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Adaptive choice of patient subgroup for comparing two treatments
Adaptive Choice of Patient Subgroup for Comparing Two Treatments Tze Leung Lai, Philip W. Lavori, Olivia Yueh-Wen Liao PII: DOI: Reference:
S1551-7144(14)00131-1 doi: 10.1016/j.cct.2014.09.001 CONCLI 1071
To appear in:
Contemporary Clinical Trials
Received date: Revised date: Accepted date:
24 June 2014 30 August 2014 1 September 2014
Please cite this article as: Lai Tze Leung, Lavori Philip W., Liao Olivia Yueh-Wen, Adaptive Choice of Patient Subgroup for Comparing Two Treatments, Contemporary Clinical Trials (2014), doi: 10.1016/j.cct.2014.09.001
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M)*('& )+/6; )/75&'+) ,"& &T/,--; -*Q&-; +! "&'&*3& +%& (&$ +"&,+2&(+ !" '!(+"!-O ,(6
+%&"&7; ,11"!B*2,+*(8 +%& ",(6!2 3,"*,7-&)
ni
,(6
n0i
7; +%& '!()+,(+
ni + n0i ≈ npi #
npi /2 ,(6 ni n0i /(ni +n0i ) 7; npi /49
α(0) ',( +%&( 7& '!21/+&6 ,) , )/2 ! *(+&8",(6)# !3&" '&"+,*( )&+)# ! +%& 2/-+*3,"*,+& √ µi − µ ˆ0i )/2σ 9 F%& '!3,"*,('& 2,+"*B ! +%*) (Z1 , ..., ZJ ) /(6&" θ = 0# $%&"& Zj = npi (ˆ
F%& &""!" 1"!7,7*-*+; (!"2,- 6&()*+; !
2/-+*3,"*,+& (!"2,- 6*)+"*7/+*!( *) 1,"+*'/-,"-; )*21-& *( +%& ',)&
@
Π1 ⊂ · · · ⊂ ΠJ #
$%*'% $& ,))/2& *( +%&
ACCEPTED MANUSCRIPT !"#!$ %!&'# ! () *+! ,(*-.'*-/0 &$-/-&'$ *1-'$ 2! &1-%!2 -/ *+! !&(/2 3'1'01'3+ () 4!&*-(/ 56 7/ *+- /! *!2
*+! !"#'*-(/
p pi /pj
)(1
i ≤ j6
α = α(0) = P0 (ZJ ≥ cα ) + φi (x)
J−1 Xˆ ∞ i=1
P0 (ZJ < cα , Zj < x
&'/ %! 2!*!1,-/!2 %= ($.-/0
j∈ / {i, J}|Zi = x)φi (x)dx,
)(1
cα
- *+! 2!/ -*= )#/&*-(/ () *+! *'/2'12 /(1,'$
cα
Zi 6
SC
?+!1!
<+!1!)(1!8 -/ *+- &' ! *+! *+1! +($2
PT
≈
:>;
RI
&' !8 9(.:Zi , Zj ;
!"! # $%&'( )*+'*,-./0 1*).$, 2.-3 /1/(-.4* (/-.*,- )'5$%&'( )*0*6-.&, -A!
µJ − µ0J
#/2!1$=-/0 *+!
',3$!
-A! &'$&#$'*-(/ -/ ' B9< -
*=3-&'$$= %' !2 (/
NU
<+! !@!&*
(,!
1!$'*!2 *#2-! '/2 '$ ( (/ &(/ *1'-/* (/ )#/2-/0 '/2 *#2= 2#1'*-(/8 ?+-&+ $!'2 *( *+! /(*-(/ () C-,3$-!2
*+! &(#1 ! () *+! *1-'$6
<+-
+'
MA
'$*!1/'*-.!D -/ EF8 36G5H6 <+! (% !1.!2 !@!&* -A! ,'= 2-@!1 #% *'/*-'$$= )1(, *+! ' $!2 *( '2'3*-.! 2! -0/
9+'3*!1 G () EKH6 4-/&! (#1 2! -0/ '$ ( '$$(?
?-*+ ,-2I&(#1 !
*1!'*,!/* &(,3'1- (/
(.!1
#,!2 !@!&* -A! 2#1-/0
',3$!
