ANALYSIS OF RANDOMIZED BLOCKS DESIGNS
BASED ON WEIGHTED RANKINGS
by
Claudia Silva
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1137
July 1977
ANALYSIS OF RANDCiiIZED BLOCKS DESIGNS
BASED ON WEIGHTED
~XINGS
by
Claudio Silva
A Dissertation 3ubmitted to the faculty
of the University of North Garo"lina. in
partial fulfillment of the requirements
for the degree of Doctor of Philosophy
in the Department of Biostatistics.
Chcl.pel Hill
1977
Approved by :
Adviser
Reader
---------_..-
Reader
CL~·.!.DIO S~GIO Hz....~lA..'1
~esi~
a.~ ?
based on
SILVA ZAl·iO?_.\.
~eighted Ra~~ings.
Analysis of Randomize-:t 310C-G
(Under the direction of
:'~CA Q~~~
f:. :;;;:::'1 )
ABSTR\CT
The tra.!itional non-parailet:ic a.'1alysis of randomize:" ~oc:cs
(?::-i~"18.Zl,
1937) uses, in essence, on..... the intra-bloC::-c info:=:Jation.
~:O':1-:;ara.mei:.....-ic
hi;hly
tests that also use the interblock info:r-wa.t:on
~esi-~ble.
a...~
A conditional distribution-free test of this tY]e
was proposed. by Hodges and Lehmann in
1962 and further d.eveloped by
Sen, :'!eh.--a. and. 5arangi during the period 1967 - 1971; a.n asy:rgtotically
~i3t~bution-free
~.!.'o?Osed
?rocedure to recover the
interbloc~ L~o~ti~n ~~s
by Doksu::n 1...'1. 1967.
31ock-scores associated with the variability
each block can be successfully used to recover
in tr.e
ar~:ysis
1m.
Quade in
cOAbining
of
~ndomized
ObS3~~ with~
interbloc-~ ~~~o~tion
com?lete blocks(RCB) designs as
A fa:nily of distribution-free tests can be
measures of
~ifferent
v~iability,
~10wn ~J
~neTa:t.ed I:y.f
different sets of
~loc~-
scores and different sets of treatment-scores. The aSY4ptotic rnJ1, and
no~-null
distributions of these tests are derived in this
Co~parisons
~
non-parawetric
n,~ber
within such a family of tests and
co~petitors
3~~nsions
agair~t
;a--a:net=ic
are conducted for designs with
of blocks using as criterion the
Com~isor.s
~isse~~tion.
e~cted
signi:icance level
for large number of blocks are basei on the
to balanced
inco~plete
e:(?erLuents are also considered.
~1l
as~to~ic
blocks designs and g=ou?S of
TABLE OF CONTEJ.'ITS
.....
AC.~NO'iL3DGEM&.¥l'S
...
···
· · · rl
· vii
···
v
LIST OF FIGURES
LI3T OF TABLES
LI3T
I.
0:;'
..
APPEi'iD I CZS
oX
LITSRATURE RE'IIZ'II
....
·. ·
..
···
i
....
1.l.- Independent Samples
...,
l.l.0.- Introduction
1.1.1.- Two Independent Samples ,
1.1.2.- Several Independent
.
Sam~les
••
2
.,
...,.
1.2.- Dependent Samples • •
1.2.0.- Introduction
1.2.1.- Two Dependent Samples
::>
1.2.3.- Randomized Blocks. ?riedman's Test
7
Info~tion
•
1J
1.3.1.- Ranking After Alignment ••
1J
1.3.2.- Doksum's Test ••
13
1.3.- Recovering Interb10ck
1.3.3.-
II.
,
1.2.2.- Randomized Blocks. Median Test
S'J:'U")Y 0;;'
~eighted Ra~~ings
m RAN'lINGS
~BIGH'l' ..
2.1.-
15
?0R S!1ALL N
Significance Level.
::'9
2.2.- Expected Significance Level (discrete cas~)
2J
~x,ected
2.3.- Monte-Garlo Bstimation of the 3x;ected
Significance Level
• • • •
:.:.:;
2.4.- Results and Conclusions •
25
-,-
III. ASY1'1P1'OTIC DISTRIBUTION OF WEICdU'El) RA..~'fGS
3.1. - Asy:nptotic Distribution Under the
Null H~thesis
3.2.- Asymptotic Non-null Distribution
.... . . . .
EVALUATION
ot' THE ARE • • •
..
...
..
4.1.- Cox-stuart Procedure
...
4.2.- AJ.ternative Approach
4.3. - Ex:gerimental Set-up and Results
4.4.- Discussion
v.
53
............
..
EXTENSIONS OF WEIGHTED RANKINGS
5.0.-
Int~oduction
....
..
..
5.1.- Extension to Balanced Incomplete Blocks
72
~eS~6L'5
5.1.1.- Analysis of BIB Designs • • • • • • •
5.1.2.- Analysis of BIB tesigns
72
72
Usir~ Ra~<j~g
After Alignment • • • • • • • • • • •
5.1.3.- Analysis of BIB
Rankings
~esigns
Using
Weigh~
•• .• • • • • • • • • • • • •
76
5.1.4.- Evaluation of the ArlE when BIB designs
are used
. . . . . . . . . .
5.2.- sxtension to Groups of
79
~~ents
5.2.1.- Groups of ExJ)eriments • • • •
5.2.2.- Estimation of the
VI.
~~s
for
Grou~ ~=ime~~
CONCLUSION3 AND RECOHMENJATIONS 3'OR ?UTURE.: fu.-SSA.-=tCE
A?3~1)I!.
A • - A NOTE ON RANKING A£I"l'.,"':.H Al.IG:lM3riT
.. ..
86
•• • • ••
:J?
A?PE1mIX C .- TA3IES OF ESLs, THEL':l ~L'3~CES .fu~
STANDARD ERRORS • • • • • • • • • • • • • • • • •
~J3
AP?SNDIX B .- ?ROGRAi'i ESL13
A.~
R3LAT30 StJa10UTllTES
APPENDIX D •- PROGRAM ARE40 AND RELAT~'D ~OUTINES
APPENDIX E .- TABLES OF ESTIMATED Rr.i::RS A.lQ NON-cENI'R.~"',T':'Y
PA.."W1E'!'ERS FOR RCB DESIGNS • • • • • • • • • • • • 1l4'
~CES,
•••••
• •
•
•
•
•
•
•
• •
• •
• •
•
•
•
•
• •
•
151
v
I wish to express here my sincere g:ra.titude to Professors Dana.
Quade and. P. K. Sen for their invaluable advice and constant supyort
<:uring this research. I must also thank each of the other mem1:ers of
"JIy advisory committee: Drs. carol Hogue, Peter lachenbruch, ;'.a.::y Koch,
Cb.irayath Suchi.nd.ra.n and. Michael Symons. Their help was essential for
successful completion of my program.
Hy doctoral program was financially supported by a. ::ellowshi?
fro!l1 the liorld Health Organization (1973-1975) and by a resea..--ch
assistantship with the Highway Safety Research Center,
UNC~~apel
Hill
(:975-1977). War.nest thanks are due to both institutions. I want to
t~<
particularly Dr. Donald Reinfurt from HSRC for his
:lexi~le
leadership and friendly encouragement.
Above all I wish to thank my wife, whose love, :;atience a.nd.
courage have been the main sources of energy for me during this step
of
~
professional
fulfi~~ent.
vi
LIST OF FIGURSS
.. NO:i.'t:\L PROBABILITY FLITS · ·
4.1
-
IT" vs k
4.2
-
ITVSK
4.3
-
t!,vs.
"
k
.•
T
4.4
" VS K
r
.- ~
4.5
. - ti
,.
vsk
2
CHI-SQUARE PROBABILITY PlOTS
2
=- IFFERENT S.t:E;) FOR EACH K
2
2
• - 6." vs k
4.8 .-
.. 0.-
"
{).
vs k
NOR.'iAL ERRORS ( m
=4 )
61
ER..110RS ( m
=5 )
UNLti'ORl"i ERRORS ( m
=2 )
Liffi'OR."t ERRORS ( m =
4.10.- ~ vs k
2
UNIFORM ERRORS ( m =4 )
4.11.-~
vs k
2
4.12.-~
vs k
}. 13 .-6.,.
...,.,-
vs k
.- tJ.
vs k
4.1!+.-~ vs k
. 1).-/1...
vs
A - 1.-
X
t.;..
~
2
2
2
._0{ 2
2
.56
60
2
1\
l.!,
····
)
:iOR.~.AL
2
51
59
ERRORS ( m
liO?...'lAL ERRORS (
2
=2
m =3
5G
)
~m?."iAL
?
4.6 .- ~" vs k-
4.7
···.
3 )
:. lJ1ITFO&."1 ERRORS ( m
=5 )
j.AP!'...ACE ERRORS ( m
=2 )
:.APLACE ERRORS ( m
=3 )
::A?LACZ ERRORS ( m =4 )
lA.?""LACE
ER.~ORS
A?~OXI?1ATION
( m
=5
)
·• ·
····· ··
·
····· ··
····
···
62
63
C4
65
66
67
68
· ····
·
···
FOR THE JISTRIBtrrION OF Ra.
69
70
·...
q~
,,)
LIST OF TABLES
2.1
2.2
-
2.3 •• 2.4
-
ESLs when
n = 3 blocks
ESLs when
n = 4 blocks
ESLs when n
ESLs when
....····
····
····
=5 blocks
n =
....
6 blocks
en,J. [.].
e.J
1
3.1 . -
Estima.tes o'f
4.1
Power Esti:na.tes Under HI: ~ = k(-
4.2 .-
Coefficients Used in Different "Linearizations"
of Equation (4.2.2a) • • . • • • • • • . •
4.3 - Values of vn and wn for n
and
= 20,
27
29
fr
n,J•
t, - t, o. t. t:
40 and 60
4.4 .- £stmted (exact) Values of ARE for Three
Important Error Distributions • • • • • •
5.1 .-
~sti:na.te1
Powers and Non-centrality Parameters Under
Dif'ferent Alternative Hypotheses : BIB (a) • • • • •
80
.5.2 •- Estimated. Powers and Non-eentra.lity Parameters Und.er
Different Alternative Hypotheses : BI3 (b) • • • • •
81
5.3 .- Sstinat&i ?owers and Non-centrality
~rameters
nifferent A1terna.tive Hypotheses : BIB (c)
5.4 .- Estima:te1 (exact) Values of AHE for 'Ihree
Under
.••.
L'llporta..'1t.
Errors Listributions. BIB Designs • . • • • • • • • •
92
·..
5.5 . - £stb.a.:te1 ?oiofers and Non-centrality Parameters for
of 3x?8riments Under Different Alternative
rty,otheses : Normal Errors • • • • • • • • • • • •
Gro~
5.6 .- EstL"Iatai ?owers and Non-centrality ?arameters for
Grow;.e of ::Xperiments Under Different Alternative
Hypotheses : Uniform errors • • • • . • • • • • . •
87
5.7
-
~stL~ted Powers and Non-centrality Parameters :~=
Groups of :::Jq)eriments Under 'I:'ifferent Alternati"l~
HY?Otheses : Laplace Errors
• • • • • • • •
...
5.8 - ::;stiL:La.ted "falues of ARE for Groups of Experhents
with Three Important Error Distributions • • • • •
C-l • ESLs, Nor:na.l Errors, n
=) ,
TIl
=) • •
J ,
III
=4
C-2
ESLs, Normal Errors, n =-
C-)
£S18, Normal Errors, n
=)
m=5
c-4 .- ESLs, Normal Errors, n
=4
III
-5 .--...
C
~~Ls,
E:SLs, Normal Errors, n
=4
=5
c-8 .- ESLs, Normal Errors, n = 5
C-9
·.
·.
=)
III
= 5
III
=)
·.
1 -~
~,
·.
m=4o
1.:2
=J
III
C-II. - ESLs, Normal Errors, n = 6
m=4
C-12 •- ESLs, Normal Errors, n = 6
m=5
=J
C-I). - ESls, Uniform Errors, n = )
r!l
C-14o. - ESLs, Uniform Errors, n = )
m=4
C-15.- ESLs, Uniform Errors, n
=)
c-16.- ESLs, Uniform Errors, n
=4
=5
m =)
:n = 4
C-17.- .83Ls, Uniform Errors, n = 4 ,
C-18.- ESLs, Uniform Errors, n
=4
C-19.- ESLs, Uniform Errors, n = 5
~Ls,
Uniform Errors, n
III
m=5
m
=J
=5
:n=4o
5
:n = 5
C-21. - ESLs, Unifon Errors, n =
~,
1:'1
ESLs, Normal Errors, n =5
C-10.- ESLs, Normal Errors, n =6
C-20. -
1:.4
Normal Errors, n =' 4 , m = 4 •.
c-6 •- ESLs, Normal Errors, n
C-7
..
C-22.- :::SLs, Uniform Errors, n
=6
!ll ::: )
C-23.- ESLs, Unifo~ Errors, n
=6
11 :.:
4
·.
, - i,
_-r
·.
.. .
·.
·. ...
1:.6
..
..
·..
·..
·....
·......
·..
·....
·.
12.5
1:.7
1:.8
1.:9
120
121
122
:2.5
:25
::.x
c-24.- 25Ls, Uniform Errors
n =6
I!1
=5
C-25.-
n =3
m
=3
c-26.- ZSLs, Laplace Errors , n = 3 ,
m
=4
C-27.-
~SLs,
Laplace Zrrors
la:place Errors
n =3
m
=5
c-28.- ESLs, La.:place E.'rrors
n =4
m
=3
~SLs,
·······
····
n =4
m
=5
C-31. - ESLs, Laplace Errors
n =5
m
=3
m
=4
C-32.- ESLs, Laplace E.'rrors , n = 5 ,
C-3J.- ZSLs, Laplace Errors
n =5
m
=5
c-J4.- ESLs,
n =6
m
=3
m
=4
C-J5.-
La. place
Errors
ESLs, Laplace Errors
,
n
=6 ,
c-J6.- ESLs, Laplace Errors , n =6 , m = 5
..
. RCB Designs, Nor:na1 Errors,
E-l
II anI fl
r.::-2
~-J
-~
E-4
E-5
E-6
E-7
- II-,
- IT
/\
-,
and
fl
and
fl
·- ..." and "
!J.
IT
~CB
1\
IT
..
,~
·• ··
1:;7
· 1;8
·
'-:0
-~ -'
Tn.
=2
1L 5
····
F ...5
I!1
1,.;.7
····
=5
· · · · · . · lL.8
15J
RCB Designs, Uniform Errors,
m
=4
· 1.51
fl
RCB Designs, Uniform Errors,
m
=5
152
"-
RCB Dasigns, Laplace Errors,
m
=2
' .::::-'
.
fl ,
,.
• - IT and !J.
E-11.-
····
=J
r.::-9
..
..:.;-0
;n
.- IT
IT
,~,.
RCB Designs, Uniform Errors,
B-8
3-10. -
· 135
1'::"9
and
..
1 ;,...,.
-I.
=2
• - IT
ani
133
!it
and fl"
..
····
····
····
RCB Designs, Uniform Errors,
• - IT
..
-..-/-
Designs, Normal Errors, m =4
, RCB Designs, Normal Errors,
• - IT a.nd. fl
~
, '")?
nCB Designs, Normal Errors, :n = 3
1\
l?O
_._/
· · . · 130
· · . · 131
··
·
C-29.- ESLs, La:place Errors , n =4 , m =4
C-30.- ES1s, Laplace Errors
·
.
and.
fl
and.
fl
.
..
E-12.- IT and t1
RCB Designs, Laplace Errors, m = J
, RCB Designs, Laplace .t:rrors, m =4
,
~CB
Designs, Laplace Errors, m
=5
... ../)
~i/
....:;-f
'
· 135
· 1.56
x
LIsr OF AP?8NDlCE8
A•
A NOTE ON RAN:<ING A.~ ALIGN:£NT
3.
P::iOGRAl1
38113 AND RELATED SUBROUTIN'3S
TAB~ES OF ~X?ECTED SIGNIFICA~CE LEVEL,
THSD1 L'IFFZRENCES A.JIf"l 8TAW;ARD &1...QO:1S
....
..
.. ..
-.;.
...
BOGRA11 A.~O AID RELATED SUBrtOUTINES
3.
TABLES OF ESTHL\TED roWERS ANJ NON-CENTRALITY
FArlA!!ETZR3 FOR ReB DESIGNS • • • • • • • • • • • •
97
• 103
...
• •
• •
• 144
CHAPI'3R I
1.1.-
L~d9~naent Sa~p1es.-
1.1.0.- Intro:luction.- Because of their insensitivity to gross
errors and
over
t~e
extre~e
obser;ations, rank methods possess clear
classical normal theory procedures for
co~yarison
~1~~~ta5SS
0:
two or
:-10re t::-ea.tments.
1.1.1.-
Indeoendent samnlas.- For two
~iO
( X.. , i = 1, •••
J.J
let
Y •.
J. ,1
j ::: 1,2 ) we can mention the ~iilcoxon (1945' a.nd.
,0.j
J
t~oIO-sa:np1e
.3r()>ffl-':1ood (1951)
indepen~9nt sa~p~e3
tests. If lfe rank the :;:>oole-:l sa:J.p1e a.nd
be the rank corresponding to X.. then the wilcoxen test stanl
tis tic is
T
.
=~
J.,1
an~ has a symmetric . distribution
r.,
i=l
J._
with
th3 3rOim-:'iood statistic is (Hajek and Sraak version, (l':?5t, :?38~::
"'..1
\~
'l'
= _/ +
5 si.crn(r.
~ l
\'1
i=l
0
...
- +(n
w
1 + n 2 + 1))\ + 1 tf
=
numoer o:~ o:se:!:'\T3. -
-:'ions :':ro:n the fi=st sanple exceeding the media.n of the
ard in8re3.Se':l by
'jelon~
1
"2
'oC'
J.~
,3
an~
~
on 1Y'J.I
n
1
+ n
to tie first samo1e (the ?rimitive
in~r,}"'l~mt.!.
The')
::JOol·~.i
s2..'1.-;::le
,
2
.
is od'3. and this meG.J..an
fo~
does
n~~ L~~lu:le t~is
and
:foc.
+.,
n
"1
"2
~
These tests are, respectively, asy;aptotically oDti:lU..'
~~derlying
f density is logistic .and double exponential.
are asy:n:;rtotically normal for
°1 ,
n
~
2
2o~
t~~
-='
tests
Ll1 an arbitrar.r 32.:l!2e=.
00
1.1. 2 ... Several !nde pendent Sa7' ",es." For several
in::'e:;end.e~t
samples we ITlay mention the Brown-Hood (19.51) and Kruskal-Wall3.s (1152)
m-sa.aple tests, Quade's analysis of variance of scores (1S"66) ap..d. :'he
Sen - Govindarajulu weighted rank-su:n tests(1966), among otbe::s.
Let X•.•.
beme i-th observation of the 'j-th samDle
a..."1d :=:q
.• its
l.J
..
rank within the ?Doled sample (i
= l, ••• ,n j ;
j
= l, ••• ,m;
N
=E n~
).
Then the Bro-..m-Mood test statistic (Hajek fon) is
T=
4[ Ln. (A. - in .)2 = 4\Ln.L A~ .. N
J
J
J
J
A. = [t{sign(r.... t(N+l»
J
i
lJ
and the
Kr~kal-~allis statistic
= N(i~l) f~.
j
J
+ l}
j
where
= 1, ... ,m
is
([ ,j)2
i
J
3(N+l) •
ri
Both statistics are, under the null hypothesis ,
x
2
The
distributed ;lith (m-l) degrees of freedom ',(hen
Bro;m~1ood
as~p+~tically
min { n . ~ ~
•
J
test statistic is asymptotically optimum
.J
:z: •
J
fO=::':l U!:·.ie::- -
lying density f of double-exponential type, and the Kruskal-'riallis tes:
statistic is
as~ptotically optimum
for the density f of
(Hajek and Sidak, (1967), pp 104-5) •
lc~stic ~7?6
0: scores
L no CY. _ y) 2
......1-..-;.:..,,
_
Quade's ANOVA
= (:r-ij1;:
"I'
L: L (y.
(.:n -1)
0
:::
Y=
where
Yo) 2
_
.
1 ~\
':='LLY·
~ i j
l.J
0
.J
~.J
as before. The "score" Yij' correspondL'1..g to -t..'-le o':::sar-
N=2:n.
J
j
is 1:ased upon the statistic
--ration X. 0' is such that y .. = fN(X .. , (N-I) x's other than :C •• )
lJ
~J
fN(~'
Nhere
~~
~J
~J
•.• '~r) is a function of N arguments which is sY1~etric
the last (N-l) of them.
This statistic
If i-re take
obtain a
y..
~J
= r.
~odified
0
is asymptotically distributed like ?(m-l,N-m' .
(r..
1
-
~J
~J
= ranI-<
in the pooled sample)
we
Kruskal-Wallis test and the corresponding F a?p=oxi-
:r:ation to the exact distribution is better than the X2 approxi.:natio:t
Qentioned above. If we take y ..
1J
M~od
= (r..
- l)(N - r .. ) we
1J
~J
obtaL~
the
(19)4) squared rank test.
In the Sen-Govindarajulu test procedure we must consider
(n+~)
non-overlapping and contiguous cells I. defined by
.J
1. = { x
J
r:~ere
II(O)
= -co,
I Xl ( J.)
<
X
XI(nl + 1)
$ Xl' +I) }
j = 0, .•• , nl
0
. lJ .
="'0
and
= l, ... ,n
Xl(j) , j
criered observations of the first sample. Let rh,j denote the
are the
n~~:e=
of observations of the h-th sample belonging to the j-th cell (~
.• .,Q; j.=O, ... ,~); r
j
= nl
l
~ =1
, .J
for
j
=O, ••• ,nl
.
Ja:ine the sequence of real
each positive integer nand
c~~aLi
nQ~cars
aSSlL":le
f
(.
)
._
(a :J, n., .; -
:·, ... ,n}
that this sequence satis:ies
given conditions. The proposed statistic is
= 1,
T =
tr; Ilb (SN,
La ( .i , n r
nl
~f =
r
c-~l
=
nh , SN h =
h
• ,
I
n
-11 i
h -
SN)
2
where
h
l
. , h = l, ..• ,!:l,
h ,.J
)
I
This statistic is distribution-free and its
2-
bution is X em-I). It can be shown that
~)
a constan± tem and. a constant factor
of the
fL~t
S'r
.
H,:l
lb
as~?toti~
iist=i -
represents (a:;;<i--t f::-O:!l
a weighted
SU2 C:
-:'he :::::-3.rj:s
sa:np:e observations with resr::-ect to the h-t:"l 5a.:'1.:?:e .
fact, i f we take
a( .J. , n ) -- n J+ 1"
l
1
.
.J
= 0, ... ,n
l
,
.:-.:."l
then
to the usual Wilcoxon statistic for the h-th sam:ale Hith res::ect to
the first
sa~ple,
h
= 2, ••• ,m.
=0
T
If we take
otherwise,
reduces to a median test previously consiiered by Sen (1952::; •
1.2.-
~e~ndent
Samples.-
1.2.0.- lntroduction.- To gain precision we can
strat~y t~e
~pulation or divide the experimental units into homogenecns (:::::-3.cic:il-
ized) blocks. With respect to this type of eX?eri:1lental d.esi5!l we ::ru.s-;
nention Wilcoxon's signed rank test (19+5), the BrOWTI-:'!OOO ::nBiia.>; t.as-;
and Frie~an's two way analysis of variance
1.2.1.for m
=2
~.-i'o
Dependent Sam'Jles.-
(1937) .
The ',dlcoxon sig:1e-l-::-ar.:k
treatme~ts is based on the test s~tistic ~ =~
L..... 3.~.
~ ,
s..>
1
=
t~st
(=n~~ber
t=ea:=en~
of pairs). Under the null hypothesis of equal
effects we have 31T]
= n(n+l)/4
and V(T]= n(n+l) (2n+l)/Zu . ~~~3
statistics is asymptotically normal and its asymptotic
efficiency
(~qz)
~.ired-t
with respect to the
rela~i7~
nev~r
test is
0.864 over the class of symmetric distributions (Conover
1:35 ~~~
(~97:).p.Z~J\.
In particular, the A.'TI( T, t ) is J/1T for an underlying no~..a.l lis-:.=-:'bution, 1.0 for the uniform distribution and 1.5 for the
eX?C-
icu~~~
nential distribution.
Wilcoxon
(1949: ,roposed to drop difference-scores
before ranking and the test to the remainder. Pratt
to rank all the differences and drop the
ra~~
(195:;::~ ~posed.
associated
zero differences. Tables for this new test have been
of :e~o
~i~ th~
provi~ed
by
?ahe
(1974) .
In
1946
one observation per cell
have
(rank-s~m
test with grouped data;. A3sume
~,e
n experil'llsntal units in the i-th block which are raclo:tly
divided into groups of sizes
s.~
and
t l... One of these
A and the other group receives
celve~ treat~ent
the observations within each block we can find a
~ilcoxon
Van Elteren (1960) improved this statistic by
weights
=L: w.
~
w.~
= 1/(
T.~
with
\
s.~ + t.l. + 1 ). It is
. _L
~
s.t
.
]. ].
s.+ t.+l
].
].
grc~~
treat~ent
(i.e.: the sum of the ranks for treatment A ) for each
T
~o~e ~~an
Wilcoxon proposed an extension of his test to
3.
T
5.].
~~r2r_~
s~t~ti~
bl)c~. ~~al~y
introc~cL~
L~teresting to co~~~
( a.~ -
re
6
a. and. b.
1
1
b~ock.
.L'
Ass~~ing
s.t.
1 1
si + t i + 1
Tr
=2
It
3,£
ARE( T,t )=
For k
1
1.---:':1
no=mality of thede resyonses and common -r.ariance 1
we have ( Hodges and Leh.:'1lann, 1962 )
If
•.
denote the average of the A and B responses in '.owns
s.1 + t.1 = k
for all i
=1, •.• ,n
= 2/ rr
we have ARE( T,t )
Sl·t l·
si t i
then 1L1E( T,t ~= ~.. _-{~
_ k
l
i.e., inferior to the
)_~
for
the signed-~~ Wilcoxon test (3/rr). The reason for this inferio=ity
ap?€ars to be essentially that the last test pays attention
toce~~in
interblock comparisons which are entirely ignored by the other
tes~.
He need to incorporate interblock Ut:o.r::nation.
1.2.2.- Randomized Blocks.
m> 2
T
= 4(~
=4
TO
- 1) T8 /~
=Lj (Li A.:l.J•
-
~) 2 =Lj
2
([i AlJ
•• )
.
- t
mn
2
=
1
if X > median (X , ••• ,X J
ij
i1
im
=
1
2"
if X••
=
0
if X. . < median (X. l' ••• , X. '; •
lJ
=
median (X. " •• ,X. )
:I. 1
.
~
1.J
1
CD
,
T
N
1.11'
i·(m~l). ;fnen Ai.j can
only the values 1 or 0, we have the test proposed by
Nood •
test for
for :n odd
Under HO' for m fixed and n-+~~e
~median
for :n even
To/n
Aij
with
Test .-
treatments (Hajek and SIdak, 1967, p. 117) is basec on the
statistic
where
~edian
3ro~
and
1.2.3.-
Blocks.
?a~io~ized
is based. on iniapen:ient rankings
~ie~~a~'s ~3St.- ~ie~~~'5 t~3~
R•.
l.J
within each block (senarate ranking
=. n."Il (m12+ 1)
,,2
(
)
~~.
t
J - 3n m + 1
Droce~~~:.
Yne
tes~ sta~ist:~
is
where
Let us assU!Ile :
(I) The random vectors ~i
={XiI ~ , •• ,Xim ]
, i
= l, •.• ,n a....-e :"n-u.tl.:a'.:.ly
independent, and
(II) p( X..
J.J
=X..
,
J.J
)
=0
r j'
for j
(no tias within blo~~
Let Fi be the joint distribution of XiI' ••• ,
~i~
j
:hen our
hypothesis of interest will be :
~o treat~ent
HO
effects, or
XU' ••• , Xim
F.
J.
are interchallg"'-a.ble f::lr each i
is symmetric in its m
argu~ents
(i
or
=1,
Under this HO' for· m fixed and n .. co , T is disi=bu-:.ec as
1
X (m_l). This approximation can be
improved by introduc~ a correc-
tion for continuity (Kendall (1970), p. 99)
T ==
12n(1ll - 1)
n (m -m) +24
(I (tR.. - ~(:n + :)y" -1)
j
i
J.J
-~, (n-1 )(:J-l-~ )
or by usL'lg the relationship
ifith
(1'
n
n - 2
+
-
::l.
1 )
-
l
N
n - 2
(Jm-l;n(n-l ')
(n - 2/~
\
ex:a.~t '.l"J:':> T.h~
(Quade (1972a), p. 32). This last approxbat:J:::l is
third moment •
Tables for the exact distribution of
are
~
ava;la~l= io~ ~
< 7
(Xichaelis (1971), Quade (1972a, 197~J ).
If we have an underlying logistic
dis~ribution
test is asympt.otically efficient, Le., its
l"ith
.!-.CS.
then
~e~~'s
res~ct
to t:le
classical ANOVA is the maximulll (Sen, 1960'0) .
Notes.- 1.- To treat tied observations we can use
ave~-e :-e-~~
and compute
T =
n:t(711°
where
R.
J
g.J.
=L R.•
J..J
=Li,i
, R••
i
= number
t. .
J.,J
= size
I".
t/ - m)
bloc~<
i, and
+ 1) - m
: 1
(L
o
.: -J
1
R•• ,
J.J
of tied groups in
of the j-th tied group
L~
within a block are counted as ties of size 1
block i,
~~tied val~~s
(~ollander
ani
io~e
(1973), p. 140) .
mij ( >- 1)
2.- If we have
....
·
0 b serva"J.ons
:t.D
(i = l, ... ,n; j = l, •.• ,m) and mI'
.J
j = 1, ... ,m; Nt
=L:
m.
.)
,2
T = ~'(N;+l)n
R. =L~"k
J
i,k
J.J
J
= mn.)
'. -
'r'
-'• .j
fo::" each
then
'\
1 (
L- m:
j
'-lhere
= n2~ = ...
...h
~, "'h ~_
... 1""
u e (.
2.,.);-".
J
(Hehraand
:::l
Rj -
.n ,
-i-(
saran.:s-i
)
2
N'+l )
(2.)67), p. 102;.
3.- The average internal ran.< correlation based. en S;ear-,."an's
9
rho is linearly relat.ed to the Fried:nan's statistic (Quade, Ij12 a)
T = (m - l){ 1 + (n - 1)
p} •
4.- The maximum value of T is n(m - 1) (Lehmann, 1975).
Friedman's test was extended to balanced incomplete blocks by
Durbin (1951) and to general blocks oJ Berna.rd. and van Elteren (1953).
Its asymptotic efficiency relative to the parametric F-test is
3m/n(m +1) as was proved by van Elteren and Noether (1959) •
Both Friedman's and Durbin's
block-effects are not additive. Let
tests are valid even when the
e*
n
be the ARE of
Frie~~an's
test with respect to, the F-test and
e.
i = O, ••• ,n
J.
(a) i f 0'1
=
= (Yn
but
F , ••• ,Fn
1
(0) if the c.d.f.'s F , ••• ,Fn
l
e
l
=... = en =eO =e~
are not all identical, we have
-< e* ~ 1. '\ e.
108 m
125(m + 1) -
n
then (Sen, 1967) :
n L..
J.
, and
differ only in scale factor, then
, i.e., Friedman's test has higher ARE for
heteroscedastic experiments. In particular, if the F's are all
CI.'s,
then by varying the
J.
pared to J:n/1\'(m + 1)
no~~l,
e* may be made greater than 1 (as com n
which is attained when all the F's are :'d.enti -
cal). These results hold even when the errors are symmetric-dependent
( Corr [X. ., X. .,] = p. , i = 1,. •• , n»
J.J
J.J
J.
(Sen, 1 971a) •
Asymptotically efficient tests (maximu:l1 ARE with respect to the
F-test) using separate rankings have been proposed by Sen (10;:68 b) and
lem.'11er (1970). Sen used the same function of ran..1{s in the different
1 -
blocks and
Le~er
used different functions in the
di::er9~t jl~c~.
An inconvenience of this approach is -':.hat (for :n ~ 4) ~he ::nci~e :::
those functions is dependent upon the
1. 3. - nanking After Alignmen-'
Durbin's) test relati~e to the
"rjerlyj~
erro=
Qi5t_:buti~~.
- The A.iE of Frie-i..- ar 's (an=.
F-test is displayed ~ t~e fJll~.~~g
table for diffe=ent block sizes
m
2
J
4
5
10
A.BE 2/rr=o.6J7 0.716 0.764 0.796
15
co
c.868 0.895 -;/1(=0.955
"The efficiency re:n.ains unpleasantly low as long as t1".e ·oloc£.s a=e
small. This is unfortunate since it is often desL.-..a,ble t::> :1.Se :::a:.hc::r
small blocks because the natural blocks are sc.lall
litters ) or because small blocks are required to
block honogeneity. L, such cases, tests DaseQ on
achiev~ ~i~~~
-
inde?Sn~ent ~~~;-~s
leave much to be c.esired " (Hodges and. Lehma..w (1962). -p. -+35:~ •
This low efficiency
can
be attributed to the
abse~ce ~f
block comparisons. We must note that the signed-rank
tes~
.
.., -0---~w
__
uses
between-blocks i.'1for""...a.tion and its ARE. is 3frr Le., large= -:han the
A..'qE for independent rankings with m
proposed a new
ap~ach
based on the
= 2.
Hodges and Lehman.n ::1;.62;
.~oL,t-~~i~
of al:
the~bse=-
vations after "aligIPent". This means 1'.0 remove the bl::>c:':::-ef:ects Cy
subtracting from each observation in a block, some ade'Iuata
:·m.cti~!l
of the observations in the block (e.g., ::lean, ....insorized 2ear.• :=.ef.ia...'1).
~is
ap:Pro&ch was extended by Hehra and 3arangi (1967~'. Se::l (::. ?S.:: a
and Sara.ngi and ~·!eh...--a (1969) •
11
--'-
(I) and (Ill
Let us accept, along with assumptions
the following
F. (X. , ••• ,X. )
1.
1.l
]Jll
= 1,
7)
ass~~ption:
, P~n (block
\
~l'
(III) There exist quantities
i
(~-e
=
effects)' such
F (X._ - ~., ••• ,X.
2.
1.
2m
tr~t
P.)
,
2
-
.•• , n •
Because of these unknown (and possibly unequal) block-e:fects,
"no worthwhile information would be contained in the ra.nks based on
.joint-ra.nk.ing before alignment. However, i f the block-effects are
removed before ranking, it is clear •.• that the ranks contain rele vant information. Also, it is desirable (although not necessarj) that,
under HO' the aligned observations in each block have a
synmet~ic
.joint distribution" (l1ehra and Sarangi, 1967). To remove these b10ckeffects we will use functions
b.
1.
such that
b.2 (x.l+a,
1.
• • •d• r.v. ' s ana.
- b (X '1' ... ,....
are 1..J.
2
2
J.:l
If
"T
).
1.S
a sy:nmetric function of these variables, then the aligned obse::.-va.tions
Y.•
J.J
= X..
J.J
J.
J.
J..!j1
also have a symmetric joint distribution.
of Y.,
].J
j = 1, ••• , :n
- b. (X. l' ••• ,X. ) ,
Let r, . denote the ran.<
J.J
in the combined ranking of all the N
= nm
observations. Then
the test statistic will be :
=
T
(m-l)( L T.
j
2
,1
- mn (:nn + 1)
/4 )
2
Lz,.
.. J.J
1,3
l'rhere
T.
.1
= Lr..
i
J..1
and
z ..
J.J
= r 1..1
..
-
l [ r ..
m j 2.1
(Quad.e, 1972'0)
?or cO:1puta:tional :?lll':"oses ·"e Hill use (if no ties)
1.5 ::lCn-li( 4 2:T; -
i:l.11
(fin + 1) )
m n(mn + 1) (2mn + 1\ - 6 [)~
·0
......
J.
-\
- L-r . .
1..J
j
"Under H the joint c.d.f. of
O
variant under the
m:
selves, for each i
(=l, ••. ,n)
ordered observations
t:'1e
~lJl
r~mains in -
permutations of the coordinates among them -
•••Thus, conditioned on the n sets of
(Yi(l)' . •• , Y.]. ( m) , i = 1, ••• , n , the con-
Qitional distribution of (Y
(m!)n
(Y.]. l , ••• ,Y. )
il
, ... , YL~)' i
= 1,
.•. ,~ over the
intra-block permutations will be equally likely, each rA'rip-6
COffi.;':lOn
_/( Jl.,J
"n •••
..L
per:nutational (condit ional) probability
For
s3all values of n (and m) the exact permutation distri~~tion of
C3J1
be computed. ( by reference to the (mn
n
T
equally IL"-<ely intra -
block ra.n.T..cperm~ltations), a task Hhich becomes exceedingly laoorio'.1s
f:>r large values of n or mIt (Sen, 196&) •
The asymptotic distribution of T, under H,], is
~
X (n-I) .
If we assume a normal error distribution, the AR3 of the
~~<ing
after
ali~~ent
=
test relative to the separate raJL<ing test
j
~
2.
2m-J .1.2
",+1
4(m"--I) 1 - n arctan 2m-l)
!-1
~ 1
The corresponding ARE relative to the F-test is
J)-%- (1
\j1 - 1n arctan (2m2m-I)
1j
= 1.5
for
m
=2
and iecreases to 1 as
~
>
0.955 .
i.e. , the
e::ficiency advantage of the joint-r3....tidng procedure ove=: the
?r0cedure decreases nith
separate-~~
trea:'nents;
for
!J.
= 3/lf = 0.955
e2
for
= 3 treat::lents and decreases
i~~rease
in the
0=
n~:er
111
to 3/iT as
:J.
-+
It can
00 •
-re
shown (Sen, 196& ) that for any continu;ms error c.d.f. (ani even
when the errors are symmetric-dependent ) e
nv.~
O~/,
2
cannot be wo:::se tha.:!
•
~operties
All these
of ranking after alignment ho:d also
under interchangeability of errors (Sen, 1968a) .
Note.- If we have m•• (~ 1) observations in the (i,5)-th
-
~J
ce 11 and "1.j
=::l2j = .•• =mnj = mj
for each
=1, ... ,::1,
j
t:'len N"e
can use a more general form of this test (:1ehra. and. Sarano-i,
19(7)
T = N' - I \" .l:. (T. _~ (nN' + 1») 2
N'O:z~) '-rmj
~
-
where N' =
LJ :n.J
1.3.2.-
z~J.
,
J
<i
2
= LJ '[ (r. '1 - r.1
~oksum's
~J
J.
2-
and
) / 11'
as~ptoti'~lly
test.- An
procedure that uses the interblock
informati~n
i
J.J
= 1, ... ,n,
L e t ":)i
~ .. ,
0
0"
( .J<.JJ.
JJ
{ Y'~'}~-l
JJ
~-
if
where j and j' correspond to an::!
.be the rank of
and define
., i •, <:: 7.•••
i ,
.....
JJ
JJ
( =0
U
J.
o .,
JJ
=Li
Y i
jj'
(R o~,
JJ
~ir
JJ
J
~
J ,~
....
-
hr'
•
• , / .. j
J.J-
distributiJn-f=ee
pro?~sed
=I XiJ·
- I iJ·,
by
I'
of trea-t:::;ents
within the
1) rj.~,
:; \
l
has been
Doksum (1967). Let us consider the difference Y•• ,
i
_
T".
co119~ion
where
=1
, otherwise ) .
U 0'0' is equivalent to the numbar of :;airs (i, i ') su::;i tha. t
JJ
•
:. <i'
X.. - X.. , +·X.,. - X.,., < 0 •
lJ
lJ
1 J
1 J
and
I(n) ' U .. -- 0 and U .
U .. , -- U'
. . ,1.2
JJ
JJ
JJ
J.
Finally define
=1, ..• ,m.
J
=
T
2.'1l
(m-l)
V
'L (u
j
A = p( Xl <
7
G. LehiJl.ann
~.2879
and
1)
Xl' .•• ,X are a random
~here
·where
1 )
2m
j.
3mn(n -
...
'A is an estimate of
X
2
+ X - X4
3
s~~ple
=s. <
and
:(5 + X6 -
x7 )
from a populaticn Nith c.d.f.
(1964) showed that A depends onG( A= 0.2902, j.29J9 and
for normal, uniform and cauchy distributions, respectiYe~y~
~ ~
and
2
_ m-
" + 13 - 6n 1
2n-l + fCm-2) 24(n-2)).
=
A
V
The test statistic proposed by Doksum is
A
~
if;. . The
test statistic T has asymptotic d.istribution
:l.
X (m-l)
and
ARE( T,F )
=
m
2 + 6 (:n-2) (4\ -1)
Lehmarm (1964, p. 731) proposed an unbiased consisten~ 9sti ~ator
AG
for
derivable from
x.lJ_, -
of
with i, i', in distinct; j, j', j"
sextuples (i, i', i"; j, j', j")
distinct; X•• - X., • <
lJ
1 J
=n~Oer
AG m(m-l)(m-2)n(n-l)(n-2)
X.,.,
1 J
A nore convenient consistent
and
X•• - X." .
lJ
estL~ator
1 J
of
~~
< X. ' I t
lJ
can te
-
X. I t ' U
~ J
•
ieriYe~
'-l
:fro:n Puri and Sen (1967) by taking
..A
-
-?
- :1(0-1)(:71-2)
LL
. . ...
L
j
~'71"
j; j',j"
Hith
.:)
~.
•
~
=n
2:-<.:
=lf~
n __... ,
I
j:l:j'" j"
'le:::..n.Lng
L.
.
J; J', j"
1
+ 1 - -
= expected
. a...
rt •• ,. _
JJ,l JJ,l
value of the k-th oreer
statistic in a sample of size
(-1,1) and R.. , .
tion over
. ..
JJ
, xnJ. -
,~
n
dra;{TI
= rank of
fro~
the rectangular
(X 1J
.. -
~is~riju-
X.. ,) a:nong {:::., ~ lJ
~J
x... .,
~J
X .,}.
nJ
A simpler but conservative test can be obtained ta:-<il"g
~= 7/2~
= 0.2917,
a value that is not too far from those ~entioned
on page 14 .
Notes.- 1.- The procedure proposed by Puri and Sen b 1907
provides estimators for any treatment-contrast that are at least as
efficient as the least squares estimators for any underlyjng
iis~ri
-
bution •
2.- If we define
S .. , =\(R.~, - 1: sign(X .. - X.. ,)
JJ
~
JJ
.
lJ
2J
1
can give to Doksum's test a form computationally more conveni9nt
T
where
e
=
6LS~
,]
,..
e
(m-l)n(n-1)
is the nu:nerator of
V
and S.
