "\'/e det"T'l'line
null hyr.1othe:1is is true
f
P ",~n ~
as
n
ffi,
f(x)
=Ii
-}'otJ
j)
,.\("1,")
J'"
_,_,.,
,",) +~
O(m,
2.a.b
("'''b)··
>.(m,n) such that. 'rlhen the
- ; > Co<
).. (m, n) \'/ill ~ ;z.. ... b
Th,;rei"ore,
~lhen
("'+b)'-
["(x)
e f-.!L
~
fYli"h
" ). (,." ,,) 1:/
•
and so the t'.,~t i~' conf'istent arain"t any e,ltcn'nr~tiv0 f
•
valuei' of V/(ffi.+.n) less than
!\
i [;,
':lhen
A (m,n) are taken as 2i/Onificant,
table of' the d istri bution of U fo r
IT! ~
"\'1ed and "';isenhart. r,.tLS. (19LJ3), pp. 66-87.
n
~
2C it: f" iV" en by
,iald (lnd 'Iolfo,'litz:
shol" that ,-/hen the null hypothesis is true the dittribut ion of U is
asymptotically
•
no~al •
•
u·:,3 .
"unnose that th0 frequency functions of x,y are respectivel"
jeX/d,8,;9J.
~heree~
)
Q
Je+...)
ar8
, .•
•
'._
"'e "ish to us·:? th
":3
•
if
djrr5~en~i:;tion
.' •
:
rc::+i]3
fc><"
_dO
f
(.,
' ..
..
f""'
-00
(Xl 8)
{(x.,qe,/)~) ",J J
") j
~e e!lall
to
~ctI
u\;;.::.
...
under tile
\f'(O) ~ ~ei..
si~ce
j
unknor:n.
..k""c.~t
t.........
~nteC?':l
:·~1:">
U.:._~·
',IF-se i(x,e~
111r1)
i_~ .• ..,r t·11~~1·~
.,:;;~
a.J
e-o
_,
(rt.,oj.
;lU-I.
...
siGn
P' (r. 0) ~ ';:.
0
\i!:"1.0
0
I
F#- =. (
[:ote thGt the :Jrimes denote alr;G:Ts differentiation \;ith reD,Pcct
to
eJ
~incer(ej
h:19
.~ muxil'lUJ:l ntS", D .IJ/'W) mus~ he 0
of U is (is''!'lptotiC.'ll:v l1Om)'.,l vThen
n~<>o.
tT"uo. ":l1t-n\3* 0; but this has not b'r.n
of +hc test fo'!" vl'.luGS of
If
9/11'70
as
e
-;> stcnd:·,::-d nOl'll1al "nd
E
:'raGunc.bly th:'s is still
~rovcd.
For the illvesU(';:;t>Jn
infinit'Jsi=l':,I fron 0 tho
eN dL;tT'ibution of
H-tc:oo the
U-
•
t~iff()rin.7,
",
G
(U)
i f the
)
-
•
-7
1,
.,
10 (1')
,.J
T'
U
-
,.Ie
.,
N
J
'! 0 (tT)
lie
c:·'lcul(~tc:
tlH' efficacy of t,\: ,-tati. tIc , U,-
\1';
.iin(~
r"'-.t it
~his 1'10r.U:J
•
) {u ::. I.", Ie = 0]
~'~u~
i06'1
-?
-. -=-h5~; i:~
VNCv. .. e:)
O()
ineff'ectiv .. uG:,.;.in::t i.:1C
~;::
in cont:Oi.i.st to
~ir~itln.:-: nrob~'bilit",:r
of :..
fG_~"
f(x)::.f(x, &, Elj>&2..~ ._,),
•
-} ot
i
vinen !; is . .·r:; . :t th3 te::;t
9:: - ~
t1:?
A",19N (
:: <><I\J
-:;~l~J
s~Gnifl.cl.:nt
u.3ual
r(~\
I
.AJ
•
~~tru!1(;t:,ic t·~ .. t::,
re::l~lt
-:lie :.:.1 t ~ r:! '. t-l V6
- 1"\'-=.
."!0
)'
t~l"('.::rn.:~',:v
frQn
~;;l':ch
C:.n
70.