-A! 1!I! *-,'*-(/J
,'$$!1 3'*-!/*
#%01(#3
!! '/2
*#2-! '%(#* *+! !@!&* -A!
D
*+!1! - # #'$$= $-**$! -/)(1,'*-(/ '* *+! %!0-//-/0 () *+! *1-'$ )1(, 31!.-(#
TE
-/ *+! #%01(#3 8 '/ '2'3*-.! 2! -0/ *+'* &'/ %(*+ &+(( ! *+! 3'*-!/* #%01(#3 '/2 1!I! *-,'*! *+! ',3$! -A! - 3'1*-&#$'1$= '**1'&*-.!6
L'1*1(@ '/2 M'- EGH8 ENH +'.! 2!.!$(3!2 ' *+!(1= () !O&-!/* '2'3*-.! 2! -0/ )(1 ',3$! -A! 1!I! *-,'*-(/
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
*+'* -/.($.! PI *'0! QMB *! * 6 R* *+! S1 * -/*!1-, '/'$= - 8 *+! ',3$! -A! )(1 *+! !&(/2 *'0! - ! *-,'*!26 7) *+! QMB *! * 1!T!&* *+! /#$$ +=3(*+! - (1 *(3 !'1$= )(1 )#*-$-*= '* *+! !&(/2 -/*!1-, '/'$= - 8 *+! *1-'$ *(3 6 U*+!1?- ! *+! *1-'$ &(/*-/#! *( *+! *+-12 *'0! ?+-&+ &(11! 3(/2 *( *+! ,'V-,#, ',3$! -A! () *+! *1-'$6 <+! ! '2'3*-.! 2! -0/
&'/ %! '331(V-,'*!2 %=
*'/2'12 01(#3
!"#!/*-'$ 2! -0/
*+'* 2( /(*
! *-,'*! *+! ',3$! -A! )(1 *+! !&(/2 *'0!J !! EGH8 E5WH8 E55H6 X! /!V* !V*!/2 *+! ! '331(V-,'*-(/ () *+! '2'3*-.! 2! -0/ *( ' PI *'0! 01(#3 !"#!/*-'$ 2! -0/ -/ ?+-&+ *+! $' * *'0! &(11! 3(/2 *( *+! ,'V-,#, ',3$!
-A! '/2 *+!
',3$!
-A! #3 *( *+!
!&(/2
*'0! -
/!'1 *+! ,-2I3(-/* () *+! S1 *I *'0! '/2 S/'$
',3$! -A! 6 R -/ *+! 31!&!2-/0 !&*-(/ )(1 SV!2 ',3$! -A! 2! -0/ 8 *+! ,'V-,#, ',3$! -A! - *+! SV!2 ',3$! -A! )(1 *! *-/0
HJ 8
(1 (,! -/Y'*-(/ *+!1!()8 2!*!1,-/!2 %= *+! 3(?!1 '* (,! !@!&* -A!
3(3#$'*-(/6 R 3(-/*!2 (#* -/ EFH8 *+- ,'V-,#, ',3$! -A! +' (12!1 ?! S1 * *! *
HJ
'/2 &'/ *+!/ 2- &(/*-/#! *! *-/0
HJ
8δ 2 | log α|/σ 2 6
δ
)(1 *+! !/*-1!
R* '/ -/*!1-, '/'$= - 8
!'1$= )(1 !O&'&= (1 )#*-$-*=6 7) !'1$= *(33-/0 )(1 !O&'&=
(& 8 ?! *!1,-/'*! *+! *1-'$ '/2 &$'-, *+'* *+! /!? *1!'*,!/* - %!**!1 *+'/ *+! &(/*1($ (/ '.!1'0! (.!1 *+! !/*-1! 3(3#$'*-(/6 7) *(33-/0 (& )(1 )#*-$-*= () *! *-/0
HJ 8
?-*+ *+! ,( * 31(,- -/0 3'*-!/* #%01(#38 *+'* - 8 *+! #%01(#3
>
*+!/ ?! '&&!3*
i 6= J
HJ
'/2 &(/*-/#! *+! *1-'$
*+'* ,'V-,-A! QMB i 2!S/!2 %= :P;8
ACCEPTED MANUSCRIPT !" #$"% ni &'( n0i )*+,&-*( . "%* -/))*0+/'($'1 0&2+,* 0$3*0 &" "%* "$2* /4 $'"*)$2 &'&,.0$05 64 "%* "*0" 4/) HJ (/*0 '/" 0"/+7 "%*' #* -/'"$'!* "/ "%* '*8" 0"&1* /4 "%* 9:0"&1* (*0$1' &'( )*+*&" "%$0 +)/-*(!)*5
nℓi nℓ0i (ˆ µℓ − µ ˆℓ0i ) nℓi + nℓ0i i
,
σ
>@?