J
=Lj' SJJ
.. ,
(Q;ua.de, 1775) .
1.3.3.- Weighted Rankings.- A different approach to
the interblock infornation is to combine the "separate
recove~
r~~~'
procedure with a "block-weighting" procedure. "Under AsstLll?t;ion (III)
(page 11), all blocks are equally variable; and if some appear more
variable than others, they are perhaps better referred to as :lore
discriminating. It seems intuitively
reasona~le
that these blocks
should receive greater weight in the analysis" (Quade, 1972 b) •
Let us associate to each block a ;neasure of "a:pparent vari ability"
v. = '¥ (X. l ,
ar~ents
and satisfies
~
~
••• ,X. )
~m
such
t~At ~
is
s~lletric
in its
:l
'fI (x l'1+
c, ... , X.lm+ c) = 'f (X.l_1
These conditions a--e :ulfilled by the
'
•••
,
:~ l::1
• )
v~riance,
for a_,_'
~
range,
difference, mean cieviation and mean diffe::::ence. E'or
~.
in~e~··a-t~=
-;.;e
conYeni~mce
will· assume
(IV)
p(
V.l = V.l ,) = 0 for
Let
%.
i
f
(no ties among the V's).
i'
the relative
be the ran.T.c of Viand
given to the i-th block (i
=1,
,n) (J < sn,l ~ .•.
Some examples of {sn,lI
.l: (a) s
.
·n,l
= a , sn,l. = 1
all i ; (c) s n, I
=i
weigh~ ~o ~
s., ~
$
--,-
for all i ; (b) s n,l..
=:.
fs::-
i = 2, .. ~,n ("drop the l.::>ast
for
variable block").
Let R•. be the within-block rank of X.• and let :'1"'. ~ -:
lJ
lJ::l
be any constants such that
{t.)
: (a)
,)
j
=j
t.
J
= l,. .. ,m
t.
; (c)
J
and
j
=
= EN3(R lJ
•. ),
j
t(m
-
Lt.J = 0
+ 1) ,
L t~J > 0
1, ... ,:1
;
= 1, .•• ,:1
•
to def:..ne
We will use the "observation-scores"
statistic
"Under
H~
be tabulated L"l
(-
=
T
'J
\-
. -::;
~e
-;
..,'._
,.
....... )
,'
this test is completely distribution-free ar:C. can
a.dvan~e.
Some block must have
"observation-sco::-es~
s n, It l , .•. , s n, It::I
in some order;
s n, 2tn in
etc.; all orders within blocks ar9
so~e o~sr;
anothe~
by the hy,othesis of interchangeability
a~d
block must
~AV9
al: orierings
s n, 2--_ ••••
equa~ ~~e_:
0: ~he ~~C!~
are equally likely l:ecause they are indepenient and the va.:d.a":Jility
measures are identically distributed ..
Actually,
tefines a family of old
T
including: signed rank test (take t
=j
all i ); Friedman's test (take t.
J
and
::. -1, t
l
1975) .
(~e,
2
new test- statisti8S
=1
and
s
:Or-.-.-n,i = i ---
t(:n+l) for all j an:: 5 n,l..=1
for all i ); BroWll-t.'1ood test (take t. = sign(j - t(m + 1») for all .~
J
and s n, ],.
=1
v
Let
and assume
for all i).
n
Lsl
([s~)~ -
::.
,
(LS~)2.
Lt: = 0
W
n
«LS~? -3Ls~Ls~
::.
«Ls~?
+
6LS~ \2.
- L S{ -;3
then the asymptotic distribution
~f
T '~der 3~
\.
oJ
is given by:
2.
T
(V
~n
N
X (m-l)
(one moment exact) , or
X(m;~)
(two moments exact) • or
T - (m - 1)
.(
)t + m w- 1
v lI'
-n
X(:11-1)
2.
N
-
(
three monents exact) .
nnw
n
We can also perform an ANOVA on the observation-scores a.ni
the F( m-l, (m-l)(n-l) )
~s~
distribution as ap?roximation to tee
corresponding variance ratio.
Different question related to this test are still
Which measure of variability shall we use?
and
'lhich block-scores (5
which treat:nent-scores (t.) are the most convenient?
J
If we
(i
~~ere~
= l, ••• ,n)
UL~e
t.J
=j
- f(m+l)
then T reduces to
(j
= l, •.• ,m)
and s n,],.
=i
:
.j
n,J.
T
72 [H.
=
2
I. -: -:;
.1
? "
\--.j • ..." .-./
~(~+1)n(n+1)(2n+1)
where H. =[~ (R i4 -!(m+1»
J
i
J=l, ... ·,m
•
oj
and its exact dist.ribution under H
o
for
m
= 3.
n
= 3(1)7
; m ~, n
'~';.s
= 3(1)4
been tabulated. (Q'1af!.:>,1972-::»
and
~
= 5,
n -- ...,..,
~
.
CHAPI'R.'t II
STUDY OF "'IEIGHTED RAl.'nCINGS FOR SM..4.LL N
2.l--Exnected Significance Leval.- When we have two or
?rocecures to solve the same hypothesis testing
to
be able to
ra~<
?roble~ ~e
these procedures using some adequate
~ore
would
~L~e
crite=i~n
For large sample sizes we can resort to the ARE (as we will in
•
f~ture
cha'?ters ) •
For small
o~
sa~ple
sizes we can compare the
po~er =~~ctio~
procecures for a given significance level. The
0:
of this
draw~c~
a?proach is that we would end up with a very large two-•.ra.y tab:"'e
because we must consider different
~
- levels •
Alternative approaches are based upon such indexes as : ESL
(ex~cted
(1?65)~, ACt
significance level, Dempster and Schatzoff
(average critical '1alue, Geary
(1966», MeV
(median critical
v~lue
,
Gaary(1966», MSL (median significance level, Joiner (:969)) ani ~~;
(significance level of the average, Joiner
SU7pose we are testing
u-;>~r-tailed
d..
"J
8
=e
0
vs H :
l
e
= G~
test based on a test statistic T. Assu,ne
T has a continuous p.l.f.
0: size
H~:
(1969») .
is a value
t~at,
~~lue
an
gi'ren .:. ,
:or a
tes~
tc( such that
?( 'f";> t
\-
feet) ; then the critical
usj~g
d.
I
60
T:-:e l.. C.V of a test for a given d-is the'vaJ:ueof e sl1c~,tha.:t. B(:-Is':=tc£.
20
(The MCV nay be defi=ed in a
sL~ilar
way; it nay also be
as the value of 8 at which the test has ~cwe~
test and. the ;'lC'/ a.re functions of d
for hypothesis
testj~g
0.5) .
~nc~t ~_
The A~l
and "are not unique=-y
problems in which
set of
t~e
J:
~
::~=:..n&:.
alte=nat~7es
corresponds to a vector para:neter" (.T,-iner (1969). p. 973) .
If we define the observed significance level X
co_~es?Ond"~5
to the (observed.) value of our statistic T by
X = I«Jfo(t) dt
=1
- ? o(?) ,
T
a random variable with p.d.f.
g(x) , then t~e NSt for u?~r-ta.i=-~
one sided tests is defined as the median of the distributi:m
under the given
p( X ~
alte~tive,
re Ie) =-1-.
i.e., MSt
=f e
a: _.
such that
TJ:1.is condition is fulfilled. if So = 1 - F/~~)
=
I
p( T ~ ~& eo}. whera ~9 is the median of the test statistic u::d.er ~~e
alternative He' Anal~ously the SLA can be defined as p( ~
with
?Pe\G }.
1"'a = expectat.ion of the test statistic under the a::"terr.a.t::'ve ::.""'C' •
The ESL for upper-tailed one-sided tests is
ESL(e)
=Jf~
g~(x)dx =
o·
= p(
T,... ::> T" )
_'
':J
"The ESt i.s equivalent to the co:np2.ement of the
:;:0'",,-13:::- f-.l."1:t~O!)
averaged unifornly over all values of d" (:empster and Scha~za:: •
"For a test statistic with a monotone likelihood ::-at:), -:':,,"e
33L is strictly ::"'Sa.ter than the NSL since the distributi:m
significance leve::" is J-sha-ped. \fuen the :?O.... er function
c:
:>: -:ie
ene -::?s-:'
21
.,
.~
-",
lli~ifo~ly
better than that
c::-i":.eria alway3 '3:"'Ie the saIlle
~wer
:y
criterion.
~ore
~owever,
o~
2
C03_~ting
rank~ng
of:
test, the ;':5:'
the bro tes'ts as ::':>es t:-:e
in situations where neither test is
··~j~O::'3-
powerful, there may exist al'ternatives for which
criterion will
values of d
ra~~
the two tests in one order (at least
) .....hile
co~pute
5c~e
the HSL and/or 351 criteria ran..T.c tne:t in t::e
other ••. In general, it would seem that the SLA and MSL
easier to
:0=
analytically than the 33L while the
wO'~d
be
~SL wOl~i sce~
to be preferable for Monte Carlo computations" (Joiner,
1;6~;.
To estbate the ESL of a sta.7.istic Twith cdf Fa ane:. F-:
'"0
~nd H~
resyectively, we can consider three different
a) :4e know analytically
,?le {Tg,l' ••• ,T.::.,n} from
?(i
~
}
from F
9
3a:a-
{To , I'
.-n
,.L_
}
_~
_rc:l.
L' ,::
{J: e " ... ,
,~
, and
c) :'1e have t . . . o dependent
and {T_ l'
"',
:::':':c).::;:1
F-B'
:'." and ,fe have an independently simulated random sample
-.' ,n
si~uat~o~3:
and we have a siJmlatec.
b) INe only have a si.llulated. ra:Jdom sample
,.,...
.1."
:.m:..e~
simulated
random samples
{To " ...
,..!.
, T.... } basec. on the same collection of rar.::':r:.
':T,n
:.eviates.
The corresponding unbiased estLiators for ESL are,
(a~7t~-~
to our present ~roblem some results fram Dempster and Scb~tzo:=,:?55)
22
OJ =
2:.n )'
( 1 - Fm(T ~,l
~
1\
= 1.n [nu:nber
F. (:.)
~
w=l
n
c)
v =
Jl
Lj
L
<i.~cO
•
l' =
then:
Var!U J = n! ('0
·2 - l-)
...
, ..
here
= 1, ••. ,n] .
Z..
1,]
= 1
if TO . > .. ~
= J
if TO ,1° <: ?:.-:0,,:;.
. ,1
- ,.J
Z ..
.1J
be the joint
denote
F;;C
whe:r. .
lJ
j:#'\.
)
of TO,J. <:. t, 0.;
Z..
(1 ) [
n n-1
i
Let H(, )
f
)"i
.)
f[
P1 =
distributi~n
l-FoC
f
iF
~()
of TO and
• 1'2 =
~~ ~nQ
fF~ () iJJ'"O
It is interesting to note that, if Y is calculated f=03
sa~ple
of size n and Wis calculated fro3
tw~
indepenQent
~
~Lp:e3
,
each of size n, then for large n
7his Lllplies that usually Y would be prere:-red instead
c:~
'iT as
estUiator of ESI. (in most practical situations
From the sa=e point of view,
TW and
rfl> ,
~et
us
~onsider t~o s~:ist:cs
whose ESLs are estimated by /ll and. y(Z,) • T::e:l
:~:1
•
.,l,a:: f'r
(I)
-
C?~"1 expect a smaller value of
var[v to -
V(21
J when T
(i)
anrl
T (~,
a:::"e
si:!lulated from the sa.:ne random sample than Hhen they are si::m:.a::'ee.
f~03 in~ependent
s~~ples
•
The use of co;nmon random nurnbers as a variance
~eiucti::m
teC:"1 -
nique (1rt~) in general simulation problems is reco~~ended ~f Kleij~e~
(:'974) .
Unbiased estimators for variances and covariances
v (~
a",';
0-: V ll;
are given by
,.
- 2(S <. ''. :1) + S ll' .....
')
1
J'
and.
(the dots indicate
sQ~ation
.. ....
~'"
over the corresponding subscri?ts
1-'
Schatzoff (1966), p. 431 ).
2.2.- EXJected 3ignificance Level (d.iscrete case)
If
t~e
aSS1l.;t ?:'ion of continuity rioes not hold for F0 and Fe' i:. -:1ay :·cc:c
,..,- 0 • = "r
t '''at
--e,j
,1
"1
~i.i
s~ch
for some i and j and it
beco~es necessa-~
tc
r
-T'y'"- this situation. Let ). denote the value to be assig:::le .
~efi~e
:~)r
cases, and consider the consequences of different 8hoices of
If the significance level
co~esponding
to an
(,')·cserv~
~
. . F?=...le
t of the iiscrete statistic T is defined, as usually as
then it
follo~s t~~t ~
t
the ES1 will exceed
should equal unity. But then, as ni2l
when
e = 0, and
seen,
~
in comparing dif:e=ent statis-
tics those "ith relatively few distinct values will have
i~:ated
SSLs
and thus pay an extreme penalty for their discreteness.
S"clppose the possible values of Tare
estL~tor
l
be the number of times t.1
W, let A.1
sample of m observations from FO' and B the
i
observed in the
t
~ple
of n observations
< t
< 7.
3
< .• , ~... i:.h
is obserIe.i
n~~ber
fro~
2
of
i~
the
tL~es:'i
Fe' Then it is
is
20t
difficult to show that
w=
Since
E[A.1
1
=
and
m:>. (\O:i
"1·
1
n m
( 1:
l>.J
E [B j
.
.
A.1 B.J
J = np j (e),
+AL:
.
1
A.1 B.)
1
and the
A's
are i..'1d.e::erl:~e~t
of the B's, it follows that
E[W]
=L
.
.
l'"J
p.(O) p.(e)
1
J
and a little algebra shows that, under Ho
Thus the choice of A = t would have the desL..-able property
that
E[.oiJ
t
ass·~~
Ul'lc.er HO' as is true when T is continuous.
Considering now the estimator V, let C.1 be the nU3be~
=
t. is observed
1
0:
L~
the single sample of n observations
TZlen it nay be sho·.m that
wha~ ~.
V=i+(A
~~in
~~~li
the choice of A = "2 would be desirable
a331J_-e that V is identically equal to
1
"2
~n
under
T just as for continuous.
In view of the preceding considerations, and in vie" of the
~t~itive
7.
appeal of the rule, we shall in the sequel ce:ine "'i;
wtenever
TO ,1.
2.3.-
~10nte-Carlo
Estimation of the Expected
=
Si.:;:n~i~ce ~e'''e::'
He are interested in considering several statistics (13
i~ ~otal)
(a) I'xo fa:nilies of weighted ran..idngs
(r) 1'1 through T defined using Vi=variance, rar~e. i~te=q~r -
5
tile difference, mean deviation and difference as :::ea.3ure 0:
variability ana
s
.
n,1
=i
as block-scores ;
(II) T through T defined using the Vi'S already ~en~ic~~~ and
6
IO
s
.=0 for the least variable block, s n,J..=1
n,J.
(b) ?rie~an's statistic (T
ll
(c) ~an..~ing after alignment (T
(~) T~e
= Fr)
I
12
othe~~ise'
,
= Ra)
~nd
standard ANOVA test statistic (T
13
= F)
•
To be in position to use the ZSLs to rank these
need
esti~ates
cet~een
of the variance (ho?8fu1ly small) of the
pairs of ESLs. As mentioned earlier the VRT
o·~ ~oblem
s-~tistics
s~tis~i~
~:f:e~~ce
H~
for
reco~~e~d~
is the use of the same random deviates to generate
under
we
t~e
and H .
l
For designs with m
=3. 4 or 5 treatments and n = ~~,
-r
lit
_,
or
,..
::>
25
ll.:J.d.er
I
'
,and
H_);
a vector of 13 statistics (las
H~
co:npu~ed
:0:each
J
experL~ent ap~ :~ally,
ences and
standa~
0:
estimates (type V)
errors of these differences were
ob~~~i.
All this ..-as accomplished by 'sip-€; the FORTRAN
listed in
A~penCix
th~i- ~iffe~-
the ES:s,
?:::-ogr...a :3:'13
B. The three error distributions used were
Normal (J, 1), Unifcr:n( 0, 3.46) and Laplace ("J, O. TJ71); thei:=' -;CL..-a.;:'teters
were chosen to
5~tee
equal variances.
generated usbg the subprogram VARGEN
treat~ent
~ha ~ndom
Cruce,
of
~e:::'e:
0, 1 )
for m
= 4,
b =
(-1, -0.5, 0.5, 1 ) and
for
=
5,
6 =
(-1, -0.5, 0, 0.5, 1 )
r;l
i;e~ 5e~eratec
'lec~rs
1?71). The
effects (see also last part of section 2.4)
for m = 3, 6 = (-1,
deviates
.
2.4.- Results and Conclusions.- Detailed tabulations
0:
3S:s ,
differences and standard errors are reported L'1 Appendix C. '='he nost
relevant pattern in these Tables is the absence of major
~if:erences
wi thin faailies (in the sense of being ESL(r. - ESL(T.) greater than
.J
~
2 s.e.(~L(Ti) - 3SL(T j )), Schatzoff (1966) ? 422). This is combinec
with im?Ortant between-families differences.
In 7ao19s 2.1, 2.2, 2.] and 2.4 an extract of
tion is dis?layed using ESL(Fr) as reference. In
t~is ~n~o~a
genera~, fa~ily
I
(1'1, ..• ,T ) has s::Jaller ESLs , but not always significant:y sna,1=..e:::,
5
than FT. The op?Osite is in general t~~e for t~e f~ily I: (T"a
Ll1 partiC".llar, when there are n = 3
b:!..ocks, T ,
l
. .. ,
-
e
e
TABLE
2. 1 •-
E3Ls when
e
n
1:' blocks
Error Distribution
0
oM
+>
Normal
....+>
III
Uniform
~
m=;
m=4
m"'S
T1
0.2274*
0.2511
0.1602
0.1689*
12
0.2285*
0.2572*
0.1557
T)
--
0.242.5
TI.j.
a.23G8*
m=)
m=4
Laplace
m=S
m=)
m=4
0.1688** 0.2376*
0.1536
0.1582
0.1800*
0.1699·
0.172.)Jt* 0.2442*
0.l5t7
0.1591
0.1864*
0.1636
--
0.1719** 0.2674
-
0.1514
0.2021
0.2398
0.1661
0.1719*
0.1731** 0.2;48*
0.1"'9
0.1.558
0.1887
0.228.5*
0.2486
0.1638
0.1699*
0.170)** 0.2369
0.1"'7
0.1,562
0.1840*
0.2738
0.3°95*
0.2242** 0.2214
0.2041
0.2788
0.2302*
0.2317** 0.21.j.Q4*
T
0.2716
0.)183** 0.2221** 0.2210
0.2121
0.2999*
0.2288*
0.2))3**
0.260~.;·
T
8
--
0.29)4*
0.2371**
0.2012
0.)3()lf**
--
0.20.58*
0.2787....
T
9
0.2848
0.2868*
0.2477** 0.22.5.5
0.20y.;.
0.272.5
0.2339** 0.2216** 0.2.520*
T10
0.2716
0.3010*
0.23.58** 0.2210
0.2001
0.2832
0.2288** 0.2200** 0.2.536*
0.2077**
0.2237
0.1446*
0.160.5.... 0.1669** 0.2317*
0.1323** 0.1383*
0.1962
0.2022**
0.21.50*
0.1339*
0.1410** 0.1)60** 0.1999** 0.12.58** 0.1397*
0.2108
0.2462
0.2375
0.1649
0.1958
(J)
T
S
T6
7
Ra
F'
Fr
m=S
"
Ui :;: 1::5L(T1 ) - ESL(Fr) ,
*
--
0.2002
0.2.586
S.El.(d i ) < ldil~ 2s.e.(d ),
i
*.
0.16.50
0.2002
0.1.5.56
Id \ > 2s.e.(d )
i
i
.
N
""-l
TABLE
£318 when n: It blocks
2.2.. -
Error Distribution
(J
o.-i
~
II)
Normal
o.-i
~
.:J
JA=j
T1
Uniform
m=4
111=4
m"".5
0.1711
0.178)
0.1279
T2
0.1701
0.1820
0.1278
0.14)l U
T)
-
0.17G8*
C.121.J7
--
T4
0.17)7
0.17~
0.1701
Laplace
.=4
m-S
0.1429** 0.122S** 0.1619** 0.1274*
0.1071
0.1;40
0.1282 11
0.1120
0.1292*
--
0.10&i-
0.1435
0.122.5
0.1408** 0.1261** 0.1618** 0.1263** 0.1108
0.1441
0.1759
0.1291
0.14)1** 0.1246** 0.1;41** 0.1282*
0.1078
0.1302
0.2133*
0.1897
0.150.5
0.1724
0.1498
0.1748
0.1623
0.1.508** 0.lS20
7
0.2090*
0.1918*
0.161}6*
0.1707
0.1483
0.16;4*
0.1)67
0.1483** 0.15/:'
Ta
--
0.1606
0.1406
0.171.1
0.199'1-
--
0.1502** 0.1605*
T
9
0.21)4*
0.18)6
0.1442
0.1639
0.15'+8
0.1730
0.155'+
0.1482 u
T10
O.209J*
0.1820
0.16.51*
0.1707
0.1556
0.1614*
0.1567
0.1521** 0.1429
0.1432**
0.1.5;4** 0.1078** 0.1225**
0.10;4*~
0.1356° 0.1078** 0.09+5*'
0.1)e8**
O.1418 H
0.09.54** O.lOYtKlt
o.O~)JO""
O.lZ(3[lM'1l
0.1062 ll1l 0.09+0*
O.W.!.'/
o.lI.lU~
0.1)2:J
0.1 /190
Cl.1907
0.1526
(J)
T
S
T6
T
Ra
F
Fr
11
1
"l:::JLC'l.'i) - II::JL(I;'&),
m=)
--
() .1.00:1
mil)
milS
0.121~)1111
O.l~OItll
C.1298*
O.17e6*
• ".tl.Cel ) < Illil:S ;z".o.(1l1 ),
i
u
0.1611·
0.1)07
0.1410
0.1126
Id i \ > 211 ••• (ll )
1
0.1146**
•
1'\1
0::
e.
e
e
e
e
TAIlLIi:
2.
J .-
li:rror
(,)
E31,8 when
e
n
=5 blocks
Diatriuution
0.-4
+'
Normal
III
Uniform
Laplace
'M
+'
m=4
1:1=5
0.1164
0.07.51*
0.0810
0.0933** 0.0613** 0.07.56** 0.1085
0.05.56*
0.0724
T2
0.1180
0.078't*
0.0828
0.0971** 0.0587** 0.0791** 0.1068
0.0589
0.0691
T
3
-
0.053}*
0.0677
T4
0.1164
0.0797*
0.0809
0.0927** 0.0610** 0.0811** 0.1061
0.o~6*
0.0791
T
5
0.1180
0.0724** 0.0869
0.0971** 0.0614** 0.0772** 0.1068
0.0.565*
0.0668
0.1.562*
0.0841
0.1014
0.1168*
0.0835*
0.1083*
0.1206*
0.06.51
0.0984*
T
7
0.15.56*
0.0896
0.1031*
0.1211
0.0151*
0.1115
0.1109
0.0630
0.0896~
T8
--
0.0771*
0.1124*
--
0.1033
0.15~*
--
0.0670
0.0786
T
9
0.15)8*
0.08~
0.1008
0.116)*
0.0899
0.1127
0.1144
0.0634-
0.0981**
T10
0.1556*
0.0824*
0.1070*
0.1211
0.0880
0.1025*
0.1109
0.0655
0.0950*
Ra.
0.0892 4'"
0.0631il '" 0.0592** 0.0820** 0.051-0** 0.0825** 0.09-t6
0.050)*
0.0617*
0.0928«-*
0.0.516** 0.0526** 0.0776** 0.0443** 0.0663** 0.0074*
0.0484*
0.0734-
0.1219
0.10981
0.1O'-t6
0.0661
0.0714
I dil >
2a.tl.(d i )
.:1
m=)
T1
C/)
T6
~'
,
F'r
"
d
i
m=)
--
0.0724** 0.0937
0.0884
= ESL(T1.) - ESL(Fr),
*
0.1331
m=4
m::3
m=5
-
0.0656** 0.1229
0.1029
0.1246
s.e.(d i ) < Idil~ 2s.e.(d 1 ),
11=4
n
m=5
•
N
'0
TABLE 2.4. - EST,s when n. 6 blocks
,
Error Distribution
...u
or!
Normal
~
U'I
II";
Uniform
m"4
illeS
mr4
rn"')
Laplaoe
m-S
m=;
m"4
."S
Tl
0.0968*
O.OS74
0.0535*
0.0411- 0.0462** 0.0654** 0.0600
0.0595*
0.0746*·
T2
0.0922*
0.0619
0.0.537·
0.0390*· o.d()Oltll 0.0618 0
0.0628
0.0621 1t
0.0662*
T
3
--
0.05.52
0.0646
--
0.0557
0.0760**
T4
0.1015
o.oS74
0.0537*
0.0374- 0.0459** 0.0670- 0.0598
0.058.5*
0.0745-
0.0922*
0.C.9l-8
o.os34*
0.0390- 0.0460- 0.0642** 0.0628
0.0573
0.0711*
T6
0.1176
0.0.520
0.0587
0.0642
0.0722
0.09.50
0.0736
0.0537
0.0892-
T
7
0.1173
0.0.91-7
o.06In
0.0639
0.07116*
0.09j1.
0.0746
0.0557
0.08', ...
T8
--
0.05.56
0.0708
--
0.0901
0.1162-
--
0.0.582*
0.0861**
T
9
0.1221*
0.0543
0.0562*
0.0627
0.0759
0.0910
0.0771
0.0,:1t2
0.00.52--
T10
O.U7)
0.0489*
0.0.567*
0.0639
0.071)
0.08<)'+
0.0746
o.O!.M
0.0853**
0.0717**
0.o'H5** O.O.9l-J** 0.0429** 0.0373** 0.0501- 0.0642
0.0309- 0.o4J8*
0.067.5 lt*
O.0298*lt 0.cJ+J5 u
u.loU7
0.06)6
T
S
RE.
F
Fr
0.06'/0
til • Il;:.iL('r l ) - b::JL(I'r) I
•
-
0.0516- 0.0981*
0.0'10211 It
U.02UUu
0.0'165·" 0.0690
0.020Ullll 0.05.50
O.07:Jf
O.O,/lJ
n.o060
Q.dl!)l
g.". (u i )
, I uil::;. 21l.u. (u i ) I
....
0.0(,(36
Ielil
> 211.". (til)
0.0536
•
LJ
o
e
e
e
bette::' than Fr only with u..,,.liform errors and m
~hen
relation improves
4 and 5. :'lhen n
=),
!T:a.l e:::Tors with m
4 and 5. However,
~~d
n
= 4.
For n
n
=4,
=4
treatillents. ='his
5 or 6 because then it holds
:0::' A
=),
T6 , .... , TIC have larger ESLs than ?'T fe=, no=,-
= 5 treatments and. for Laplace errors with :Tl =),
when n =4, this ha~pens only with Laplace errors
= 5 and
6 mostly minor differences are
p=ese~t. ~e
must notice that with uniform errors, and beginning with n
of thesa ESLs are smaller (but not significantly smaller)
= 1..:.,
so:ne
t~an
ES1(Fr) •
A
sL~ilar
study was conducted for treatment
effec~s t~~ce
those described in Section 2.J. Within a pattern of resul:s essentially in agreement with previous expectations, some of
the~ ~e=,e
"out-of-
line". t1hen n was increased from) to 4 or 5 it beca:ne c::'ear that the
alternative hyPOthesis there involved was far enough from
H~ ~o ;n~~e
'.I
the power of most of these tests too close to 1 and the ESL teo close
to 0 , producing a great loss of sensitivity in the whole
~ocedure
•
For further illustration of this phenomenon we can refer to ?ig,33 in
~empster
and Schatzoff
(1965) .
In summary, the superiority of weighted rankings (fam.:..ly I)
-IS
n, is clear for uniform errors and n
~
4. Selection of a :::I.easure
of variability'appears' a'matter of taste or convenience.
CHAPl'E..'t III
ASTIPr':n'IC DISTRIBUTION OF
3.1.let
ASym~ot1c
RANKINGS
D1str1but10n Unier the Null
HyPO~~esis.-
be the j-th observation in the i-th block, R..
X..
1J
its ran.<
1.J
within that block and.
j
;';3IGHT~
= l, ... ,m.
F.. (x)
1.J
=
P{ X.. ~ x ) , for i
1.J
1! = IR ij ] , ~ = [F .. J ,
Let
nXt.i
:uc:.:J.
F .. E
1.J
1. J
= 1, ... ,::1
S
a::1d
= {univariate
continuous c.d.f'.' s } •
=p(
Let
for j
"I
(. .,)
P ~,J
l.,rr
= 1,
j', r ;. s
R..
1.J
(.
=r
. I)
= p(
) and 'p ~,J
1.,rs
••• ,m , i
= 1,
...
,!l •
==
R..
P
1 . J ' "'ij'
= s)
Also let
( ;)
= &j j '
II -"Sage 7- '"ri'e have no
J:'l,r (because of ASSUlil"Ption
- -
'!'\...
ties w~thin blo~<s )
Let
t=l\t.
m
TJ
'
t
r
, r = I, ••• ,m
2
A (t )
=m-l:i'
1
L. (t.J
be non-eonstant known scores
- t) 2 (::> 0 ). Then we can iefhe t.":e
average score for the .j-th treatl!lent as
-
and.
,tn,m
nE
J•
t
Under HO '
.
=l[R..
,
n i 1.J
3
[t~
n, J
] =t
-
l'
j
.
=l
.... ,!ll
a.nd
2
- !m.""""
1 I'}
[(~n-tl
)(t'
-tl')]=A
(t){I
_
....
""n
-m
which is a singula.:r covariance matrix because
t l 1 - t 111
-n -
=
a
33
u. •
Define 'n
,J
m;
Ij;
n
f
l'
~ r
J.
r'
=1 ,
n(j,j')
s - i,rs
= [0'n , j j , ]; 'ir;- ={~*:
n
mxm
Yl\
=1.\'I't t
n,jj'
t
i,l"'
" m
~
POn,:!. ).
=1.n '\"'
1)~j)
~-~ r
rank
'tn
••• ,m;);n
= (POn,l'
P"n,j ll'n,j' , j,j'
"2
••• ,
=1, •••
I} ; A;- ={.z.: a.·Ea>o}.
,... Nn-
Lemma.- (Puri and Sen (1971)).- For all { F* I e{a-*}
and
n
Nnf
.
1
{ ~} ~ {~n*} , the sequence of distributions of {$( t -}l) 'a/(a' it a) a}
.... n .... n - ... nconverges (as n + (0) to the standard normal distribution •.
Lt.
. . n,J
?roof.-
=.! I[L
=:n
n·J!.-R••
J.J
..
= -t
assume
-
\l
.)
rn,J
tV
=[
=L ~1 v.~ +). '[1. v.J.
Let z
w. = 4=J a j
~
(t-R.
n
•
J.J
= rn ,...a' ("'n
t
-
L ~~... = 0
Xt~ v.J. with
p~j)
J., r
1: t r
yo
.....n
) = 1.
n
), i
=1,
- u
1. [E: I w. 13 S {:nax
n
j
J.
i
a. } [range t
r
J
r
= [(A.
1.
~
- ). )v.~ + ).v.:J. =
I
).
L w.
1
where
~
••• ,n. Then WI' ••• ,W n a.re
independent random variables with E [~i.]=
0
J.
and
we can a.lways
N
(rv.
= 0 implies that ['x.v.
i
~
i
1. 1.
a.1l
I1U
= 0 and, when considering a' (i -P.:J
'" . . . "
;
. Then
!l\
Lj (£n,J.
t
and V [W.]
= tv
a'It
a :;>0,
J..
,.., n'"
} '} <: co because
range t
r
r
..::::
00
•
Then the Berry - Sseen theorem implies the lemma. •
Coro11arz.. - U-.:3
uuer H0
uniformly in
FiE ~
, i
S
n
= 1, ••• , n
=
•
n '" (t- . _ t-)~])
IV 2~ " ( 1)
A2.(t) j
n,J
:n-
Jt n
?roof.- "Jnder H\.in ,
and.
;n
a -11
"""*
a'Itn""
a
-
IV""
= A2 (t)
a'a > 0
...,
"'n
oJ
.
{rN!ll - 1.
:n ...lIt}
,..,
=;n
(t
t
-
..n
film
H
A (t)
{I
-
m
r..t ,..0
:hen
0
m '" '"
is idempotent
1
1)' 2
AI
a
,.J
IV
(f - t 1) -+ ~(O , A2 (t) {r _ l I l t } )
since
sn
i:roly
N
= A2(t) {I
- 1.1 I'} • Then
.. i l l . ; n - ..,
,..,
wj
of rank (:1 -1:, ar:d
'
th trace (m. - 1), we ha'Fe
1.
1 1'} In (t - t
m '" ,..,
-n
X==r
1) -+
1\ •
"',:::l- .
Theore:n,- (Quade,l972b).- Suppose Assumptions I -':.h....'""Ough .Ll
hold, and in addition the sequence of weights s
and n
=1,2,
satisfies the
L(
s
. - s
1. - n,1.
--
(v)
= ~ ~ sn,i
sn
l
n
(m-l) [.
T
.
= O(n1 -tr)
• Then, under HO' as n-+
=
J
U: sQ.~
1.
"'i
«)
.0'
,n
condition
= ~ , 4-., ...
for r
.....
)2}tr
.-s
{ L(s
1
n,1.
n
where
Wald-Wolfo~itz
= I,
. for i
n,1.
the statistic
•• } 2
1.J
"2-
is asympt.otica.lly distributed as X with ::a-I d.egrees of freedo:n.
Proof.- :Jefine treatment totals H.
J
and consider a contrast
(L >...=0
J
L
n
="T ~.H.
J J
=[. so. L
1
r,\.
=0
- J
with
,
~ ;tij
j
and [.).::
.;
d.oes not represent loss of generality because
LH .,=C
= 1, ••• ,~
> C
i:aplies
u
that any linear combi.na.tion
Then L n =
L >...J
J
H.
J
=LX.
, J
•
La.H.
JJ
[s~tR
1
•
••
1.J
equals
LCa J.-a)H.J
+
a>=;t.
~.:
= [,.
= [s~
L )...J;t
L
i
"
..
From Assumptions I-III (Cha9ter I)
l.J
j
;(9
1.
. . - '\
=.L~ ~
J--j-"
s,., '11.
"4_
1.
can conclude the "i's a--e
35
i.i.1.
rando~
variables.
~or
each i
= l ••.•• n
the~ .. 's
have zero
1J
S'Xil
and. u."l:ler HO' they are interchangeable, so corr[t R.. ' t:EL .
1J
Therefore. under
and V[ :ol i] =
L~ V[~.1J.J + L: >'j ~,
CC'
tR• .' t R. .,1 j
:- [
1J
h-.l
=p=t.,
1f L>-j _.l. L Aj A
l mil
m-l 3+t
j '}
J
LA.J = 0
bounded. E
.1
HO' E[W.]=[>.. r.:[~ ]= 1I>..(I,t.) = 0
1
J J
. ij
m J J i J
1
because
1J
implies L}.~
J
J
[t
~:l
0.1.1)
V R• .] "/ [ t R• .
1J
= {[ tjH~)j} >
m- 1
= -L
). ~., . Also,
J~I' J .J
=
1J
0
1.J
J
0.1.2)
because the W' s are
[lwI2~&I<a) for any 0>0. Then, by Hoeffdingt s theorem
(?uri and Sen,
(1971) Theorem 3.4.5), the W's satisfy the Noether
condition.
From Assumptions I-IV we have that the Qts are a random per ~utation
of the
of the integers 1, ••• ,n , and under H they are independent
O
Wts. Then
E[Ln ] = E[[So.
W.] =[E [so.1
E [U.]
=0
1~J.
~
J.
1.
=[ {v[S~]vtNiJ + v[wilE[s~]~
=
L {v [s%] V ['Hi]
=
L tv[s~]
+
V['l'1i]C;[s~t}
+ v[s%l E [w i
l
Z
}
(because of 0.1.1) )
Y1V[:-1J = LE[ s~ ] V[~f i 1
+ E [s%
= -L
(l,s
2.)
m-l
n L.. n,1.
0·1.4 )
(\).~) ('t~)
L J
L J
In summary, [Wi} satisfies the Noether consition and
i s,~.l
"l.
•
satisfies the HaI1-~0l:owitz condition (AssU3~tion V). ~an, ~y ~n=
",'iaB,-"flolfowitz-Hoether-Hoeffiing theore;n (~i and Sen, (2..97-) 2eo~n
'3.4.1)
L}..
b
=-----"'
-
H.
L
_ _ _ _......ll.--....
.J -------"T= ..1!
i
( L,>-j
[Sn~ i
L tj
for all ~ ; consequently,
/(m-l»)"2
O'n
:J rJ N( g , t
H
N(
-,~),)
1 '.
i
--r-
)
Therefore, reasorrl.ng as in our previous Corolla.ry with
A2
)
= (L S n._,'--1. (L t:"",J_
n
:n _ 1
2 .)
we can conclude that
T =.l:... H'{r - 1:. 1 I'} H -+A 2.
'"
,.,
n
m '" '"
I'J
X2.
(:a-I)
3.2 .- Asy?ptotic Non-null Distribution
~ithout. loss of generalit.y we can aSSmte LS:::l.
"'i
=L5n,1..
= n
(divide the orio~l block-scores by their mean value); then 3[Sg.1 '1.
0< V [SO-i ] <eo (in fact, V[S~] <
3eca.use of our Assu:l'ption V, the
~ition,
s~.s
1/3 ;.rhen sn,i
Let us consider the sequence of local
1.J
i
=1,
= 7.1. (
••• ,n , j
x -
e*n,J.) = F (
=1,
every
i
= I,
••• ,n
~~
rn
(l~)
?
alternath~
x - .-L - [31.' ) ,
L S~J = 0
•.• ,m • Associated to each block
vector of trea t::1ent-scores
U
72 ;.
:-"Y?Othese:s
,
~~ ~ve
t. = [ 't-;:, , . •• ,~ ] 1 such tha:~
1\il
u
---1.
.
satisfy the Tt'lali-fiolfO'rl.tz; con -
and then the Noether condition (Puri and Sen
:en: F. . (x)
=n 21+ 1 )
a
37
Let
= Et [-L
Is~, = s n,l..]
l{,.
en,J, [.l. J
denote
the e:q:ected j-th
J.J
t~sa~~ent-score within the i-th least variable block
;
[s~en,j[i]]the
expected weighted
and
en,J. =
j-th treatment-score. Even under
HI' the alternative hypothesis, all orderings of blocks are equally
li.'<ely, so
en,J,= -1n [1
In -na.rticular, under
n,
Kn :
8"n, J• = n31,.... \"
'-r s n, l..
and
s
ie"
n,J
= 0 (1)
l.
-+ 8 '['
en, J'['1
= nz
J.
8, [i J = 0 (n
J
(J.2.1)
,n
J
J., n
l' Is,J r'
m=0
ll.,
(3,2.2)
-t) , \'
lr 9n, J. = 0
(J .2. J)
Because blocks may differ in ltdiscriminability",
.en,j[i]
5
•
n,l.
=1
- e.J
-I 0
T
and consequently,
en,J. - e..1
J T
(exce'P't
if
-
0
for all i) • These ideas are illustrated in Table 3.1 •
= ).'
We must note that W.
boWlded random variables withE (w.1 J
tt
'rie know that, under 1i Kn
},, .1.1, t
(1971) 7.2.24 ) J
and
then
E[t'ii J
t. , r:~, = 0, i = 1, ••• ,n
e and V[',of.1I. =
)"
,.,
= 'A'
#10/1.""
LtlZ.
-+ m
-\
{
L >'j
1
I
101m
=L Xj OJ/Iii
0 < V[Wi]-+Ltj
B.-"""e
J
~"'J.
J.
- -m ""
1 ,.,
l'
}
It.. ~ .
,.":.,,,IW
(?uri and. Sen ,
-+ 0 ,
(3.2.4)
/(m - 1) ,
(3.2.5)
and, by the Hoeffiing theorem, the ',1'5 satisfy the Noether ccndition.
Also Cov [ S%.,W i ]
>;
=
=L:
~
=[
J
},j
tE [s~ t Rij 1- E [s%] E [tnij ]}
~tE[So.1l'j]-ej}
~
~.{E:[s<t:t
J 5
•
J.
(Erso.J
-
~
[L I s%• = sn, l..11
E
~ -E..
J.J
=~ Aj{~[S~ 6n ,j(i1]-
8j }
=l.E~. .J:ej)
1J
-
4?,J }
':ov [s CL ' 'N.
~ ~
1=\'L- A.J {9n,J. - e.}J
whiCh is bounded and non-null
( -) - ?- .
exceut for
-
- belonoaing to
~
5')
so~e (~-2)-
dimensional hn>erplane of measure zero •
TABLE 3.1 .- Estimates of en,J.[.1'
9.J and en,J.
~
(*:
"
en ,2 (i1
e" n,J(i1
A
,.
&n,4li)
';;n, 5 fil
-0.8920
-O.51JO
0.0)60
o.46Jo
=.901J
2
-1.1J!+O
-0.5140
O.OZ+o
:>.5980
1.:J7;O
J
-1.2170
-0.6290
C.OOJO
o.6C4c
1.2J;(:
4
-1.J260
-0.6.560
-0.0310
0.7050
1. JCcO
5
-1.43JO
-0.7080
-O.Q178
Q.78JO
~ .3750
-1.2c44
-0.6094-
o.oeJo
0.6316
1.2792
-1.2880
-o.64Jo
-C.OC78
0.6808
l.2.5tJ
-0.6740
-0.JJ95
-o.oc66
0.3591
c.66C9
-0.7Tl4
-0.J910
-0.01)4
0.4257
C.7:pZ
-0.0075
-0.0008
-0.OCJ5
0.0248 -: .·:'1;:)
-0.0114
-0.0037
-O.OC91
0.0329 -: .':161
"
i
9n ,l[il
1
1\
9.
(a)
(a.)
J
A
8 .
5 ,J
"
9.
...'"
(b)
J
9 2O ,j
A
9.
(0)
J
~
92O ,j
~
(*) One thousand experiments. , nomAl erro:-s and 3 different
(a)
~
(b)
§
(0)
Z
=[-1.0,
=[-0.5,
A --
[f)"
.v,
~
-0.5, 0.0, 0.5, I.e],
n
-0.25, 0.0, 0.25, 0.5],
0 .0, 0 .0, 0 .a,
r.]
r;
v.\'"
,
= 5 bloc~ ;
n = 20 ~lo~<s
1 n -- 2"v .D_OCKS
•
...,-
.