•
u
'f'l(O' - -
,0.
)-
.2. i? J,.
~
r
"'"
~'(
:z...
)(,0) (L ::.
(c..+-tJ"_<>o {-(X,D) 7-
since
'.'Ut
•
·ihen
eo
=:.
0 the limiting dIstribution of V is standard nOrmli].
lim]'
··.'hcn
B::..
' [1\ e- .!.."l..
... nu
f V II\1 =C\ uhere :r.;:;
~
@"I)
\1hlch
•
-7
T)
1
I
l
0<-
the lini tine; distribution of V+/A. k:;'" is stQndard
norm~:l, th8refore
lim
()t.j{.:'
v;l.Tj __
1';TId
v~).1
Sf)::
as
~
k
~
k
"J.1f
oe ..
f
" ...).1"'.!.
-00
e-lx'-Jr-
710
•
2!
22. Consistence! and .;cyrmtot1c "'om,r
·.":e have n volues of
G
Fishor' s
~
'psin.s
:I:J.n1~s.
continuous chsnce 'l:lrj&bL, x of r:ll.ich
PJ..:' :,(x<. Ol~
whers :; is e randoM ')ositiv'. value of x and.
-Y a r2.llGon
l1e~:;.tiv:
valuo.
--r :::.
SUT.'!
of
ran}~s
af nosi t i ve v' .1\I" 3
-
sum of ranks of
neCfltiv,; v,,1ue5
- 2 sun of :',:1111:s of nositive vulues 2 (~J1" ~ n
Wlv~re
•
I
(
n 11-1 ))
-
1. n(
nT- 1 )
,;,. n ( n Tl ),
U =:::. nU'7lber ')f times H neeati va t1er:foer of the
It
ob.<.'.)rve\.J
vnlucG of x is GJ1l"ller. than U --'ositive ffi.3nber (~17).
-;(n):> n, n~p. therefore
Fer fixe:l n I ' n J..'
",?(',') :: 2 -;
"':(n, n,)
_
f.
n,11... p;- i n
t
(n,'r
ll] - i
:::. .; (n I (n-n,)) .:: no.o/
- nUl
\)
•
.. n
-2. - n-'-r:l.l..
'\ :n(n-1}D
"'-J.
"S(n, (n ,Tl)) -::. np, P.l.. 't" n'"-p~ -+ np"
,\CT)
",=n"'{
E( ~~--j
n(n't-l).
P'Pl. (2p-l)+ Pi
PI P ~ (2p-l) ;- PI -
It cnn be shoun that
to P'!>l. (2p-l) -"P,
V (,':}*Jas
- 1
,,2.'
Hence
-±}tn{p,pl.
i
n~
Vi hen
Q<:l.
rence
(1-2p)-r P,
n --">
ti,
"""
-±]
<>c>.
~ in probability
',:'hen the null hypothesis is true, the
distribution of x is s;vt'.J'lletric1l.1 about 0, and PI ~p... "i:. p , and so
w
~~
•
--1
,
~
in probabi~ity.
Eence the test is consistent ::r,ainst the alternative
•
:'l1e left hrend ex"')reRsion nay be ,'.':oitten
"
';;1)
D P ...... u :J.
- I - J..- • • I J
which shaY1S th,:t i t is the Clrobubi.ity tl!Elt i" a GciJ.1)le. of tuo
-: :Ji'lS
0 lJosit.: v:: vC!lue , i.e, tLe :>robubi.:'.ity th:.:t c,
con::;ists of f'?:lther t . . .;o
0.
snallor
neCi~.t5,vo
po~.:!.tive
v:-:lue"
():2'"
0113
8<11'1p10
of
pos.tt:tvc value
~\!O
i.::...
vclue.
';uJ:>pose no,,: that t11" di"tribution of x
i~
~Jym;:etricr
1
abou~
&, so thp..t the fl'ei'jUency function of x is f (x-&j \rho:") f (x) i13
evr;n in x,
':hen
~ ~O
n _ n -!,-1)
" -
' .....- J..- .
•
=0
at
since then
p-o_·L.