No;r consider
i
n
s
t
1'\
\ ' ~ R..
I s ~.r. = [
~. /----=:..-..=1'0;...7
n _ Q. ~
.
J 'n
=-
1
1
"].
and let?
i
_\ A
-L
•
J .J
T
\.3.2.7.'-.
(
•
n. J
G and H be the cumulative distribution functions of'
(v. ;.1). respectively. Also let
F
n
, G and H
n
n
l{,
Tr
be the corresponding
empirical distributions. Because
=1n =increment
dHn (v.w)
= 0
and
G (v)
n
at sample point (v,w)
otherwise
=
~/ n
=
0
for
v < v[ll
=
1
for
v ~ YEn]
v[~] 5 v <: v [%.+1 J
for
the bloC:<-scores can be expressed as sQ .
"'1
oS
= a nn
(G (v))
= a (u.) •
nl.
u S 1, and then
i
= ItO]CO a (G (v») w dH (v.w)
n n
n
-<I:>
0.2.3)
_CO>
The a's a-~ conditioned by the following assumptions (?u=i
and Sen (1971). section 8.4) :
lim an(t) = aCt)
a)
n~
exists for C <: t <: 1 and is not constant.
co
b) f]COran(Gn(V)) - a(Gn(v))]
_r:P
c)
W
dHn(v,w)
=op(n-!)
~;()
IaCt) I <:
K > J and 0 <
.~. <:
K[t(l - t)]
t,
-~ ,
la' (t)1
~ K[t(l - t)]-l for SOl:le
and
d) g [Wlt]<:oo.
Notes. 1.- These assumptions hold. for exam?le, when
sQ.
"].
= 2%./ (n+1). If we take
t
= ~/
(n + 1), then
and
Sel ••
~
= a (t) :: 2t. ,
n
-
:: a(G (v)
n
t. < t < t. +1
~
~
i :: 1, ... , n.. -'-
'
~
1
r'
'-'
, and so
(a)
and
are satisfied. Conditions (0) and
(b)
(d) are obviously satisfied.
2. - Using the Cauohy-Sohwartz inecl'Jali ty and assu:"U;rtion (d)
we have
(G (v; - a(Gn(v»
{lJ w ra nn
J dH n (v,w)}2<{![ann
(G (v)-aa(Gn (v))]2d.G:n (Y)}
,
and. (b) can be restated as
By
an(Gn(V)
1
n
L (a (1.).
observing that dH
nn
n
G(v)
[ .
n
.
2
2-
= 0p(n- 1 ) •
- H)
:: a(Gn(v»
+ {an(Gn(V»
:: a ( G (v»
n
+ { (Gn (v) - G(v) )}a' (G ( v) )
1? n = re:i!ainder
•
(an (~)
- a(~»)
x 0p (1)
n
n
- a(-n~))
:: dH + d (li
and
- a(Gn(v)}
+ {an(Gn(v»
with
:: ~
n
+
1t n
+
- a(Gn(v)} ,
in the Taylor series ex;ansion of a (G (v)
n
arounC.
we obtain :
in:: fJw a ( G(v»)
dB ( v, w) +
JJ
;or
a ( G(v)) d (En - H)
+ JJw[Gn(V)- G(v)]a'(G(v)dH + f/w[Gn(v)+
Jf w [an (Gn (
Y
» -
a (Gn (v) )] d.ttn
+
G(v)]a'(G(v»i(Hn~) •
(3 •2. 9)
The right hand s11e of (3.2.9) is compose·1. of th.!:'se parts
of different nature. The first integral is a non-random finite
quantity ~i~ (:~~ 1 below); the secon~ an~ third integra13 :o~ a
quantity 0 tr.at p~s a limiting normal distribution (Le~A 2 below;;
\
finally, the last integrals form a negligible quantity C
= open-i)
(I.e:nma. 3 below) • Therefore
2
0'--
.1::"
=ff w a ( G(v))
\
(::.2.1J;
)
1e!IlIlIa 1. -
~n
Proof.
using the Cauchy-Schwartz inequality we have
By
{11"'w a(G(v)) £-i(v,w)}2
~ {[w 4
eli! ( v, w)
is finite .
cU'(w)} {[a2. (G(v)) :lg(v)}
-q, - -
:0=
{I"'w~ dF(w) } {Ii a2 (u)
-~
dU}
<00
0
because of Assu:nptions (c) and (d) •
le~
2.- B has a normal limiting distribution •
?roof.
= B1
~
= ~ [w a(G(v))
i
=
~ [{Wi
1
3
2
= 1'\
nL
a(G("l
d[C(V-Vi , w-W i ) - R(v,w)],
i )) -
E[a(G(v))
flC>w
_ a.' (G(v)) [c(v-v.)
i ...
1.
w]},
c(t,s)
=! ~
~J
+ 3
or
s~O
other.nse
ani
- G(v)] dH(v,w) ,
c(t)
={~
t~0
t< J
=~[{wi at(G(vi » -E[wat(G(v» g(v)]}.
1
Then, both parts are averages of LLd. ranioJ} variable3
and the normality of B (adequately
2
stantariize~)
follows from the
42
Central LiJlit Theorem after veri..:ying
<
0
V [ B i]
<
00,
i = l, 2 •
UsL.'1.g the Holder inequality and conditions (c) a.nd. (d.) we
E[iw
have:
a(G(v»)I'"l =E[{W}1{a.(G(V))\2]
~ {E[W 4]}t{3[fa(G(v»}i)} t
t
:5 0(1)
t..~e~fore,
C
< V(31 ]<
Lem:na
3. -
00
c
=
{;(4f l;.(l
- t)],,\f}} < GO
c)
,
0
<f3<
Similarly we can verify that C
•
S
= 01' (n-
+ C
2
t ;
< v[ 32 1< 00
•
1
'2) •
Proof. From condition (b) on page 39
let us consider that , given
such that (Puri and Sen,
p
such that
I
s'
~t
I = { (v, w) :
per)
:> 1 - ~* and let
boundei, a'(G(v»)
0
< S' <
i ,
there exists c ( s;" cS' )
(1971) 10.2.12)
ntlc (v) - G(v)
n
1
{G(V)(l _ G(v»}Z-
sup
[ v
0 and
E'>
]
> c(€ ,8')
IwI <
r
:s.,
Iv I <
<
(3. 2 .11)
£
K2 } be a subset of JR2
be its complement. L'lsi'ie I ,
11'
is
is bounded and.;n {Gn(V) - G(v)} = Opel); then, by
the ReIly-Bray theorem (Tucker, (1967) p. 84)
l.;n
W
J
Inside
)' \-,,;n
I'
r'
(Gn (v) - c(v»
a'(G(v»
d(H n - H)
~
0
(3.2.12)
the same integral is less than or equal to
(C (v)-C(v»
n
a'(G(v»\clli
n
+
flw;n
I'
(Gn(v)-G(v»)
a'(G(v»)I-::~
(3.2.13)
'. -.
-'
~ater
1- ~
than
£ Iw Iii (Gn(v)-G(v::) a'(G(v»
dH
j.
n
I~
~Kflwl(G(V)(l-G(y))i-S"iY.
.....
l'
.•
c(e ,S'
By using Holder's i.."1equality ,
fI'
Sit
1
dH S ('w"dF)~
1
(G(v)(l~(v»)r-
Iwl
t
~ +
Because
l'
n
Z
~
(fI' (:;(v)(l-G(v))f~-~
n
b" < 1 and because -inside I
1:'
G(v) is close to '::
I -
.1
c:;)1;
n.
-
-"..
......
-,
"'::
the last factor can be arbitrarily small;· :::: [j,(It] < ~ i:nplies a si:nila..r
.!.
(f w~:iF ),. • Therefore, for any
It'
n
conclusion for
p(liln
W
It
(Gn (v) - G(v»
a'(G(v»ldH n <ilJ)
l']>
0
1 as n ~cc)
-+
(3.2.14'
"
.I
Suilarly
PC!.II llii W
IdH < tY! )
(G (v) - G(v») a' (G(v»
n
~ 1 as
n ~ (Q
0.2.1.5)
!
From
(3.2.13), (3.2.14), (3.2.15)
This completes the proof of the
.0
~
n
=1n \'
L- s~
ill..
'1
:En = [ Tn,j
=L'3 '\.J
T .
•
~
n,J
have
A2
~
:z-
\'
n
= ~/ sn,l.• L..\ t..)
( m - 1 ;n
Since
L. t
J
will have
3 ij
as~ptotic
C2 = ~?(!:-Z:) .
normality of
f or every 'A (~\
re z~o"""
L.~' - 0'\j • Th
• e_
.- ,
.....J
"'1
\'
"
0.2.12)
1= [ ~ [ So.~ l....J ] ~ N( ~T ' Ifr )
1.
ne
a.:rl
=0
LS
=
0
2
\'
2-
. Lt.
nil.
J
m- 1
and
0
L n,l. = n, )'- s"".
= 0(:"1)
n.l.
5
= 0(1). Let us denote Q=
.
, after defining
,..,
•
, r
=-,r
A· .... ::1
,
_.=.
ll'
:n.'" ..,
}
(3.2.16)
! = In T =[n-.'2L s",~l t.:,-'i~ 1j,
""::1
1
-.. e
~erefore
T will be asymptotically chi-square distributed with k
c.egrees of freedom and non-centrality pa.:ameter b
= limlin
= li=. fJ-., Q, lL or
Q ..... q>
t\-;"Qj)
Q. 't~
if and only if
=an,J.,
.]
n,J
{Kn })
en,J2,
An =I!!...2
J
lp",:.
is idempotent of trace k.
Because E [T
(under
IN(. IV
. A
we have
1
A
IZ'E
=
[/it
en,.),],
l' Ii- = C ana.
"."
J;.~
= 2 2 \L(\ s
n
J
) )2= 0(1)
L1 n,l..b.[,
J l,n
(3.3,18)
Along with its vector of conditional expected treatment scores
= b.J (.l.,nl/;n],
[ Sn,J• [ 1.. 1
covariance rna trix
we have Cov r~ .,
- J
cov[ej ,
and
t
the i -th
least variable block has var:a.nce -
c' ] =[0", ., ['1] • Using this notation
:aX:ill.
JJ
and
1.
e J.. '/"s ] = n! L s n,l
2..
0" ... [ '1
JJ
1.
1
~j.l = E(cov['gj''C"j,l z]]
~
•= [ ~
lI1l1·J
1
··ij-
=0(1)
+
cov[E['ejl~J,3[6,j,J~]]
Var [~ . ] = 1 \' s z • C7' .. ['1
J
n L- n,l JJ 1
(3.2.20)
C3 .2.21)
3ut, as
n
~~,
the t •
,IS
R
lJ
will "tend to
This implies identical marginal distributions (1. e.,
= o-~ = 1.mL t~J
,
V i,j
intercha~-eabi1ity"
0' "["1
JJ 1
~ C" ~' =
,iJ
) and equal correlations. Because the t ' s
a fixed sum for each i (
a
when we take t
= R.. R
. , .
lJ
t(~ +
1)
have
~, that
l.J
com;non correlation is P
0"
~
00'['1
n,J.J
1.
<r
.:' •
n,~J
=-
1
--1- •
m -
=P';0" JJ, "'
Therefore ,
0-
(Y., "
J J
2
m - 1
=
Lt7
.1
- m(:n - 1)
and.
45
-1
m:r
1
L. t~.J.
{ =:rn.Lt~
J I
- 1 .... m
n - 1
-1
;:r
......
:; ??)
(.....'" --.--
1
From (3.2.16) and. (3.2.22)
= [ sem~_- [t~
1),{ {T _.!. II'}
Zm
m
('J
n,J.
1
}
m:" '"
- 1 l'
N
.J
which is ide!l11X>tent with trace
<:......::1.
for
n 4- co
1" s:L It
--);r'T
n,i,., [il
i I - 1-.
l --TIl ::l
1
l' 'J
-
-'
.:b
~
(m-l). Then T ~ X (:n-l,~) .
CHAPI.'ER IV
EVAWATION
4.1.- Cox - stuart Procedure.- When the non-null
6(;·~e
dis~ribution
of
test statistic proves to be too difficult to deal -.dth, an a:;:prox-
i~te
procedu.-e for estimation of the
.~
becomes a necessity. Such a
proced1L"'"e was p:::o:?Osed by Cox and stua.rt (1955) and recently usee. Cy
Lee et al. (1975) •
en)
tn)
Gi ven two consistent tests T1 a.nd T2 of a
the
JL~
is the
reci~Tocal
~:.:
0,
req~i--eQ
to
hypothesis:l :
J
of the ratio of the sample sizes
attain the sa:ne power against the alternative hypothesis H , ta.ki..-,g
1
an d ~'b\J H-1- H,~ (to :.cae} 7-he
the li.:ni t as (a" t.'1e sample size
power of both tests away from one) •
'f th
!pe
.e~ 'lr'\5
\ 7 ~-)
S h owe d
ing diStrioot.i:ons under.H
O
tha,
t ~~
. ..,. T(n)
h
1 and T':l") 1........th
~u
. a~
2
and HI' the ARE of T
In;
I
no~
..
1
r"l"".n)
compared ... 0.1.
2
1·
•.
_~~~~-
is
given by
-
. -\
( "'!'·~·.l.1
([tr E(TrN ] ~.o)'
.-.-?)..
( i.;, -:
v(T~")IA=O)
~
proviied tr.a.t
li~
l'\
The quantity
increasing
the salle
n
r
'-'00
R2( T.(n») n-r
~
=
R.2 , for both i
~
= 1,2
(~. • .,..i...)"')
illeasures the rate at which the power increases ·..i-th
and "comparable tests of a given hypothesis ;iil1
r '" (~ox ~..,d stuart (1955), p. 87) .
~a.-.re
47
If Tty,\ is norm.a.lly distributed with raean E [T,nl\ >.1 and. stanC..a....""ii
deviation
(Yn
(TtJt) '~l
the null hypothesis >.
=.
a
a~
is =ejeC"'"..ad
the
(4-.1.+ )
deviate • Then
p' ()
0
=
E'[T(nll>']O'n[Tlnll>t] -
{E[t"llAl-
E[T·llo]-z~O'n[r~lcn7~::''''I~J
-----.;--------------....;~---..;-...---
:'
O'~rT(")1
E ' {T''')lo] + z 0"
" n
0'
. n
and, whenever
I
).]
~,:O
(T''''IO]
[T(") I 01
0':'
n
[TIll)' 0]
.. 0, we will have
'
Then, in a neighbomood of ).
=:
0, p(A)
=:
(using 4.1.2) ,
AR('r) -
z~ + O(:>\~)
.,.
but, :from (4.1.3) R(Ti )
~ R
i
n2
as
n -+ 00
•
Therefore, for each given n, the relation between the ?Ower
deviate
p(") and A will be essentially linear. From (4.1.4) "We
Imow that a plot of the power function TI (>.)
p(~)
vs the power d:!viata
on no:r.mal probability paper would be linear for n 1al:ge. Co~b-'...n-
ing these two facts we expect that, for each test statis'tic :?,
~
":;rrobabilistic" plot of ~ (~) vs
A would
be (asY')lptotica.l2.y)
a
:.be
straight line with slope H.• These lines would intersect the proba.:oi:'
J.
lity axis at ~, the ratio Rl/H
Z
for the sa;ne T. but different
Ct
J.
t
= 1,
would be independ.ent of
a. atd
plots
would be pa....-ra.llel lines. Then, for
can be approximated by (Rl /RZ)2 •
ARE(Tl'TZ)
For T asymptotically non-normal we can expect "pa.ra.llel
i
curves" corresponding to different values of
attained only in a narrower neighborhood of
Linearity would be
(1.,.
nO
(and for a l~"'ge:!:" n)
than in the asymptotically normal situation.
Cox's approach was used in an m = 5, n = ZO set-up for normal
4.~
errors. The most relevant results are displayed in Table
and
Figures 4.1 and 4.2. Experience with this and other set-ups was
influential in the development of the alternative (also gra?hical)
procedure described in the next section.
4.2.- Alternative approach.- In our problem the statistics of
2
interest are asymptotically distributed as X (",lL)
with ')::; m - 1
~i.
We are inter -
J.
degrees of freedom and non-eentrality parameter
ested only in local alternatives
.,m • For any fixed (large)
theses
~k
= k ~, k
= 1, ••• ,N
n
e =[e.J
'J
-
with
we can choose
e.;=
0 J~
.1mn ,
J
j
= 1, ••
N alternative hJ?O -
• Then for our test statistic Ti we h..a.ve
2
N associated distributions that can be approximated by :x. (v,.lik; ...-ith
non-centrallty parameters proportional to
k
2
,
say
2
6
ik
= Y k
i
Therefore, if we can estimate
Yi
for two test statis"tics7.
and T., , we will be in position to estimate
J.
(4.Z ..l)
, k = l,. •• ,N
J.
their
A.R3
as
A
TABLE 4.1.- Power Estimates (ITik) Under HI: e:'k( -t,
-i,
i, i)
0,
0
1
2
)
4
0.058
0.207
0.72)
0.98)
1.000
0.058
0.217
.727
0.984
1.')00
T
3
T4
1).050
0.205
0.687
0.970
1.JOO
0.051
0.201
0.685
0.968
1.JOO
Fr
0.059
0.212
0.672
0.978
l.JCr.l
Ra
0.055
0.2)6
0.762
0.998
1. ')0')
F
0.c49
0.252
0.796
0.997
:.JC~
T1
T2
0.012
0.077
0.461
0.9C6
'.999
0.012
0.078
0.457
').908
J.999
r-f
T)
0.013
0.067
0.458
0.905
J.999
0
T4
0.013
0.070
0.456
0.9c4
J.999
Fr
0.')14
0.069
0.443
0.888
').999
Ra
0.013
0.083
0.517
0.~9
).999
F
0.015
0.093
0•.566
0.966
·J.999
k
T1
T2
.
\1"\
0
.
0
8
0
.
"
d
Note.... In this .Chapter we· will .use . the following notation fo=. the
weighted ranking
statistics :
T1 : s n,i
::
i
and
v.J. = variance ,
T : s
2
n,i
=i
and
V.
J.
= range ,
="7aria.nce
T : s n,l :: '0, s n,i :: 1
3
i=·2, ••• ,n
and
V.
-1
i=2, ••• ,n
and.
Vi=~
T : s
4
n,l
=0,
s
n,i
J.
.
A
l>-
FIGURE 4.1.- IT vs k
NORMAL PROBABILITY PLOTS
n
11:0.05
1l=Q.Ol
o --x ---
T1
Fr
+ --- F
1
2
3
k
FIGURE 4.2.-
fi
vs k
2
CHI-SQUARE PROBABILITY PLOTS
:
n_--,...----------,.--------------,r-.,
o ---
T
Fr
4 - - - Ra
+--- F
)C - - -
1
4
k
2
b..
/Il.,.
_y./y.,
L.-t
~.-t ~
1
(Hannan, 1956) .
=
The OOlfer
function of T.~ , 0
.en-ven HI : .....8K
8,
T
~
,k. ,
TI(A
ik
)
=P
::
( Ti >
1 - P
:x~ IHI )
2
2
P (XOIJ:> Xc) =-
with
is
rJ.,
(Ti~X:IHl)
1-
( Xc iV, 6. ik )
:: 1
_"l?
-1
- F (l(iv,kY )
2
:1
(It.2 .2b)
i
Equation (4.2.2a) can be solved for 6.
ik
by using a direct sec.rch
procedure that cO:Jlbinas the subroutine RTMI from the IBM
Sci.enti-~ic
Subroutine Package with the function CNCP from the CFTDIS :;rr=g:ra.::a
(T~e Sharing Library, Department of Biostatistics, UNC). 3~~
requires a preliJl.inary guess at an interval to include the sJlution i
to do this we can express the right hand side of (4.2.2a) as ~ Poissonco~bination
weigthed linear
of central chi-square c.d.f.'s
:1
4
C Y+2h ::: P (X(l'+2h)~"Xc ) , h
=1
- e
_.A
Z
~
1 (A)h
'2
~ h~
/).
b.'L
n ... 24 -
=1
2"
- C" +
!:J.
Cv
114-
!:J.3
; i.e. ,
(writing 6. for ~ik)
C V + 2h
:: 1 - 11 -
=0,1, •••
b7B" + 24 .16
- ... t x
at.·
- '""'8 C ~
!l
6.1..
- 2 cY",,:L'" 4 C \1",,2
At.
-"8
t:?
-
C"'+4 +
Ib
~3
Ib
b~
6.4-
C vH +
"9b
C"'1"~ -
b4
- Zj:8 C Y + il
t:/r
.
CY -I-'2
-
C"V+~
+ ......
064-
+ 9b C y+6 6."
- 3m
••••••
C .... -r Ii + ••••••
53
or IT(t.ik )
where
=
0(,'"
Al \' ~ ... A2 ~
Alv
=~
A2v
=- ~
A)y
=
A4~
: -
t/ ....~J
y
~3
...
(C v - CV + 2 )
4i
(C y -2C v +2. + CV +4-)
(C y - )C V + 2 + )C ...
)~
CCv - 4C v ","2. +
4 -
Cy + 6 )
6C\I"'~
In Table 4.2 we display values of C
v: 1, ••• ,4; i,h
= 0, ... , 4
(~=
- 4C Y + b ... C,/+S)
V+
0.05 and
2h
and A.
~=
fer
J.\I
0.01 ). 3y "trun-
cation" of (4.2.)) different polynomial equations in
~
can be defined..
Solutions to these equations can be obtained using, for
e~p:e, ?O~~
(an S.S.P. subroutine) and they are well -behaved (in the sensa of
equati~n)
showing only slight differences for increasing degree of the
As described later, the smallest positive root of the 4-th aegree
'" r
eauation
proved to be an excellent guess for the final 15..
1K
4.) .- Experimental set-un and :!:"8sults.- Because of
•
~o!lputer
time limitations only randomized block designs with n : 40
and m· : ..'2,'), 4.·~or
blocks
5 treatments were sblulated for the three 1tost
interesting error distributions : normal, uniform and double
nential (mean -= 0, standard deviation
Some preliminary exploration
values of Nand
k -= 1,
/2, /5,
2.
=1)
~s
'JXpc> -
•
required to find
The resulting decision was: take
N
adeq~te
=9
, 19 ;
for m -= 2, 9 -= (-0.0625, 0.0625) ,
for m .: ), ~ -= (-0.0625, 0.00, 0.0625) ,
for m -: 4, 9 -= (-:J.C625, -0.0)125, O.CJ125, 0.0625) and
and
TABLE 4.2 .- Coefficients Used in Different "Linearizations" of Equation (4.2.2a)
a
."
C'I
C\/+2
CY+4
C,,+6
A
O
C"+8
1 0.95000 0.72090 0.42754 0.20215 0.07847
0.05
Al
2-
x
A
3
xl0
3
A4 10'"
l(
A
.a"-ni1 0.114550 0.803250 -2.7:J+792 2•.564840
-fti2
2 a.95000 0.80021 0.57585 0.35181 0.18402
a,
3 0.95000 0.83325 0.65079 0.44708 0.27020
"
a, ..IT
4 0.95000 0.85205 0.69717 0.51)48 0.33920
2 10
A
0.074895 0.932125 -1.560208 0.493750
i3 0.058375 0.821375 -0.926250 -0.094271
'" 4 0.048975 0.711625 -0.585833 -0.263021
, oc. - IT.
.
~
"
-rs.1
. 0."037245 1.128250 -0.758125 -0.752Y¥+
1 0.99000 0.91551 0.75076 0.53214 0.32493
0/,
2 0.99000 0.94395 0.83791 0.67514 0.48774
"
oc. -rr
i2 0.023025 0.749875 -0.067917 -0.751042
3 ()'.99':>00 0.95504 0.87573 0.74719 0.58516
d-
0.01
II
0.99100
0.96115 0.89712 0.79119 0.65075
-ii3 0.017480 0.5:J+375 00.101667 -0.536979
1\
oc. -IT1/l- o. 011¥~25 () );.172,50
o. 148"31 -0.17786 5
'{}
e
e
e
55
for m
= 5,
2:
(~J.:625, -0.03125, 0.00, 0.03125, 0.0625) •
One thousand independent experiments were simulated ::or each
A
situation. To reduce the variability of Aik ' the same
kZ
dO:ll deviates (same seed) was used for
= 1, ••• ,9.
basi~
sat
o~
=an-
What ba:;I:ens i.-hen
we try to use a different seed for eaC:l value of k Z is depi=t.ed in
Figure 4.3 ( in that illustra.tion, m
=2
with nomal errors) •
A FORTRAN program (ARE40, listed in Appendix D) was 'written to
sinulate the
exper~ents,
to compute all the statistics
0:
~~rest
to tally the rejections of H at two levels of significance.
O
::0
anQ
elL-ni-
nate unnecesary repetitions we report in this Chapter only ':.he resul-:'s
corresponding to
C(.=
0.05 •
The critical values for Friedman's test and for the
rankings tests were derived from the 3-moment-exact
_e~~ted
approxi-~tions ~en-
tioned in Chapter I (pages 8 and 16). Values of the consta::lt3 v
:l
3.I".i;.;
n
involved in the approximation for weighted rankings are given in Tab:"e
4.:3 for n : 20, 40 and 60 • These values confirm the
conjec--:.u-~
tr..at
Table 4.3
n
v
n
and w
n
-to
1 as
20
0.91226.5
0.751025
40
0.955565
0.870260
60
0.970251
0.912338
n'"
eo
(Quade, 1972b) •
It is interesting to note that for most situations (a.bo~t &J ~I
the "4-th degree guess" for
Awas,
up to four decimals, iie::rr.:'::a.l to
the "exact" value of .& • Only for high values of k i (7, 8 0::- s:) did. we
" vs k 2. : DIFFERENT SEED FOR EACH k
FIGURE 4.3.- 6.
D.,----.-~-~-___r---r-----r-.....,.----r--..,....-.,
"
4
3
o ---
T
'IC - - -
Fr
Ii. - - -
Ra
+--- F
2
1
1
2
3
4
5
6
7
8
9
57
find discre:;:a.nci9s in the
"
3rd or 4th decblal.
1\
Values of IT and 6 for these situations and seven test 3tatistics
- TI' T2 , T , T4 (see Table 4.1), Fr (Friedman chi-square),
3
:sa.
(ran..'ting
after align:nent) a.:J.d F (standard varia.nce ratio statistic) - are displayed in Tables 1 through 12 in Appb:.iix E. The values of
and T
2
TI:.::< for :'1
(similarl] for T and T ) are very close to each ofuer. As t..'1ese
4
3
statistics differ only with respect to the measure of variab;lity ('/.]. \
involved (variance for Tl' range for T ) we can conclude t:."la.t d.iffer 2
ences in
".1 are not likely to "induce differences" in the A?E of these
statistics with respect to any other third statistic, at
lea~~ ~ot
=or
the small values of m used here •
..
On the other hand, the values of lt
for T and Ti.;. are, usually,
ik
3
very close to those corresponding to Friedman's test statist:'c. I.e.,
t.'1e use of the second set of weights or block-scores (s
least variable block and s
n,1.
=1
for all the other
be a source of differences in the ARE of weighted
Friedman's test (except, perhaps, for m
= 2;
.
n,1
=J
bloc~"'),
ra~.:o-s
for the
ioI'ould. !lot
versus
which is, cer-...a.inly, not
the most interesting set-up) •
Because of these reasons, it would be redundant to ,:acLy out a
detailed analysis of more than one of the weighted rankUg"· s:.atisti-::s.
I:1 Figure 4.4 we display plots of
Fr, He. and "5' for
~.
!:l :
6ik
versus k 2 corresponcnn: to T ,.
1
2 and i. i. d. normal errors. ProceedL"lg as in Lee
(1975) we can fit
by eye a straight line to each
0= t.::.ese sets
of points. The slopes (Y ) of these lines are good a ppro:d.:Jlations to
i
the Y. 's d.efined in (4.2.1) ; we can verify this by
1
com~'1.S
SO::1e
ratios (Y if ~ i' ) against the corresponding theoretical A..'Bs C"reference values") •
In Figures 4.5 through 4.15 we apply the same procedure for normal errors with m : 3, 4,and 5 and for uniform and double ex:xmential
errors with m :: 2, 3, 4 and 5. (The
distributions were chosen to· give
pt. .
a: =
..IJIleters of each of these error
1). Estimated and t.~eoretical
AREls (when available) are displayed. in Table 4.4.
4.4 •- Discussion. - After an overview of .Table· 4.4 co:nparing
esti:nated AREs with available "reference points", we can safely con clude that the approach described in Sections 4.2 and 4.3 is certainly
a good aU"OrOximate procedure to evaluate the ARE of statistics that are
asymptotically chi-square distributed.
It seems clear that with normal errors, T is 3.Symptotica.lly
I
more efficient than Fr and its performance is similar to na (or Fl for
increasing m • We can conjecture that with normal errors,
Under uniform errors, Tl is asymptotically more efficient than
F.r for all m, more efficient than Ra for m ~ 3, and more efficient
F for
In. ~
t~~
4 •
For double exponential
erro~,
Tl looks slightly less efficient
than F.r which, in turn, is almost equivalent (in the ARE sense) to Ra
for m ~ 3 •
These results are basically consistent with those presented in
Chapter II, i.e., when we compared the performance of these statistics
for designs H'ith small number of blocks •
,..
FIGU~E 4.4.- ~ vs k Z
A
NOR~lAL ERRORS (m = 2)
b.r---~-~---:--~--r--...,.------.....--..------I
1
2
3
4
5
60
/'\
~
_ _--,__....-_....,..._---..__.---..
FIGUR:: 4.5.-1:::. vs k2.
b..~-_-_--_-...,.....-
r;ORMAL ERRORS (rn
=
3)
3
2
+--- F
.1
2
3
4
5
6
7
8
9
61
~ r_~
__r~I_SLJ_'~_,E....,4_"
-~~~V_S_kr-2._: _~_WT""R_M_';_L_E.,.R_R_O_R_S"'l"'(:...m_=_.~~.:..)
_
_6_"
4
F
Ra
T
Fr
0--- T1
1
L---
Fr
Ra
~---
F
x---
i
2
4
5
5
7
8
9
62
~
nl-__
'" vs k
FIGU~E 4.7.- ~
: IIO~~·'.4L ERRORS (m = 5)
~-~-~-....,....--:---..,.---,.--~r--~---,
2
4
T
Fr
3
1
o ---
T1
A ---
Ra
X ---
Fr
+--- F
2
3
4
5
6
7
8
o
oJ
•
'"
Z. r-_-:-__F.,.I_G_U;:(_E~4_.8_._--r-!1_VS--rk_2._-l-U..;.;~1:.;.F..:O~?~~'i
..::E:.:.:.R:.:.:RO::.;.R~S~(:;-~l~=.....;::::../' ~,--_
3
2
0---
x--t1---
Tl
Fr
Rc
+--- F
2
3
4
5
6
7
8
a
=
~,
2
1
0---
'X - - -
A---
T1
Fr
Ra
+--- F
2
3
4
5
6
7
2
9
,,
!:
" vs k 2.: U~JI:=OR~'l ERRORS (:~ = .: ~
FIGCRE 4.10.- ~
3
2
1
0---
Tl
:Ie - - -
Fr
6---
Ra
+--- F
1
2
3
4
5
6
7
8
,. "2.
"
66
FIG L: ~ E 4. 11 . -
A
,.
6
vs k 2.
:
UNI FOR 1·1 ERR0R5 (m
= 5;
A_--r---:---:--~--r-'--:---~--'----:--~-~
4
3
2
1
2
3
4
5
6
7
o
u
9
67
FIGU~~ ~.
'"
12.- Ll
k2
VS
:
LAPLACE ERRORS (m = 2)
x --- Fr
A ---
Ra
+ --- F
1
2
3
4
5
6
7
8
..,
I-
t<
68
A
Do vs r, 2.. •• LAPLACE ERRORS (m = 3)
~_-r--or-----r--~:-----r--"------~"""'----I
I_
FIGURE 4.13.-
A
4
'"'...
l
o --- T]
x ---. Fr
A---
1
Ra
+--- F
2
3
4
5
6
7
8
9
69
A.
A.
F1GURE t.14.- (). vs kZ
: LAPLACE ERRORS (m = 4)
u.r--~_---r-.....,...---r----r--...,.---r---~-----
II
2
o ---
11
X ---
Fr
A --- R~
+--- F
1
1
2
3
4
6
7
8
9
70
'" vs k~ : LAPLACE ERRORS (m
FIGURE 4.15.- fJ.
=
5)
2
----A --+ --0
::!
1
')
<-
3
4
5
6
7
Tf
Fr
Ra
F
e
e
e
TABLE 4.4 .- Estimated (exact) Values of ARE for Three Important Error Distributions.
Error Distribution
2
e( T1 ,Fr )
(1 •.500)
1.52
)
1.18
4
1.19
5
1.2)
2
1.47
3
1.))
--( --( --( --( ---
L~
m
(:L)
1.52
(1.500)
)
1.3)
)
(1.570) (])
(LJL~9)
1.39
(1.397)
1.25
(1.26)
1.30
(1.)09)
)
1.24
(1.210)
1.28
(1.256)
)
1.47
(
---
)
1.59
(1.500)
)
1.22
( --- )
1.)3
(1.333)
1.40
( --- )
1.12
(
---
)
1.25
(1.2:'"')
5
1.41
(
)
1.14
( --- )
1.34
(1.200)
2
1.32
--( ---
)
1.32
(
)
1.07
(1.000)
)
0.95
( --- )
1.05
)
0.95
(0.889)
4
0.91
( --- )
1.01
--( --( ---
)
0.85
(0.833)
5
U.81
( --- )
1.00
( --- )
0.86
(0.8nO)
Laplace
( 1)
.._,_..,______.___.____._._____._A..
1.58
Normal
Uniform
e( F ,Fr )
e( Ra,Fr )
(
Puri and Sen (1971) p. 279
lIollan(lor and Wolfo (197)) p. 18)
(l)
( ---)
(1)
(.'2)
(2)
Not available
~
EXTENSIONS OF WEIGHTED RANKINGS
5.0.- Introduction.- In this Chapter we present preliminary
results on two possible extensions of the weighted rankings approach.
Both problems are analyzed here in terms of ARE; analyses for a small
number of blocks will be undertaken in the future.
5.l.-Extension to Balanced Incomnlete Blocks Desig<l'!ls (BJJ3).5.1.1.- Analysis of BIB Designs.- The usefulness and/or necessity of using incomplete blocks designs is well documented in most textbooks on design and analysis of experiments.
Let define the following notation:
t
= number
of treatments to be compared,
k = number of experimental tmits per block (k < t),
b
= total
r
= number
number of blocks,
of times each treatment appears (r < b),
x..
= result
J.J
of treatment j in the i-th block if the t...-eataent j
appears in the i-th block,
R..
= rank
Y..
= observation
r
= rank
J.J
J.J
ij
of X.. within the i-th block (if X.. exists),
J.J
J.J
aligned on the i-th block mean, and
of Y within all the observations.
ij
Let us assume that: (1) every block contains k experi::nental
tmits; (2) every treatment appears in r blocks, and (3) e-.,ery treat-
ment appears with every other an equal nu;nbe:: of times, )... Then
kb
= rt
and A(t -
1) = r(k
-
1) ;
i.e.
A = r(k
- 1)
t - 1
Of course, i f k = t = m and r = A= n , we ha°re an nxm ::::'3.ndOClized
cOllplete blocks (ReB) design.
Under standard assumptions a parametric ANOVA can be used to
analyze this design in terms of an F-test (Kempthorne (1952), Chapter
26; Anderson and Bancroft (1952), Chapters 19, 23 and 24; Cochran and.
Cox (1957), Chapters 9, 11 and 13) •
A non-I&ra.lJletric test for BIB designs
'WaS
proposed by Jurbin in
19.51; it uses intrablock rankings and the corresponding statistic is:
D =
~f~2:
i, [
t'
(R j - r(k 2 + 1 2)2
(5.1.1..2)
j:l
with
R.
J
~
=1:
R..• When r'+
1=1 J.J
00,
2
DI\J X. (t - 1); however, in ~ctice
this asyptotic approximation is also used fo:: small r (Conover (1971),
p.280).· The e%&ct distribution of
found in
D for sOlie small BIB can be
van dar Laan and Prakken (1972). Van Elteren and ~oether
(1959) showed that, under alternative hypotheses
e=r
c.n-t 1
I'
l J
,..
~
the
ARE of Durbin's test with respect to the parametric Al'iOVA test is
where
0
2
is the variance associated with f( ). This last expression
matches with the ARE(Fr,F) for RCB designs with m
to
ABE (D,F)Normal = k
when f( ) is normal.
2k1
=k
and it reduces
(5.1.1.4)
"",
(~
5.1.2.- Analysis of BIB Designs Using Ranking After
)~i~ent.-
As Durbin's test is the natural extension of Friedman's test,
also "ran.1dng after alignment" can be naturally extended from RCB to
BIB design.s. This is in fact the special case of Sen (1971 o) -..here_ ?=l.
(univariate res:ponse), n
= 1, ... , t )
r. = r (j
J
and
::I
1 (no rep.L.L.cations of the basic design) ,
r .. , = A (j., j'); then v = t, N = rt = bk
and
JJ
,,_r(k-l)
t-l
•
1\-
If I'll' . = r.. = rank of X•• after alignment, then proceeding
~J
~J
~J
from (2.8) through (2.16) in the reference cited, we have:
1'\1' = r.
~.
=
~.
1 t
-k c: r .. I.. , ( I. . = 1 i f the j-th treat;:lent occurs
.
J.J
'::1
J-
~J
~J
in the i-th block, I.. = 0 otherwise) ,
~J
Ill .. = r .. =
1
\\
bk L L r ij
I ij =
(r ~1.. ' ... , r Ill.)
lxt
Finally, after defining
bk+l
2
r
= ¥bk
+
l
=
TN,j
and
1)
k•
r ij
•
Iij = Tj
1.
and
!~
= [T j ]
l.xt
we will have :
=
\'
v....
~
a
(2)
=
v =
N
1
bk
~[
1.
\
L
\'
L I..
~J
i
j
( r·~. - r
L
J
( Tj - ~(bk + 1) )2
lj
T~
..
e r lJ
••
J
~
,
- T ( bk + 1 )2
- r.~. )2
,
(5.1.2.1)
(.5.1.2.2 )
)2
,
.
1
=
-k
1
tXt
(1)
A
[kr O.JJ"
- r .. ,
JJ
]
=
rt
l'
{I - -t I I } .t _ 1
- ::. I
,., c::)
_' ,• .A. &:._-/./
(
= J:!.-~ ~ ylN + II!..~ ~ YNC%) =
-~
-"N
t rt
~ {
'=-r-:l'
!-t~~
with"
11
w
-
{I _ !'t 1 I'} {Vii) + b - r
- N
b - 1
1
v, l:':: }
If
txt
1
I. .
~J
Because {
inverse of
! ~
' }
(
r ~J~.
.. - r.
t ~ ~'}
) 2
+ k ( b b - - 1r )
is idempotent ,
2.
(r.
_
1..
~
If- = ( t~ ) ~
-= ••
is a.
)2.
ge~eralized
and the statistic (2.16) in Sen (1971) will reduce to
Ra = ( t - 1 )( T _ TL )' I (T _ 1'L )
~
-N
-N
-N
-N
( t - 1)
=
· I •.
{ L
J
~J
{2J
T~
- b
J
~
r ( bk
+1
)2}
(.5.1.2.6)
( r.. - r. )2 + k(b
b - 1r)
~J~.
-
2 (r.
J.
J..
- r
)2
••
Under H this statistic is asymptotically chi-square
O
. ad
with
distri~t-
( t - 1 ) degrees of freedom. For normal errors ,
ARE ( Ra.D )
=k ~
1
( Sen (1971b), ~. llll) •
76
5.1.3.- Analysis of BIB
Desi~s
Using Weighted Rankir;&s.-
The tlincompleteness" of a BD design may be well exploited by
weighting the different blocks according to their apparent
v-a.=iab~l ity
(discriminabUity). It is likely that blocks involving the !!lore "dis tant" treatments will show the larger observed variability. Therefore,
by adequately weighting the intrablock rankings, we should surpase
the power of Durbin's test.
the indicator function I. . defined in Section 5.1.2 ,
By using
1J
we can write the generalized weighted rankings statistic as
b
t
(t - 1)
=
T
b
( E
b
E s 2.
. 1 b,1
Sh
2 •
)
,1
= j - t(k + 1) , j
tj
(L t .2
jd J
=
)
l, ••• ,k and
k
= b ( b + 1 ) ( 2b + 1) and
then , if
L
6
1=
H.
J
I
=1
we can also write
)2
k
id
wben
J~ {i~I ij 5~ llij
.,
J~
I.. sQ ( R. . 1J
i
1J
T
= 72 (
t(
t - 1)
rt (k 2
-
sb,i = i
t~
J
=k
i
= : •••••
( k + 11 ) ( k - 1
2
m + 1 ) ), j
} Hj
= 1, •••• t
= ,.
)
;
,
(5.1.3.2)
l)(b + 1)(2b + 1)
Naturally (5.1.3.1) and (5.1.3.2) reduce to (1.3.2.1) a.~d (1.3.3.2) ,
respectively, when k = t = m and r = b = n •
As in Section 3.1 we can state the following
Theorem.- SUppose that the Assumptions I through IV (Chapter I)
hold, and
s. . =i, i=l, ... ,b and
0,1
Wolfowitz condition
b
= 1,2, ... satisfies -the
',jald-
77
(v)
for
= J ,4, •••
r
.
1 \'
w1.th sb= b t... ~,i • Then, under HO '
1
contrast
Ie
=
[)
so.
Li
1.
LA.
J=l
1.. :: \ '
= ~•
H.J
Proof.- Let
t
~
H.
J
J
I 1.J
.. sQ • t, j
-R ••
= I, ••• ,t
L. >... = 0
L.A~ /'
1.
with
and consider the
1J
and
J
0 • Then
J
\A.
•• t
= \' So w.
R.• L
Lj J I 1.J
. 1.
1.J
i
~1.
•
From Assumptions I - III (Chapter I) we can c.onclude the W's are
independent random variables. For each given i = 1, ••• , b, the non-zero
I ij "taij's have a constant sum, and, under HO' they are int.ercha.nesa.ble
then
Carr
[I..
t1.J
,1. ., t
]
1..)'R.. ,
1J
j,1. •
1.J
- -
1
if
k - 1
1. . =
lJ
I~~,
.J..,)
=1
(set to zero otherwise) •
Therefore, tmder H
O
x:J VfLli
tR
0
1· J.
0)
J + Jf.r
\'.• , >. .J.1
.)..,
.,.J
corr[1. ~ t R ,1. 4' t..,
1..].
o.
1..
_1. 0 0,
1.)