, •• - ~;. -
(y-&
•
nd
d
= -3)
73 •
•
~herofo~e
v(:.') , = ~- n (n 1" 1)
bi=o
'"
(2n r- 1 )
t
f(X):"'dx! '
-r;c;
-j
"
t _ ~ "r;;:;-
For the t test,
~','1us uSY1~"'L()tic
'. ~
.0.:-..
.f J
Hence (:f'ficacy' .~~, l2 n
and efficacy
)
-"'I
--d, ',J
S/t".-I)
releti VB efficiency of Fisher's test by ranks
compared to the t tcst for te::tinS
9- ~ 0
)
dx (
•
,~
;,.
,
.,
For a normal distribution
and so in this case
t~o
Analysis of Ynri.lnce
~
/2.
asymptotic relative efficiency is -
Of
.3
.... - 'iT - 7/
Randomized Blocks.
7he Jrinciples of this test may be briefly summarized us
follows.
ta~en,
:;over~~l
batc!ws, each consistinr; of n individuals, are
pnd the individuals of
~
batch subJccted to n different
treatmentr,. the allocation of the treatncntr, to the individuals
of
Cl
bntch ':icing c1.etf:!"r.!iwod by cha.nco.
l1ea"u:'crl., =nd
',1'3 \ii
';ach indi.viclu:-.ll is then
sh to dete!'lldne ';:]letllfl:::- the
differenc:J~
in
tl"catr.nnt have nroduced un,'l 1"0')1 difforences in the charucter
•
meo.Gu~ndo
-:-'hp. batche.:> r:i;--:h::, for ::lxan~loJ be the blocks in an
at:rlcul turnl
eX~(~ritlent,
tlnd t}:c
j
ndi vil:ualc t.ht") plot.: into
~111ich
.
.
?4.
-'yield" from each ]lot.
For conven¢iencc
C:lse, '.,hich is in no UflY S!1ccial.
Ehc.ll consicL:r this
that tilCre <,ro m bloc::,';.
SUl)yOSC
X1jk ')e the yield fron the i. "',treatment in the?<."Q., plot
Let
in the.j
process
YIe
block to iihich it ,;as aSl'icnGdhy
t~le
rc.l:doriization
'.Fe assume thct
0
Xijk '" 7 1 T 13j 1" Ojk
where
T~
of block
denot():3 the "effect" of tr8atncnt i,
J, find tile third
t'lrl'1
Y
jk
B·J
the "effect"
e.rlses from v"riabili ty uJ;10nr3
nlots in .:} block, erc'orE in meusuroI1ent, ant; othec- tJccidents
flffectin,~
Darticular ylots.
r:'he null
hynoth;;~si3,
which '.';e
~jinll
to
_.~
-
l. ''', .
-I ...
n '7
x13
?lit
• Xij = yield or 1th traatmant in jth
1
block.
The nnnlysis of variancG is
S
where
.3r""3~.
I;:.
3.1""
,~
=1.1. (Xij _x)2., is
S.:J ~ n I (X~3 _x)2,
the total squariance,
,3
t11e squurinncc due to blocl{s,
ani!. is inCi.ependent of T,.l T;.)·:
ST:: 11l
L (x~:.. -xf.
thE squariol\c,e duo to treatr,;Emts,
and is independep.t of 13,."J
S,:; :: ~
l
(:Y1j
-Xl. -x.j
T
n~,.
.....
'"
x):2, the ~'eddual squariance,
and is independent of both 1'3, :> 13-.,~) .. , and ':.'.I ) '1'."
J.~
Differences i.n the values of the T tend to increase the value
of ST wi thout affect;lng the value of
It can be shown that if the
variables, each
~ith
mean 0
y Jk
~nd SoDo
",E'
~<re
\~~
indcncndent chanci'
•
?3 •
m
1. (Ti.. -'~).:, t-
"
(n-I) ;r~
'" • n-l L.Lt..
~ ..,
where
.L
-
••
::: (l'l-l)( n-1) o-.;~
E (,;~) - r:r.1. T
r.
T)~
"t'l
(T:·
-.:....;;...;..;.;-.-
"""'-1
- ( v,,,=0'(,..,-0 •)
(-
t
'i'herefore to tent the null hynothesis we consider
F = 5:r/V'h-1)
I
.
-5~ ('Y>1-1)(1YI-1)
In.1'r.;G vn1uQr; of :lhi ch ere si('nific"nt.
•
If, furt 1:01', \;;\8 y c.ro
nomCll Vl:!ri~b1C:3, than ',1hen the nu),l ily.nothcsis is true, .:'rlo-;)..
and :'f£ 1t;rJ- 8.rEl in:1e,-)enctent cha:lce v,;.rli:. ble1i Iii. tributed likc
\lith
n-l
:);1(1
?ron "thic the
C(\.l! b~
(m-l) (n-l)
e:'-:~lct
Jt.;l.
decrees of fr38'lom re:,\)cctiv"ly.
d.i stribution of F
un~le~'.i~le
null h:,nJothcsis
obtained.
~~i th:Jut
U!ly 0 £:·Uf.'tpt 10:1
of normality
01' (', ven
ind.ependcncc of t.he :,..
If the !lu11 hY1")r>tJ1.esis is t!"J.e. t!ie observed vIl1ue of F is the
result of t:le chance allocation of the :liffGrent treatm.ents to the
different plots in the blocks.
\7e.yS,
but these occur in sets of nl w11ich differ only by inter-
chanGe of treatment nmnes.
•
'fllis (illoeation muy bc done 1.!'. ~Il!~ .'".
jTencc the~a are in g.::nerul ("1!)tm.-1
values of F, some of ,:!r.ie h moy hUPTlen to coincide <lith one nnother,
A.
~
tte allocation of
t~entments
to rylots is deternined by chance,
all such ullocot ions are equally ryTobnble, nnd so, the refore, ure
•
t:le corr8sry:.,.'ndinr:: v:..:lue:: of Fa
t~e
:lc can obtain u test by
cdlllparin~;
obc:erved vulur. of ? ,';ith the other ::JoGsiblc v;·lue".
--;de~l e.~
Yates (J•. \.6ric. :3cL 1:133 VoL 23 99' 6-16), lJy u
smrplinc :Jroces::
a~)l)li()c1
to data from n unifrr;-;it;{ trial consistine;
of 8 blocks each of 5 nlote,
sho~!e<1
that in that cc:.::e tl1ere was
r:;ood l1"'r8'?1'l'mt between the ratl'l=izo.tion 'ii:::tribution of F !lIlc1 tr,e
snmplinr; distribution of F on
t~:o
assurption of nonnallty.
3ee oleo -:elch (p,jol'lctriR:a 1937, 29 pp
~;1-52)
and "itmu!1
(Biometrika 1-:'3f.\, 2\J pn 322-:535).
:":e obtain
fl
consirlerablQ
siI:l~lifi(~ut.ion
if, instead. of FPc
1.7e US'3
•
for
1101'J
blocko
if
YJe
the denoninator 1s const',nt for "ermutationo uithin each
. .~ is e r:.:.onotonic increusinr-: funct.iozl of 1? und. thereforo
tak0. larGO values of . i
a~:
siGnifj cunt the test be.sed on .f
will be eQuivalent to the test based on F.
Hate that in the usual test assuminG nO!'llw.li ty. 1'Iflen the
[cUll hynothcsis is true J hus u
c.istr1bution and that
•
t~(n-l).
1.<3;-1"5,.:: )!~ is a
:. (m-l) (n-l))
r( """'(;~:JI)
vtlrlablc.
in probability as m .-,,;;.::;.
Dmce
~herefore
B
the
tends to the
sum~linG
'A.'A.
dictribution of
distribution ',lith
n-l
de~ri}es of f:-ecdor.;.
•
n'-1.>_
.,
9..
,'fa
"J
... <.
_..
"
find It enn be
n
_
J~
J
.~
,1.
thut fo-:- t le rene.o:"::"z. !tion dist.;,."ibutioD.
sh0~1n
., ('" )
y
••
'T")
v . ..
11'('{'~'
~.... ........
._.2. (m:.U..