.
l.]
j
x
={Lt j }{>j';J, >.~
I. . _
J 1J
It
=['[tk-I . f'r Ii·J A~J
.Z'
J
}
-
IlL
k -
j:t=j'
II (I.. 1V[I;.,
1J
L
~.
.
1.J 1J
~kl 'T
I . .).: +L\ 1..>'.,1..1..,
1.J J .i#j' J J 1J 1J
x
t....
-J.:1 •.
1.]
A .A.,I. .1. .,}
J J
1·.
}
)
1.]
,1'1
78
then V[WiJ> 0 ( except for the zero-measure k-dim9nsional 5ubs9t
R t defined by
A.
J
= constant
for all j such that I ..
1J
(I'll 11+ cS ] < co
because the TN's are bounded, E
Theorem
3.4.6
in Puri and Sen
=1
0=
). Also,
for any &> C. 'I'h9n, applying
(1971), we can conclude that the W's
satisfy the Noether condition •
The Q's are a
random permutation of the integers 1, .... , b and
under H they are independent of the W's. Then as in Chapter III:
O
E[
11,] = E [ L s~Wi J ==
C"~ ~ V[~l
=
r
-bk
::: I:. V[sQ.J.
0
wi1
and
(blocks are independent)
{[s.2.tJ{Lt~}{[A~
. . 0,J.
J
J
--t:l j;f;j'
L A·~·,1
J J ,
implies \
>,:
L J
but
LA.J =0
= - [~.J >. J.,
because rt = 1:k •
Then by Theorem
3.4.1
in Puri and Sen
(1971) we conclude
that
79
L-o t:J
-
C'b
-----+ N( 0 , 1) for all
~. Consequently, H IV N( 0 , ~H)
- , . ,
AI
and
""-
with
With some additional algebra we can show, as in Chapter III, that.tt
the asymptotic non-null distribution of (5.1.3.1) is 'X2(t_l,~) •
5.1.4.- Evaluation of the ARE when BIB designS are used.To study the asymptotic performance of the weighted ran..1dngs
statistitic we applied the procedure described in Chapter IV to three
BIB designs. These designs were chosen as large as possible within
computer time limitations and they consist of replications of the
ing 1:8sic designs (Cochran and Cox (1957), p. 469) :
(a) Plan 11*
(b) Plan ILIa
(c) Plan 11.2.a*
1
2
:3
1
2
:3
I
2
:3
4
I
2
4
I
2
5
1
2
:3
5
I
3
4
I
4
5
1
2
4
5
2
:3
4
2
3
4
1
3
4
5
=3
3
4
5
2
:3
4
5
1
2
4
1
3
4
I
3
5
2
3
5
2
4
5
t = 4, k
r o = 3, b O ': 4
t
t -- 5
t
= 5,
k
= 3, r o = 6,
, -k-lJ.
- .
r o = 4 , bO = 5
b0
= 10
~ollow-
'l'ABLi:
5.1.- E:3t.i.':1&Wd Polfers and
:lo!1-<:z.:lL~':.7 ?a..-uel:ers
uY?Ot.bues • 31:3 ~sl6:l
......c ........
"".3 ...
...
I::~ ....
.... n,
0
0"-
~
~2
0
~
Tl.
OJ
01
Uoie= D~~e ll~,"
(a).
2
)
4
5
6
7
8
9
0;"8.5
,'0.10)
0.125
0.141
0.148
0.167
::1..290
0...2:)1
1
'Q
1.')71 UI
'-1'
'>.9'
U\
'>.5)81lo
0.8217"
1.12~
1.)4J4
1.43.57
1.6815
1.:Mr;
2.:..-;87
1''-'166 '
1).75
').')9)
1).107
0.121
0.129
0.1.51
') .!-52
C':'73
').261+5
0.4215
'J.6m
0.8786
1.0rn
1.1e21
1.4619
1.5175
l . .:zt.=.
1).1')1'2
').'le'7
0.101
0.125
0.147
0.1.57
0.186
3.~
O.zl9
'l.)y;4
1.5883
').8'786
1.1420
1.4226
1..5.5)0
1.922:>
2.:'457
2.:297
1).')8'7
0.11)
0.1)2
0.1!j...5
0.161
0.l8l
0.2:)1
c
0.J5S4
O•.52B)
'.9629
1.22:J2
1.)90)
1.~
1.8.592
2.1~
2.;"393
O.O~
0.:)8'7
0.100
0.111
0.142
0.1..56
0.131
0.201
0.22.l
0.)9'6
0.5883
').7187
1.018.5
1.3..561
1 • .5400
1.8.592
2.::'087
2.;~1
0.'J6!I.
o.m
'.>.78
0.088
0.101
O.lll
0.127
0.:)7
0':55
0.2124
0.)2~
').4.52)
1).60J2
0.79')1:)
0.9)49
1.15..56
1.2m
1 • .:;Zn
1.'>66
0.'178
0.091
0.103
0.124
n.12?
0.147
1.l71
O.Zl.5
1').261+.5
0.4.523
1.64?6
0.8217
1.114a
1.1.556
1;4226
1.7]27
2':'jC3l.
I).-nt+
0.1JS9
0.10';
0:119
O.l~
0.1.5)
0.17)
').2::2
O.2ZJ
".)9"'-6
1).6J.a'l
0.8786
1.()t;()Z
1.))01
1.Sl10
1.7.,53J.
2.l.2.lJ
2.;735
').069
I).
'.>.106
0.124
O.l!j.l
0.1.5Z
0.1&.
'J.2.J7
O.::z?
').)1.22
1) • .5~
O.~
1.1148
1.34J4
1.~
1.8.592
Z.'122:7
2."Z'72
0.068
I).
0.104
0.12)
0.151
0.160
0.190
0.Zl9
0..!J8
0.296)
0.,50.3)
0.9068
1.1011
1.473'
1 •.5917
1.972:>
2.;291
2.~
r'J.c67
o.o~
0.10.5
0.118
0.14)
0.169
0.197
'J.2Zl
O~
0.2~
0.61.&l
O.f.>Y'YZ
1.0)24
1.:3699
1.1172
2.0.591
Z.j419
2.;)23
.).068
0.08)
'1.1)98
0.11)
0.1)6
0.1.57
1).171
,>.!-5)
0.::1.7 .
r'J.296J
0 • .5233
').7498
O.S!629
1.2769
1.55JO
1.7J27
2.~
2.3'.5J
0.J4J7
D
...
2
:z:
0
P.a~
,..' ".m
1'"1
D
rl
....0
....
c
Ra
::>
.
0.99
""'3
1.=3
1.00
J.6a
0.37
F
'J.97
'1'1
..
').75
'J8.s
')87
1.06
.
1.:.1
D
<>
oS
~
:! Ba
F
(1)
Est1_t8<:l ?n«t:,
(11
Est.1.uUd lfon-eeat:-al!.
e,. Fan>teter
.
1.1.~
1.00
TABIE 5.2.- E:3Uat.cl Pa1rers aDd Non-e:el1traUty Pa.n:Mters UDder Di!:'e::'!lC"':. ..o.:.~-:.:.­
Hypoth. . . . . alB Deaip (b) •
..... ....
B
c
0
........ .....
0
Zb
w ..
h~
0
1
:3
2
~
4
"t
.s
6
7
a
i
9
CI)
Q
0.048
T1
.
O.~:3
f
0
lI:
1).:61
0.071+
0.096
0.110
0.113
0.1)0
O.H.5
~.:..50
O.nao
1).461)6
0.84'5
1.0m
1.2007
1.)871+
1.6",4.2
!_~
'l.'l68
".07)
0.Oa5
0.09110
0.107
0.117
0.1.."
:.J.TI
'>.4424
1).65.58
0.&:198
1.024a
1.1849
1.2791
~~l
0.2,562
').,'"
a.l),,:!
1).'l71
1).084
0.096
0.114-
0.120
0.1).5
0.151
:.2;5.5
0.0000
0.4),58
o.6JSf+
0.84).5
1.1)n
1.2322
1.46)7
1.7.)26
:.9=73
O.O!J+
0.:&
1).079
0.09110
0.1.16.
0.l2'+
0.141
O.lU
:'.1.:2
0.()8)8
0.21.52
0-.5.SOJ
o.Bo98
.1.1691
1.2~7
1.5~
1.56)?
Z.l.~
0.')81
O.CS)
~.09)
0.106
0.121
0.128
0.141
0.1;9
:.1.75
'>.5851
0.6209
1).7929
1.008.5
1.2479
1.)566
1-5.114
1.~
z...05)6
0.070
~.C81'
1).083
0.088
0.090
0.104
O.ill
0.122
:.135
').)~
fJ.~57
,.15m
0.7077·
o. 74:r~
0.97.59
1.1r;~
1.2635
:-46]7
1).')6)
').m
').087
0.096
0.108
0.118
0.129
0.142
::.156
':I.2,sz
').j",
').6~
0.84).5
1.0410
1.2007
..•
1.)720
1 •.5654
~'7756
').061
').".8.5
').099
0.109
0.122
0.1)1
0.144
O.lOo)
D
...•
IU
w
Ila
:::.85
:::.65
~.?
P
::.;0
T1
D
~
~c
::>
Ra
,
T1
D
8
;S
""
~
.
(U
,I
I
I :;.3:3
I
:,•.sa
I
I
L-5a1
I
:.Bl!~ I
I
1).6,,,,
':1.89).5
1.0,572
1.26'5
1.4021
1.5993
U3J>9
~5 f ~-3J t
~.?71
O.~
0.1')2
0.115
0.1~
0.1'9
0.161
O.ln
:.295
0.41>,58
'J.~57
0.9110)1
1.1,532
1.448.5
1•.52'.)
1.849110
2.1m
2..r:5.5
0.1)1")
O.ca.
0.106
O.llB
0.1)1
0.14)
0.158
~.172
:...l.29
I).~
0.6_
1.0085 .
1.2007
1.4027
1.~)
1.Bo58
2.0C79
~
1).')71)
a.aT!
0.104
0.110
0.12.5
0.142
0.1.57
0.1~
.:.1';73
?~
?.51~
?
1.om
1.3102
1 •.569110
1.7912
2.~
2..37.D
97.59
I)."'"
I).
"'89
o.n98
0.112
0.124
0.138
0.1.56
0.171:-
~
I').~
').7Zl'8
'.8769
1.10"
1.2~7
1.5)92
1.7766
1.993.5
2-~
Ea~bated ~
Ifat a.,.Uable a1J2ce ~.,. obeerwd power is lesa than the slg:UIlcance le~l (~::
I
I..
!-
:::z
1·-,.1
i f
I
,
·
;!...::2 ~
(U !!:stbatecl :'Ioa-celr'::'aU ty Pa1'llllleter
•
·I,
".))11
Ba
or
I
::.::.5).
I
i
J
•
IJ.~
J
i
I
i
i
82
'l:AilL3 5.).- E:5t.i:1&ted
~n
and
Non-eea'='-ll~,. Fa=-.t.ars
Code:-
~:-~ __ ~::::a:::a~_
HJ?Cl'=-,e,," : 30 Daal.6n (0) •
...... ...
~.a
...C.=...., .....=t
<:
-
0
b
0
T
i
Q
7
a
9
0.15.3
0.181
0.2C1
:: ::>""'9
1•.5J93
1.7)26
2.1J50
2. 4 1:0]
2...~1
0.1:J8
::;.1;2
0.140
0.156
c.l26
e-z=5
l.dolO
1.4180
1..5J93
1.7766
2.2:6)
2..~
0.106
0.127
3.1.52
0.164
0.188
0.2.2.3
::.z4.J
0.67)1
1.C08.5
1.)412
1.7179
1.8929
2.2)40
2.6477
z..~
0.071
0.092
0.112
0.1))
0.152
0.176
0.206
Q.2Zl
C..z.:z
0.40.58
0.7160
1.10.54
1.4)::n
1.E6)9
2.0648
2.48:30
2.~'"
,]...:!:J7'+
0.069
O.tJ92
0.118
0.144
().176
0.188
0.228
0.2.5]
::~
1.)689
0.7750
1.2007
1.5993
2.004a
2.2J!1O
2.78)2
).17'2-
3..]~
').~)
0.065
0.071
0.099
o.lZJ
0.126
0.1~7
0.l66
C.!.-~
').21~
0•.5146
0.8935
1.2791
1.)257
1.644Q
1.92l7
L~
').~.56
O.C72
0.091
0.110
0.14-0
0.1.5)
0.179
0.2;)5
O,,22?
·3.120.5
0.4241
0.7.590
1.07))
1•.5393
1.7)26
2.l:J7.3
2.4659
2.7';67
.s
6
0.115
0.140
0.9101
1.15.32
0.::81.
0.093
0.1<;87
0.5357
0.7929
O.:J!:#
0.086
0.00'..8
1
2
)
4
') •'J6lI. fl)
0.~80
0.100
').27.,52lU
').;68l.
3.~
-<
0'"
2
en
Q
T1
11.2Z
D
.E
....
0_
:z:
2a
D
•
e
...
.....0
<:
.-
.'
').0.58
0.C85
0.108
0.127
c.156
0.178
0.214
1).23.5
C.z~
0.1.598
0.65S8
1.0410
1.)412
1.9217
2.0932
2.5'1.3Z
2.37;f:.
;..:97S
0.066
0.087
0.118
0.127
0.1.53
0.166
0.207
0.2~
C25.5
0.)130
o.6~
1.2007
1.)412
1.7326
1.9217
2.49('t1o
2.Clc:.
)":!:.:::.j
').049
0.075
0.104
0.114-
0.17.3
0.189
0.2)9
0.257
C.2";7
•
0.4737
').9759
1.5.544-
2.C22l
2.2480 •
2.9311
) .:;0;.5
)":'??2
3.:66
0.087
0.12)
0.146
0.l89
0.20)
0.2)2
').2.5)
C
1.)1)")
0.6sr:4
1.2791
1.6291-
2.1779
.2.442)
2.8)71
.
J.lla:
J.3ZC
().~)
".~
0.114
0.1)7
O.lS+
0.17.5
0.208
0.2)1
C...z:57
').2562
I).7Z4a
1.1)7)
1.4901
1.8929
2.0.501
2 • .5lll
2.22)5
., -......5
?
T1
..
!)
~
-, t
-J
f
I
!l
i
j
11_33 i
1-1=.)
1_~
I
1
I
2.~) t
,
I
0
01
....
:!""
I~~J
l..zl
!!&
-::>
I~
1.)0
?
T
1
I
!!a
?
E:3ti:::ated 'cn-eentrallty Pa~~ter
.. l:ot aV&il.Abl. si=e to'\. observed
pow8:-
is less to""n
to".
signl..fics.~ce l '.....l ; ~
::'•.5Y
= :.:5'
I
I
,I
I:.~5 i
_'J"_ J
(:) E.3ti.» ted ~:(:J
I
~sa
:
83
Ten replications of (a) and (c! and five replica~ions of (bJ
provides reasonably large values of r (:.r
o
X
nl.Lllber of re:;>lications).
Our FORTRAJ.'i program ARE40 (Appendix D), adequately mcdifhd
(introducing for example, the use of the subroutine ABI3A...'i :ro:n the
IMSL package), will simulate the ex.;ci~ents, compute all tr~ statistics of interest and count the number of rejections.
As. in Chapter IV, four statistics (modified T , :'l.odi.:ied Ba. D
l
and F) and. three error distributions (normal, uniform and double
exponential, all with
theses
0"2.
= 1) were studied under alternative hypo -
~ = b.2 ( h = 1, 12,
...
,f9).
~
In Tables 5.1, 5.2 and 5.3 we show the TI
estimates of power,
ih
~
A
the corresponding Llih'S and the Yits (estimated slope of /).ih
VS
h::! )
for designs (a), (b) and (c).
Estimated values of ARE, along with some theoretical values
are displayed in Table
5.4. Except for the double exponential errors,
the weighted rankings statistic is consistently superior to Durbin's
statistic; the corresponding ARE seems more dependent on k than on t.
5.2.1.- Groups of experiments.- The second extension of the
area of application of weigthed
rank~ap
to be considered
chapter is the analysis of groups of experi"!tents. As an
i~
this
il1'.1Strati~,e
situation, for example, let us assume that the tobacco yie11 in fields
located in Virginia, North Carolina, Kentucky and South carolina S;''"I.OW
different variability from state to state. Let us also aSSU29 that
~
=3
fertilizers are tested using similar ReB designs in each of
these states. A joint analysis of these experiments in a parametric
TABLE 5.4 .- Estimated (exact) Values of ARE for Three Important Error Distributions.
BIB Designs •
Error Distribution
Design
Unifom
Laplace
1.36
(1.397)
(1.333)
1.45
(1. 397)
1.21
(1. 250)
1.29
(1.3 0 9)
( --- )
1.25
(
)
1.38
(1.333)
( --- )
1.17
--( ---
)
1.43
(1. '"''3)
---
=3
1.18 ,(--- )
.1.30
(1.333)
(b) t :. 5, k:; 3
1.29 (---)
1.30
(c) t ::; 5, 1< = 4 1.13
( --- )
(a) t :; 4, k ::: J 1.45
(b) t ::: 5, k ::: 3
(a) t ::; 4, k
Normal
e(F ,D)
e(Ra,D)
e(T1,D)
1.43
(1)
(c) t ::: 5, k -= 4 1.30
(
)
1.11
( --- )
1.20
(1.250 )
(a) t ::; 4, k::: 3 0.95
( --- )
1.03
( --- )
0.90
(0.889)
(b) t ::; 5, k ::: 3 1.08
( --- )
1.08
( --- )
0.95
(0.889)
(c) t = 5, k ::; 4
( --- )
0.9i-
( --- )
0.86
(o.S)))
0.83
(1)
Sen (1971 lJ) p. 1111
(1)
Van Elteron and Noethor (1959), Hollander and Wolfe (1973) p. lS3 .
(2)
~
( --- ) Not available •
e
e
e
framework is studied by Kempthorne (1952, Chapter 28); we
wi~h
to
whether
~he
ncn-
wei~~ed
pa~~etric
rankings will have a good performance in
~cv
framework.
To study this situation, we have
sL~ulated
a
J-treat~e~t ~d
40-block design, where the first 10 t .l.ocks have error varianre
the second 10 b10c..:.cs have error variance
o-i '
Cf ~ ,
etc.
In relation to the weighted ran."dngs statistic two ?OS3ible
selections of block-scores were considered: (a) s n,~
0
i
= l,. • .,n and (b) sn,i = n - i
The s% ' i
= 1, ••• ,0
=i
(~~
+ 1 (antL..-ank of Vi)' i
of Y;),
= l .... ,:J.•
arP. Li.d. random variables 'tJithin each
of the q (=4) groups of blocks. They may have different dist::i;:".lti~ns
between groups, because the i th and i' -th blocks are ele::xe:1ts of
different experi:nents. Thus it is no longer true that
E [s~];:
The magnitude of
different the
E[S~.l:=
(3 [s~ ] -
*[
0"1 ' .R = 1, ••• ,q
*
[sn,i
Sn,il, i :: l, ••• ,n , will depend en no...
are.
and from (3.1.4) and (3.1.2)
V[Lnl
= )"
_!(sQ~
..
]V[W1o] =
""1
[
t~ I)~
,1
:n - 1
.J
I
1= 2::: t~.J [ )..~.l)- S 2 ..>0.
E \SZ
L ~
m - 1
£- 11,1..
Therefore, the derivation of the asymptotic iistribution of
weigthed rankings statistics under the null hypothesis,
~~e
?rese~~e~ ~~
Chapter III holds also in this particular set-up. Also unc.er the
native hypothesis we can apply a similar decomposition as in
III but using the central limit theorem for non-identically
a:;'~e:::'-
Cha~~a=
distributed random variables.
5.2.2.- Estimation of the AREs for grouped experiments.- Using
the procedure described in Chapter
alternative hypotheses
2 = (-0.0625,
~k
r: 1000 experiments und.er each of
= k 2 ' k = 1,/2,
•••
,19 were
s~ulated.
0.0000, 0.0625) was the basic alternative hypothesis .
The usual three error distributions were used along with each of the
{OOl}: {I, 0.75, 0.667, 0.50}
sequences
In Tables
and {l,
5.5, 5.6 and 5.7 we present the
1.25, 1..50, L 75} •
est~~ates
of
.6ik for T,_ (weighted rankings with V. = block-variance and s
~
,
Rik and
. = i),
n,~
Tl (weighted rankings .with V~~ = block-variance and s n,~. = n - i +1),
Fr, Ra and F. The corresponding estimates of the AREs are sum::larized
in Table
5.8 •
Against
prL~ry
intuition the statistic Ti has a consistently
poorer performance,across the different situations, than T ; thus,
l
until more "intuition-a.ppealing"sequences {s~} are studied (see
Chapter VI), we should prefer Tl • To explain this situation we may
say that both ~T
1
and 8 ,
T1
should be functions of
A = (l:.sQ.:!fi(x)d.x)2
1.
if {Q'i} is increasing with i,
and to use
strong
s~
=n
,
.J..
{j fi (x)d.x}
will be decreasing with i
- i + 1 would further reduce A, resulting in a
"penalty" for Ti •
37
'.S.- :::stbateli Powers aIl4 Nonooe:ent:n.Ut,y ?a~ters for G=oc;::s :~ .2x;lr='..=::-::s
'I:A3I.a
um.r Di£rerent Alternative l'iy?ot.'uws
.....
oc
....
II
0
.....
:>I~
>
-0
I~ar-...al
Erro::s •
k,2
A
~
'Y.
~
1\
1
2
)
4
.5
6
7
3
9
O.~:l\
0.122
'J.lS5
0.185
0.217
0.240
0.28)
O.;c.9
:).352
0.~J9U)
O.~~
1.2~.57
1 •.526.5
1.e620
2.l6Z5
2.5.;41
2.?~5?
].2.5;5
O.m
0.·J82
0.090
0.100
0.106
0.U5
0.1)1
() .:.';.6
J.::..51
0.26c30
0.)71'
0.47)1
0 •.5906
0.6600
0.76})
0.5426
1.::73
1.2'n5
".~
:l.ll)
0.144
0.172
0.200
0.223
0.2,52
0.22
3.320
".461)
O.~
1.0860
1.)884
1.6Sll4
1.~J
2.2245
2.;))7
2.;-252
~.lOl
0.1)1.
0.164
0.205
0.242
0.28)
0.313
0.:;;03
J.378
".(mz
o.~26
1.)028
1.7'...68
2.l2l2
2.:J;41
2.8.5)7
J.:2.:rJ
) •.5J)J
~.102
O.lJa
0.168
0.210
0.2.51
0.275
0.)12
c.;~
J.J9J
0.61)8
1.'::201
1.JI+.56
1.7891
2.2142
2.462.6
2.e4)4
).8:b-
).:S9)':
0.'76
O.~
0.108
o.UB
0.141
0.15~
0.171
'J.".:.:;'1
J.2.11
0::J:P7
0.:.6~
0.6830
0.7909
1.05.31
1.24C9
1.4417
1.:;5JC
~.7795
O.'J64
0.0.59
0.064
0.071
0.083
0.09)
0.101
O.:.~
).1.~7
0.1.5"
0.0926
0.1.;60
0.24))
0.J895
O.seas
0.6022
C.:l)3
J.6715
O.~
O.ca,
0.098
O.llO
0.126
0.144
0.161
O.~
J.~9.5
0.101:36
0.,5673
0.7060
0.8869
1.0860
1.270.5
1.""34:!.
1.6]1::1
0.082
~.m
O.ll1
0.1)0
0.1.50
0.1'7!!-
0.192
0.?'.J9
J.2'·n
O:m.5
0.5789
0.18,56
0.9315
1.1.51.5
1.4c97
1.6adl-
1.726
2.!.l'::9
'J."9'
0.118
0.126
0.141
0.155
0.171
0.190
O.Zl)
J.2jLr
0.4721
O.6a)O
0.8869
1.0.5)1
1.2057
1.)777
1..579)
1.:.z::s::-
2.'=385
ll')
...
'I'
-1
.....
S;
0
T"1
r.:
~.
c
?r
~
0
....
--.
.....
-
<:
:!a
:J.~
:: .. 2,;
.
¥
Tl
..,:.
-.p.
:.~
:..s:;
").71
.'
--'"
...
...~
T"1
~
...'"
~
....
.....•
to
?r
~.Z&))
Ea
.
:I.)::
::l.~
...J._..
-,,",
..
:J.~
F
til
Estbat8cl .",..~
11) EsU.ted .!Ion-cen~lt1 Paraaetar
·38
TA3T3 5.6.- ::sti:l&tad
Ocler
..
<>
... u
? :::
.. IS
...........et
",
>
k
...........
2
2
)
4
.5
6
7
2
9
0.120
0.160
0.197
0.245
0.280
0.)24
0.;.58
:.;"JJ
).44S4{1) 0.619.5
1.2597
1.6.5)0
2.1522
2 • .51Jl
v~j'n
).'-'2?)
3•.:r.cz
0.0,56
0.::62
0.078
O.O~
0.092
O.lli
0.112
O.Lry
:.:25
0.,:)"'1
0.1~8
0.)291
0.4016
0.4958
0.71~
0.7288
0.5.5JJ
:.::559
0.'::68
).197
0.125
0.1.57
0.188
0.218
0.2.50
0.2"',4
}.~"'9
0.2::lO1
0 ..5556
0.87.57
1.~
1.5.582
1.872'+
2.20)9
2.'''51)
Z.~-t;.
".079
'l.U,)
0.146
0.18)
0.220
0.261
0.289
o.m
:.375
".YH)
0.7060
1.1')78
1.5053
1.89:32
2.)113
2.6059
).:.512
; •.5015
'l.087
0.122
0.164
0.198
0.24)
0.282
0.)15
O.:r9
-:.JSl
1
~
Ti
el)
0
~
:g
'!'"::
C>
'"
...•
.......
:-..
0
..
='1
on
'1'1
~
llon-e"ntr.Llity ~ra-t.a~ for G=u:;:s ~ ~c<;s
A1ternaUn IiJ?OtlleS8s : UnifOr:l1 Errors •
~
.,.t;08
-
Polf&r3 .. <Xl.
Di!fe~n~
Ea
po
'1"1
J.4P.5
O.O+2!l
1.)028
1.6635
2.1316
0.C?6
0.096
0.117
0.1)5
0.151
.
2.5337
2.8744
).213:
;.62:0
0.176
0.196
0.2:.5
:.z:>5
o.50J9
0.78.56
0.9870
1.162)
1.4)10
1.6425
1.:.c12
2.~
0.0,59
0·1)52
0.0.56
0.061
0.070
0.079
0.072
0.::79
-:.:i:J.t.
I
3..69
:.451I
,i
~-30
I
I
2.-.561
2..61
If
i
I
- ;>9
- ' . -..J
O::P~7
I
j
,~
I
-...r-;""'
T'1
....~
Fr
.r;
...."'!
r.:
1).C925
0.X24
0.0"'2
0.1181
0.2)09
0.)41)
0.2.5.56
0.:;:;1)
:.~J16
:).051
O.~
1.0&+
0.098
0.117
O.IYr
0.147
0.l.5)
:.1.n
0.1181
O.zao:,
0.4')16
O• .56n
0.78,56
0.9759
I.U88
1.2~
:.<":;'17
').::68
O.CfTl
O.O~
0.119
O.lJ)
0.151
0.178
0.1;)
:.Z!.O
n
"....... po
1:)
'.2"'61 .
0.Jl69
0.5204
0.8082
0.$64a
1.162)
1.4523
1.6::9
:.7391
".C72
0.::88
0.106
0.1JO
O.1!l-9
0.166
0.190
0.2:9
:.1)4
0.25.56
0.l;4~
0.6600
0.9:l1.5
1.1!l-06
1.J242
1 •.5793
1.77'=6
Z.:';35
Est L.,.ted ?oooer
t» Zsti:l&ted ~Qt.J:allt1 Para:aeter
j
I
0.57l
~
",.22
l!a
-
t
:.18
I
:.881
TA3!..E 5.7.- E.st.b&ted Powers and Non-otl1t.n.llty Pa.:::a..:Mtars for CrQu;::s :.: 2:t;e=-=-...3
.. ...
... ....
co
()
'"'~
>
II
..-. Tt
1
~
<:>
~
"T•
-.....
Ra
•
I>
-
2
3
4
.5
6
7
a
-;
0.!.29
0.162
0.218
0.255
0.)03
0.7+'+
'J.:..a7
J -,..,
0•.5204\11
O.92cNo
1.2813
1.87Z1+
2.2.5.5.5
2.7504
).17.58
).62:;J::
;.:.:;.r3
0.0&>
0.112
0.1)4
0.1:;'
O.l~
0.209
0.227
1.2'-'-6
~.E8
0.).534
0.7298
0.9759
1.19t9
1./;.JI0
1.71&5
1.9059
2.:52.'
2 • .c..;2;
'l.C~
".1'+3
0.17.)
0.22)
0.20)
0.)14
0.J68
0.4->9
-:;~7
0.4')),6
1.0750"
1.)991
1.9244
2.)379
2.8641
J.4276
J.~
~~=r:
O.1}8)
".131
0.181
0.2)5
0.28:1.
0.)26
0.:376
0.420
:!·-n
'.:-.-'396
0.~26
1.~1
2.04S8
2.5Z)4
2.988.5
).5l22
).~l
:'.;7,;;
'.076
0.131
0.16.5
0~201
0.2.52
0.29)
0.)31
0.33:)
0.3cM
0.~26
1.3155
1.69t9
2.224.5
2.6472
).()!;Q4
) •.;ao:..
19
'01:'.::0
J •.-7.;2
0.070
0.CiJ7
0.11)
0.1J2
0.14B
0.19l-
0.220
:~~l
0.2)09
0.4375
0.';402
0.9537
1.1297
1.3m
1.6211+
1.39"';2
2"::"':9
0.06)
0.085
0.102
0.115
0.l29
0.141
0.1~
'0.:57
:.:.""..a
0.14)4
0.41:36
0.61)8
0.76)0
0.9204
1.05)l
1.1m
l.~
!..';s;:2
0.069
0.09+
0.12l
0.145
0.162
0.18)
0.212
0.2)2
:..::51
0.2185
0.5?fl4
0.8)08
1.0909
1.281)
1 •.505:3
1.8099
2.0177
2.~73
0.075
0.09l
0.115
0.1)8
0.159
0.19t
0.222
0.25J
-:.m
0.2925
O.~.Y>
0.76)0
1.0201
1.24a9
1.6214
1.9140
2.ZJJ?
'Z..-~-:;
".076
o.C96
0.109
0.1)0
0.151
0.177
0.201
0.217
: ..::'.8
o.)~.47
0.:M9
0.69t.5
0.9)1.5
1.162)
1.4417
1.6;49
1.SCZ::
2.:'799
1
11)
Fr
Vi
t'o
..f
.
It
o.c9t
0
lApace e:rro:s •
111
Tl
~
0
I
.2
0
L. .........
"
~
C::d,r Different. A1t8rnatl". E]?Othu••
F
.-.--
~.~
:.CJ:;.
:'.Sl
? ......
---J
.
Tl
~
T'1
rot
~
rot
.
Fr
Vi
N
- .,
rot
~
.......
JJ
~-t
00.171
.-
:.9>-
~.6?
:='.0)
Ra
J.89
0
(:) Eatiated ~
(II Eat.bated lIoft-ettIlt..-allty l'lLnmeter
"l.es
TABLE 5.8 .- Estimated Values of ARE for Groups of ExperLdents
with Three Important Errors Distributions
Error Distribution
(Variance1? ) .
e(Ra,Fr)
e(T1 ,Fr)
(1) (2)
(1)
e(F ,Fr)
(2)
(1)
e(Ti,Fr)
(1)
(2)
(2)
"Nor:nal
1.12 1.12
1 .....)
1.20
1.24 1.14
0.39 0.43
Uniform
1.30 1.33
1.20 1.22
1.24 1.31
0.35 0.26
Laplace
0.95 0.91
1.06 1.02
O.~
0.54- 0.64
(1)
{ 1, 0.75, 0.667, 0.50 }
(2.) { 1, 1.25, 1.50, 1.75 }
As mentioned earlier the use of s n, :L.
block-scores, results in a weigthed
0.86
ra~~ings
= n-i-l,
i
= 1, ••• ,n as
statistic with very poor
performance. On the other hand, when
using s n,:L. = i, i:: 1, ••• ,n ,
..
the originated statistic T performs, except for Laplace errors, much
1
better than Fr and for uniform errors even better than Ra •
It is interesting to note that the AREs are practically un affected by the use of one or the other sequence
{O'i.} . However,
in
order to conclude that these AREs are invariant with respect to~O'i
further investigation would be necessary involving other vectors of
alternatives
~
and other sequences
{O'i} .
1
CRAPI'ER VI
COYCWSI01G AJm RECOMMENDATION'3 FOR Ft.rl'URE RESEARCH
To summarize the results discussed in Chapters II, rl
~,d
V
~e
can state that :
a) -..Ieighted. rankings can recover much of the
interbl~c~
mation lOiDt when using tests based only on independent
b) In particular, T with
I
v.l.
= variance
and
s
ini'c::--
ran:-::i~~
.
n,l.
=i
;
,,'as
found superior to ?riedman' s test for normal and unifom e.:.--rt.-::5, and.
for Laplace errors if
n
is small
c) The results do not appear sensitive to the choice of mea.;;u:=e
of variability :
d) But the efficiency is sensitive to the choice of ~eights: in
particular, the "drop the least variable block" method did net
per±~~r::
well :
e) These conclusions still hold for the extensions i~ C~apter
We
~ust
recall that only treatment-scores of the
v.
fo~
t. = j - ~ (ill + :i.) were considered .
.J
The gra;hical procedure used in Chapter IV to estbate the A?.3s
snOi,s acceptable
a-~nce
a~curacy
despite its simplicity. However,
i.~ t~e
of any estimate of its precision, any comparison betveen
t~a
esti:nated. ARSs for different statistics must be· made cautiously. furtunately, ,the one::-ing between the esti!Jlatas obtained in :::-:a;ter 17
is,
L~ ~ost
situations, the same as between the
corre3~~r.d~ng ESL3
92
ciscussed in Chapter II. This fact can be considered as an external
nlidity check.
An
L~portant
area for future research is the development of
some procedure adequate to estimate the standard error of the AREs
j'.lst mentioned. Such a procedure might be (ideally) analytic, or
nU:llerical (linked to the computer simulation as was the standard error
of ESL(T.) ), or perhaps graphical •
~
A najor goal to be accomplished is the analytical evaluation
0:
the .l\...'tE for weighted rankings. The relationships between the
weighted
~~ing
statistics and the weighted average internal
~~
correlation may be crucial to work out this topic.
Conclusion Cd) suggests that further research on the choice
0:' weights or bloc:-{-scores may be rewarding. Such an investigation
could be combined with the study of treatment-scores
we have been using (Le.
t.
J
mal scores is an interesting
=j
othe~
than those
- t(m+l) ). The use of expected nor-
~ossibility.
Because of (c) ,
fu-~her
research on the choice of the measure of variability should be
deemed secondary •
As this work has been limited to only normal, unifor:n or double
exponential errors, it may be of interest to study the
t~e
pe~o~ance
of
statistics Cased on weighted rankings for other error distribu -
tions (U- or V-shaped distributions would be most interesting) •
The first extension discussed in Chapter V is also a field
that calls for additional study. For example, how does T1 per:orm
wten it is used to analyze BIB designs with a small number of blocks?
The expected significance level seems again the adequate
the
pro~~
SSLIJ
C=l~a=2J~
can be adapted to generate estLilates
): 3S:S
8-~d
standard errors in this situation •
For the second extension an
al~ernative
weighting
sche~e ~~at
can be studied in the future is
(!)
S o.
~
=s
.,,",
J.. -1
]I...
,
ni,J.
i.e., a combination of some
••• ,
intragrou~
n t ' ,{.[)-l
- , •.. ,q;
block-variability wei;3ts
a~
the group variability ranking, this last being known a l:.dor:'.
We look forward to handling personally at least some :;f
research topics.
th9~
APPENDIX A
A NOTE ON RANKmG AFTER ALIGNMENT
2
The "poorness of fit of
X
as an approximation to the distribu-
tion of Ra -the ranking after alignment test statistic- for small n is
illustrated in Fjgure A-I, where we d:Lplay the empirical distribution
(actually
1 - F ( ) ) for underlying normal, double exponential and
n
uniform error distributions. In fact, we have significant values for
the Kolmogorov-Smirnov statistic under all the error distributions we
have tried (normal, uniform, logistic, exponential, lognormal, Cauchy
and double exponential). These results are consistent with those
reported. by Gilbert (1972) and can be attributed to the fact that Ra
has a. finite ma.xi:mun.
Lem:na.- The maximun possible value for Ra in an
man
block
design is n(m - 1) •
Proof.- Let us consider the two most "extreme" configurations:
(a)
r ..
, ...... '"
1
B
1
0
c
k
s
:n+l
•
•
•
J.J
.m
.... ..... m+m
•
•
•
•
For (a) we have .Z Tj
=rln~
1
••••••••
(m-l)n+l
B
2
1
••••••••
•
(m-l}n+2
•
0
•
•
c
k
(n-l)m +1 • •• •• (n-l)m +m
(b)
Treatments
r ..
Treatments
J.J
s
•
•
•
n
••••••••
(m-l)n+ n
m (4m 2 -I) + in3 m2 + ~zm ,
J
E B~ = d-nt zn(2 +
•
J.
1.
6run + Jnzm z + n Z )
and consequently Ra =n(m - 1). For
e
e
e
FIGURE A-lo- X2.APPROXIMATION FOR THE DISTRIBUTION OF Ra (ranking after alignment)
1.0..-.....:: .......
r
1
+---+ 1
0---0
1
~---~ 1
.8
-
F(t) for
Fn (t) for
Fn (t) for
Fn(t) for
t= X,t. with 4 dof.
Ra with normal errors
Ra with uniform errors
Ra with laplace errors
(n :;: 1000 experiments with 3blocks and 5 treatments)
00
.4
.2
'l)
3
5
6
7
8
9
10
11
t
'-"
96
After perfor:ning an !\NOVA on the aligned ranks we can write Ha
as n(m - l)SST/(SST + SSE). Because SST and SSE are non-negative
quantities, n(m - 1) is an upper bound for Ra. But we r.a.ve just sho"lffi
that such upper bound is attainable. Then n(m - 1) is the maximun f02:'
Ra. •
(If we were willing to tabulate the exact distribution of Ra
we would have to consider (mn - l)t/(m - l)t(n - l)t basic and not
ec;.ually
lL~ely configu..~tions
for an
sets more than 1.8 billions for a
for a
)x)
nxm
)x5
experiment. This repre -
experiment, and 10080 even
experiment (the corresponding for weighted rankings
are 14400 and )6) ).
97
c····
C
C
C
C
C
C
C
C
C
I
c
c
C
C
C
99
199
C
C
C
89
c
c
189
C
79
179
C
C
c
c
c
22
23
24
21
AP!'E:mIX B.- PROCHA)! EStl) ANll !!ELATED SlJEROtrrI:es
PGII lSI. 13
(2/10/77)
ESL
ESt
Est
TH~S PBOGBIIl COIIPOTES
THE EXPECTED SIGHIFICAliCE lE'~
ESL
OF SEVER II. TEST STATISTICS THAt CAN BE 05ED TO ANALYZZ A
ESL
8l11DOIIIZED BLOCKS DESIGIf. VAiIUlJ. A~D COVARIANCE:S OF T::tES!
Est
ESTIlllT!S A8E ALSO OBT1INED Bt OSING THE SUBROUTINE ESLCOi.
ESt
THE OMtX EXTEBNIL SUBROUTINE BEQUlaED BT THIS P~OGBlll
ESL
lifO BOT LISTED HEBE IS YIB~EN (TeCC, 1971).
ESt
Zst
DI!ltlISIOIf 1(100,13),T(100,13)
ESt
bIIlElfSIOll Z 1 (100 ,100) ,Z2 (100,100)
Est
DIIlEliSION ZB 1 (100) ,ZR2 (tOO) ,ZCl (100) ,ZC2 (100)
ESt
DIIIUSIOM 'Ul (100) ,TTl (100) ,TI2 (lCO) ,TI2 (100)
ESI.
DIIlElISIOIf E (6,5) , B(5) ,8 (5) , SW (&) , EA (&) , Q (61 ,0 (6) , DEL (6) , BII (6) , C (5) ESL
D1IIENSION SrX(1J),SIt(lJ),TX(&,5),TY(b,5),ra(5)
ESL
DATA STX,STT,TX,TY/1l6*0./
ESt
D1TI X,T,Zl,Z2/22600*0./
ESI.
I,TEGER*2 OIST,PI.AG,DISTII
ESt
I.TEGEB 1ST
ESt
ESt
DEFINE NUIlB!8 OF TREAtllENTS,llOllBE8 OF BtOCKS PEa EX?EaLIIEIT,
ESt
BOIlBEB O~ EXPERI!ENTS lND BOIIBER OF STAtISTICS ( Plal~!T~a
ESt
CUD 1 ) •
ESt
ESt
REID (1,99) !,TS,B,TJ,HIf,tlfN,HS
ESt
FOBIlIT (3(I~,F~.0),I4)
ESL
RRIt! (3,199) 8,H,NI
ESL
ESt
rORlllT (lX,14,' TRE1TIlEUTS,',I4,' BLOCIS,·,I6,. EIPERI&E!TS',/)
ESL
DEFIlE TR!lTIl!BT EFIECTS (P1RI!ETER C1RD 2)
ESt
ESI.
REID (1,89) (TR(J),J-1,!)
ESL
FOBlIlT (10PS.4)
ESI.
1I11ITE (3,189) (1'R(J),J-l,lI)
ESt
ESt
rOBU.1' (1X,· TRUT8EIl1' EFFECTS: • .10PS. If,/)
ESt
DE1'IN! ERaoa DIS1'RI8U1'I01f lifO ITS P1RAIlE1'EBS (PABIIlETEB CliO 3) ES"
ESt
IIBAD (1,79) (DISl',FLAG,IST.1XU,BXBX)
ES!.
PaRIlAT (2I2,IS,2P8.1I)
Est
WRItE (3,179) (DIS1',IST,AUX,BUX)'
ESL
FORIIIT OX,, DIST: ',12,' SEBD:·,110,' P1B11I:·, 2FS. II,/)
ESt
ESt
DISTII-DIST-11
ESt
GElfBRATE THE SAIlPtES
ESL
DO 60 K-l,1I1f
EST.
ESt
GEHBRATE ON! PAIR OF SIIlPLBS
ESt
ESt
DO 50 1-1,1
GEHBRA1'! ONE BLOCK
ESI.
ESt
DO 1 J-1,II
11' (DISTil)
20,21,22
ESt
DIST-2
Est
ESL
CltCI'"1.
ESI.
CALL VARGBI(DIST,XST,FLIG,lllX,CXCX,TX(I,J),D)
IF (1'I(I,J)-0.5)
23,23,24
Est
ESt
TX(I,J)- BXBX*lLOG(2.*U(I,J))
GO TO 25
ESI.