•
m---(:I-l:,
',:1:e B·j)·
:n,-11,
• rI··
for all j,
,
distributiun has the
It:·,:, yuriancc is
2,'[.1-1)
---~-='--
FJ~il1!l-l'l 78 )
011 E;,hout t.he- S.R.l1C
of
",r
vH.lue~
l1en~e
HE
r:o.~,
(:'~:lJ0Ct
this dist:r:1Jut.io:1.
n
I /1 J "
.· .. ,i-J.J
... ~)~
to be c p.')'C'ox2.TI3"tcly a
\.
.
under c8rte.in ''lEmk conditicns ,m t,l'c OJ t'le 15.J!liting Cli<;tributl')n
•
or
m::,n-l) 1:.
is u
;(;<'
dj,str:i.h··ltior. \71 th
x.,
the yinld of treatr:.ent i
t..) .1
Ct·.),.1.
i
1
!t.:"" • . ,. '. 0- ~~T"
'1lr.'
•
....
bleck j
\1·-1
t~l:ms
decr96s of :reed.)il1 •
tho v01111es
•
78 •
-::(x .. x· '
"'1 J
1
'. 1...
-
1'1'\_/
-,J.'
_
/"'" I:. ('.It ~. - Il.) J..
-
'> '):" 1)(, ..
< " ' . _ \ ,...j
.. -r. \ ):. .
v "Lv
Let
uh(::
. . .....
- p.- I:~
1~
- Jj..1
~
. \
..,
).... ) 1 - -
cov
•
;~er:.cc
l/no
each '.:i th probubi2.i ty
X .. -
L.'
·.ili
-x:.)';....
../
(i:l,"',n)
VJ
is an ortho',·.'onal ru:trix :.iith
\. •. v ,
f\
\
\
::\
='''~f\~."I~
1',...,
f\""..;..
_.1.
=I\n~ .~
'f'
so that
-,
..,
,(x ..
':'hcn
\J
) ;; ($ A~ ..) t:. j
~
V(x . ; )
0
if i -:..1,2,
.. ,n-l
=(x~l..:..;<.,)
::.
-,
-;
. =(~ A~v)E(~;;) T";lt("\i.v,A~vJE(Xvl):VJJ
~ 4; ,... (0 -I) (;:~)
€ov (x·t..,j. x L.~ ). )
A
•
rut
-
x.,. _
v
-I '\
111.
LX
J
•
...
••
~
= t:; 1'"\:', .
E [C L~,
At'" X",-j)(i. Ai. II A~.)l
~
v.l.
i=.l,'l.,n.
J
•
':'he
rcll3t~o:l
of tho
..
is
t~~e
the reIn ti:m
SaYle
.. =- JU'
n.
..
"" .e.
V(x. I::.
of the x .. w~ .. an::\
t~
l.J
X
J:
Jr\~
J
<;
,...",J:
l*
"\7here
'"
1..
Fh'lt(X(.-X)
-
..
~~ (o.j~-~UJ.
rm
~
... .2-
I
-,:.
""'"~ IP'\
•
i. <
&-
m(n-I)} _
..
.~
-:'Ut
XJ{ ::: (x I~ )x"-"tJ •.•
•
X
'"
=, ~:'-'..2,....:~:::."'~·:·=;:::-,T'
Ix,
-vo.
.,.
r .: 1,
~
X~
I
X I l' Xo·~-oj-
="'2::":'"':.!-
~
I
,m
+ X M'\.
1'Yr\,J ~
J ::..~
x:.'1.
• ..
t
"" -I
arc i:ldeoondent
chance verinol"'::'
.
sntLfy c8rtain concitions, the di,tribution of X nil" tend to
'\1h3n
ril..,oo.
:'~f;nce
the
dis~T'ibution
~:I.
m(n-lt:
of
,/:"J..
x, • ............x
-,:
r
1\1-110
nb
n::r
w11l tond to a
•
;{~ rlistribution ?Ii. th n-1 de :::_'::.e" of fr·:,odom.
8imi)le suffi.cient condition,: are
for all J. b l and c' belnc; fixed.
•
BO.
cost
~
Indepcnderlcc.