1'X (1 ,J) - (-1.) *BX8X.ALOG (2 •• (1.-rx (I, J) ) )
ESt
GO TO 25
ESt
DIST-2
ESL
CIcr-l
ESI.
CALL VIRGEI(OIST,IST,FL1G,lXIX,CXCX,TX(I,J),DJ
ESL
TI(l,JI'" BI8X*TAN().14159*(TX(1,J)-O.5)
ESL
o
10
20
30
0110
50
60
70
80
90
FJO
110
120
130
'''0
150
150
170
lao
190
2')0
210
220
230
2~0
250
25Q
270
230
290
300
310
320
3 ''''
HI)
JV
3'SO
HO
370
380
390
400
1110
1120
430
1I1l0
.50
1160
no
480
1190
500
510
52')
530
SilO
550
560
570
sao
590
600
610
&20
630
6~O
98
APPENll!X B (continued)
GO to 25
20 CILL 'laG~'(DlST.IST.PLAG,AXIX,BXPI,TX(I,J).D)
25 tI(I.JI=~X(I.J) + TB(~
1 COIiTIlIOE
50 COliTIlIOS
CALL 81CaO(TI,E,8,»,STI,NS,C,B,a,B8,Si,BA,Q.D,DEL)
CALL nACaO(TI.E,8.N,SrI.NS,C,B,&,83.SV,BI.Q,D,DEL)
DO 51 JJ a 1, liS
X(It.JJ) a5 rx (.1.11
Y(It, JJ) aSTY (.1.1)
51 COIiUlIO!
60 COIlTUU!
DO 71 JJ-1.13
Jl·14-JJ
DO 72 lt a l.II'
1'11 (It)·x (It,Jl)
Tn (It) "I (lC,Jl)
72 Conu!)!
,)1I=J1-1
DO 73 J2-1,JII
DO 74 1t-1.111l
tx2(1t) -x (K,J2)
Tt2 (It) al (l< • .12)
7' COllUIOB
CALL ESLCOV(TX1,TI1,TX2,TY2.1II1,&sL1.ESL2.Vl.V2.C12.012.SD12)
WRITE (3.59) J1.J2.ESL1,ESL2,V1.'2.CI2.D12.SD12
71 connOR
71 COllrIllOE
59 rO&lUT (/.2l:3.8F15.5)
s1'ol,>
no
C
C
C
C
C
C
C
C
C
I:st.
lSI.
l5L
ESI.
!SL
I:~L
lSI.
lSL
Est.
ESL
ESL
ESL
ESL
Est
ESL
Est.
lSI.
Est
Est.
Est
ESL
ESL
lSI.
ESI.
r;SL
ZSL
ESL
ESL
ZSL
BSL
ESI.
ES:'C
SOBROOT1!E ESLCOV(TX1.Tfl,TX2.TI2,lI,ESL1,ESL2.Vl.V2.C12.D12.SD12) ZSLC
ESLC
»
1I!J~3~B OP EXPERlftEnTS
ESLC
D12 = D~FPEaENCE BET~EEN THE EXP. SIGN. LEVELS
ESLC
SD12
S1AlIDARO Ea~08 OF 012
ESLC
Txl " 'ECToa OF VALUES OF TH~ FIRST SEATlSTIC ONDER 60
zstc
TIl
V~CTOR OF VALUES OP TH! FI~sr STATISTIC UNDE& Hl
ESLC
Tx2 " VECTOR OP VALUES OF THE SECOliO STA~ISTlC rNOER HO
EstC
~Y2 = '~CTOR OF VALUES OP TH! SoCOND STITISTIC UNDEk 61
ESLC
EstC
OBElISIO' TX1(1) .TT1(1) .TX2(1) .'l'Y2(1)
!SLC
OII:EI/S10' Z 1 (100.100) .1.2(100.100)
EstC
DI!lEIiS10ll ZRl (100).1.82 (100) .1.C1 (100) ,1.C2 (100)
!SLC
1.1. 1"0.
ESLC
ZZ2=0.
!SlC
S10=0.
ESLC
S20=0.
EstC
5]0=0.
ESLC
SIIO=O.
ES!.C
511=0.
ESLC
S21=0.
ESlC
S]l"O.
ESI.C
541=0.
ES!.C
SI2"0.
EstC
S22=0.
ESLC
532=0.
ESl.C
SII 2= O.
ESLC
=
=
650
660
670
680
690
700
710
720
730
740
750
760
710
780
790
800
810
820
830
8110
850
860
870
880
890
900
910
920
930
940
950
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
20(1
210
220
230
240
250
260
270
99
APPENDIX B (continued)
SOB800111! BlCBO(T,E,",M,STl,MS,C,B,B,B5,SW,8A,O,O,OEL)
DIIIE1I5IOli T (N,Ii) ,E (N, II) ,S1" (IIS),C (Il),tl (1I),i (II)
OInnSIO.f Sli (1),81 (1l),O (II) ,0(11) ,D;;L (Il) ,BII(II)
DI!Il!:IlSIOIi !I (30)
L8 a ll*1I
TBS"O.
TBsaO.
SSC-O.
ssa-O.
TR"FLOn (5)
T"-PLOA l' (!I)
DO 35 0101-1,15
35 ST1(JJ)-O.
00 45 1-1,11
SIl(I)-O.
81(1)=0.
QU) =0.
0(1)"0.
DELlI) -D.
'5 COlirINUe
DO 55 011-1,5
55 C (JA) =0.
DQ 50 1-1,_
C
C
paOCESS Ol! BLOCI
85..0.
8B"0.
DO 1 J"'l,B
B(J) "''1(1,01)
BB-BB+B (01)
BS"'BS+B (01) *8 (01)
C(01)=C(J)
B(01)
COnUD!
85 (I) -SB/T!!
SII(I)"B5 - (BS*B8)/TB
+
TBBaTBB+88
TB5-TBS+BS
SSe"SSB t eS-aB
C
'"lll THIS BLOCK lID STOBE ITS BlllKIWG
C1LL B181(9,B,B)
00201·1.11
or (I, 01) -8 (01) - (Tn+ 1.) /2.
!(I,J):B(J) -&n(I)
2 cOllruop:
THIS BLOCK IS SORTED
CUL 50ilT (a, !I)
COIIPDTE 8111GE AIID IB't!8QDA8TILB DIPF. (5-4,5,6,1)
C
C
III (I) "'B (If) -8 (11
11 (8-4) 11,12,13
13 Q (I) =B (ft-l) - B (2)
GO TO 11
C
12 0(1) "'(B(~ltB(3)-8(2)-B(1)t/2.
COIIPor! IIEll OEVIArIOI AWD BElli DIFPERENCB
11 DD"'O.
DEaD.
'''0.
DO 30 o1 a 1,1l
DO-OD+ABS(B(J)-B5(I»)
'''P+1.
DE=DEH'*S (01)
30 COIITUUE
0(11 "DD/l'!!
DP;L(I)=(~*DB-2*(Tnt1.).88)/(T5.(T5-1.))
~O
COIITIIl.,P;
CALL HII(r,Si,5,II,H,HH,ST1(1),STl(6)
CALL H!l(1'.a1,II,lI,H,H~,STA(2),STl(7»
CALL HY (T,O,II,II.lI.Hff,SrA (3) .STA (0))
CALL IlN(T,D,:I,lI,fl,lIH,SfA (4) ,STA (9)
CALL HIl(T.O::L,Il,ll,H,IlIl,STA (5) ,STo\ (10»)
CALL
S~B!I(T,E,U/~,lI,LR,STA(11).S~A(12))
DO 3 J
l
a
l,:I
SSC=ssc ~ (C(J,*C(J))
FH=«TH-1.) *(Tn*55C-TB8.T88))/(T5*T!I*TBS-Ta.SSC-TIl.SS9+TB8·~BB)
STA(13'''?lI
R~TUR!l
END
IllCR
IllCR
!lIC8
IUC8
IIICB
5lCa
1I1CB
IlleB
B1CR
UCR
BICB
nlC8
B1CR
IIICR
laCR
IlICR
IUC8
!llCa
IIICa
IllCR
1I1CR
IIICB
!lICR
IIICR
IIlCR
1I1Ca
IIICR
IlICa
IIlCR
BICa
IIICR
1I1CR
IlACR
IIlC8
IUell
nlCB
IIICR
1I1CR
1l1CB
IllCR
IIlca
IIlCB
IlICR
IllCR
IIICR
BICR
IIlCR
illeR
IIlCR
II1CB
IUCIl
1I1CR
IIICR
IIICR
1I1C8
IlAC8
IllCB
IUCB
B1CR
o
10
20
30
1i0
50
60
10
80
90
100
110
120
130
1 .. 0
150
160
110
180
190
200
210
220
230
240
250
260
210
280
290
300
310
320
330
340
350
360
370
380
390
.. 00
IilO
420
430
IIIiO
450
460
470
1180
490
500
510
520
530
540
550
560
570
580
UCB 590
IIICR 600
IIICR 610
IlACB
lUCR
IIlca
Illca
5lCR
II'CII
IUCR
IllCR
IIICR
lUCIi
:lICa
IIACR
IIACR
620
630
640
650
&60
67()
6130
6'10
700
710
720
7J0
740
lI"ca 150
IIICR 760
100
APPENDIX
B
(continued)
lSLC
l5LC
ESLC
},;SLC
ESLC
l5LC
ESLC
ESlC
ESLC
TN-fLOAT(!!)
DO 90 I=l,1lI
DO 90 J=l,1lI
Z 1 (I, J) =0.
lo2(I,J) =0.
ZC1(J)=O.
loC2(ol)=0.
ZR1(I)=0.
ZR2 (I) =0.
90
~SLC
COIl'IINO~
DO
DO
It'
IF
1P'
IP
100 1=1,11
100 J=l,lI
(J. EQ. I)
GO TO 100
(TXl (I) .CT.TYl (J»
(TX1(I).EQ.TY1(ol»
(TX2(I).GT.n2(J»
Il' (TX2 (I) • EQ. TY 2 (J) )
lol (I,J) =1.
41(I,ol)=.5
lo2II,J)"'1.
lo2(I,J):.5
100 con IlIUE
DO 150 I" " III
DO 150 J=l,]I
ZR1(1)=ZR1(I)+Zl(I,J)
ZB2 (I) =loil2 (I) +lo2 (I,J)
150 CO~'l'IIIUE
DO 210 J"l,'
DO 200 1=1,11
loCl(J)=loCl(~)+Zl(I,J)
loC2(J) =ZC2 (ol) H2 (I ,J)
200 COlIlUUE
ZZl=ZZl + ZCl (ol)
ZZ2"ZZ2 + ZC2(J)
210 conn':E
E5Ll=Zlol/(I'1I" (TN-l.»
l5L2-ZZ2/ (1' II" (1'11-1.»
012=E5Ll - ::5L2
DO Jeo J:l,!!
DO 310 I'll,!'
510=510
L 1 (I ,J) .Z2 (1 ,J)
530-530 + Zl(I,J).Z2(J,I)
511=51'+Z1 (I ,J) *::1 (I,J)
S31 c 531tZl (I,J) "Zl (J,l)
S12=512+Z2 (I,J) "Z2 (I,J)
532=532tZ2(I,J)"lo2(J,I)
310 COIIT IN UE
520=520 + (Z81(J)+tCl(J»)"(ZR2(ol)tloC2(J»
521= 521 + (2C 1 (J) +ZB 1 (ol) ) "(ZC 1 (ol) +LR 1 (ol) )
522= S22+1ZC 2 (ol) tZR2 (ol) "(tC2 (J) +<:82 (ol»
300 COnINCE
Ql=(~.*'!'!1 - 6.)/«TlI-2.) "(TN-J.))
Q2=TlI*TlI*(711-1.)*(TII-l.)
520 " 520 - 2.*510 - 2.*530
S40=zlol*ZZ2-510-520-5]0
C12: (510+520+530 - 01":i~1)) /02
521 = 521 - 2.*SI1 - 2."531
S41=ZZl*ZZl - 511 - 521 - 531
Vl= (511+:>21+531 - Ql*:;41)/Q2
522 = 522 - 2.*512 - 2. a jj2
5Q2=ZZ2*Zz2 - 512 - 522 - 5J2
Y2 a (512tS2l+532 - 01=542)/Q2
S012=son IV 1 + V). - 2.=C12)
+
P.ET'JlU
Ell 0
E5LC
ESLC
ESLC
ESLC
lSLC
ESLC
ESLC
ESLC
ESLC
ESLC
E5LC
l!SLC
ESLC
lSLC
lSLC
E5LC
!51C
E5LC
lSLC
ESLC
EstC
ESLC
lStC
ESLC
I5LC
ESLC
ESLC
l!SLC
E5J.C
ESLC
ESLC
lSLC
Esr.C
ESLC
ESLC
ESLC
ESLC
ESLC
l5LC
ESLC
ESLC
lSLC
ESLC
lSIC
ESLC
ESLC
ESLC
ES1.C
ESLC
ESLC
ESlC
280
290
300
310
320
330
J~O
350
360
370
38')
390
400
410
420
430
440
450
460
470
"ao
490
500
510
520
530
540
5~O
560
570
580
590.
600
610
620
630
640
650
660
670
680
690
700
710
720
730
7.. 0
750
760
770
780
790
800
8tO
820
830
8'<0
850
860
1370
sao
1(:1
APPENDIX
C
C
C
C
C
C
C
C
C
C
C
Is
(continued)
BI
s,
SOBPOUTINE SM(T,V,ft,N,H,HB,C,CC)
coaPOTE QUAD!"S HM STATISTIC T~ 'MALTZ! RAMD08IZED SLOCES USIMG aJ
B:I
UliltIlIGS
all
8· NOSBEB OF TBEATaEWTS
1 2 IOft8!S O? BLOCKS
3'
!!'
'(I)- VECTOS OF nEASOBES OF 'ASIABILITY, l~l, ••• ,N
a,
T(I,ol)= BATRII OF RANKS WITHIM BLOCKS ALIGNED ON THE aEAI
SI
DIIIEIISIOli T (II,!!) ,V (II)
SJ
DI!Il;ISIOM ST(10),SS(10I,Vi(6)
lU
TUE 5 (I) ~ RANK OF V (I)
SJ
C08POTZ THE STATISTIC HN
BI
CALL SAIK(Y,VB,N)
SI
SaO.
HI
00 62 ol 2 1,11
al
st(ol)-O.
81
DO 61 1:01,.
al
ST (ol) -ST (JI i-T (I,ol) *U (I)
BJ
61 COllTno!
BI
S~BtST (ol) ·ST (.:I)
BII
62 COIlUNO!
BI
P.PLOlT Ill)
QaPLOAt (a)
B'
BI
DEll-0*(Otl.)*P*(Pi-l.)-(2.*P+'.)
BlI
C.. 72.-H/DElI
BI
CHAMGE TSE aEIGBTS 5(1)
Sl
00711-1,11
BI
IF (VI(I)-l.) 72,72,73
BI
72 n (I) -0.
BI
GO TO 71
RI
73 'f8 (I) -1.
SI
71 COHIIIOE
HI
conpOTE T~! STATISTIC CI WITH THESE HEW WEIGBTS
BI
BIt-O.
31
DO 82 ol-l,8
HI
5S(J)-0.
B!!
DO 81 1-1,11
Sll
55 IJ) -55 IJ) +1' (I,ol) -U (1)
!!.lI
81 Conno!
liS
BH'"HH+SS (J) -55 (ol)
BJ
82 COUIIIO!
SS
DES- (P-l.) *0* (Q+l.)
U
CC·,2.-i1B/Dilll
HI
In:)u
SS
lID
5DSPOOTI'E SORTI8,1I)
SORT A GITEN VECTOI or LEHGTH
DIl'J!IISIO!l 8 (Il)
!J:I-1I-l
00 20 ol-l,1S1I
'UIIP-B(ol)
IIS-J
r-oltl
DO 25 L-lC,!!
IF (8(L).GE.TBIIP) GO TO 25
TEISP-a (L)
IIS-L
25 CONl'IlfDE
IF (ftS.EQ.ol) GO TO 20
8 (!lS) "BIJ)
81JlzTEftP
20 con no!
UT!JU
EMD
II
:I
10
20
30
'H)
SO
6\l
70
80
900
100
11')
12~
130
140
150
160
170
laQ
190
20~
210
220
230
2i4:J
25;)
260
270
2BO
290
300
310
320
330
3,.0
350
360
370
380
390
'00
'10
420
'30
no
SOilT
o
SOiT
SOIl'
SOiT
son
SOilT
SO_T
10
SalT
SOU'
SOiT
SORT
soaT
SOiT
son
SOIT
SOIT
~jr
20
30
110
50
60
70
80
90
100
110
120
130
1'0
1:>J
160
salt 17"
SalT 190
SORT 190
102
SIBN
SNBN(T,e,D,ft,N,LB,SN,bu/
SIIBI
COapUT! SN (FRIEDK1N STATISTIC) AND BM (aAHKIlI~ APTER ALIGllllEIT)SlIBlI
azNOKBER OF TREATftENTS
SWBN
SIBN
N=NoaBER OP BLOC~S
T(I,J)=aATBIX OF RA~KS WIT9I1 BLOCKS ALIGHED UN THE ~EAI
SliBl
!(I,J)aftAT8II OF OBSERVATIONS WIT311 BLOCKS ALiGlIED Oll T3! ftE1ISSSM
SIiDlI
DISE~SI03 T(~,ft),E(N,~I,O(LB)
DlftElISIOIl PR (5) ,TB (5) ,S (6) , U8 (30)
SHBII
coftPor~ TH! STATISTIC SI
5MBII
SNaO.
S!lBIi
SNBM
DO 12 J"l,11
SIfBN
PR(J)"O.
DQ 11 I a l,1
SSDlI
SIIDII
PR(J)"PR(J)tT(I,J)
SIIBN
COlirIHO!
SUI
SHwSlltF8(J)·PR(J)
SIlBM
COlirIllO~
SIIDII
P=PLOlT (II)
SIIDN
g..rLOlT(!I)
SIIBII
SN-12 •• S!/(Q*(Otl.)*P)
COapOT! TH! STATISTIC BM
SI8N
SIBil
00 22 I-l,M
DO 21 Ja 1, II
SIBM
SIBil
1(-1I. (1-1) tJ
SlIBN
O(K) -!(I,J)
SIBI
COn'1I0!
SIlBII
COIiTIIIU!
5MBM
CALL dAlIK (U,08,L8)
SNBIl
Tr"'O.
SIBN
DO 32 J-l,1l
TR (J) "0.
S'BN
SI/BIl
DO 31 1"'1,1
TB (J)"TJ (J) tU8 (II. (I-l)+J)
SUN
SIfBN
CONTINOE
SIIBN
Tr=rrtT8(J)·TB(J)
SIBil
COllnNU!
SDBROUTI3~
C
C
C
C
C
C
11
12
c
21
22
31
32
20
30
110
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
210
220
230
2110
250
260
270
2dO
290
300
310
320
330
340
350
360
SIBil 370
SS"O_
DO 112 1-1,11
5(1)-0_
DO 41 J-l, II
5 (I) -5 (I) tUB (S. (1-1) tJ)
111 COll'rIIU!
SSwSStS (I) ·S (I)
112 COITINUE
DN w l.S*O·(Q-l.)·(Q_·rT-o·p·P*(Q·Ptl.)*(O·Pt1
DD a O·O·P*(Q·Ptl.)*(2.·Q·Ptl.)-6.*SS
BNsDH/DD
a
10
.»
SIIBll 380
SlIPM 390
SIIBI "00
SIBM 1110
Sliall "20
SUI "30
SIIBII 1140
SI/Bif "SO
SIlIBI 460
SIIBI "70
UTDU
SIIB. 1180
END
SIlIBIi
Q90
APPENDIX C
TABLES Oli' EXPECTED SIGNIFICANCE LEVELS,
THEIR DIFFERE...lIlCES AND ST&'IDARD ERRORS
In each of the following tables the last row contains the 3SLs,
the u:;>per triangular matrix shows the differences between
t..~esa
ES!.s
and. the lower triangular matrix displays the stand.ar1 errors of such
1i.f'ferences •
TABLE C - 1.- ESLe, their differences and cOITespondiJlg standard eITore
T
Ti
T1
j
T1
0.001
T
--
T
'1'4
J
S
T6
'1'7
0.003*
0.001
0.046-
0.044-
0.002
0.000
0.045-
o.o4J*
-
--
'1'2
1'l.003.57
T
J
--
-
-
'1'4
"1.11)269
"1.00447
--
T
5
"1.003.57
0."10000
-
0.0047
T
6
1).02101
0.02162
-
0.02163
0.02162
T
7
"1.02275. "1.02171
-
0.02332
0.02171
0.01137
-
-0.002
--
0.04)*
0.041*
0.045**
0.04)*
-0.002
-
--
--
-
--
--
-
'1'9
"1.')1<]95
0.1'l2060
-
0.01982
0.02060
0.00637
0.01300
11.1')227.5
0.02171
--
0.02332
0.02171
0.01137
0.00000
Fr
0.012115
1'l.'11257
-
0.012))
0.01257
0.03(8)
0.0311l4
Ra
1.1'l12f))
f).onBl
--
0.012')1
0.01181
0.f)2.514
0.02616
r
'1.01.587
O.015JO
0.01624
0.015')0
0.02588
0.02633
ESL
0.227'37
0.2281f8
0.23081
0.221l48
0.27379
0.271.57
Idj11 '>
2e.e. (d j1 )
•
I.e. (d j1) <
I dj11 '!a
--
2a.e. (d j1 )
,
..
..
J, •
Eo
3).
T8
'1'9
'1'10
--
0.0.57-
0.044*
0.019*
-').020*
-0.025*
0.0.56-
0.043*
0.018*
-0.021*
-0.026*
-
,-
--
-
T8
'1'10
e
'1'2
(norul BITers, n;:
-
---
---
-
--
lr
--
Ba·
---
F
--
0.0,S!l-**
0.045*
0.015*
-0.023*
-'1.029*
0.0.56-
0.04)*
0.018*
-0.021*
-0.026*
0.011*
-0.022
-0.028
-o.:l66- -0.072-
0.013*
0.000
-0.025
-0.064- -0.069**
-
--
--
-0.013*
0.01300
--
-0.039
-0.077** -0.083**
-0.025
-0.064- -0.069**
-0.0)8** _').'14 /1"
0.02717
0.0311l4
0.02/119
1).02616
0.014')4
1).02529
0.026'))
a.018')1
0.00891
0.21l480
0.27.517
0.211621
0.20m
-0.006
0.20222
.....
~.
d J1 ;: ESL(T ) - ESL(T )
j
1
e
e
e
e
e
TABLE C - 2..- ESLs. their differences and corresponding standard errors
T .
Ti
T
2
T1
j
T1
0.006-
T)
T4
T,5
T6
T
1
T
0.059"
0.067-
0.042*
8
(nona! errors. n ~ 3 •• ':" ).
T
9
T10
Ra·
0.0)6*
0.0,50" -0.014
-0.027-
-0.0)3- -0.d.2 u
-0.009
-O.OU n
-0.015
-0.017" -0.009.
0.0,52**
0.061**
0.0)6*
0.0)0*
0.044*
0.006*
0.061**
0.016"
0.051-
0.044*
0.0.59" -0.00,5
-0.019*
-0.027"
0.009*
0.010-
0.076-
0.0,54-
0.047-
0.061" -0.002
-0.016*
-0.02.5.
0.061"
0.070-
0.045-
0.0)8*
0.052** -0.011
-O.O~*
-0. oj4 II
-0.002
T
2
0.00450
T)
0.00697
0.00,569
T
4
0.0052)
0.006'74
0.00441
T
5
0.00626
0.00479
0.00)39
0.00511
T
6
0.02146
0,02299
0.024,52
0.02272
0.02)6.5
T
0.02126
0.02095
0.02268
0.02246
0.02170
0~00981
TO
0.02151
0.02123
0.02011
0.02047
0.0200,5
0.01599
0.01311 .
T
9
0,0225)
0.02)86
0.02339
0.021.51
0.02296
0.0108,5
0.01421
0.01140
T10
0.02253
0.02222
0.02205
0.02202
0.02098
0.01399
0.01019
0.0090)
0.01226
Fr
0.01506
0.01470
0.01441
0.01495
0.01471
0.0)462
0.03350
0.03269
0.0JI~'71
0.OJ384
Ra
0.01.597
0.01.597
0.0147.5
0.011.79
0.01509
0.0)0)2
0.029.52
0.0280.5
0.0296.5
0.02889
0.01539
r
O.OHY~o
0.01879
0.01199
0.01162
0.01191
0.02012
0.02772
0.02100
0.02779
0.02'139
0.02092
0.01212
ESL
0.2.5106
0.25717
0.2424 7
0.2398.5
0.24864
0.J0965
0~J18J)
0.29)43
0.28671
0.J0101
0.23747
0.22)69
7
*
8.e.(d j1 )
r
1r
-0.00)
< Idjil ~ 2a.e.(d j1 ) I ..
0.009
Id J11> 2a.o.(d j1 )
-0.020·
-0.040'
-0.016*
-0.023" -0.009
-0.072*
-0.02.5·
-0.0)2- -0.017
-0.081" -0.09.5....
-O.lO)u
-0.007
0.008
-0.0,56-
-0.070"·
-0.078--
0.014
-0.049·
-0.(6)-
-o.onn
-0.064*
-0.071- -0.086*'
dJl '; ESL(T j) - iSL(T1)
-0.086** -0.095 ....
-0.014
-0.022 0
-0.010
0.21.50~
.....
•
o
\.JI
TABLE C - :3.- ESte. their differences and corresponding standard errors
'1'1
'1'
'1'1
j
T1
'1'2
-0.005
(noI'll&1 errors. n ~; •• :5").
'1'6
'1'7
'1'8
'1'9
o.~
0.064**
0.062**
O.077 n
0.087**
0.076**
0.005
-0.016-
-0.026·
0.010*
0.008*
0.069**
0.066**
0.081"
0.092**
0.080"
0.009
-0.011
-0.022-
0.002
0.000
0.061**
0.058**
0.073"
O.O~**
0.072**
0.001
-0.019*
-0.0)3.
-0.002
0.058-
0.056"
0.071"
0.082-
0.070" -0.001
-0.021*
-0.0)2-
0.060-
0.058**
0.013-
O.O~-
0.072-
0.001
-0.019*
-0.0)0**
0.01)
0.023-·
0.012
-0.0.59- -0.080- -0.090-
0.01.5
0.026*
0.018
-0.0.57** -0.077** -0.088**
0.011
-0.001
-0.072- -0.,,/.3- -0.10)**
-0.012
-0.08)** -0.10)- -0.114-
'1')
'1'4
'l'5
0.003
0.006
0.008
'1'10
1r
Ra
'1'2
0.'lO6')8
'1')
0.')15)9
'>.01379
1'4
0.01060
0.00974
0.00775
'1'5
0.'1'>711
f).007l)
1).01292
0.00729
'1'6
0.01865
0.02081
0.02788
0.02171
0.021)4
'1'7
0.02183
0.01980
0.0)0Ii7
0.02AA3
0.02265
0.01602
'1'8
1).02864
0.0)072
0.01860
0.02552
0.02801
0.0).599
0.04001
'1'9
0.02406
0.02562
0.02602
0.02017
0.02352
0.020)2
0.02577
0.0)051
'1'10
').02144
0.'>2172
0.02706
0.02148
0.01880
0.01572
0.01833
0.0)469
0.01895
Fr
0.1)1025
1).01009
0.01098
0.0104)
0.00952
0.026&.
0.02742
0.028)2
0.02887
Ra
'1.01269
0.01)09
f).01479
0.01)15
0.0116)
0.0211)9
0.02,571.
0.02761.
0.02685 . 0.02))2
0.01261
r
0.(16)')
0.01672
o.016'1f
0.01595
0.01.l~5
0.021.96
0.02659
0.02708
0.02629
0.02271
0.01699
0.00926
ESL
').16')20
0.15.566
0.16)64
0.16606
0.16)79
0.2~19
0.22207
0.2)712
0.24768
0.2)576 . 0.W.90
0.14460
-0.002
F
-0.071- -0.091** -0.1020.02651
-0.020-
-0.0)1-0.011·
0.1))1.
I-'
•
e
•••• {d j1 )< Idj11~ 2s ••• (d j1 )
I ..
IdJ1 1>
2S.8.{d j1 )
e
dJ1
~
()
ESL(T ) - ESL(T1) •
J
0\
e
e
e
e
TABLE C - It.- Este. their differences and corresponding standard errore
T
Ti
T1
j
T
2
1i
T)
-0.001
T2
0.~158
---
T
J
14
-
O.OOlSS
'1./)()22S
T
S
0.00158
'1.00000
T
6
'l.a13')S
0.'11324
T
7
"'.')1148
0.01)28
T8
-
T
9
0.01308
0.01327
T10
0.01)48
Fr
Ra
T
7
1
1
8
T10
9
lr
,
Ba
-0.001
0.042-
0.0)8-
-
0.042"*
0.038-
0.011
-0.028- -0.032-
0.004-
0.000
0.043"
0.039"*
-
0.043-
0.0)9"*
0.012
-0.026"* -0.031-
-
--
--
--
--
O.dJ.O-
O.OJS-
0.040-
O.O)S-
0.008
-0.031- -o.03S-
0.04)-
0.0)9"*
0.04)-
0.0)9"*
0.012
-0.027- -0.0)1·
-
T6
0.(0)*
--
1'4
'l'S
(nonal errore. n "" "1- •• "" ., ).
-
I
-0.004"
O.OO22S
__
-
-0.004
--
--
0.000
-0.004
-0.032*
-0.070- -0.074*
0.004
0.000
-0.027"
-0.066- -0.070-
--
-
0.01336
0.01324
0.01)76
0.01328
0.00406
--
-
--
--
-
0.01)))
0.01327
0.00290
0.00504
0.01328
-
0.01)76
0.01328
0.00406
0.00000
0.012)0
0.01244
-
0.01226
0.012AA
0.02172
0.02200
--
0.02155
0.02200
1).01269
1l.0l')l)f)
0.01261
0.01)00
0.01849
0.01881
0.01871
0.01081
0.101306
,
--
1).01496
0.01500
0.01406
0.01.500
0.02018
0.02081
--
0.020,56
0.02021
0.01651
0.00067
ESL
0.171'l6
O.17f110
0.17J7I1
0.17010
0.21'}))
0.20699
--
0.21:M
0.20899
0.18167
0.111)18
-
1I ••• (d
--
)< Id
J1
1-s:
J1
-
21 ••• (d j1 )
,
..
Id j1 1>
2ll ••• (d j1 )
d j1
--
C
--
-0.004
0.00504
iSL(T j ) - i3L(T )
1
-
-
--
-0.0)2"
-0.070- -0.075-
-0.027*
-0.066- -0.070·
-0.0)8**
-O.~I)*
-O.O~I
O.DU'I
~
(.)
'-...J
TABLIi:
Ti
T
T1
j
11
C -
5.-
T
2
0.004-
ISLo, their differences and oorreeponding Iltandard errors
T4
T)
T
S
T6
T
1
T8
(noJ"llA1 errorB , nil:" , .,. ~ ).
9
T10
1r
T
Ba
po
-0.008·
-0.00)
-0.002
0.011
0.01)
-0.018-
0.001
0.004
0.010
-0.023-
-0.0)7**
-0.011-
-0.001-
-0.006-
0.008
0.010
-0.021-
0.004
0.000
0.006
-0.027"
-0.040....
0.005
0.00.5-
0.018"
0.021-
-0.010
0.015
0.11
0.011-
-0.015·
-0.029·
0.000
0.014
0.016-
-0.01:5-
0.010
0.007
0.01)
-0.020-
-O~O34**
0.014-
0.016"
-0.015"
0.010
0.006
0.012
-0.020"
-0.0)4....
0.002
-0.029** -0.004
-0.008-
-0.001
-0.0)4-
-0.048....
-0.031.... -0.006
-0.010-
-0.004
-0.036-
-0.050-
0.021"
0.028
-0.005
-0.019"
0.003
-0.030"
-0.044....
0.006
-0.027"
-0.040-
T2
0.004)4
T)
1.00705
0.00691
'r4
0.1)())81
0.00.575
0.005)2
T
5
0.00431
0.00440
0.00464
O.0cAA6
T
6
1).01339
0.01338
0.01632
0.01464
0.01)74
T
7
0.01385
0.01305
0.016)6
0.01504
0.01391-
0.00539
...'8
'1.'11489
0.01429
0.01303
0.01464
0.01356
0.01231
0.01253
T
9
0.01340
0.01)43
0.01595
0.01386
0.01367
0.00532
0.00758
0.01197
T10
0.01)68
0.01326
0.01592
0.01482
0.01)42
0.00.508
0.00522
0.01142
0.00711
Fr
0.01395
0.01437
0.01325
0.01379
0.01)85
0.02382
0.02401
0.02191
0.02337
Ra
0.01422
0.01379
0.01)08
0.0134)
0.01)41
0.02133
0.02071
0.01956
0.02058' 0.02125
0.01351
F
0.01596
O.Ol.5/YI
O.Ol~/Il
0.01.558
0.01"'2
0.02177
0.021111
0.021176
0.021119
0.02202
0.01628
0.009<11
ESL
0.17833
1).18197
0.170Ul
0.119+5
0.17586
0.189'10
0.19177
0.16056
G.16556
0.18202 -, 0.18823
O.15!1W
0.025....
-0.004
0.02)61
-0.033.... -0.046-0.014"
0.14162
I-'
•
e
•• o.(d J1 ) < Jd J1 1~ 211 ••• (d J1 )
( ,J
I .. IdJ11;, 211.CI.(d j1 )
d j1
e
K
£UL(T j ) - ~L(Tl)
CO
•
e
e
e
e
TAIlLE
T
T1
Tl
j
T1
6.- ESte, their d1fi'erencee and corresponding standard errors (nol'll&1 errors, n; It-, •
C-
T
2
-0.000
T4
T)
T
5
T6
T
1
'r9
T
a
TIO
z
S ).
Ir
Ra
-0.00)
-0.005*
0.001
0.02)*
0.0)1"
0.01)
0.016*
0.0)7-
0.004
-0.020-
-0.0)2 U
-'l."lO)
-0.005
0.001
0.02)*
0.0)1-
0.01)
0.016*
0.0)7-
0.004
-0.020"
...' ).0)2-
-0.002
0.004
0.028*
0.040"
0.016-
0.020"
0.040"
0.008
-0.011-
-o.029 u
0.007
0.021*
0.042-
0.018
0.022*
0.(4)-
0.009
-0.015"
-O.021 u
0.001
0.0)6-
0.012
0.016*
0.0)6**
0.00)
-0.021*
-O.oY~-
0.014*
-0.010
-0.006
0.015
-0.018
-0.048** -0.0.55-
-0.024*
-0.020*
0.001
-0.0)2"
-0.057-- -0.069"
0.004
0.024
-0.008
-0.0))"
-0.045**
-0.0)6*
-0.049""
T2
0.'1'l4 52
T)
'1.'l1<'21
0.01199
T4
O."'l451
'l.'l'lYl5
0.01116
T.5
'l.(\'1514
0.00429
'1.0115.5
0.00)64
T
6
1).01199
0.01)08
0.018))
0.01280
0.01)0)
T
7
0.01)6)
O.012n)
0.01865
0.01).5)
0.01251
0.01011
1a
0.02005
0.01999
0.01211
0.01904
0.019)6
0.02268
0.02)80
1
o,'l1)0)
n.01)92
0.01814
0.01187
0.01274
0.00824
0.01266
0.021SO
T10
1).01466
0.01440
0.01699
0.01)82
0.0125)
0.00996
0.00901
0.02282
O.OO~O
Fr
0.n114Z
0.01244
0.01086
0.0115)
0.01241
0.018,52
0.02042
0.01909
0.01881
Ita
().'l1l?1
0.'l12.55
0.012)0
0.01Z12
0.0125.5
0.018,56
0.020.58
0.01889
0.01909· 0.02144
0.01060
,
0.01)21
0.01168
0.011t12
0.01))1
0.01J119
O.OIOn
0.020,51
0.01902
0.01!1f.s
0.02111
0.011161
0.0094)
tSL
0.12,,00
0.12'10)
0.12470
0.12247
0.12909
0.15051
0.16460
0.14061
0.14424
0.16510
0.1)221
O,lOn8
9
F
0.021- -O.OlZ"
-0.0))"
0.0213)
-0.051- -0.070-0.024- -0.0)7-O.OlZ-
0,09:115
,t-')
-
..... (tl Jl) < J.l Jl1 ~ 20.11. (tl Jl )
I ..
IdJl1 ~
2,,,,,, (tl jl)
dJi
K
EaL(T j )
- ~L(Ti)
,
\{>
1.- ESLs. their differences and oorresponding standard errors (nonal errors • n .. $ ..... 3 ).
TAIlLIl: C -
Ti
T
T1
j
T1
T
T)
'1'4
0.002
--
-0.000
0.002-
O.(~W"
O.OJ9-
-0.002
0.000
0.oJ8"
0.OJ8**
--
-
--
-
O.~O"
O.OJ9"
0.OJ8··
0.OJ8**
2
--
T2
0.00091
T
J
--
--
'1'4
1).'10152
0.00189
T
S
1).00091
1).1)0000
T6
').014-")7
1).01)89
T
7
0.01)98
0.01)81
'1'8
--
-
-
T
9
0.01)81)
0.01)66
T10
0.01396
Fr
-
'r.5
0.002
0.00189
'1'6
'1'7
T8
'1'9
T10
-
0.OJ7··
0.OJ9H
0.00.5
-0.027.- -0.024·
0.0)6*·
0.OJ8-
0.004
-0.029-- -0.025·
--
--
0.037"
o.OJ9"
0.006
-0.02.,,, -0.024.
0.0)6"
0.038--
0.004
-0.029" -0.025.
--~
-
Yr
--
Ra
--
0.01375
0.01389
0.0136.5
0.01J81
0.00046
-
--
--
0.01797
0.01353
0.01)66
0.00)1
~.01381
-
0.01365
0.01)81
0.00046
0.00000
0.01249
1).01256
-
0.012)1
0.01256
0.01828
0.01821
--
0.01797
0.01821
Il&
o.nl197
0.01196
0.01171
0~01196',
0.01'119
0.01710
-
0.01681
0.01710
0.01142
r
1).01195
0.01187
0.01)76
0.01'}O7
0.01917
0.01908
-
0.01079
0.01908
0.Ol'}87
0.00561
~L
0.11641
(J.11798
.." .
0.11636
0.11796
0.1.5616
0.15556
-
0.15379
0.15.5.56
0.12107
0.08924-
20.0.(4
> I ..
J1
....
--
-0.001·
---
-
0.002J5
F
-
-0.002·
-0.001·
-0.0)4·
-0.06"" -0.063"
-0.002
0.000
-0.0)4·
-0.066- -0.06)"
--
-
..
0.002·
0.002)4
--
--
--
-0.0:)2 . -0.065** -0.061··
-0.0)4-
-0.066** -0.(6)"
-0.03)** -0.029"
O.O~
0.0928)
1-'
•
e
•. 0.(d J1 )< / d
J1 1'"
IdJ1 1> 20.0.(I1 J1 )
d J1
e
;I
i:51,(T J>
.. i3L(T1 )
I-'
o
e
e
e
e
TABLE C - 6.- ISLa, their d.1.fferenc8s and corresponding standard eITors
T
T1
T1
j
T1
0.003"
T
S
T4
't
T
2
J
T6
T
7
Ta
(norlllll eITors , n" 5 , • ~ 4 ),
'1'9
'1'10
Ir
Ba-
-0.003
0.005*
-0.003*
0.009*
0.014"
0.002
0.014*
0.007
0.024*
-0.012*
-0.023**
-0.006
0.001
-0.006*
0.006
0.011*
-0.001
0.011*
0.004
0.020*
-0.015*
-0.027**
0.000
0.012"
0.017*
0.005
0.017·
0.010
0.026'* -0.009
-n.021*
-0.007·
0.004
0.010
-0.003
0.010*
0.003
0.019*
-0.028"
0.012"
0.017*
0.005
0.017*
0.010
0.026** -0.009
0.006*
-0.007
0.005
-0.002
0.015
-0.021*
-0.013
-0.000
-0.007·
0.009
-0.026·· -0.0)0--
0.005
0.022·
-0.014
-0.025"
-0.007
0.009
-0.026*
-0.0)8**
0.016*
-0.019*
-0.1)31**
T2
0.00253
'l)
'l.OO;fi7
0.00631
1'4
0.1)04)6
0.00529
0.00325
'l
".1)0268
I) •
oo)lJ5
o.O()/J7)
0.oor167
T
6
'>.00821
0.008'74
0.01122
0.01079
0.00924
'l
0.00896
0.00856
0.01152
0.0112h
0.00965
O.Q()/J82
Ta
0.01117
0.01190
0.00765
0.00913
0.0106)
0.0128)
0.01347
T
9
0.01002
0.01081
0.00908
0.008)6
0.01084
0.01038
0.01141
0.00727
'1'10
0.00891
O.oo9tl
0.01135
0.01119
0.00928
0.00430
0.00607
0.012)6
0.0109t
Fr
0.012'37
0.012)7
0.01300
0.01)16
0.012)6
0.01529
0.01592
0.01698
0.01717
0.015)1
Ila.
0.011)76
0.010)7
0.01066
0.01148
0.01040
0.01277
0.0121)
0.01439
0.01409
0.012.58
0.01058
,
1).01066
'1.01051
0.01002
0.01142
0.010)0
0.01419
0.01345
0.01510
0.01S01
0.01)78
0.0129t
0.0060,5
ESL
1).07511)
0.07843
0.07237
0.07970
0.07237
0.08409
0.08960
0.07707
0.08939
0.08237
0.09869
O.06Jl~
1I ••• (d
Ji ) < /dJil ~
)
Idjil»
5
7
r
0.007"
0.012*
-0.017*
-0.021""
-0.032"*
-0.036"* -0.047"*
-0.012"
0.05162
1-'
"
2ll.0.(d
J1
I
••
2u.o.(d ji )
d J1 ~ E9L(T j ) - ~L(Ti)
•
~
TABLE C -
Ti
T
T1
j
T1
T
2
f),OO2
ESLa, their differences and oorresponding standard srrors
T)
T4
T
5
T6
T
1
T
8
(norJAAl errors, n .. S , •• S ).