Su,p1. 1£'37,
(pit'ilun, .Tourn. liOy. Stuti,.t. 30c.
225-23<').
1)'1
;3u9Pose that vie are 'Given n pairs of observ::tions from
a bi V(lr.i.a te distribution,
(X1.Yt ),
• (x'" ,Y", ) ,
".
and that ue ,:ish to test the hypoth8ciQ Ho that
K and Y
elre
inder.1endent c!1(lllce variables.
Consider the
n!
pos:ible ussociQtions or r~irincn
(x,. Y", ),
UI
i
07hore
•
(X'"1 Y ~,,, ),
,
isa neTI'lutation of t
qp'''.iq'''l
Under Eo all· such
•
,..
uSGo~i:::.tions
;y~
2.rc cqual.l.;r
p~~ob:l.bleo
For :iny
such es;:aci2tion we nay calcul:.L' a correlation coefficient r
defined by
r -
Z (x:..
-
X ) (Y,&7 .~)
~
-JI.(X..-{}A..I..l}1i;j/
where x is t!1e !'lean of the
va1u"ls.
'~e
A
VGlu"l'·
-
".110.
y
the mean of the Y
may then deflrle a critic.':.l reg.ioll of size
to test Ho by decle.rinG the
critical, or
siGnlficen~.
denomim,tor af the
r- larcest values of
Jinc~
exprG~'"'iO!l
for ull
II'I
r!/n~
=p
to be
~s·ociations
the
for r is C"", same, the siGnificant
associations are those \I/1lcl1 :.;iva the lalTest value,: of
I txy '"xamnle.
•
niil
';:'he followinG puir8 of values r:-f the variable s
are observed.:
X
1.1
1.2
1.3
1.fi
1.9
)'
1.7
1.6
1.9
1.3
1.0
X and Y
•
81.
Is there any evidence of dependence?
Since r is inde::>endent of
scales and ori;;ins, ':m may take the oricins r,t tIle lowest ob:::erved
valu8s of
Xanil Y• and
then drop the drscil'ial noints.
X
0
1
2
4
8
x=3
'j
7.
U
9
3
o
y :::. 5,
':'he ?~irincs 17l"dch Give the larcest v,;luos of
•
)(
0
1
y
':J
7
"7
(;
7
9
(;
6
9
7
o
3
6
7
7
U
9
3
9
7
3
6
6
7
9
o
3
7
2
4
8
3
o
o
o
o
3
32
Ilx.v- 75 I
are:-
4.
-- rz'-'
35
40
9
115
40
36
3 """
37
38
3
o
o
o
37
3G
6
9
113
38
'.L'he pairinc; determined by the
obsorvv.tions C;i ves the sixth largest value of
siGnificant, and ue conclude that
--
nxy '::. 75.
42
r,;':'5~/20=6.
If we take P::O.05,
',:e have
X and
,r"
and is therefore
yare not indopendent.
For the test distribution of r
~(rl ~
0,
;;'rom this it is easy to shon that the tnst is consistent if '}. and
~
have finite v~riances and a non-zero correlation caefficient.
For laree n we can use an approxir.£te distribution.
Let
Y
•
8 "'.
be the
'{lp
(~
. . r:10mcnt allQut tne nae..n of
be the (,.;'~ mOMent about ohe mC8n of tho y vulu88
)At3
_
Then
and
I..{~~·
q-
ere snull,
Po
(';ood an:JrOxim:ltion to the distri.bution of r is the
continuous diDtrlbution vith frequeacy function
1
)
( 1 -
~-.2
If this wore the exact distribution of r.
•
r Gl.\7oul(l j·,i3.ve
frequency function
1
B(!.,
.....(r~ 1)
and 1.;
~-,,2.
1.
x·~(l-x)-
--
O~x"l
fm-I)
liould have frequency function
1
o r..x,," 1,
--
The use of anyone of these is exactly eCluivalent to the ordinary
test for
si~njficance
of u correlation ceefficient based on the
Dssunntion of normality.
I think i t can be sl1o',m th:lt if
~.
Xand Yhave
finite variances
the test distribution of oJ n-l r tends in probG.bili ty to standard
normal when
•
n..,
=.