'r9
T10
'r
Ba
F
0.01)·
-0.000
0.005*
0.020*
0.022**
0.0)1**
0.020*
0.026**
0.007
-0.022- -D.OZ8"
0.011
-0.002
0.004
0.019*
0.020-
0.0)0*
0.018*
0.024**
0.006
-0.024** -0.0)0-
-0.013*
-0.007
0.008
0.009
0.019*
0.007
0.01)·
-0.005
0.021*
0.022·
0.0)2**
0.020·
0.026-
0.008
--0.022- -0.028-
0.015*
0.017*
0.026*
0.014*
0.021-
0.002
-0.027·· -0.0)4"
0,002
0.011
-0.011
-0.006
0.009
-0.002
-0.012
T2
0.00)70
T)
').011&.
0.01195
1'4
0.00468
0.0(6)2
0.01025
T
S
1.0(1)7
0.00)87
0.010~
0.004)2
T6
1.010)4
0.010)1
0.01'305
0.01107
0.01052
T
7
"1.')1072
0.00715
0.01)6)
0.01162
0.01046
O.OO.sao
T6
0.01570
0.01581
0.00716
0.01455
0.01488
0.01600
0.01655
T
9
0.01147
0.01185
0.01145
0.01001
0.01127
0.00828
0.01017
0.01)98
T10
0.0109)
0.01041
0.01271
0.011)2
0~01002
0.0056)
0.00616
O.OlSO)
0.00924
Fr
0.010SO
0.010)7
0.01072
0.010)0
0.010)9
0.01428
0.01)88
0.01566
0.01502
0.01478
Ra
0.0067U
0.00886
0.01126
0.00080
0.00911
0.01)0)
0.f'lJ29
0.01612
0.01376
0.01)67
0.00881
r
'l.OO~H
0.0092/1
0.01174
0.00928
0.OO9J7
0.01261
0.01275
0.016')0
0.01333
0.01)15
0.009.58
0.00301
!::lL
0.011101
0.0U278
0.09)69
0.08086
0.08641
0.10141
0.10)1)
0.112)7
0.10076
0.10702
0.08842
0.05919
*
e
9.-
0.006*
•. e.(dJl)<ldJ11~2s.e.(dJl) I u
-0.0)4--
-o.da"
-0.()I~9··
-0.01)
-0.042-
0.004
-O.OlS*
-0.044- -0.051"
-0.005
-0.024*
-0.0,)) .... -0.060**
·,).006
-0.012· ,
-0.042·* -0.040**
-0.019*
-0.048** -0.05-+-o.02g*it -0.0)6-0.007-
0.0526)
I-'
I-'
dJ1
IdJ11> 28.0.(d j1 )
e
S
I\)
iSL(T ) • iSL(T1) •
J
e
e
e
e
TABLE C -10.- ESLe. their differences and corresponding standard errors
Ti
T
T1
j
Tl
T
2
-0.004·
T2
:>.01)269
T
J
--
--
14
').1)0219
0.00313
T,S
:>.01)269
1).00000
T6
').0104,S
0.0101,S
T
7
:>.01070
0.01014
T8
-
--
T
9
).010'14
0.01043
T10
).01070
0.01014
Fr
).01106
0.01107
Il&
1.(09)4
0.00877
F
1.010.51
0.01013
ESL
:>.09682
0.09217
T4
T)
--
S
T
6
T
8
T
1
T9
D:: ,", • '::.
T10
J).
1r
Ba
F
0.004- -0.004*
0.020*
0.020-
-
0.02,S-
0.020*
0.011-
-o.02,S- -0.029**
0.009*-
0.02,S-
0.02.5-
--
0.029**
0.02.5-
0.016-
-0.020- -O.024n
--
--
--
--
--
T
(noZ'lla1 errors.
0.000
-0.009**
0.00)13
0.016*
0.01,S-
0.02,S-
0.02,S-
--
0.0104)
0.01015
0.010.59
0.01014
0.0029)
-
--
--
--
-
0.010.57
0.0104)
0.002n
0.00321
-
0.010.59
0.01014
0.0029)
0.00000
0.01137
0.01107
0.01306
0.01)40
0.00974
0.00877
0.01141
0.01169
--
0.01077
0.01013
0.01331
o.013n
-
0.10146
0.09217
0.11763
0.11727
-0.000
-
--
0.020*
0.01,S·
0.001
-0.029** -o.O:n-
0.029-
0.02,S--
0.016*
-0.020- -0.024-
0.004*
-0.000
-0.008
-0.04.5- -0.0.50-
0.004*
0.000
-0.008
-O.ot•.5'"
-
-
-
-0.004*
-
--
0.00)27
-
-o.049 u
--
-0.01)*
-0.0.50- -0.0.54--
-0.008
-0.04.5- -0.049**
0.01308
0.01)40
0.01168
0.01169
0.0121.5
--
0.013.5.5
0.01371
0.01376
0.004.59
-
0.12212
0.11727
0.10874
O.On72
-0.0)7- -0.041-0.004
0.06747
t-'
•
I 1:;
•• o.(d j1 )< d j1
1-1
ZQ ••• (d j1 )
I u
Idjil> 2•••• (d J1 )
d j1
&
E3L(T ) - ESL(T )
j
1
•
\....l
TAllLE C - 11.- ESLs, their dif'ferences and corresponding standard errors
T
Ti
T,
j
T
2
0.004"
T1
:).00307
T
2
T)
T4
T.5
T6
-0.002
0.000
-0.00)
-0.007"
-0.004
0.002
(nonal errors , n
6-, .,.,,).
9
T
10
Yr
-0.002
-0.00)
-0.008
0.006
-0.016"
-0.028....
-0.007
-0.006
-0.008
-0.01)"
0.002
-0.020"
-0.032-
-0.003
-0.001
0.000
-0.001
-0.006
0.008
-0.014"
-0.025....
-0.005
-0.00)
-0.002
-0.00)
-0.008
0.006
-0.016-
-0.028**
-0.00)
-0.000
0.001
-0.001
-0.006
0.009
-0.01)-
-o.02S"
0.004
0.002
-0.00)-
0.012
-0.010
-0.022-
0.001
-0.000
-0.006-
0.009
-0.01)-
-0.02.5....
-0.001
-0.007
0.008
-0.014-
-0.026"
-0.00.5
0.009
-0.013
-0.02.5-
0.01.5"
-0.007
-o.Olg-
7
T
8
-0.00.5
-0.00)
-0.007"
-0.010"
-0.000
-0.00)
T
T
%
Ba.
1).0~9+
0.00607
T4
o.00J'~7
0.005J2
o.OOJ)6
T
5
0.00271
0.0039)
0.00372
0.00)68
T
6
0.0')869
0.00934
O.OOm
0.00958
0.00895
T
7
0.009+2
0.009+2
0.0102.5
0.01039
0.00969
0.00269
T
8
1).01140
0.01232
0.00972
0.01021
0.0113.5
0.008.51
0.00869
T
/).010)6
/).01108
0.00918
0.00910
0.01042
0.00717
0.00757
0.00466
T
10
0.009)1
0.00980
0.00976
0.0099.5
0.00921
0.00248
0.0030)
0.00814
0.00718
Fr
0.0131.5
0.01424
0.01)76
0.01)7.5
0.01')0.5
0.011'76
0.01255
0.01)09
0.01)24
0.0116'7
Ro.
0.0110)
0.012)6
0.01097
0.01156
0.010)6
0.01006
0.011'70
0.012J73
0.01339
0.01086
0.01017
F
0.00060
0.00982
0.00905
0.009)6
0.0082)
0.00950
0.01019
0.0119'~
0.01206
0.009'f7
0.01151
0.00lf7.5
tSL
O.I)'jT)7
0.0610"
0.0.'5.52.'5
0.057J7
O.O~fO.s
0.0519"1
0.054'70
0.055.56
o.o~Jt:YI
0.0489'~
0.(6)64
0.04152
T)
9
•
e
o.e.(d j1 )< Jd j1 1s 20.Cl.(d j1 )
0.00)"
dji ~ ESL(TJ)
, .. /dj11,. 2l1.0.(d j1 )
e
- ~L(T1)
•
F
-0.022** -0.0)4"-0.012"
0.02<)00
I-'
I-'
..(::-
e
e
e
e
TABLE
T·1
T
T1
j
T1
c
-12.- ESLs, their differences and corresponding standard errors
0.017"
0.00)
0.00)
0.014·
0.001
-0.010·
0.017·
0.002
0.00)
0.01)·
0.001
-0.010·
-0.008
-0.008
0.002
-0.010·
-0.021 u
0.017·
0.002
0.00)
0.013"
0.001
-0.010·
0.011"
0.017·
0.00)
0.003
0.014·
0.001
-0.010·
0.005"
0.012"
-0.00)
-0.002"
0.008
-0.004
-0.01.5"
0.007
-0.000·
-0.007·
0.00)
-0.010"
-o.021 u
-0.01.5"
-0.014"
-0.004
-0.016"
-0.1)27....
o.on"
-0.002"
-0.0l)"
0.010"
-0.002
-0.01)·
1).00,5
0.011"
-0.000
0.00,5
0.010"
-0.011·
-0.006
-0.000
0.005
0.010"
0.005
T6
0.011"
0.000
-0.000
0.011
0.000
-0.011·
'l.000
'1'10
T8
T,5
T)
'r9
T
7
T4
T
2
(nornwJ. errors, no,: 6, 11-:'!».
Yr
Ra
T2
1).0'"'229
T)
'l.Ol'112
0.01147
14
~.1)'l297
0.0'Yn6
0.00908
T
5
'1,O'l224
0.00266
0.0'1907
0.00)05
T
6
0.00818
0.0088)
0.009)7
0.00811
0.00712
T
7
0.')0892
0.0089)
0.01090
0.00929
0.00857
0.00507
T8
::>.00985
0.01066
0.00805
0.00984
0.009.5.5
0.00812
0.009.55
'1'9
~.I)OOOI
0.00877
0.00892
0.00756
0.00760
0.00440
0.00680
0.00856
TIO
0.00790
0.008.59
0.00910
0.00779
0.00747
0.001)2
0.00.501
0.00805
0.0042,5
Fr
0.1)0891
0.009)1
0.01028
0.00990
0.00879
0.00887
0.00822
0.001322
0.00929
0.00887
Ila
I).008~
1).00911)
0.00998
0.001389
0.00782
0.00801
0.00829
0.00801
0.00085
0.00807
0.00627
V
O,OI1/}'IO
o.O'lO??
o.O](){,O
0.00027
o.no'n?
O.OO9lJ
n.oo,)75
U.OIOj2
o.ouy)
0.00')10
O.OlOW
o.ou;iJO
t:.lL
O.O~)jIIU
0.0.5.169
0.06....55
0.05369
o.Orm
O.05(~9
0.06414
0.07076
0.05616
0.05<'572
0.06707
0.0;1.,'.
-0.000
00.006
0.001
F
-0.01)·· -0.02)"
-o.oU*·
O.ljlJ'}l
f-J
•
1I.1I.(U j1 )< IdJll~ 20.0.(d ji )
I .. Idjil> 211.0,(d ji )
dji ~ t3L(T j )
-
~L(Ti)
I-J
VI
TABU:
T
T]
T1 j
c -13.-
.T2
T1
0.001
0
T2
0.00334
ESLe, their d1N'eronces and corresponding standard errors
TS
T4
T)
T
6
T
7
,
-
0.057**
0.0.52-
0.027·
-0.008
-0.028·
-
0.002
0.000
0.0.51-
0.0.51**
0.0,56-
0.0.51-
0.026*
-0.009
-0.029*
-
-
-
--
--
--
--
-
0.049**
0.049**
-
O.O~-
0.049**
0.024-
-0.011
-0.0)1-
0.0.51-
0.0.51-
-
0.0,56-
0.0.51-
0.026·
-0.009
-0.029*
-
0.004
-0.000
-0.026
-0.061- -0.080-
--
0.005
0.000
-0.025
-0.060*- -0.080.·
-
-
--
-0.005
-0.0)0
-0.065-
-0.025
-0.060- -0.080"
-
T
5
O.M)J4
0.00000
-
0.00)93
T6
0.02116
0.02(6)
-
0.02143
0.02063
T
7
1).02088
0.01978
-
0.02111
0.01978
T8
--
--
-
--
T
9
1).02061
0.02010
-
0.02041
0.02010
0.00599
0.00817
T
10
0.02088
0.01978
-
0.02117
0.01978
0.00,564
0.00000
Fr
'>.11422
0.01498
-
0.01436
0.01498
0.0)10)
0.0)072
IlA
O.0141)1f
0.01428
0.01414
0.01428
0.02575
0.0251)
,
1).01779
0.0170,5
0.01761
0.0170,
0.02019
0.02737
ESL
0.1689+
0.16990
0.17187
0.16990
0.221')6
0.22096
Id j1 1"7
28, •• (d j1 )
e
Ba'
0.052.....
0.00)93
Id j1 1~
Yr
0.0,52**
1).00204
•••• (d J1 ) <
T10
.3).
0.001
1'4
•
T
9
Ta
1\
0.00)·
--
.
errors. n .3,
--
--
T)
(un~fol'lll
-
-0.002
2u.o.{d j1 )
I ..
.
-0.000
0.00564
-
--.
---..
0.00817
--
-
--0.035-
0.0)060
0.0)072
0.02505
0.02.51)
0.01624
0.02696
0.027)7
0.02140
0.010'12
0.22550
0.22096
0.19,581
0.160.50
--
-o.o~-
-0.055·-0.019·
0.14101
f-'
d j1 • IC:JL(T j)
e
- E3L(T1 )
,
1...1
0\
e
-
e
e
TABLE C -
T
Ti
T1
j
1i.
14.-
ESLs. their differencee a.nd corresponding standard errore
0.004
'r9
'1'6
T
7
0.002
0.035**
0.043--
0.0)2*
0.0)5-
0.0)1*
0.0)1- -0.002
-0.013
0.001
-0.002
0.032*
0.040-
0.029*
0.0)1*
0.028*
0.028" -0.005
-0.016-
0.001
-0.002
-0.0)2-
O.UfO*
0.029*
0.0)2*
0.028*
0.020" -0.005
-0.016*
-0.003
0.0)1"
0.0)9*
0.028*
0.030*
0.027*
0.027- -0.006
-0.017·
0.0;4*
0.042-
0.031*
0.03)·
0.030*
0.0)0- -0.00)
-0.014*
T
3
'1'4
'l'S
0.003
0.004
-0.000
T
2
(uniform errore. n .3. m .4).
'1'2
0.00599
'I')
0.0052)
0.00748
'1'4
o.OQli66
0.00697
0.00)8)
'1'5
0.00)97
0.00524
0.00518
0.00461
'1'6
0.0171)
0.019)7
0.019/)0
0.01889
0.01862
'1'7
'l.0215')
O.OlCn
0.02)0.5
0.022)2
0.0~O07
O.OlTJO
TO
O.'J1990
0.02161
0.01778
9.01805 ' 0.02(J115
0.01809
0.008
T8
TID
Da
i1r
F
-0.00)
-0.001
-O.QQ/j.
-0.004
-0.0)7*
-0.040**
-0.011
-0.009
-0.012
-0.012
-0.<)115"
-0.056"
0.002
-0.001
-0.001
-O.OY+*
-0.(JI15"*
0.02/fOO
-0.00)
'1'9
0.01911
0.02048
0.01859
0.01729
0.019)2
0.01518
0.02090
0.01186
'1'10
0.01788
0.01G15
0.019-10
0.01854
0.01695
0.01162
0.01)40
0.01898
0.01515
Fr
'1.01)22
0.01)64
0.01)71
0.01))5
0.01)15
0.028))
0.0)029
0.0299-1
0.02897
0.02824
lla.
0.01214
0.012.7'~
0.01201
0.012)4
0.01211
0.02)01
0.02.7'fl
0.02))2
0.02)5)'
0.022g~
0.01421~
~'
o.01'j6'/
O.Ol'J'TI
0.0111/ 10
0.01lH6
0.1)1)'19
0.02J50
0.(2)'/',
0.0226)
0.021';10
0.021)')
0.01';1.55
o.ol06u
E3L
0.16379
0.172)2
0.17192
0.17)06
0.170)0
0.2Q1n4
0.21212
0.20121
a.2o'M
0.20010
0.20015
0.16687
M
····(l1 J1 ) <
Id J1 1SO
2s·.·(l1 j1 )
, •• IdJ11"7 <!s.e.(l1 J1 )
dJ1 :
~L(TJ)
-
-0.00)
0.000
~3L(Tl)
•
-0.0)7·
-0.0))*
-0.047'"
-0.044**
-0.0)3- -0.044**
-0.011*
0.15596
1-'
1--'
~
TABLi: C -15.- ESLe, their d1f'ferenoe8 and oorresponding Iltandard sITore
T
11
Tl
j
Tl
T
z
0.007
.
T)
'4
0.0)0- -0.00)
(W\1tOI'1ll errors. n
.3 ••• 5).
T6
T
7
T
8
T
9
T
lO
1r
0.001
O.dH-
0.062-
0.09)"
0.0)5*
o.QIf6*
0.021*
-0.006
-0.0)8--
-0.00,5
0.0)5*
0.056**
0.086-
0.oz8*
0.039*
0.014*
-o.OlZ
-0.044-
0.011
0.0))1
0.(6)0
0.005
0.016
0.044
0.06,5-
0.096-
0.0)8*
0.048-
0.024*
-0.003
-0.0)5**
0.040
0.061-
0.091-
0.034*
0.044-
0.020*
-0.007
-0.0)9**
0.021*
0.052*
-0.006
0.0)0
-0.027*
'1',5
Ba'
T
Z
0.OO!X)8
1)
0.01)8)
0.011171.
'4
0.00703
0.011)8
0.01214
5
0.00550
0.00920
0.01188
0.00.570
T
6
0.02019
0.02358
'0.02915
0.0229t
0.02261
1
7
0.02440lIl
0.02158
0.0)001
0.02667
0.02533
0.01775
T8
0.02421
0.02560
0.01680
0.02221
0.02)05
0.0)026
0.0)196
T9
0.02161
0.02456
0.02712
0.019))
0.02117
0.01677
0.02409
0.02651
TlO
0.02011
0.02216
0.02597
0.0200)
0.01876
0.01518
0.01996
0.02n5
0.01196
Fr
0.014)8
0.01)79
0.01305
0.0147)
0.01408
0.0)19t
0.0)20)
0.02m
0.03172
0.03062
Jla
0.01704
0.01649
0.0162)
0.01619
0.016)5
0.0)147
0.0)370
0.026.55
0.029)4'
0.02911-
0.01485
,
0.017.57
0.01991
O.O190?
0.016'10
0.01717
0.02921
0.0)272
0.0200'.
0.026))
0.02697
0.01971
0.01236
0.2)758
0.21W19
0.21m7
0.2;)48,5
0.2')009
0.27879
0.29995
0.3)035
0.27247
0.28)2)
0.Z5859
0.2)172
1
~L
•
e
0.023*
-0.009
-0.0))" -0.020 1*
0.004
•••• (d J1 )< 111 J1 1" 211 ••• (d )
J1
I ..
IdJ1 1'»
21l ••• (d
J1
e
-0.0)6** -0.067"
0.004
-0.020
-0.047
-0.079....
-0.017
-o.O'n*
-0.060*
-0.100"
-0.058** -0.047*
0.011
dJ1 • ~L(TJ) - tDL(T1)
)
-0.009
r
-0.072- -0.099**
-0.1)0"
-0.014
-0.041*
-o.07)u
-0.025
-0.052*
-0.08)-
-0.027*
-0.059-0.032**
0.19990
f-'
I" J
0.>
•
e
e
e
e
TABLE
'r
T1
T1
j
T1
16.-
C -
T
ESLe, their differencee a.nd correeponding etan<1a.rd errorB
2
O.uoo
T
2
·1.')')Zlfl
-0.002
0.000
0.02ylt*
0.028*·
--
0.021"
0.028"'
O.OJ7""
-0.021"
-o.OYI"
--
--
0.01')85
0.014)6
T
0.01/l] 0
0.1)11 56
Fr
0.012)5
0.01209
rw.
0.01273
0.01264
).
0.01455
0.011118
l!::JL
O.1~20tl
0.14]OU
F
--
T
6
0.01)56
na
-0.0))""
-
'1.01410
Yr
-0.020·
0.00000
10
'1'10
0.0)7--
0.01)2111
T
'rg
O.OZr1'
T
5
0.01486
8
0.021"
-
0.014'))
T
--
0.0045)
9
7
0.026"
0.00)71
T
T
0.029"*
'1'4
--
6
3 ).
0.000
--
-
T
m ..
-0.002
--
Ttl
'1'5
4
4,
--
T)
7
T
T)
(uniform errore, n ..
---
0.002
0.0045)
0.01464
0.01436
0.01500
0.01156
--
--
--
0.0)1-
0.0)0**
--<
0.029""*
0.028-0.002
----
0.00756
--
--
0.02)*
0.0)0-
0.0)5**
-0.018*
-0.0)1-
0.021·
0.028*"
0.OJ7"
-0.021"
-O.oy,"
-0.002
0.008
-0.050"·
-0.06)"*
-0.007
0.000
0.010
-0.0'10"
-0.001"
--
--
--
0.007
0.016
-0.041-
-0.055*"
0.010
-0.048*"
-0.061-
-0.0.58-
-0.071-
--
-0.010·
--
--
--
--
--
0.01406
0.01486
0.0060)
0.00952
0.01.500
0.01356
0.00756
0.00000
0.0130)
0.01209
0.02236
0.02173
0.01192
0.01261.
0.01900
0.01851
-
0.01009 . 0.01851
0.0150)
--
0.011~18
o.OllnO
0.02055
0.01')03
--
0.02001
0.0190"
0.(17)6
0.00037
--
0.140'/6
O.14)Otl
0.1'1'1. J'l
0.1'/0'/1
-
o.16)SJ1
O.l'/U71
0.Hl025
0.12251
-
--
--
-
0.009.52
0.02278
0.02173
--
--
-0.01)" .
0.lO')j9
I-J
•
•• <I.(d jl) <
IdJ11 ~
1-'
4:l.l.0.(tl l.)
J
I
n
Itlj11'1' 211 ••• (d j1 )
d ji ~ ~L(Tj) - g~L(Ti)
'0
TABLE C - 17.- ESLe. their dlfterencee and correeponding lltandard eZTon
'1'
'1'1
'1"1
T1
j
:2
0.002
-
T)
'1'4
"5
'1'6
T
1
T
a
(unlfon errore • n • '\ • • • 4 ).
'1'9
T10
lr
Ra
.,
0.007*
0.004·
0.002
0.027*
0.026·
0.~9**
0.0)2**
0.0))·*
0.027·* -0.017·
-0.029*·
0.00.5
0.002
0.003
0.025*
0.024*
0.~7·*
0.0)1**
0.0)1**
0.025** -0.019*
-0.0)0··
-0.004
-0.00.5
0.020·
0.019*
O.dn··
0.025·
0.026·
0.019·
-0.002
0.024*
0.022*
O.~S**
0.029**
0.029**
0.02)" -0.021** -0.0)2**
0.025·
0.024*
0.~7**
0.0)0**
0.0)1"
0.025** -0.019*
0.021*
0.005
0.006*
-0.001
0.023·
0.006
0.001
0.001
-0.016*
-0.016*
-0.022
-0.066.... -o.077 n
0.001
-0.006
-o.~9**
-0.007
-0.0.50** -0.062**
T2
0.()'))51
T)
0.00690
o.OOm
'1'4
0.00)42
0.00496
0.00607
T.5
0.00218
0.00406
0.00674
0.00)86
'1'6
0.01374
0.01443
0.01,520
0.01359
0.01440
'1'7
0.01391
0.01282
0.015J9
0.01411
0.01451
0.00921•
'1'8
0.01702
0.01778
0.01445
0.01638
0.01698
0.01209
0.01500
'1'9
0.01)48
0.01431
0.01432
0.01277
0.01372
0.00539
0.01066
0.0103.5
'1'10
1).01339
0.01415
0.01442
0.01)07
0.013)6
0.00,568
0.01069
0.01083
0.00486
rr
0.01162
0.01087
0.01211
0.0113)
0.01131
0.0209.5
0.01904
0.02284
0.02051
IIA
0.01062
0.01086
0.01025
0.01120
0.01069
0.01611
0.01729
0.01891
0.01798· 0.01778
0.01440
Y
0.01285
0.01)17
0.01414
0.0133.5
0.01)21
0.01939
0.01892
0.0218)
0.019~
0.0191.5
0.01681.
0.00681
~L
0.12247
0.12429
0.1297.5
0.12606
0.124.5.5
0.1497.5
0.14833
0.17106
0.1,5480
0.15.5,56
0.14904
0.10.:1<0
-0.001
-0.024·* -0.0)6··
-0.0)1"
-0.044" -0.0,56"
-o.~'3**
-0.055**
-0.061-
-0.044** -0.055-
0.02~0
-0.012*
0.09)81.
1-'
•
e
I 1s Z•••• (d j1 )
•••• (d j1 )< d j1
,
u
N
Id j1 /> Z•••• (d ji )
o
dJ1 • ZSL(TJ ) - J:3L(T1) •
e
e
TA5LB
T
Tl
j
T1
T1
I
e
e
e
C -18.-
T
ESL13. their differences and corresponding ntanilard enora
2
-a.OOO"
T
T4
J
1'5
T
6
T
7
(uniform erroru • n .. 4.
T
'f
8
g
In ..
TID
Yr
5).
na
0.017·
-o.oeo
-0.000·
0.01)·
0.001
0.OJ7"
O.CJll
-0.001
0.029"
-0.026·
-0.0:3)"
0.025·
0.000
0.000
0.021·
O.oog
0.0115"
0.019·
0.007
0.OJ7**
-0. olD·
-0.025"
-0.015
0.021·
-0.006
-0,017"
0.012
-0.(4)" -0.050"
0.01)
0.008
0.038"
0.011
-0.000
0.029- -0.026"
-0.033·
0.021"
0.009
0.04.5-
0.019"
0.007
0.037- -0.010·
-0.02.5"
-0.011
0.025"
-0.002
-0.013·
0.016
-0.039*
-0.046**
0.0)6"
0.010
-0.002
0.027"
-o.a20·
-0.0)5·
T
2
).<yI6YI
'r)
).11259
0.01186
'1'4
).00810
0.00938
0.009)6
T.5
).0~J9
0.00612
0.01071
0.00729
T
6
:>.01222
0.013.52
0.01666
0.01448
0.01284
T
7
).0129')
0.01128
0.01702
0.0149')
0.01296
0.01056
T8
':1.017'17
0.019')0
0.01126
0.01(,1;7
0.01670
0.01861
0.01923
T
J.Ol/IOl
0.0153.5
0.01584
0.01237
0.01420
0.01133
0.015)0
0.01764
T10
).01164
0.01282
0.015<9
0.01)60
0.01135
0.00684
0.01042
0.01700
0.01096
Fr
J.012~1
0.01217
0.01 70
"
0.01' 129
0.0121a
0.01979
0.01829
0.02026
0.02133
0.019)9
lla
.1.'11511
0.<)1'161
0.019'11
0.01917
0.On87
0.02095
0.0202 •
'
0.02002
0.0:>105
0.01905
0.01)63
~.
0.(11(>62
0.01615
n.0l710
0.(11671
0.01 'j/O
0.07115
(1.02((,')
O.O;I='2~
n.o;,:o.'j~'
O.01nn
0.(n6nr.
0.00111')
II;:jL
').161'/£
(J.l~J',1Y
U.1'/IJ()/t
U.161('1
0.1:>4U9
0.J.'/'II.15
CJ.J.6JJO
O.lY:JJY
().17J03
0.16141
0.190'/1
0.13)61
9
•• .. ·(JJ1)<
l uj1 1~
F
-0.017"
-0.025** -0.001.
-0.000·
2D.O.(l1 j1 )
,
..
Id j1 1>
2u.O·(l1 ji )
-0.026·
d
jl
~ ~L('T)
J -
-0.0)0- -0.009
-0.064- • -0.071**
-0.012·
0.018
-0.037"
-0.044·"
0.029*
-0.026"
-0.033
~~L(T)
l'
-0.055"" -0.062**
-0.007
O.lWN
\-,
t'J
IJ
TA1ILIi:
T
Ti j
Tl
T1
ESLer their dif'ferenoee and oorrosponding standard errors
T
2
1
2
0.002~
T
J
-
-
1'4
1).'>0292
0.00414
T
S
/).()I)2~
0.00000
T6
0.01026
0.010~
T
7
1).01038
0.00997
T
8
-
-
1
9
0.01110
0.01152
T
10
0.01038
0.00997
Fr
0.01140
0.01182
Ila
O.OHII6
0.01077
po
I).OIl02
O.fl112)
ES1
O./)9"JJ3
a.Orn07
'r.5
T
6
T
7
-0.001
0.004-
0,02)-
0.02S-
-0.004-
0.000
0.020-
0.024-
T
T4
J
-
0.004-
it
e
C -19.-
--
-
0.0040.00414
T
9
T
10
0.02)-
0.028-
O.~O-
-O.OU-
-0.016*
0,019*
0.024-
0.037- -0.015-
-0.019*
T
S
--
-
lr
aa·
--
-
I-
0,024-
0.028-
0.~1**
-0.011-
-0.015*
0.024-
-
0.019*
0.024-
0.037-
-0.015-
-0.019*
0.004
-
-0.001
0.004
0.017-
-0.035** -0.039**
--
-0.005
0.000
0.013-
-0.039** -0.044**
--
-
--
0.024-
0.02S-
0.020-
-
0.010~
0.01091
O,oom
0.O()l}59
-
-
--
--
-
;-
0.01006
0.01152
0.00839
0.00957
-
-
0.01091
0,00997
0.004.59
0.00000
-
0.00957
0.01125
0.01182
0.01676
0.01701
-
0.01576
0.01701
0.00989
0.01077
0.01576
0.0161fl
--
0.01493
0.016111
0,01)07
O.Olfl51
0.O1l2)
0.01520
0.01604
-
0.01'111"
0.0160'1
O,01J91
0.00}1"
O.092n
0.09273
0.09707
0.11682
-
0.11631
0.12111
0.13369
0.08202
-
-
8.e·(d'ji)< Idjil.s; 2a.ll.(d ji )
r
--
--
0.01077
.
-
(unifon errors , n .5' r ... 3).
dj1
I .. Idjil> 2a ••• (d ji )
e
I:
0.005
ISL(Tj )
-
-
-
-
-
0.017·
-0.0)4- -0.039**
0.013
-0.039** -0.044-0.052** -0.056-
-o.od}
0.07758
t-'
N
N
iSL(T1)
e
e
e
e
TABLE
T
'1\
j
Ti
C -ZO.- ESLs, their differences and oorreeponding Iltandard orrors
-"1
T7
(uniform erroro • n -S. m -4).
T10
T
2
T;
T4
'1'5
-0.00)
O.()()II*
-0.000
0.000
0.022**
0.014*
0.()I12**
0.02')u
0.027**
0.~2'"
-0.007
-0.01'1"
0.007*
0.002
0.00)
0.025**
0.016*
0.()I.15**
0.0)1**
0.029"
O.()I.t4 u
-0.005
-0.011/"
-0.00.5*
-O.O~*
0.010·
0.009
0.0)8""
0.024**
0.022**
0.0)7**
-0.012·
-0.021"
0.022'"
0.014*
0.~2**
0.029 u
0.02'1**
0.042**
-0.007
-0.017"
0.022**
0.014*
0.()/.t2**
0.028**
0.027**
0.~2**
-0.007
-0.017**
0.020*
0.006*
O.O()/.t*
0.019"
-0.029"*
-0.0)9"
0.028**
0.015*
0.01)*
\).028*
-0.021*
-0.0)1**
-0.01)*
-0.015*
-0.000
-0.049'"
-0.059"
-O.OOZ·
0.01)
T
6
TO
'f')
l'r
T
2
'.OoW'
T)
o . ()'))0.5
0.004130
4
0.00279
0.00515
0.00)08
T
5
0.00190
0.0~27
0.00).59
0.0025)
T6
').00859
0.011\4
0.00982
0.00881
0.00897
T
7
J.oI064
0.OOB78
0.01070
0.01159
0.01109
0.01181
T
8
0.0121.5
0.01295
0.01124
0.01124
0.01256
0.01042
0.011))
T
9
O.()O[l~
0.011)2
0.00')64
0.00088
0.00909
0.00Y~7
0.01218
0.oos;t;6
().OO9~1
0.Qll112
Q.OOs;t;8
0.00900
0.00911
0.00)72
0.0121/2
0.009'll
0.00157
Fr
0.0110)
0.01069
0.01052
0.01108
0.01099
0.01.574
0.01472
0.01.582
0.0152)
0.01522
Ra
').00700
0.00606
0.00702
0.00000
0.00769
0.011/25
0.01260
0.01)05
o.Oll ill)
0.01/151
0.01062
n,(){1'/711
O.()(lGV
O,nfl'/GO
0.00{'/12
0.00'l19
O.IJl)n
0.0121/1
0.OJ51l'l
O. (Jl ]66
o.oun
O.I)];'o.~
1
T
10
~'
--_._.I!::JL
__ ..
_~_
.•.
~
__
0.061)1
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___
._.~
_ _ _ _ _ _ _ _ , ... _ ••.
O.o.'Jlj6y
0.~~1
O.OOC
_ .•_.... _
.•..
~,_
0.06101
...... , . _ . _._
0.061111
-0.008
0_·_."
-
-"
0.OUj18
'
..-.-.....-.-
0.0'1)05
-_...
"'-'."'~
-
0.10]];.1
_"
~.,
•••
~
O.oUYW
-0.0)6"
0.015
....,
• • _',",
... _
. . . . . . . _ •.
~
0.00'/')8
••
~_
.• _
.................... '
0.102YJ
po
11&
-0. ()I.f6 u
-O.OYI"
_O.()IIJI~·
-0.()/.t9"*
-0.059"*
-0.010"
O. (0)'/0
- . . . ." , . _ . . . . . . . .,-..0 _ _ _
O.O~j(jl
O.cJillj.
1--'
t\)
•
B.o.(d ji ) <
IdJ11 ~
2G.o.(d j1 )
J
*" Id ji I >
2B.e.(d,ll)
d JI
= ESL(T J ) - 1I:3L(T\) •
\ .-.l
TAllLE
T
T1 j
'1'1
T1
c -21.-
ESte, their differences and corresponding standard errors
T
2
T4
T)
0.004
"s
T6
T
7
T8
(unlfon errore, n .5,
'r9
T10
III
.5').
Ir
11&"
0.047"
0.006
0.002
o.O:n**
0.0)6**
o.ofJl,"
0.0)7**
0.027**
0.049"
0.007
-0.009
0.044*-
0.002
-0.002
0.029**
0.0)2-
0.081**
0.0)4-
0.02)-
0.046**
0.00)
-0.01)"
0.042-
-0.046" -o.OlS
T2
0.00109
T)
').')1)'0
1).016)13
14
0.00612
1).01067
0.01214
T
S
0.00436
0.006,.
0.01427
0.007.39
T6
o.oIdn
0.01)16
0.01487
0.01016
0.01095
T
7
0.01247
0.01198
0.01716
0.01)10
0.01214
0.01078
TO
0.019")0
0.02191
0.01097
o.oum
0.01?m
0.019)1
0.0212)
T
9
0.01080
0.01)34
0.01411
0.01010
0.011)9
0.00538
0.01169
0.018,.
T10
0.(10)1
0.0121)
0.01")
0.01085
0.009lJl.
0.00)4.5
0.01040
0.01g..8
0.0062.5
rr
a.012')9
0.01168
0.01481
0.01)90
0.012).5
0.0160)
0.01.5.50
0.02007
O.OlSg..
0.01216
0.012)1
0.01710
0.01)91
0.01211
O.Olsao
0.01664-
0.02187
0.01617 "0.01479
0.01280
0.01276
O.01)Y.
0.01~f6
0.01397
0.01)09
0.01675
0.01700
0.024)2
0.01704
0.0161)
0.0152.5
0.00699
0.0'15.56
0.01906
0.12293
0.00106
0.07722
0.100)3
0.1l11~
0.1.5960
0.11213
0.102.5)
0.12465
0.082.53
Ra
r
I:IJL
r
-0.004
-O.Oll
0.OJ1" -0.010
-0.020-
0.002
-o.dIO" -0.0.51""
0.021**
0.0)0-
0.079"
0.0)2-
0.021-
0.044**
0.001
-0.0105-
0.0)1-
0.0)4-
0.082-
0.0)6-
0.0205**
0.047-
0.0005
-0.011
0.00)
0.0.5l-
0.004
-0.006-
0.016-
-0.026-
-0.042"
o.d.9**
0.001
-0.009
0.01)
-0.029*
-0.()I~5"
-0.041- -0.0.57** -0.0)05"
-0.010*
-0.0'/, " -0.091"*
0.012
-0.0)0"
-0.046"
0.022
-0.020-
-0.036-
-0.042- -0.0.58-
0.01539
-0.016"
._---0.06626
I..J
-
e
•• 1l.(4 J1 )< l u
J1
1lit
211.0.(d
J1
)
I ..
IdJ1 1~
N
2U.o.(d
J1
d J1 ~ ~L(TJ) • EtlL(T )
1
)
e
'c:'
•
e
e
-
e
TATILE
T
Ti
T1
j
T1
T
2
---
--
J
'1'4
T
5
T
6
7
O,()O269
~.00l~
0.00'100
0.01077
8.01099
O,rllOOS
o.nll02
-0.00+*
-0.002-
0.02)"
0.02)"
-0.002
0.000
0.025**
0.025-
--
--
--
--
0.027**
0.027-
0.025"
0.025**
----
5
0.002
0.0269
0.01125
0.01099
0,01112
0.On02
-,
----
-0.000
0.000111
--
--
0.01105
0.01104
0.00227
0.00249
--
0.011)2
0.01102
0.00111
0.00000
--
--
0.01129
0.01093
0.0l1f25
0.0142)
o.oafM
--
0.00717
0.00644
0.012)4
0.012)0
0.00669
0.00621
-
0.00735
0.00621
0.01219
0.01218
O.()lfl06
0.0)901
-
0.O)~~2
0.0)9d~
O. ()(jf2J.I
o.CJ6J~
--
T
9
0.01067
0.01104
TIo
0.01085
0.01102
Fr
1).01057
0.01093
0.007d~
F
E:';L
-
--
.
T10
9
Fr
Ra.
F
0.022-
0.02)-
O.OYt**
0.022
-0.001
0.024-
0.025-
0.036**
0.004
0.001
--
--
--
--
--
0.025-
0.027-
0.0)8**
0.006
0.00)
0.024H
0.025**
0.0)6»
0.004
0.001
--
--
--
'f
T8
-
'fa
Ra
---
T
7
'r
--
o.oo~62
T6
T4
T)
2
0.001B'~
T
(W'lifonn errore. n .6, m .3).
ESLs, their differences an:J. corresponding etandard errore
-0.002
'f
T
C -22.-
-0.002
-0.000
0.011
-0.021* -0.024-
-o.OOl
0.000
0.011
-0.021"
--
--
--
0.001
0.01)
. -0.020*
-0.02)*
0.011
-0.021-
-0.024*
0.00249
-
-o.()~II"
--
-0.0)2** -0.035**
0.01427
0.0142)
0.012)4
0.012)0
0,00907
--
0.0l21~8
0.01218
0.00)61
0.00'159
--
1).06273
0.06J9~
0.07535
O.d.293
-0.00)
0.0'1020
I-~
•
B.o.(d ji ) < Idjil
s
ZIl ••• (d ji )
I
..
Idjil ~ Zs.e.(d j1 )
d ji
=ESL(TJ ) -
N
~L(Ti)
•
\.;\
T
T1
T1
j
TAllLli: C
-:13.- ESLe, their differonoee and oorreeponding
Tl
T
2
-O.OOZ-
.
(unHoI'll errore, n .. 6, II .. 4).
T
6
T
T
T
9
TIO
-0.000
0.oz6-
0.028-
o.dl4-
0.030-
0.025-
0.025**
-O.OO~
-0.017-
0.002
0.0288
0.031"
0.046"
0.0)2··
0.027-
0.027-
-0.006
-0.015"*
-o.OO6U
0.021--
0.02)-
0.039-·
0.02J.I-
0.020 n
0.020 n
-0.014-
-o.oz)n
0.000
0.026**
0.029-
0.044 8
0.0)0"
0.025-
0.025" -0.008·
-0.017-
0.026**
0.029**
0.044-
0.0)0-
0.025-
0.025**
-o.OO~
-0.017-
0.002
0.018-
0.004
-0.001·
-0.001
-0.0)5- -0.043-
0.016**
0.001
-0.003
-0.003
-o.OJ7** -o.cIl6-
-0.019**
-0.01~
-0.05)- -0.061**
-0.005
-0.005
-0.038-. -0.047-
0.000
-0.0)4- -0.042-
T
4
'1'5
0.005-
-0.000
0.008-
0.002
T)
lltandard errore
a
7
Yr
Ba'
T
2
0.00209
0.0
T)
0.00))8
O.OdIZ4
1
4
0.00)08
0.00407
0.00278
T
5
0.00202
0.00247
0.00:330
0.00)07
T
6
0.00939
0.009'Jf
0.00542
0.00960
0.00928
T
7
".009)8
0.009)1
0.00959
0.01006
0.00929
0.00510
T
8
0.01075
0.01112
0.00995
0.01095
0.01100
0.00811
0.00720
T
9
0.009)2
0.01007
0.00862
0.00898
0.00546
0.00485
0.00696
0.00113
T
10
0.009)2
0.00988
0.00937
0.00956
0.00925
0.000'(11-
0.00535
0.00870
0.00485
0.010'))
0.01010
0.00928
0.01025
0.00997
0.011)6
0.01009
0.01106
0.01149
0.01137
0.00793
0.007.5.5
0.008)6
0.00798
0.00760
0.01106
0.01088
0.01289
0.01140
0.01097
0.01043
0.007~
0.00742
0.00883
0.00791
0.0075l
0.01189
0.01154
0.01).5)
0.01204
0.01182
0.01212
0.00)68
o.Ql1616
O.<m84
0.05157
0.04591
0.04601
0.07217
0.01160
0.09010
0.07591
0.01126
0.07126
0.0)747
Fr
Jl&
F
~L
-
e
-0.006"
-0.014
I'
-0.0)4** -0.042-0.009**
0.02879
I-'
l\.l
•• 0·(<1 j1 )<
IdJ1 11i
2•••• (d j1 )
I
n
Idj11> 2•••• (<1 j1 )
d j1
e
1:
0\
i:8L(Tj ) - B::ll.(T1 )
e
e
-
e
TADLl';
T
T1
T1
j
T1
C - 24.-
ESLc,
T
2
-o.ow
th~lr
dl fforoncoo and corrosponding nt.&net&rd errors
T4
T)
'1'S
T6
(Wllform errors • n .6. m .5).
'r9
T
8
T?
TlO
Yr
Ra
0.0))**
0.002
-0.001
0.0)0**
0.0)0**
0.051**
0.026*"
0.024*"
0.021*" -0.015·
-0.019·
0.0)6-
O.OOS
0.002
0.0))**
O.oYl-
O.O~-
0.029**
0.02S-
0.02S- -0.012·
-0.015·
T2
a.ocMS
T
J
J, '11ul:.'
0.011')6
"4
).OeM?
0,0")0624
o.OJ844
T
5
1.0'1247
0.O()lI)7
0.00959
0.00)81
T6
a,rllO)O
0.011)1
0.01279
0.010)S
0.00953
T
1
o.01l67
0.01171
0.01)12
0.011119
O.OlOGS
0.007d+
TO
).01J9O
0.011.jJj8
0.0095'+
0.01298
0.01))2
0.01421
0.01569
T
9
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0.011)1
0.01166
0.00926
0.00931
0.00710
0.0101S
0.01)71
T10
1.00992
0.01100
0.01135
0.0095)
0.00881
0.00S)8
0.00814
0.01))8
0.00576
Fr
1.01011
0.01030
0.01011
0.00911
0.00911
0.01)10
0.011+19
0.01151
0.01)00
0.01255
Ha
:l.Ol()lfl
0.00711
0.01)61
0.01069
0.009'77
0.01511
0.01665
0.01475
0.014)4
0.01395
0.01065
-'.0115J
0.01058
O.Olll~h
0.01161
0.0111)
0.01578
0.017))
0.01629
0.01511
0.01466
0.01272
0.00519
0.06535
0.06171
0.0981)
0.06702
0.061124
0.09.500
0.095'10
0.n621
0.09101
0.089'14
0.08662
0.0.5010
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-0.007
-0.009
-O.OU"
-O.cJIO" -U.052"
0.02S-
0.028-
0.~+9"
0.OZ4--
0.022*·
0.020** -0.017*
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0.0)1-
0.0)1**
0.052-
0.027"
0.02S-
0.02)- -0.014*
-0.018*
0.00000
0.021·
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-O.OOS
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0.021*
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-0.006
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-0.027** -0.029** -0.066** -0.010--
211 ••• (d j1)
d j1
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-o.od.
0.()I16116
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N
-.,J
TAU 0 - 2;._ ESLe. thoir differenoe. Uld oorroepondlng .tanrlard error.
T
1'1
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j
1'1
T2
'2
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T4
'1'.5
1).001
-
0.001
-
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0.00)10
0.00.5.50
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0.00000
'1'6
f).019.5J
0.020'3.5
T
1
·0.020!14
0.0193.5
T8
-
-
T
9
/).01969
0.020.5)
T10
0.02eM
0.0193.5
Fr
0.01442
0.01466
Ra
0.01308
0.01306
r
0.01601
0.01512
E:JL
O.151~?
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-
0.060....
0.01.5**
0.011
-0.021"
-0.020"
o.o(CIu
--
0.019"
0.0?!...
0.010
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-0.029'
-
-
-
-
0.019**
0.074**
0.010
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-0.029"
0.019**
0.0/'4-
0.010
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0.001
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0.01.5 u
0.000
0.000
0.015"·
-
-
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0.01.5**
0.074**
0.07.5**
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0.020)0
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0.02116
0.01935
0.00902
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-
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0.01916
0.0205)
0.01028
0.01358
0.02116
0.0193.5
0.00902
0.00000
0.01)81
0.01466
0.03081
0.0)074
0.01287
0.01)06
0.02732
0.02696
0.01624
0.01512
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0.0272.5
0.19190
0.1.1175
0.2101.5
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-0.069"* -0.100'" -0.100'.
0.02980
0.030'(11.
0.02667
0.02696
0.01460
0.021112
0.02725
0.010611
0.010'12
0.22001
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0.001
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10
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0.061**
0.062-
-0.003
-0.020·
-0.019"
-0.1)/'4"
0.0'17"
0.U62 H
0.061"
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-0.021"
-(l.Ol')'
0.000"
O.OOZ**
0.0.1'"
0.070-
0.069--
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-0.(1)"
-0.012
0.076-
0.078-
0.0.50-
0.066-
0.064- -0.000
-0.017·
-0.016·
0.07S-
0.071-
0.0.50-
0.06.5-
0.064- -0.001
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-0.017·
T6
T
7
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0.07)*·
0.07.5**
-0.003
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0.00.5·
0.000
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0.002·
(Lapl&co
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nil
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0.0o'f02
0.00551
14
0.00237
0.0037.5
O.OdHS
T
S
0.002)9
0.00302
0.00421
0.00207
6
0.02083
0.02110
0.02233
0.02149
0.02113
T
7
0.021.53
0.0209'1-
0.02299
0.02221
0.02140
0.008.9+
8
0.02021
0.02')'15
0.01875
0.02011
0.0202)
0.01'.9'.
0.01776
T
9
0.'12021
o .r1~f)'15
0.0:'0')6
0.02007
0.02')'10
0.ONJ55
0.0121(j
0.(12)7
T
10
0.02:»9
0.02028
0.02182
0.02097
0.02017
0.00677
0.00.599
0.01650
0.01032
o.Oll:n
0.(11)1
0.01052
0.01092
O.OlCO!!
0.02957
0.02965
0.02737
0.02853
0.02i375
o.n12fYl
0.01:'01
0.01101
0.01297
0.01701
0.0:'(,79
o.onoO
0.0:' ~N'
O.075'jf
O.O;>(,l(l
0.011:'1
O.nl/l rj7
0.0]/172
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o.Ol/vIII
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0.0;>7(,0
0.0701(,
0.02 S'x,
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0.0;'7]/;
0,01'>011
0.00-(11
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-0.026*
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0.014
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T
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T
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(Laplaoo orron • n .) ••• 5).
T
9
TIO
Pr
Ra
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0.009"
0.00.
0.060--
0.080"
0.099"
0.072"
O.O~"
0.020-
0.016-
0.0)1-
0.016-
0.002
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O.O~. .
0.092"
0.066-*
0.067 n
0.014-
0.010
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0.030'
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0.077"
0.0.50"
o.05l n
-0.002
-0.006
0.009
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0.052-
0.072**
0.090-
0.063-
0.06.5-
0.012
0.007
0.022*
0.056-
0.076-
0.09.5-
0.068-
0.070-
0.016-
0.012
0.027-
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0.012
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-0.040-
-0.044*
-0.030-
0.018
-0.000
-0.001
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-0.049"
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-0.02.5
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0.002
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-0.057"
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T2
0.01579
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0.011110
1'4
0.00727
0.00928
0.00914
T.5
0.00.513
0.00791
0.01106
0.00.537
T
6
0.02000
0.02130
0.02591
0.02278
0.021,56
T
7
0.0i'009
0.01837
0.0267)
0.023.57
0.02246
0.012116
T
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0.02999
0.02009
0.02513
0.02708
0.03058
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0.0219?
0.0i'1t'?
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0.020fJl
0.01776
0.02162
0.0:'172
T10
0.1')2199
0.02)44
0.02451
0.02147
0.01998
0.01561
0.01988
0.02648
0.01011
Fr
0.'11273
0.01219
0.01124
0.01282
0.01289
0.029'.8
0.02862
0.0)199
0.03010
0.030)3
Ra
O.01If75
0.01576
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0.01139
0.01400
0.02713
0.02805
0.02000
0.02558
0.0266)
0.01500
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0.01610
0.01765
0.01698
0.015")1
0.01525
0.02707
0.028.57
0.020(19
0.021.05
0.02.576
0.011169
0.01030
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TULE C -18.- iSLe, their differencee ILIld correepon'ling lltanrlArd errora
T
T
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Tl
j
T
1
T
2
0.001
T
2
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T
T6
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0.029**
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-0.020*
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0.000
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0.027*
0.020**
0.0211*
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T6
0.01)78
0.014))
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0.01)92
0.014))
7
0.01257
0.01166
--
0.01276
0.01166
TO
--
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T
9
o.on')'1
0.01 11111
--
0.01)57
O.ol l ll n
0.00';09
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0.01257
0.on66
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0.01276
0.01166
0.00858
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0.01110
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0.02124
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0.01105
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0.00058
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--
0.029-
0.0)0**
0.026-
-0.019*
-0.020*
0.027*
0.028-
0.024-
-0.020*
-0.022*
-0.0.56**
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-0.000
-0.010
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0.000
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--
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TABU!: c -1<).- ISLa, thoir difforenool and oorrooponding atandard erron
T
TI
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j
T1
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0.001
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0.0005*
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T10
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T
9
0.001
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O.()I11··
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0.045--
0.006
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-0.011"
-0.001
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O.OJ,]"
0.0)6"
O.O)OU
0.0)6"
O.()I10"
0.001
-0.017"
-0.010 1
0.002
-0.001
0.042-
0.040-
0.042....
0.040"
0.044'"
O.O()l1
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-0.01/.·
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0.040-
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0.0)9-
0.0)7*
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0.002
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-0.017-
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0.041..... 0.042....
0.040-
0.044....
0.005
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1'2
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0.~.57
0.00670
0.00)S5
0.01l612
0.00)01
0.00214
0.00529
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0.00)52
0.01196
O.011sn
O.Ol)6S
0.01)36
O.Ol21S
1'7
0.01187
O.OlllS
0.01)79
0.01)50
0.01208
0.00499
T
S
0.01)09
0.01290
0.0116.5
0.0125)
0.01)16
0.01129
0.01267
0.0121S
0.011~
0.0119S
0.01109
0.01191
0.00969
0.01115
0.00S17
0.012)6
0.012))
0.01)80
0.01)22
0.011S1
o.o<Mo
0.00647
0.01199
0.009))
Fr
0.0fl86.5
0.00889
0.00865
O.ooS51
0.ooS7S
0.01729
0.01745
0.016)2
0.01581
0.017)1
Ra
0.009'39
0.0099+
0.0097.5
0.00959
O.oosns
0.0161S
0.016))
0.01576
0.01.50)
0.016.50
0.00936
r
O.OIO'/'}
0.01151
0.01109
0.01076
0.01091
0.01692
0.01670
0.01667
0.015'·5
0.01710
0.01290
0.00715
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0.10707
0.11197
0.10&1')
0.11001
0.10778
0.15076
0.11181')
0.15015
O.llj(l2")
0.15"12
0.11250
O.O'illl']
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1'9
1'10
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0.001
0.001
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-0.0.57....
0.002
-0.000
0.004
1'.004
-o.o~O"
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-0.002
0.002
0.002
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-0.0.56-
0.004
0.0005
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-0.0'1"
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-0.019*
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<1 j1 .. !'UL('J'j> - II::SL('J'l)
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e
e
TABLE
T
T1
T1
j
T1
C - ~O._
T;>
-0.005
E~T.B, their differencoe II.nd oorroopon<Ung IIlMlUuU erroru
T
TI~
3
T6
'1'5
T
7
TO
(lJ\pl.. ~o llrr'orll
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n
=4.
111
TID
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nil
0.009
0.010*
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0.018-
0.019*
0.026-
0.027-
0.009
0.007
-0.019-
-0.001
0.01"-
0.015-
O.oul
0.02,)·-
o .OZ'!if.
0.0')1-
0.0')2"
0.0111-
0.012-
-0.(1)-
0.002
0.001
-0.01)11
0.009 11
0.00')
0.017-
O.oW-
-0.001
-0.00)
-0.02')"
-O,Ol)
-0.014*
0.000
0.009
0.016
0.017*
-0.001
-0.00)
-0.029** -0.01)11
0.022*
0.02)*
0.0)0*
0.0)1-
0.001
0.008
0.000
T2
O.OQ52?
T)
0.oF157
0.011/18
T4
0.00607
0.00858
0.00976
T
5
0.00574
0.00528
0.00955
0.00696
T
6
0.00960
0.01086
0.01428
0.01148
0.01136
T
7
0.01208
0.010~
0.01.585
0.01441
0.01142
0.01025
T8
0.0169)
0.01668
0.01040
0.01650
0.01536
0.01767
0.018Y,
T')
0.01116
0.01287
0.01417
0.00961
0.012 /;6
0.00917
O.Ol)'Jf
0.01756
T
0.01116
0.01092
0.01428
0.01)0)
0.0098)
0.0079)
0.00675
0.Ol6l+)
0.01191
Fr
0.01061
0.010)1
0.01365
0.01181
0.0112)
0.01308
0.0142/,
0.0185+
0.01500
0.01)40
lla
0,0111)9
0.0l10')
0.01 1107
0.01220
0.01077
0.011.166
0.01526
0.01765
0.n16l6
a.Olillo
0.01051
F
0."1 'Yil
0.017.0)
0.<n1q2
0.012fl)
0.01169
0.0l/)'l)
o.OH16l
U.0166)
0.01/;'/9
0.01 /1'/9
o.m/IS'!
0.on el')5
1 ')IIl~1
0.17.')1 'J
O. ]lU'IO
o.lIl / ln')
0.1101')
0.1 'j:'llil
U.l(,O'jO
0.1(, \()(j
0.1 11;'\)]
o.l'lO'Xi
0.1 LiI(,5
10
l;.qr,
1-..•.-
fl.
•.
F
..
...... (,l j1 ) < Illj11:.. ;'n.".('ljl)
..
••
__-
0.1
jj:'()~
..
1·1 Jl1 ., 2" .... (,1 Jl)
-
-0.016 11
0.001
0.01)*
0.011
0.009
-0.009*
-0.011
-0.0)7" -0.021 11
0.008
-0.010*
-0.012
-0.OJ8-* -0.022 11
0.001
-0.018"
-0.020*
-0.0'16 l1li -0.0)0-
-0.018-
-0.020-
-0.0'16"* -0.010 11
-0.002
-0.028..... -0.012
.
d jl .. J:::ll>('l} - U;:';L(T
-0.026- -0.010
0.016"*
U.l')O/1
,_.
',.J
\ .•l
1)
TABLB C - 31.- ISLe. their differenoes and oorresponding etandard errors
T
T1
'1'1
j
T1
'1'2
-0.002
T
2
0.00111
T
J
-
-
1
4
0.00)205
0.00421
T
S
0.00114
0.0001)()
T6
0.00992
0.01028
T
0.00967
0.00966
T8
-
--
T
0.010)1
T10
--
"05
T6
T
1
T10
Ir
Ra
r
0.012*
0.002
--
0.006
0.002
-o.odt
-0.014*
-0.021·
-0.001
0.000
0.01:4*
O.Od.
-I
0.008
0.00.
-0.002
-0.012-
-o.Ol?"
-
-
-
-
--
--
--
0.008
0.00.5
-0.002
-0.012*
-0.019*
0.008
0.004
-0.002
-0.012*
-0.019*
-0.016*
-0.026*
-0.0))**
-0.016*
-0.02)-
-
0.01028
0.00912
0.00966
-
--
--
-
0.0107)
--
0.009)0
0.0107.1
0.odf82
0.006213
0.00967
0.00966
--
0.00912
0.00966
0.00420
0.00000
Fr
0.0126)
0.01)11
-
0.01112
0.01311
0.01481
0.0146.5
I1a
0.01)970
0.01006
--
0.00872
0.01006
0.01)62
,.
OJJ1l49
0.0116)
--
0.01069
0.0116)
ron!.
O.lO[Ylll
0.106fl2
--
0.106]]
0.10602
---_.
'1'9
3 ).
-0.002
0.009)8
9
'1'8
lr
-0.002
-
7
0.001
0.00421
0.014*
0.0005
0.014*
0.004
-0.010
-0.006*
-
0.00420
-0.010**
--
--
0.004
0.000
-0.006
-
--
--
-O.Odf
-0.010
-0.020-
-0.027-
-0.006
-0.016*
-0.02)-
-0.010
-0.017-
-
--
0.00628
0.01417
0.0146.5
0.01)21.
--
O.Ol)dt
0.0l)2tt
0.01121
0.0153J
0.011147
--
0.01.5513
0.01/147
0.01466
0.00690
0.1~061
0.1l0()(,
--
0.11111?
0.110(16
0.1(j155
O.O'/I(jO
-
-0.007-
0.OIJTI7
1-'
-..,)
,..
,.
.
•
e
T4
T)
(La!llace errore • n , 5 ••
•• 9·(<1 J1 )", ' ,t J1 1..
;:... *.(lt j1 )
I
..
l aj1 1)o'
l ...... (<1 j1 )
e
d
jl
.. II:;JL('I' ) - 11::11.('1' )
J
l'
e
e
e
e
TABLE C - 32._ ESLe, their differences and oorreoponding Ilwdard OlTore
T
T1
j
Tl
T1
T
z
0.00)
T4
T:)
'1'5
T6
T
T
7
B
(I.a:,la~o elTon. n
'r
T
9
:t
S,
II
IO
rr
.4).
nIL
-0.002
-0.001
0.001
0.010*
0.007*
0.011*
0.000"
0.010"
0.010"
-0.005
-0.007
-0.006
-0.004
-0.002
0.006
o.odl
0.00/3
0.005
0.007
0.007
-0.009
-0.011'
0.001
0.00)
O.OlZ"
0.010"
0.0111"
0.010"
0.012*
0.01)"
-0.001
-0.005
0.002
0.010*
o.ooB"
0.012*
0.009*
0.011*
0.011*
-0.004
-0.006
0.009"
0.007"
0.010"
0.007*
0.009"
0.010"
-0.006
-0.008
T;>
0.0'1180
T)
0.0(jjJI6
0.0(6)5
'1'4
0.00)15
0.0'1551
0.0(28)
T
5
0.00n6
0.00)97
0.00'~35
0.00338
T
n.O:;055
0.00744
0.00717
0.00680
0.00642
T
7
o.o06oB
0.00674
0.0067B
0.00625
0.00615
0.00272
T8
0.00765
0.0')871
0.00703
0.00761
0.00762
0.00117
0.00608
T
9
0.00609
0.00775
0.0071)
0.00690
0.00679
0.007.28
0.00)62
0.00'197
TIO
0.00651-
0.00759
0.00704
o.o066B
0.00629
0.00160
0.00200
0.00:;60
0.00273
Fr
0.00800
0.00830
0.00748
o .007l~9
0.00786
0.00870
0.00920
0.00926
0.00805
0.00855
Ra
n.') f1 v.J7
0.01078
0.00852
0.00861
0.00'1)5
0.01007
0.0098U
0.01l997
0,009111
0.00903
0.00801
r
1l.()O!JU)
O.'J()·)Ul
O.OUUlY
0.00U15
O.OOyUl
0.01lXi1
O.GOYI\!
o.ololn
0.00')'1/
o.ollil)
O.OO'N)
0.ooJ 15
~~:IJ
o.()')'!,l
o. a 51llJ<)
0.05111
0.01165
0,05650"
n.06510
0.06'}O1
0.06697
0.061111
0.(6551
0.(l6(,o6
o.()~no
6
Jl'
-0.002
0.002
-0.002
0.000
0.001
-0.015"
-0.017"
0.004
0.000
0.002
0.003
-0.013*
-0.015"
-0.004
-0.001
-0.001
-0.017"
-0.019"
0.002
0.003
-0.013"
-0.(J15"
0.001
-0.015"
-0.017"
-0.016*
-0.010"
-0.002
-I
--
1--...._-
"
lI.9·("Jl)"· "'Jilt.: ;'''''''(''Jt)
,. 1,I J1 '"
~·" .... (.IJ1)
0.0'111'\11
,I Jl
-
li;:Jl.('I} - U;:H.(T
1
)
J
\ ,)
()I
TABLE 0_33._ ISLa, their dlrtorencee and oorreeponding etandArd errore
'1'1
T
'1'1
j
'z
-0.00)
'1'1
'1')
Ta
0.026-
0.017·
0.006
0.026-
0.02)·
-0.001
-0.001
-0.001
-0.002
0.029-
0.020**
0.009
0.029-
0.026-
0.002
-0.007
O.om
-0.001
0.0)1-
0.022·
0.011·
0.0)0-
0.027·
o.om
-0.006
0.006
-o.OlZ-
0.019-
0.010
-0.001
0.019-
0.016·
-0.008
-0.017
-0.006
0.0)2-
0.02)-
0.012
0.0)1**
0.028-
0.005
-0.005
0.007
",5
-0.005
0.007·
-0.006-
-0.001
0.010·
O.OU·
'1'6
'9
'10
Yr
Ra
0.QI))7a
'1')
0.01202
0.01170
1'4
o.oosn
0.01)726
0.OU09
'1',5
0.00499
0.am29
0.01050
0.00697
0.')0987
0.0101S
0.0149l
0.01197
O.Olam
0.l)lu6
0.01000
0.0144,5
0.01)10
0.0097,5
0.00607
0.01711
0.01646
0.00960
0.016~
0.01~
0.01760
0.01708
0.010)5
0.0112S
O.Olyl6
0.00968
0.010,:tf
0.00822
0.01042
0.016)7
T10
0.01148
0.01101
0.01467
0.01))0
0.0099l
0.00409
0.00509
0.0170S
0.00924
Fr
0.01081
0.00907
0.0111)
0.011)6
0.00972
0.01)'1.8
0.01248
0.015)1
0.01)01
0.01)64
Ra.
0.01116
0.00995
0.01122
0.01208
0.00905
0.01)22
0.01193
0.01.527
0.01292
0.0124)
0.00809
F
0.01160
0.010'1)
0.01)6.5
0.012)1
O.OO'/,W
0.01)59
0.01226
0.01'1)'1
0.01))2
0.01250
a.OllY)
0.00600
'l.0,!2)7
0.06909
0.0677)
0.07909
0.00677
0.0')0'1)
0.08955
0.07059
0.0')008
0.0919S
0.071'36
0.0617Z
T
7
TO
"9
Il::JI.
-_.•
•• O·("Jl) « ,. I J1 11O
~"···(·'Jl)
I
••
,
'1'7
'4
'1'2
"6
e
(Lar1&oe errore. n .5• •• 5).
-0.009-
I"Jll., .:•••• (I'Jl)
-0.020·
-0.000
-0.011
0.009
O.ooS·
-0.010·
-0.020** -0.016·
0.019
0.016
-0.007
-0.017"
-0.00)
-0.00)
-0.027** -0.0)6** -0.025.
-0.005
-0.027** -0.0)6"* -0.025"
-0.024*
-0.0))- -0.022.
-0.010·
0.002
0.012·
0.07)'1)
I-'
-
\...l
0'
I
e
l1 J1 - It.11'('\'J) - IlO:1L('l'l)
•
e
e
e
e
TADLE
T1
T
Tl
j
T1
a - 34.T
2
O.OOJ·
1:81.0, their differences ILIlJ oorrullponding otandll.I'd errors
T6
T
J
T4
--
-0.000
O.OOJ·
0.014·
O. --
-0.00)
0.000
O.OU·
--
--
'l'S
(L&plllce errore, n ':: fl, III ':: 3).
Ta
'f
O.OlS·
--
O.Ol?·
0.012"
--
0.011.·
T?
9
TlO
Ra
O.OlS·
0.009
0.004
0.010·
O.OlZ"
0.006
0.001
0.00'1"
--
--
T
Z
0.OCi220
T)
--
--
T
4
0.0014S
o.OO)Oa
T
S
0.00220
0.00000
T
6
0.00989
0.00980
--
0.00962
0.00980
T
7
0.01018
0.00992
--
0.00992
0.00992
T8
--
--
--
--
--
--
--
T')
0.01012
0.01006
--
0.00775
0.01006
0.00:")?
0.001(,0
--
T10
0.01018
0.00992
---
0.00992
0.00992
0.0027lt
0.00000
-
0.00)68
Fr
0.01122
0.OU5S
--
0.011)0
0.OU5S
0.01257
0.01ZI'2
--
0.01266
0.0121.2
Uti
0.0"6 f1 5
0.00715
--
0.00661.
0.00715
0.00<)0<)
(1.0101~1
--
n.no?TY'
O.OlOo';
O.OO?(\()
F'
O.f),)(jGo
O. 0 11(, ~Il
--
0.00661
0.0(1(,')0
0.0175'1
().01272
--
0.0176'1
0.01272
o.on/')
O.I)I1(,'jl
1";:1.
1).Il~'J')j
Il.u(j~·/II
--
u.u~')(\O
o.IJ(';;''111
Il.U·/I·,')
u.(l7(1(;J
--
f),tI'I'/I;'
(J.
tI.()(,(~Ji
U.Ucll;11
-
0.00)
o.ooJoa
--
--
--
po
P'r
--
--
--
0.014*
0.015·
-
0.01?*
0.015*
0.009
0.004
0.010·
O.OU·
0.012*
--
0.014*
0.012*
0.006
0.001
0.007·
0.001
--
o.od.·
0.001
-0.005
-0.009
-0.004
--
O.OOZ
0.000
-0.006
-0.010"
-0.005
-
--
--
-O.OO;!
-0.000
-o.on"
..().()()'1
-0.006
-C.01O.
-0.005
-o.od.
0.001
O.OOZ'(/;
t)'i"I{.~
--
--
O.o()(,
t',I)(,'J!\l.1
1-. /
'. ,)
"-.J
"
... cl.(d j1 ) < Idj11 ~ 2tl.tl.(d j1 )
,
u
Idj11> Z.... tl.(d. )
11
d j1 ': Jo:SL('l} - I£SL('f )
1
•
TABLE C -35,- ISLe. their differenoeo and oorresponding otandArd erroro
'1'1
'1'
'1'1
j
'1'1
'1'2
~O.OO)"
T
'1'4
'1',
5
'1'6
'7
TO
(Laplaoe errore. n :6. a ,4).
'1'9
'1'10
rr
Ra
po
-o.02~
-0.0)1"
-0.01)"
-0.011"
-0.01)"
-0.001
-0.007
-0.025" -0.027'"
-0.004
-o.OQIl.
-0.009*
-0.028" -0.0)0"
0.001
-0.00)
-0.00)
-0.000
-0.026" -0.028"
0.001-*
0.000
0.001
-0.00.5
-0.02)- -0.025"
-0.001
-0.007
-0.025" -o.on"
-0.004"
-0.009*
-0.027'* -0.029*'
0.000
-0.00.5
-0.02)" -0.025**
-0.005
-0.02)** -0.026"
-0.004
-0.001
-0.002"
-0.006
-0.004
-0.001
-0.005
-0.005
-0.010"
-0.006"
-o.OdI
-0.005"
-0.000
-0.006
-o.OdI
-0.000
-0.006
0.00)
0.002
-0.002
0.000
0.00)
-0.001
-0.001
-0.005
-0.00)
-0.000
-0.004
-0.002
0.002
,
'1'2
0.00216
,)
0.00526
0.00592
'1'4
0.00283
0.00)79
0.00420
'5
0.00176
0.00245
0.0QIl.26
0.0024)
'6
0.00871
0.00886
0.00775
0.00829
0.00783
'1'7
0.00061
0.00852
0.00757
0.00814
0.00775
0.ooY~8
'1'8
0.O08~
0.009)0
0.00792
0.00056
0.00820
0.00268
0.00l-~
'1'9
0.00867
0.00879
0.00772
0.0082.5
0.00778
0.00100
0.00)66
0.00261
'1'10
0.00875
0.008~
0.00780
0.00836
0.0078.5
0.0009)
0.0036)
0.00250
0.00127
Fr
0.009.58
0.00976
0.00924
0.009)1
0.00902
0.00892
0.00906
0.00858
0.00892
0.000'+9
na
0.00~8
0.00977
0.0068)
0.009)0
0.00899
0.00896
0.00066
o.oo~
0.00095
0.00681
0.00719
F
O.OO9(H
0.00906
0.00856
0.00009
0.008.58
0.0102)
0.009l:l6
0.0107.5
0.01019
0.0102)
0.00956
0.od15J
Enl.
1).'15119
0.06207
0.05'116
0.0501'
0.05727
0.051'(t'
0.05'566
0.051110
0.0~1Il9
0.09rY,
0.()I1909
0.010'71
,
0.00)
-0.001
-0.004*
-0.018" -0.020**
-0.002
0.O<'fl79
1·....1
\ .l
0.)
•
e
•••• (.IJl) < JCl J1 1.. 2a···(<I j1 )
I
u
I<lJl'-r za .... (<ljl)
d Jl ... C::lL('l'j) - IC::lL('l'l)
e
•
e
e
e
It
TATILE
T
T1
T1
j
11
r: -
ISLo, thoir difforoncoo ILlld COIToopondtng utM,!a,N erroru
,6._
T
T.,
-0.006·
J
T/ f
0.001
-0.000
'1'5
Tr,
T
7
TO
(Ltl'lacll
erroro , n 1:6, m ,,5).
'r9
T
lO
Yr
nn
0.015
0.01)·
0.012
0.011·
O.OU·
-0.021 u
-0.0)1"·
-0.019"
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AP~NDIX
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FROGRA:'I
AH£40 AND ?El.An:n SUBHCUTL'lES
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liE"
(10/2S/76)
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B (5),R (~) .TR(5).Sr(51 ,55(51. C(S)
KC(l. 10), IS (10),1' (101 ,Z (101
DIIIallSIOll SII (1I0),IIA (110) ,VB (110) ,BIl (110)
DATA T,E/1I00-0./
DA7A B.~,TB,ST.SS/25.0./
DATA IC/ZO-il/
DATA SIl,BA.iR.8ft/160*0./
II1TEGE!-2 DIST,FLAG,OISTS
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PARAIIET!RS (1'ARABETEB CABO
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79 rORaAT (2I2,I8,2P8.4)
illItE (3,179) (DIST.IST,AXAl,BIBI)
179 P08SA7 (11,' DIST: ',12,' SEED: ',110,'
DISTB"DIST-l1
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300
310
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DErIJE LEiEL OF CEYSOSIIIG (PARAIIETER CABO 6)
Ie • IOBBES OP LEAST V1BIABLE BLOCKS :0 RECEIVE WEleRT
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BEAD (1.49)
It
IlRIT! (3,150) IIC
150 pORa"r (lX,' MC "',12./)
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TRE SA8PLES
DO 60 1"1.111
C
GE'!IATE AID PROCESS OIlE SlapLE
1'8B-0.
TBsaO.
ssc"'O.
55R-0.
DO 5S ol A-1,!
S5 C (J'\) -0.
DO SO I a 1,1
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340
.liZ .. 35C
AE.:4 360
DEFIIIZ OPfIOM1L OaT PUT (1'1&1I1ET£8 CARD 5)
LoaaO (al) ELIItIIIATES (PRODUCES) LISTING OP OBSERVATIONS
LST=O ("1) ELIIIIIIUES (paODUCES) LISTIlIIG OP STATISTICS
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ABE4 380
1~!4 39D
1il!4 "00
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VRITE (3,1119) 108,LST
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89 rOB!AT (101'8.4)
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189 rOiSAT (1X,' Ti EUIUIIT Erl'ECTS: " 10P8.Il,/)
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69 PORII1T (101'8.5)
YSlrE (3,169) WRS.WR1.VC5,vCl,rs5,PB1.CRS,CH1,FP5.rr1
169 POB~l: (lX,' CHIT. VALUES :',10P10.5./)
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99 1O&5At (3(I4,r4.0J,14)
LI.. 5 ••
iSITE (3,199) 11,11,1111
199 POisAr (11,14,' TREATSEIITS.',I4,' BLOCKS,',16,' EXPERIIIEsrs',/)
~iIOR
ZJ
3,:
A3E;a 10
Ai~.
12J
DErnE 1l0:l8ER OF TREATttEIITS.1I0IlBER or BLOCKS PER EXPEaIlSEJlT,
MUB,Ei or EIPERIBEIITS AIID lIUllaEB OF STATISTICS ( PABA5ETas
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DerIll
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A?cs:mn D (continued)
C
lOEMEiAT!
85=0.
O~Z
AA~. ;00
Ai!?,. bl)
BLOCK
A5.::. :iN
BE. HO
13:; .. " .. 0
60=0.
1 J=I,5
Ir (OISta)
[lu
20,21,22
22 IH:>r=2
2)
216
21
20
25
CICX"I.
CALL V13lOZllliHST.IST,FLAG,AXAX,CXCI,5IJ) ,D)
l!' (B(J)-O.:'I
23,23,2 4
B(JI" IU:BX·UO.. (2.·B(J))
lOu TO 2:'
B(JI =(-1.) ·BI3X·ALOG(2.. .. (I.-6(J)))
lO<> TO 25
0151'''2
CXCI=1
CALL 'U!lliE!I (DIST ,1ST ,F!.lG, AX AX, Clex, B (J) ,0)
B(J) ~
3X;U·t'!lI (3.1 .. 1:'9. (6 (J)-0.5))
GO TO 25
CALL V13GE!I(DISt',IST,FLAG.AIAI,DIBl,B(J),D)
B (J) =B (J) +1'3 (J)
B3=BB+B (J)
BS=BS+6 (J) "'a (J)
C(J) =C(J)
8(J)
CClI1'lIfU::
85(1) =ild/f5
Si{I)=as - (B3·B8)/71l
T3B=TBB+BB
T35=TIl3+BS
SSs."SSR
BB·Ba
BUlK TdiS BLOCK AllD STORB ITS IlUKiliG
CALL BA~K(3,Il,5)
00 2 J=I,11
+
+
C
T(I.J)=a(J)-{T5+'.)/2.
(LOil.~Q.O)
GO TO 11
nUll (3.299) (B(J) ,J=I.Il) ,Si(I)
299 FO:i:tAT (lX.111'10.4)
11 CLlllrI 1jU E
C
T~IS BLOCK IS soaTED
CALI. 50~r (J.Il)
3
13E~12 ~O
1.3<..12:0
DO J J=I,1l
SSC"SSC + (C(J)·C(J)
FN:«1'll-I.) .(!!I a SSC-TBa.TBB))/(1'llaT!I"TSS-TIl.SSC-TlI.SSB+T3S.T33)
CALL Ull(t'.SW,!I.ll,H.dH,Cl,CC1.llC)
11"
(C1.GZ.W35)
KC (1.1) =KC (1,1) +'
IF
(C 1 • G ll ... a 1 )
iCC (2 , 1) = i<C (2, 1) +- 1
IF
(CC1.GE.~CS) ·KC(1.1) =KC (1,3) +1
IF
(CC1.GE.iCl)
KC(2.3) =KC(2.3) +1
+
II"
(C 2 • G :: • ii a :. )
KC (1. 21 = KC ( 1 , 2) 1
IF
(C2.GE.lIlil)
KC(2.2)=KC(2,21+'
IF
(CC2.GE.JC5)
KC(I.4)=i\:C(I,41+1
If'
(CC2.GE •• el)
KC(2,4) =KC(2,4)+'
C1:'L SIIBlj /T.E,[J,Il,N.LR.SN,BNI
IF
(H.G::.Pe5)
KC(I,5) =KC(1.5j+l
II"
(S:t.GE.Fal)
KC(2,5) ~KC(2,51 t 1
If'
(il'.G~.e<l5)
KC(1,bl=KC(1,bl+l
IF
(Slj.G.<:.C:Jl)
KC (2,o) ~KC(2,6) t 1
I!"
(1"lj.Gll.FI'5)
KC(1.7)=KC(1,71+ 1
IF
(n.GE.l'i'l)
KC(2.7)=KC{2,7)t 1
U'
(":t.Ge.I'~S)
KC(l,tI)=KC(I,6)+1
U'
(n.G~.r~l)
KC(2,tI) =KC(2,6)t 1
II" (Lsr.~Q.O) GO TO 60
II~:-rr; p,3'l'l) K,Cl,C2,CCI,CC2,SII,911,I'1I
l~~ FOi~.\T (lX.I 3,101"10. 5)
60 cHr11lU::
IIR:n: P,59)
1I/,n,'~LPHA'.(JX,'liEJ. BASED 011 CHI-SO. APPROX.','ll,'!')
1 A'J 3111 AllaH KC CORReSPO~D TO ·~:'I~;H::D R.\NKI:.r;:;
,au J(l) : 5.(1) =
1'1.'Z5 TH~ DTfA-SLeCK VARONce.
lIil:rE 0,1:>4) (KC(I,Jl,J=I,lIS)
.
159 I'll 0 AT ( / , ' 0.05', 2X, 10 Ib)
WiI:,!': 0,259) (KC(2,J),J=I,NS)
2:'9 l"iliH1' V.' O.OI',2X,10Ib)
59 .'Oi.'U
~;JLS
.lllE.1120
lilE"1130
c::J~rlNr~
C1~ 8!1(1'.RA,!l,N.U,HH,C~,CC2,NC)
C
C
AilE41110
lil.::;11 ?O
A.H.l .. .;)O
RA(I)=B(:l)-s(l)
SO
aao
Ail2H1.0
.1aE.l1S0
J.li :;: .. 11";0
1BE41170
cu~rl!lD:;
IT
1a~.
lll:'. ,,;/0
13!" JOO
18;;: .. '910
All:: .. 920
16E. 930
.:.aE4 ,40
All:;. 9S0
A3t;. 560
I.:lE. 910
Ail::. 'de
AilE" <;90
lil:: .. 1COO
AilE.1C10
l.ilE .. 1G20
.lil ;;;.10 30
J.llE.'C ~O
lBE.1C 50
J.il.::.1Ci)0
131::.1C10
.l:lH1G"O
l:iE.le ,,0
A3:;:.,1 )0
All~.11JO
E(I,Jj=i3 (J)-S!I(I)
2
AilE .. "SO
A3!. ;. .. 0
Ali:; .. ;;;70
,,'-1)
Aa;:~12JO
l3E~12aO
lilE~12S0
A3E~1250
lil';:~1270
Aal:~1250
lft:;~1290
1ii:;"U':>0
Aili:~lJtO
AiH:-1JZO
lili:41J30
13::::;13-0
.13:::4n 50
A3E~13?0
lll!~U70
1 ii:::~l 3 ,,0
1 in:-l 3',0
A3E:.1;< :;0
AilE.'410
13ll.'''20
liiE~I~)O
13E'I
',,~J
A"E",q :,.()
A~E~l.:'O
)'3::.:;.\,,-;0
A3EA '''<:0
,\, ilE"'. ,,0
Ail :::"'5·)0
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li<~ .. 15':O
130;"5.30
),:<E,,15"0
A3E"':-'S')
lit:: .. '5,,0
13.~.':;J'
1iE41S.:.0
142
c....
A?PE:mIX
D
(continued \
SOBBODTI!! lilI(T.V.II.H.H.IlH.C.CC.IiC)
C
C05?U~! JUAOE'S IlH (C)) STATISTIC TO AIALIZ! SAYDOIIIZ!O a:OCLS
C
OSIliG iE.IGiI'l'!D fiAHKIHGS
C
lis 1I0!3!S Ci' TB!HIIEIHS
C
lis lIu!a~ 01 BLOCKS
C
'(I)~ v!=roa OF
IIEASUBES OP VABIABltITt, 1'"1 ••••• 11
C
T(l.J)~ 5A'l'a1I OF HANKS WITHIli bLOCK3 A~l~S~D ON THE SEA.
C
lies 11156EB OF LEAST VASIABLE aLOC~S rHlT .ILL BE CEHSOiZ~
DIII!.SIOli T (I,II),V (II)
DldElISIO!l 5';(5) ,SS(5) ,VB(&O)
e T A " ' ! S(I) =0 IiUK OP '(I)
C
C05i'O~E 'tHE STATISTIC HlI
psptOAt(lI)
Q=PLOAT (5)
D!lISQ.(J+'.I.?a(p+,.) .(2.*P+'.)
61
62
C
72
73
71
C
81
82
c····
C
2S
CA1.L SUK ( ' , va. II)
H"'O.
HH-O.
DO 62 Ja',5
57 (J) =0.
DO 61 1"'.1
ST(J) -S%(J) +T(I.J) .VB(I)
COllTIIIO!
B"'HtST (J) eST (J)
conUlO!
c-72 ••H/I)E.
CHUG! T3E WEIGHTS S (I)
IF (lIC.ZQ.O) i!l'DilI
DO 71 Ia'.11
IF (n(I)-'C) 72.72.7~
Ya(l) "0.
GO TO 71
VI(I)"'.
COITUO!
COI!IPOU TlfE STATISTIC CII lIITB THESE lI!iI lI!IGHTS
DO 82 Jal,5
SS(J)"O.
.00 &1 Ial._
5S (J) -S3 (J) +T (t ,.I) *n (4)
COlfTUlJE
BH"B!tS3(J).S3(J)
COlfTlIiIO!
D!II- (P-,.) ao* (Ot'.)
CC-12.*U/Dill
UTlJ U
0
1;)
~I
20
:!11
30
all
-()
all
311
all
311
so
60
7J
ao
;/0
all
~ JO
all
EI
no
120
DO
140
~!I
1! 1I
:::1I
:1I
all
SlI
31
31
aI
!iJI
ell
311
:; I
=1
:I I
511
,n
Ii II
31
IU
al
Jill
iii
Ii II
«:11
!il
~50
~ .. o
170
1.:10
130
~'O
210
;20
2.30
240
2:50
2100
270
:Z;lO
1"30
300
1'0
320
330
3·-0
350
3':'0
370
l::JI
3·~0
a II
IU
EoJ
IU
al
ii I
Ell
.HO
.... 0
450
liD
il
'.0
s.oR!
0
SlJ8800TIII SOIt(S,II)
SOIT 1 GIYEI neToa or LUGltH
Dl II US 101 B (II)
1!I1I"'11-1
DO 20 J"".1I1!1
%EII'-8 (J)
as-J
K"Jt1
DO 2S LaS,!J
Ir (8(L) .G£.T!lIP) GO TO 2S
T!lIp.8 (1)
IISaL
SOiT
~O
Couno.e
Ir (as. EQ.oJ)
8 (SS) -i1 (J)
B(J}al'ES?
20
311
3J
COliTUiJ~
BEToali
liD
GO TO 20
II
"·00
*~O
~.20
~30
son
20
SOitt
.30
JlO
':>0
S.Qst
S.QilT
SoOiI%
SOit
son
s~n
SOBY
.so
70
;l0
;'0
'.'0
SOit 110
SOit 120
SOlit 1.30
soat 140
S.OB': 1 >0
S~B':
1'!>0
son
170
SOi'!
liIO
S.oi'! 1'30
APPENDIX
D
(continued)
o
S~3j
s~
suaaOUTr~z S~5~(T,E,U,IS,~,:a.~N,B~)
c
c
c
c
CO~PU::~ S~
(Pll.lEDII.\H STATISTIC)
IS= liU1I3Eil
OF '1REAIlIl::NTS
.=liU4j~i
01' BtOCKS
T(I,J':~A:ail
Z(I,J):IIA~aIl
c
AliD
ll~
SlIli..
511« ,
30
'+0
Of HANKS WITHIN BLOCKS ALIG~ED Oll THl:: :lEAN
OY 08SERVATIONS WITHI~ BLOCKS ALIGNE~ Oll :JZ
PIl(J)=O.
DO 11 I: 1.1I
Pil (J) :fa (J) tT (t.J)
11 CO!lTIlIUE
SlI=slI+yaeJ,*YIl(J)
12
CONTI!lU~
P=YltlAT
(~)
Q=PLOAT(lI)
SR s 12.*SN/(Q*(O+1.)*P)
c
'0
.2~
DIISElISIO~ -:- (1I,1I) ,l::(lI.lI).U (1)
DIIIl::lISIOli "a (5) ,TR (5).S (0:» ,UIl (l00)
COISPUTe 7HZ STATISTIC Sli
SII=O.
DO 12 J:l.1S
c
31
(ilAJlKlljG U'T::1l lLlGlI!lE!i':) 'illl;;*
5110*
50
:1:..1 SSS"j
5 S;j
S~3 ,
,,0
70
S"~i
10
ao
SN.H • JO
110
5 liS
5115j L!O
*
S!i3J , JO
5NB' : .. 0
~ 50
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Sli 3j 150
SBt
no
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S.!i;; t
';;lI3t
t
~o
: '10
~
00
COISPUTZ ~H:: STATISTIC BlI
DO 22 I:l,lI
DO 21 J=l,1S
5li:>j ::'10
Sli::.t :20
P\=!l*(I-l)t.J
o (It) =E (I .J)
S lI3i :':'0
21 COllTIlIO::
22 counuz
CALL BA!lP\ (O,UIl,LB)
TT=O.
D032J:l.1S
TII(J):O.
DO 31 1=1,]1
Til (J) =T5 (J) tUIl (I!* (1-1) tJ)
31 COIiTIlIU:>
TT=TT+T3(J}*TII(J)
32 CUIfTIIIUZ
SS=O.
DO 42 1=1,.
S(I) =0.
s ~3j
51i,,1
3 !l3i
S!l3.i
Sai5i
SBJ
5113f
S1I3.
S!i,,1
SlIi3J
311lil.
S1I3)
230
~50
2:i0
10
:'30
.. 90
~ :10
~
:;'10
:?-20
~30
-=>10
:;';;0
'>1131 ~oO
S !lili
:no
Sri3'
.::: ao
DO 41 J=l,1S
5!131 3'10
S!l31 aoiJO
5(1) =5 (I) tuR (IS. (1-1) +,J)
SliIilJ
"1 CaNT INU!!
SS=sstS(I) -S(I)
42 COSTINua
Dll=1.5*~* (Q-l.) * (4. *TT-O*P*P* (0*1't1.) * (0*Pt 1 .))
.DD=Q·0*?*(Q*Pt1 .)*(2.*0*P+l.)-o.*ss
BlI=DlI/D:>
BETUllli
END
~10
31131 "'20
S!l31
"'30
.:ill"j <:'40
SIiBt .. so
Sl/~l
.:..";0
S !l3i + 70
5!1aJ .. ~o
SlIil) 4'10
APPENDIX E
TA3L£S OF ESTIMATED POWERS AND
NON~'n'RA1ITY
PARAMETERS FOR ReB JESIGNS
e
e
T~~LE
£-1 .-
Esti~Ated
Power$ and
lIypothuDo:s
I
Non-centrality Parameters Under Different Alternative
Rell Dou1gno, Nom"l Erronl and
.'
Ill" Z •
.
k2
...u
.'....
e
•___ ._._h
______._____
1
:i
.~
__ •.• _ .'. _ . _ _ _ _. __
~
1
.~
_'..__". ___ . _
It
_._._--5
-_.__._-_. ---.--_.-----_._-----6
7
8
\
9
III
------------- ----T
1
C.C79 (')
0.D9
0.142
0.171
0.208
0.2;.5
0.277
0.;20
0.;4.5
0.24')1 (lj
0 • .50~
0.7716
1.007.5
1.;070
1 •.52.57
1.868;
2.224.5
2.4;.52
0.079
0.109
0.142
0.171
0.208
0.2;.5
0.277
0.;20
0.J'f.5
0.';7.5
0.0')8
0.120
0.14.5
0.1711
0.199
0.2'31
0.249
0.27.5
0.05
0.(;98
0.120
0.14.5
0.174
0.199
0.2;1
0.24')
0.27.5
'),fJ 55
0.':;72
0.095
0.12e
0.1:37
0.160
0.179
0.2e)
0.222
0.'-);.:35
u.1897
0.}'338
0·5S!13
0.7)08
v.9182
1.C7Z3
1.260.5
1.42C)
",c'7?
f"\.l"e
0.11,2
0.166
0.206
0.234
C.27'-1
('.J1B
0.)44
O.2/fS:~
0. 11921
( .7716
0.')66')
1.2';\.'tl
1.5176
1.81IJ7
2.2')78
2. 11267
n
T2
1.06
T
J
TI~
Fr
0.70
ilil
1.06
0.115
[1.15"
(\.i:. 11I )
,).; r~
U.;~ul
II.;','J
0. j~/I
(l.'JUI
'.).;:,,1 1,
lJ • .'5)\,ill
(J.U;);;
.L~I~)
.L';'J,;
l.,'UIJ
.L'/I)'
~.~;'uu
':'.:n/u
U)
~:..;tlj~'lJ.toJ~l
ll.lt{ul"
(.1
S"ti:;,..toj :;or.-ccntra1ity PnrlJ.motor (not co:np~t.o I for 'f
J
,.',!,
•
.
..11
~'
1.10
I .~,
2
",)
an,l '1'4
).
(..
1..1\
TABLE E-12.- EBttmAted Powers and
Non-cent~ul1ty
Parameters Under Different Alternative
HypothooOl I ReB DOO1.gnD. 1.&p1&co Errorll &rid II • 5 •
......
k
..
'l:...
~
1Il
T1
T2
2
"
'Vi
1
2
J
~
.5
0
7
tl
9
0.067 (11
0.092
0.107
O.l;tl
0.112
0.191
0.21;
0.2~')
0.269
0.1316 (2)
~.'n58
1.02.40
1.5090
2.0076
2.2757
2.5793
2.9!J.l4
:1.3297
O.on
0.09)
0.110
C.l~
0.169
0.11l5
0.219
0.245
0.265
l.~!l
I
I
0.0l:!2
0.111
0.133
0.159
0.197
0.221l
0.269
0.30;
0.32)
0.085
0.112
0.1)2
0.160
0.19b
0.227
0.261l
0.297
0.)19
o.c-e)
:.J.11)6
0.1)5
0.162
0.200
0.220
0.25/.l
0.295
0.))0
O.62~7
1.0Clji.
1.46)6
l.IJb)7
2.4007
2.b'M
).11)4.1
3.0717
4.129tl
O.~eo
0.102
0.127
0.169
0.2)9
0.224
0.25tl
0.299
0.))2
1'1.:/.)79
0.\1f2?
l.yal)
l.~M
2,52J;o
2.72U9
:l.Uin
').7242
4.15.59
0,07°
C.f)')l
0.11',
0.1/¥~
O• .lUZ
0.195
0.225
0.257
0.2')2
n.',IlTJ
1.75UtI
.l.l~Hl
1. 59'11
2.1495
2. JJ1I;
2.71,2/;
).1709
J.6J2I~
T)
T4
Fr
nil,
P
e
til Eat1mato<l
Powur
(;:)l.;~tHl&t"'''l
Non-<:lllntnUty lilrar.lot,QI'
(not nonrputllcl fot' T , '1') ll.nd Til) •
2
e
l.U)
lotl3
1.5U
....
t-
-
e
e
e
TA Hr.',;
b:-<' •- SGtimatei Powura an,1
lion-contra 11 ty [,;\ril'1loloN Unclor D1fforont Altc.... u~ tI vo
lIyvotiluuOIJ I llCll lloo1~nEl, Non"l l!;l'1"Ol'O llnd IU •
J •
2
lJ
,,'
.'
.,"
:'I
'11
,
k
_0 _ _ _ _ _ _ _ _
• _ _ _ _ _ _ _ _ _ • ________ • __
.~. _ _ _ _. _ . _ . _ . _ . _
----'.-
_.-
·d
1
2
)
If
5
6
0
7
.--
Yl
9
-
T1
o. :'77 ,II
0.10)
0.125
0.144
0.161;
0.182
0.205
0.235
0.267
0.}464 ("
0.6594
0.9130
1.1259
1.)452
1.5393
1. 7038
2.0989
2.4323
0.077
0.099
0.126
0.146
0.166
0.188
0.210
0.240
0.267
"J66
').~e1
0.103
0.126
0.148
0.161
0.176
0.19.5
0.222
).068
c.rn
0.10)
0.127
0.140
0.158
0.175
0.193
O.nl
I;.' l:ib
IJ.U'J'~
').104
0.1)0
0.155
0.167
o.Hll
0.19)
0.216
1.)778
1. 52fJ5
1.6,566
1.0')98
1.03
T2
T
1
T4
:-'1'
0.2)33
O.432!'
0.6711
0.~95
1.2£;70
0.':72
0.1299
0.116
0.142
0.132
0.21)
0.2>,)
0.260
C.;S:!
U.2f}',2
'!.ol2)
a.OW}
1.10J7
1.5')))
1. ilGe2
2.1 1,(,7
2.:J5')5
2.7'155
O. '1:/)
U.101
'l.1l7
0.150
O.li.JO
0.219
0.21,9
0.270
0.1":'2
']. =:1j~)t
,) ,1.159
O,fJ21l1
1..1')21
1. ',17;;
1.Y)1'3
:!. Li. t,n
2.1.<,'1
2. 7~.(;1
(I)
E:jl1:r;i..~t,.:.J
r01:cr
l!)
E,.tb•• td ::(!l.-""nlrnl\ty !'a'rll'nuLl,r
RJ.
c.e?
1.16
1.:<1
I-
(nrit
0(':;::)I..J~tjil
..
for· T"
T!
/tn'
Til
t-'"
+-
>J
TABLE E-1.-
Esti~ted
Powere and
Hypotheses
I
Non-centrality Parameters Under Different Alternative
RCD Dea1gns, Norul Errors and m· 4
......
.......:i
•
.
u
k
Ul
'"
1
J
4
\
6
5
8
7
9
0.074(~1
O.OCX>
0.119
0.1'}6
0.171
o.1ti6
C.2~6
0.252
0.277
0.;9:61')
0.72,18
1.1'1.61
lo27b9
1.7;27
1.922:)
2.l1150
2.7290
).C2tll
0.073
0.095
0.115
0.1)9
0.175
0.186
0.221
0.2.50
0.282
0.c6lJ
0.091
0.110
0.124
0.1)6
0.171
0.19J.
0.211
0.2))
a. '65
O."e6
a.l0e
0.125
0.155
0.171
0.19_
0.212
0.2)6
.). "61~
';.Co5
0.11.;1-1
0.125
0.154
0.1'15
0.1\)6
0.220
0.241
n.167!'
0.55('/..
0.81:>0
1.126'+
1.51 /"1
1. 73J5
2.(;467
2.Jl119
2.5'166
':1.'75
0.096
0.116
0.149
0.181
0.192
0.225
0.259
0.)00
O,l,{,l
0.72(;0
1.orj+7
1./II'tlu
1.8592
1.997)
2 ,lt~28
2.8110
).)C2C
0.0'16
C.I01
0.121
0.1/,')
0.1'11
0.2'j(J
ll.25D
O.2tlJ
0.)16
n.4?l5
u.??y,
1.1011
lo tti,8U
1.9~,5
2.l 9~'J
2.7050
1.099'1
} ,4')211
1'1
T2
2
2
1.)2
T;
TI;
Jo'r
F."
"
e
(I) l>Liilll<ltod
Power
(,I 1~~tl'I,~l,(d
N"n-\illntrIlHty l'"raln~t.. l'
1.11
1.)9
1./111
(not n(1onI"I~'"\ for 1';11 '1') l>IV\ '1'11 )
e
r-'
-+(.AI
e
-
e
TAB~£
£-4
~sti~ated
-
Powers and Non-centrality
HypolhOG06 I ReD
~esign6.
....,
0
"
.'
-~-.
~:l
--.
-
'.r,n
'1'1
2
Un~er
Different Alternative
2
A
\
--------
--._--'
1
vI
Farameters
Normal SrroTe and m· 5 .
k
M
e
J
If
5
8
7
6
9
'J.sn
O.lC6
C.131
0.173
0.1U6
0.2J7
0.231
0.200
c. 55";2 ~1)
"..7758
1.0084
1.4026
2.0219
2.20,58
2.4972
2.82J4
3·2107
C.t;32
Q.~?5
0.110
0.132
0.165
0.177
0.205
0.229
0.253
).1:00
C.El
0.lZ0
0.11.2
0.1'11
0.19)
0.20B
0.2)4
0.265
''',.C,'1[3
'J.l !)6
0.121
0.143
0.166
0.107
0.207
0.2y..
0.26)
,.,;'i.
,j.l.14
:::.131
0.11,7
v.l'll
0. '(;5
0.~11J
0.2.1'1
O.Z;';U
'1. 7i.,.113
1.1372
1.4r:26
1.643tl
1. 99))
2.1917
2.6475
2.9;>41
3.1G41
G.e'lL;.
G.IC{;)
0.1)0
0.152
0.188
0.202
0.236
0.261
0.263
0. 4,6"'1,
1.:Y:U'f
2.8)C6
3.2239
3.5142
(,1
1.59
TZ
T)
'1',
"
1.25
;"r
1.55
f~.a.
; .'1')
(J.
t
)-/)
1.3872
1.7177
2.233B
2,112S)
u .1~~6
() .1:;6
0.1\13
C.~,l')
0.2,\)
0.26j
O.2')U
1 .12 '1)
] .TIUf
2.';" 16
2.5;:'::l{,
2.91(/)
3.2709
'j,'/lll
i,l
1.M;
" .iji'i)lj
(l)
O.I:'r\)
i::L1tb'llo.j
('I U:,.~h41.,.1
I,mor
~1.,n-()lll,lr.. 1tt:f !·olr....r.ot"r
.
(noL Qo.np"l."l ror 'J'.."
'1', I1n,l TI,, )
I--'
+.()
T~L~
E-5.-
Esti~ted P~wers
Hypothesos
I
an1 Non-contrality
Para~etQra
!lCB Dealb1l3, Unlfon Errot"$ a.nd
..
.
u
k
....III
Under Different Alternative
2 •
III a
2
~
"it
,-
:'l
',1
1
2
'3
It
--T1
5
6
tl
7
9
G.C92 :")
0.12)
0.159
0.19t
0.22)
0.260
0.290
0.))2
G.354
0.1587 Ul
0.616)
0.9101
1.19)7
1:42B4-
1.7292
1.9'75)
2.)25)
2.5119
0.';92
0.12)
0.159
O.l<}l-
0.22)
0.2bO
O.2~
0.))2
0.).54
a.e.a5
0.112
0.135
O.lb)
0.190
0.222
0.246
0.277
0.299
'l.0!:!5
0.112
0.135
0.16)
0.190
0.222
0.246
0.277
0.299
O.G.5?
0.091
0.10)
C.12!:!
0.1/1-6
0.lb9
0.19l-
0.211
0.2)9
o. C7!ll
0.)5~)
0.4505
0.6570
0.tifJ.l.)
0.991)
1.19'")7
1.))1)
1.5582
0.':92
0.12'J
0.159
0.192
0.222
0.25/l
0.290
0.)30
0.)55
').15lJ7
a. .5?11
0.9101
1.1'/75
1.420)
l,('965
1.975)
2.):154
2.,5204
')·ou')
1).1::'/
0.1(,1
O.2i]J
O. 21~o
0 ... '1')
0.;20
O.J5n
O.JtF)
0.)))5
O.UHlU
O.92b)
1.2b05
1.b151
1.OOIHI
2.22Jl.5
2.4'177
2.7460
1.12
T2
T
1
Tlj.
F'r
Ra.
1.1)
1.21
F
e
0.76
~) I::stbJl.t.Qd
Poltor
<..1 S::tlmAto:l
Il<;n-contrali.ty
I'a.ra:11tltor
(not com~uted for
T , T) and. '.'4)
2
__
~J
•
~
e
e
e
It
TABLE £-6 .- Estimated Powers and Non-centra11ty
HypotheB8lI
I
ParL~eters
Unjer D1fferent Alternative
RCB Dea1gns, Un1fonl irrora &n,l • •
..
....
J .
k2
()
....
~
1
.J
j
'
O.l')tl
1+
5
6
7
8
1
9
0.1)0
0.1,57
0.195
0.22tl
0.2.51
0.2tlO
0.;12
0.42")2 (1) 0.717tl
0.~95
1.20!:l9
1.0779
2.0257
2.2b.5tl
2.5675
2.9005
o.c84
0.100
0.1)4
0.101
0.19)
0.22!:l
0.249
0.2!:l4
0.)09
0.065
0.077
0.104
0.121
0.14)
0.175
0.19)
0.21)
0.224
".S61
:.:J7Cl
0.1~
0.122
0.140
0.175
0.194
0.211
0.22)
).r,5U
1).'170
O.l()t
O.lHl
0.144
0.104
a.1Sl)
0.21tl
0.2)4
').PS5
0.))40
0.6711
0.lJ))2
1.1259
1.34.52
1.0;66
l.92()lJ
2.0ClCl.5
DJ:71
0.102
0.1:;1
0.149
0.171
0.199
0.2)2
0.205
0.2lJ7
'1.2716
0.1;)170
O.9l:lCCl
1. lUll
1.4210
1.720)
2.0676
2.4115
2.tA02
0."''/'
').11,0
Cl.l)'1
0.16)
0.192
0.220
0.2;)0
O.2U;l
0.)11
O.2~;Q7
O.(,'Jr5
1.0400
1.)5(>0
1.6460
1. ',mu
2.)179
2•.5tkl)
2.U,/Ol
J,':ltl) I')
Tl
2
1
Vl
)
1.28
T2
T)
..
T"
Fr
He.
'I
(I)
('l
h:uU",l1tlld
1.17
l,ZU
l'owor
~4tlllllAt,,'1 Ilon-c<lntrllhty l'll.rlllllotill'"
O.~
/
(not OCilllputti l \ (or
T2 : T I.n,\ Til)
J
•
...
,.
III
TA~:2
S-7.-
Estlrr~ted
Non-centr~11ty
Powers and
Hypotheses
I
Jarameters Under tlfferent A1tern&tlve
RCB Designs, Unifom Errors a.nd m • 1+
0
k
•
.
2
• J
'Vi
L1
rl
c,"j
1
2
...
4
6
5
7
8
9
J. :;'1')
0.13)
0.176
0.217
C.22U
0.200
0"W2
0.))7
:.9209
1.2)67
1.7X>1
2.)05)
2.4J9)
2.8249
).)258
).7425
0.069
0.112
0.141
0.17)
0.216
C.2);
0.26)
0.)05
0.)4)
0.075
l).c90
0.110
0.1))
0.161
0.169
0.199
0.220
0.239
0.076
Cl.a?:!
0.113
0.1'3.5
O.1b)
0.173
0.2C1'
0.21')
0.235
0.072
O.Cetl
0.109
0.129
O.lbO
0.171
0.200
0.219
0.237
O.35S't
Cl.b"")2
0.9068
1.1tl27
1.5917
1.7327
2.0~)
2.)297
2.9<8)
o.O?}
0.0)1
0.116
O.lLfl
0.169
0.19)
0.229
0.251
0.275
C.;'/;")
O.Uf'/b
1.');)~7
1.:JI1".Jlt
1.7072
2.0o~4
2.4514
2.7170
).0.),3
0.071
,').()~
O.12U
1l.155
0.1 ?2
0.210
0.251
0.2B6
(l.ojl')
n.:}';7
0.7208
1.1t>92
1.5270
1.9970
2.29')1
2.7170
).1J,9'
J.52U1
').'X,;(3 "'
T1
'3
G.2);3
ul
lob!!
'2
1
1
3
4
?I-
Ra
e
il:~t1"llltod FOlio I'
(tl
:::,"~ll1"L~u'l Non-o"ntr"lHy
r..r .. :ng~ur (nv~
1.)4
1.50
~.
(I)
1.20
I-'
OI"lIlI!Juti>d (01'
-
T , T
2
J
~
.. n1 T4 )
e
-
It
E~.- Esti~ated
TABLE
e
Powers and non-centrality Paramoters Un<ler Different Alternative
Hypothosos
I
ReD Dooigna, Uniform Errors and m • .5 •
...
(,)
.
\(2
.
->
....
Ul
Yi
__ .___-_n._. _ _ _ _•.·
j
,,'
1
2
- - --0.070
T1
'J
4
.5
6
tj
7
9
-----(I)
O.0!:!9
0.Jcm{l) 0.7247
0.117
0.149
0.186
0.19;
0.222
0.250
0.275
1.1848
1.6734
2.205!:l
2.3175
2.7·')18
3.0778
3.40tl9
1.58
'J.071
;).095
0.12:J
0.151
0.1!:!3
0.200
0.237
0.267
0.30~
(.\.D70
0.190
0.10.5
O.ll!:!
0.139
0.144
0.1~
0.199
0.214
0.'In
0.091
C.IC4
O.ll!:!
0.139
0.14l:l
0.170
0.19tl
0.210
V.L'!)
U•..,YI
lJ.l"y
O.U'1.
0.1)U
0.1)0
C.lc,o
0.191
u.2·,b
0.£;422
O.!:!J97
1.0.570
1.1b4l:l
1..5)90
1.cI:Jtl2
1.<;215
2.2757
2.4UJ4
I 0.C05
o.om
0.102
0.123
0.151
0.159
O.l!:!tl
0.215
0.2)6
\ '').2'N)
0.5[)'j6
1).1.29
1.27').)
1.70')0
1.11202
2.:mS
2.6066
2.b906
1.'·Uj
(I.ll'.
1,1.1)'1
1·.1C,~
V.W')
v.~~~
ll.2'l
o.;n.J
I),(J""/,
1 .117;'
1 ,/IljlVI
1 • ~;(Jl7
2.;tJI:JH
i' ,','niH
1. I"}'I 1
'\, 'II V
12
T')
'1'4
~'r
I
1.12
I
Ra-
..
~
I".
'6'/
'. TII'I
(II
(.)
So tlm:ltO'l
Powur
C:8t1m~tu.\
Non-eontno.l1 ty l'ar~"ltlt,,~·
(
not
1.2tl
loyJ
oOlapl~toi\
for
T2' T
J
..".\
Til
)
~
t,J
\ /1
Iv)
TABLE:
E-9.- EaUl1IAted Powers &nd Non-eentral1ty Pa.rallletera
IlYJlOthQl3es I ReD
ne"!~8.
.........."
Laplaoe li:rroro &00 II • 2
,
------------------_._-- \
2
1
:).:'90
T1
•
kZ
j
01
Under IJUforent Alternative
(L\
J.l44
0.4:89(~ 0.7860
4
J
5
6
B
7
9
0.2(;()
0.245
0.200
o.::m
0.)63
0.411•
0.447
1.2508
1.6070
1.9588
2.337
2.1890
3.0)60
3.3)62
1..56
0.098
0.144
0.200
0.245
0.200
0.333
0.303
0.414
0.447
0.104
0.153
0.192
0.232
0.271
0.307
0.339
0.373
0.406
'J .1eA
0.151
0.192
0.2)2
0.271
0.307
0.339
0.373
0.4')0
I:.u'/l
(J.ll'
').l:;U
0.1')(>
o.~:l
o.'·'?
u.2:1f
o.J~')
0.)6
°J.len
I).
0.9020
1.2099
1.4122
1.7rJ+7
2.0083
2.3')00
2.5290
0.095
G.l'})
0.2Ch
0.2/12
0.2tl5
0.332
0.360
O,lnl
0.444
-'.)0,6
'.m"
1.27'>'
1._
1.9'.1"
2.)25)
2·5612
)."""
).)0",
'J."l'J
0.1)')
1).1j/1
O.~l')
(l.~:I17
O. )J)
n.)?:!
'l.'I()I1
1.97-'0
2.1337
2.6G67
2.'1109
1'2
1'3
TIl
Fr
l
!lB.
55'Xl
UIl.:::;'
l.Ui
O.l.ms
f).2BY'
_"_.'."
.-_.~
(')bJ~I:ialu,l
. . . _ _ .•
1.0('75
1.3960
l"UI",1
__., ___ . __ . . . . , '. • .__ •. ____•• _ ...... __ . 0 " " ........_ _ _ • _ _ _ ._ _ _• _ _ _
_._.~_.
_
.
h.luol'Q
'''::;"U.'WoLu.l lIvl-~,,"LL·lI.ltt.Y l'.. l·...lI"H"l·l; (uoJL l.oJl"I,uL".l
-
1.56
1.;)7
V
_ . _ ..
,
-
roJL'
'I'.,. '1']
t-'
1I.,.t 'I',.)
'E
e
e
e
it
TAB~S
E-10.-
Eeti~ated
Powers and Non-centrnlity Param&ters Under Differont Alternative
Hypotheses
.......
I
RCB Deeigns. LA;>la.ce Errors e.nd
.,..
---,
U)
1
.:!
O.O/jO
T1
a)
•
k2
0
Ul
III
2
J
4
5
_._--0
.. _.
__ -..
7
---
...
U
9
---
._.
0.099
0.125
0.1,58
0.186
0.225
0.258
0.295
0.)2)
0.)8)4 (0)
0.(12)
0.91)0
1.2798
l..5I:l20
1.9<;4)
2.))87
2.72)5
).Ol.7i
O.C?/j
0.102
0.12)
0.161
0.1/j9
0.229
0.2.59
O.2~
0.)20
O.G?7
0.101
0.1)6
0.1?2
O.2c4
0.240
0.273
0.)08
0.)4.5
0. r.?5
C.IQO
0.1)0
O.l?)
0.2;',6
0.2)9
0.272
O. ):-6
O.Y>3
". "c..:J
().IJ~tJ
0.1 11 )
u.10)
u.<: Ji
u.~
JJ
v.n"
V.JUU
v.'jl)
O.2J}tl
o.o')();
1.L'Ul5
1.;5'>0
1.?1))
2.C7U'l
2,IHj',J
2.e)~tJ
J .2462
'J.OtJ)
0.11.5
0.149
0.19)
0.2)1
0.20.5
0.29.5
0.)20
0.)0)
~. ?2'35
)
11\
•
\
1.)/j
12
T,
"
'1'4
l,'r
RIl.
1. 115
'l.I,? ;2
u. 'I?~'d
1. Hll1
1.b'l;6
2.(1571
2. 1,115
l J .L'll
;).10')
(1.1;:<)
() .1,/\
O.~·,O
V.;:L'/I
(',:::/)
0.):';
(\. ~ '/1 (,
(' ,i,"';')1
n,t)",,;;n
1 ,/I'j ,l '".i
1.,/,11"
;:.1 rx,.'·)
;1,'1'.1 ' 7
;1
.00IO~
1.,52
J,Lt)O'J
0.)<:1
loW
~.
,111'/"
\11 I:::,~h.t.o;l
!'ow"r
(" lC:;th:l.ld
N,)U-c~ntrtr.llty ~an.>r."t"r (n.,t. OO"I\'ut"d f,),' 1',,, '1") and 'I'll ) •
;> ,ri""l
~
r-'
'"",
VI
TABLE E-ll.- Estimated Powers arA Non-eontrallty Parameters Under Different Alternative
Hypotheoes , ReB Designs. lAplace Errors and iJl ....
..'..."
u
k
2
~
....
.>
_.---~
:s
1
U1
2
0.067
'1'1
•
0\
3
I.j.
5
0
.. -.,-----~
7
'1
1
9
0.000
0.124
0.1~
0.200
0.226
0.2b5
0.301
0.323
0.28)4(1)
O.b:m
1.1148
1 •.5659
2.0903
2.4150
2.0048
3.3139
3.5757
0.Ob8
0.0;4-
0.130
0.lb8
0.207
C.231
0.2bb
O.2'J+
0.324
a.07b
0.109
0.151
0.1~1
0.226
0.2.56
0.299
0.339
0.370
1.69
T2
,
'!'
'J
I
f).C??
0.107
0.151
0.1t})
0.225
0.25J
0.3°2
0.338
o.J71
U.U'/'J
'1.1 )2.
'
0.1/1("
0.173
0.<:19
l:.:8t2
0.291
0.331'
0.)75
!:
').467b
0.l:l')74
1.4':';4
1. 75tJ1
2.)297
2.b:)tl7
3.1~M
;.71;25
4.1979
"I
0."'72
0.107
0.1%
0.172
0.224-
0.251
0.)02
0.;40
0.379
I
;1.3.57.
f).U7tJO
1.4~1'
1.7'-f 7'
2.)')07
2.7170
3.'3258
).~5)'l
1'.21lbl
'1.'/0
lJ. JI)l
').1 /,)
'1.10)
O. J',)')
(,:.<:22
0.2,~
(J.2 'N
1I.'j2j
2,}U>')
2.Ul'jO
1.lUJ()
J,'J7Yl
l'
4
j,
IIr
Ra
pI--.,~
).'J~lj)
(j .Il'Yf/,
2.0UJ'I
1.Jb9?
l.t>5bO
' - - - _ . _ _ _ _ _ _ _ _ _.,
lil
l~o
LImatou
l~) ,;"ttl/~to'l
--
;,
1.06
1.88
\
l.)'}
!'owor
Uon-cont."t\llty lur/onvtvr
(no\. (1()m,,~:t.I"\
e
t':ll.'
T;.'.
'l'J
~
I1nl
~
T'I)
e
.11_"11arson. R. ~~i Sancraft, T. (1952).
nesearch. Ne·,f YO:-k: HcGrali Hill.
Statistical Thearr
~~
Eerna~,
A. and van ~lteren, Ph. (1953). A ;ene=alizati~n c: ~~=
::lethod of n :-a.n:dngs. Indagationes :!athe:naticae, 15: ;53-;6.?
B:::-own, G:tl. and ~·Iaod, A.H. (1951). On media:! tests fo:- .:..2.-1.5=':::hy?otheses. Pracee~~~s of the Second 3e=kele SynJosi~~ ~~
;':atheTaatical Statistics and. Proba~ilit.Y, 2.: 159-10 .
Cha.!<ra.vorli, S.?. ani Grizzle, J. (1975). Ac2lysis of cl.:!.t.a. :r:·:"!
nulticlinic ex~er~~ents. Biometrics, Jl: 325-338.
Coch...""'an, ~'i .G. a:ld Cox, G.a. (1957).
Jan..." Hiley and SO::lS.
Experi.''7l:m-tal Desi;ns.
.e'''' ~~=.-{:
Conover, ~L.J. (1971). Practical ITon:Jara:le-:.ric Statistics.
Jo~ Wiley a~i Sons.
:~a~·"':':)r:~:
Cox, j)3. and Stua.....""t, A. (1955). Some qUlCZ<: sign tests
location and dispersion. Biometrika, 42: 80-95.
:r<=n:' i::1
:0:::-
-
Dem,ster, A.P. Me. Schatzoff, H. (1')55). ~A~cted. signEi~ce
as a sensitivity index for test statistics. Journal)f t:.e
American Statistical Association, 60: 420-4J6.
Doks~,
K. (1967). ?obust procedures for 50=e linear :"!o~e_s
observation ;Jer c3ll. Annals of Hathenatical 3tatistics,
673-883.
Du=bin, J. (1951).
British journal
:ncomplete blocks in rar~:ing
?sychology, 4: 85-9C.
0:
,-
~~
.
J-'
exper~~en~s.
F=ied.'"1an, :-1. (1937). The use of ranks to avoid. the assu::l: ~i:n of
nar::la.lity ir.qlicit in the ana.lysis of V?-riance. Jo'.rr:1a.l::~ -;:.;-.~
~'"1eri~~ Statistica~ Association, 32: 6?5~S98.
"::J Q .
('-:J
,..,~?\
l 0 S vu_y ():~ anal
' 0:, 7'3.~an·:e
1_ / . A 'r
:.' onte e
ax
YSlS
C1'lber.-r" _~.
ani co::metL~; =ar_~ tests for Scheffe t s :1ix:=·i r.'.or'lel. .~2'~,a.::" ::~
the ~:"!e~ic~~ 3ta-;:'is~ica.l Association, 67: 7l-75.
"';l
_.L
l:aje:<, J. aml S~:ia<,
Acade::lic .?:!:'eS3.
z. (1967).
,
'='~.,,~
___
'I .... _i
158
Eal'L'1an, ~.J. (1~5S'.. The asymptotic pCnie:l:" 0: cer:.-ain tests -::a.sec
on mul-:.i?le co::::.-elations. Journal of the :ioyal Statist:'-:;;L
Society, B, :l.3: 227-233.
Hast~O"S, IT .A.J. a:ltJ.
lje~; Yo::=~:
Peacock, J .B. (1;;-75).
Jo1'_'1 ~";iley and Sons.
S':.atistica1 =ist=.-=-.l"::'::.::s.
H:>d.ges, J .L. ane.. :.eh:1ann, E.L. (1962..
?a.7J..~ methods for co~--~-~:':.:'c::l
of L'1Qeyenient eA~riments in the a.~lysis of 'roariance. ;:~~:s
of :·!athe::a.tica.l Statistics, 33: 482-497.
H~llancier, L
;'~athois
.
ad ~lolfe, D.G. (1974). Non"Ja..---ametric sta:'ist:'~
new Y Jr~: J olm ~filey and SO::lS.
J~hnson, lL and :{otz, S. (1970a).
tions - 1.
Continuous Univariate 'J:'s-:::'i":t:Eoston: Houghton Mifflin Co.
Jow..son, £i. and Iot.z, S. (197'Jb). Con:'inuous U:1ivariate ::J:'s-"""';xtions - 2. ~oston: Houghton Hifflin Co.
:3 .L. (l-':'~\
.
J ~1ner,
\ ~~;.
The median significance level an~ ot~e~ 2::la:'1
of t est eff i cacy • ~J...;;o..;l.Ir::!;.;;_;;,;_a..;:";;;.'.....;.o.;;;;f-.;t;.;,h;,;e;..,..;h:...;;·..,...;;_~e;;;::=.;;i;..;.::;a.::;.;;,;;-;;;...;S;;;....;;,:.c...;;;-..;-:..;::.;;;;:.3~tical Association, 64: 971-985.
~~ple ~ea5UL~S
o. (1952). The Design and .~lysis of EXDeri~ent3. ~~~
York: John 1iley and Sons.
Ke~pthorne,
Kenc.a.:u, ~LG. (171')).
Griffin a..'1d Co.
Rank Correlation ::ethois.
London: C~les
Kleijnen, J.P.C. (~974). Statistical ?ec~~iaues in Si~u2a~ic~ - !iew York: :'~e:' 'Se!<ker, Inc.
~eijnen, J.P.C. (~975).
New
Yor~:
IL-us~l, °iT.H.
proble:;].
Statistical ?ec~~icues in SL~ulatic' _
Inc.
7T
~a~el ~ek~er,
(1~52).
A nonpa.ra.metric test f::>r the severa.::' S~?.:..~
A.nna.13 of Hathematica1 Statistics, 2J: 525-~~.
Lee, :S.T., :sesu, :·!.:L and C-ehan, E.A. (1975). A ;ilonte Car::'o 3:'~:'- .....;the ?Ower of 30~e two-s~~p1e tests. Bionetrika, 62: ~2~32.
Lehna.l"lll, Z. 1. (1 ~) . Asymptotically non.?CL~i.etric in.::e~e~·~e i:l
some linear ~oQels with one observation yer cell. Anr~~s ~:
:·:athe:na:tica.l Statistics, 35: 726-734.
Le~~'1,
~a.'1ks.
Z.1. (1775).
~ -;.r
;••.
J..
tests for
-Le:a:::er,
.
Non~~etrics:
San F::=ancisco: Holden-Day,
('..1.7(..
rY'J.-
'.
•
T t e on th e construc'tJ.c:l
..
... ••
•
~
10
OJ.J:' d'~Sl-r!.:tl::."-=::-:~e
ran~o~ized
Jourr~l, ~: 21-~.
Sta~is-:'i~l Met~o~s ~e~
.Ll1C.
block designs.
S~uth .~rican S~atis~i~::'
1.59
::er-.Ia, te.:::", and. Sa:r:a:lgi, J. (1')67). As::r7:.:;:-:'::;:.:':: e:ficie!}:;y ~­
certa.in r:"_'1..:": tests lor comparative e;·:;-;;e::-:5...-:e:-:-':.s. ;~':...":.a=.s --'
::a':.her'.atical S,:,atistics, 38: 9;)-107.
".'1' c'naS-'-lS,....
I •
T
( J..)'().
' '"'71 \
C'
;,
rl t'1 cal v alUes
Bio~. Zeit., l~: 118-129
-~
0_
?ri eclna..'1
..L~~_
t,,_ _ " ' .
~'IooC.,
A.H. (195+). On the asymptoti- efli::;ie~ty of ce~...ai:l :.::;::..?3-1:'a:letric two-sa.::lple tests. Annals 'J: ::;:?tc9::latical .3tat~-:i:s,
25: 514-522.
O~fen, .J.B.
(1;;62).
::andbook of Statisti::a::.. ::-3.:'19s.
~ea:'i.:J.g.
:':as..s.,
Ad:::_ison-',~esley•
Pratt, .J.~.j. (1959~.
:%marks on zeros ani ti-=s in the -'1i::.cox:::J.
JOurT~l of the A~e::-ican Stat:s-:i~3.~
Association, ~: 655-667.
si~ed
r~'1..~ yroce~ures.
Puri, :·f.L. and Sen, ?K. (1967). On some ap-:i:lU:.:l non;era::l~t::-ic;
procedures in t~o-way layouts. Jo~~l of the Amer~can S~~~is­
tic;al AssociatiJn, 62: 1214-1229.
Puri, 11.L. and Sen, ? .K. (1971). NonJara.."'1et.::-ic ::ethoi3 :"''1
\-a.::!:"iate A...'1alysis. New York: John ~liley 8.1:d Sons.
::-.:.1.-:i-
Qtlade, J. (1966). On analysis of variance fer the :{-sa-:l?ls :-:;:--:"':J::'e::'. •
.A nnals of !!atne:1a:'ical Statistics, 37: ~7-7-l759.
Q":la.d.e, ::. (1972a). Average internal rap_~ so::-::-elation.
i-:athematische S-:.atistik, Amsterdam, ~·Iay l~,?2.
QuaC.e,"J. (1972b).
;''''::''e-:,:.=;'
l·_'1a.lyzing randomized b::"oc.:{s by weish-:ec. ~3:.b53.
Statistik, A~te~~~, July 1972.
Af~eling :1athar.At~sche
Quade, D. (l973). E~~ct Distrimltion of Fri~~~an's Sta.ti3~i~; 5
Treat~ents.
Un]Uolished notes.
Quade, D. (1975).
Effects.
Analyzin,g; Randomized 31oc:§ '",ith A:<.::'i-:i-;e
Un;ublished notes.
.>
J~_:
Pahe, A.J. (197~). ~ables of critical val~es for the :~t~ ~~tchsi
:;;air s ignec. ra~ statistic. Journal of "tte .-".:1erica.r: S7.a:':.:'sti::.3.l
Association, 69: ;68-373.
Rao, C.P.. (1965). Li~ear statistical IPJe~er::e ~id Its AJ~l~~~i:~.
r;eH Y:::>rk: John ',iiley and Sons.
Sarangi, J. and :1e!l...."Cl, ICL. (1969). Some
Eoc.ges-Leh.:.";1aJ"'.n co!:.ii tional rank tests.
Association Bulletin, 18: 25J~1.
='~her
results ~n
Stat:5-;.i::=.-
:2.- ~ltta
(1 >'-J
-,,(6 ).
\
~
. t' . t
.
.
'
~ens~ ~ Vl Y C07il~lSOnS al:lor.g ~8 5-:'3 J: :,::.~
€0ne:~~ linear hy?othesis.
Jou-~al of the A~eri~~ statis~ica:
.':'3sociation, 61: 415~I:J5.
-,
,.:;
~i1a
'r
j..",.
", ~
",:30::,
.1.
_
0_
""01
of a class of qillL~tile ~ests ~
_
Jrob1ews. Calcutta Statistica~ Associat:"o~
3u~letin, 11: 125-143.
C
, , . , :....J ......
~;1"...J.__".L,
(1,::-t:2)
\;AJ
•
On.:.
-'-'n°
Lt _
S0~9 non-par~e~ric
Sen,
?~. (1967). A note
x~-test. Biometril~,
on the as~ptotic efficiency
677-679.
54:
~f ?ri~i3an's
3sn, ?K. (1968a). On a class of aligned rank order ~ests ~n two"ay layouts. A.'1Jlals of (':athematica1 Statistics, 31: 111.5-"'124.
0:
Sen, ?K. (1968b). As~ptotica11y efficient tests by t~e nettoQ
n ~~J<ings. JOil-mal of,the Royal Statistical Soci~t~, B, :~:
:712-317 .
.,~en, :.~.
.,. , -. (1971'
-..L'
.,
~
a). A.e-~u~uner
no t e on th e asympt o~~c
Friec...llan' s X~ -test. r'!etrika, 18: 2)4-237.
.
~~..
eI=lClen~y
01~
Sen, P.K. (1971). Asymptotic efficienty of a class 0: ali~e~ ~-r~{
order tests for ~ultiresponse eX?8rLllents in SO@8 inco~p:e~e
bloc=< designs. Annals of Hathematica1 Statistics, L;.2: l:'CL-ll12.
3en, P.le. and Govindarajulu, Z. (1956). On a class 0: c-sa.:l:pl~
~eighted rank-s~ tests for location and scale.
A~~~a13 ~f t~e
Institute of Statistical (·1athematics, 18: 87-1:)5.
'=".:~x
VA.:.qGZ~'T - :=:andom varia.ble distribution generator. TU:C
1S-11;)-). Rasearch Triangle Park, North ('.a.""o~ir::>:
Triangle Universities Computation Center.
(lS-71).
~':e;:lo:::andu.ll
Van
der Laan, P. and Prakken, J. (1972). Exact distribution of
distribution-free test statistic for Bal~ced Inco3plete
~~bin's
Eloclc Designs,
a.w.~d com~ison
l·rit.h the Chi-squa......-roe c....."1.Q ? a;.J7o:<-
bation. statistica Neerlandica, 26:'155-164.
31teren, Th. and. ~ioether, G.B. (1959). The asppto-:'ic efficie:l.cy
of ).,. -test for a balanced incomplete block d.esign. B:':)::e:::-i~:a,
Lt6: 475-477.
Van
'lan :~ateren, :?h. (1;16;).
On the co:noination of
incle~nc.e~t t';.;:,-
sa,mh tests of :lilcoxon. Btrlletin of the Insti-:'uta
natio~a1 statistics, 37: 351-361.
0:
:"'1t2::--
"'1
..,::'. (1d.:..5'
ri'
.~
1 co:nna.r~sons
.
"Il~coxon,
,/,,'. I n_l.V1~Ua
?;o::1.strics Bulletin, 1: 80-83.
-iilcoxon, ?
r;.a.-l.rL<i~g
(1S"46).
nethoc..s.
L'1divi.d.ual co~:;arisons of
of 3nto::ologv, 39:
JOUIT2Q
b'=OUpe-::' data. '::-,
269-27~.
151