RI
Ziℓ =
s
PT
;/ * 2/)* 0+*-$<-7 0$2$,&),. "/ "%* 0=!&)* )//" /4 "%* ,*4":%&'( 0$(* /4 >9?7 ,*"
HJ 7 "*)2$'&"*07 &'( -,&$20 *C-&-. /4 "%* '*# ")*&"2*'" $4
NU
ZJℓ ≥ b.
SC
(*'/"* "%* "*0" 0"&"$0"$- 4/) +&"$*'" 0! 1)/!+ Πi &" "%* ℓ"% $'"*)$2 &'&,.0$05 A" "%$0 &'&,.0$07 "%* ")$&, )*B*-"0
>D?
MA
6" &--*+"0 HJ &'( +)/-**(0 "/ "%* 0! 1)/!+ 0*,*-"$/' $4
>E?
Z˜Jℓ ≤ ˜b
TE
D
#%*)* Z˜iℓ $0 "%* "*0" 0"&"$0"$- /4 "%* &,"*)'&"$F* Ki : µi ≥ µ0i + δ 7 "%&" $07 Z˜iℓ =
s
nℓi nℓ0i (ˆ µℓ − µ ˆℓ0i − δ) ℓ ni + nℓ0i i
,
σ.
>G?
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
H'-* HJ $0 &--*+"*( &" 0"&1* ℓ &'( & 0! 1)/!+ Iˆ7 #$"% "%* ,&)1*0" F&,!* /4 Ziℓ 4/) i 6= J 7 $0 -%/0*'7 "%* 4!"!)* *')/,,2*'" /4 "%* ")$&, #$,, $'-,!(* +&"$*'"0 /4 "%$0 0! 1)/!+ /',.7 #%$,* "%* 2&8$2!2 "/"&, 0&2+,* 0$3* N
)*2&$'0 "%* 0&2*5 I$2$,&) "/ "*0"$'1 HJ 7 "%* ")$&, )*B*-"0 HIˆ &" 0"&1* ℓ′ &'( "*)2$'&"*0 #$"% &' *C-&-. -,&$2 /4 "%* '*# ")*&"2*'" $' "%$0 0! 1)/!+ $4 ′
ZIℓˆ ≥ b.
>JK?
′ Z˜Iℓˆ ≤ ˜b.
>JJ?
6" 2&. &,0/ "*)2$'&"* 4/) 4!"$,$". $4
64 '*$"%*) >D? '/) >E? *F*) /--!)07 "%* ")$&, +)/-**(0 "/ "%* <'&, 0"&1*7 $' #%$-% -&0* HJ $0 )*B*-"*( $4 ZJ3 ≥ c
>JL?
&'( $0 &--*+"*( /"%*)#$0*5 M$1!)* J 1$F*0 & 0-%*2&"$- 0!22&). /4 "%* 0*=!*'-* /4 (*-$0$/'0 $' "%$0 1)/!+ 0*=!*'"$&, ")$&,5 @
ACCEPTED MANUSCRIPT
PT
Final Analysis
SC
RI
Start 1st Interim Analysis 2nd Interim Analysis Accrue !! Test "! If eff, stop(J) If fut, select subgroup #$, test "%$ If eff, stop(#$) If fut, STOP Else accrue !%$& only Test "%$& If eff, stop(#$) If fut, STOP Else accrue !%$& only
Test "! If eff, stop(J) If fut, select subgroup#$, test "%$ If eff, stop(#$) If fut, STOP Else accrue !%$& only Test "%$& If eff, stop(#$) Else, STOP
MA
NU
Else accrue !!
Test "%$& If eff, stop(#$) Else, STOP
Test "! If eff, stop(J) Else, STOP
D
Else accrue !!
TE
!! =entire population; !%$& =selected subgroup; eff=efficacy boundary crossed; fut=futility boundary crossed; stop(·) = stop trial for efficacy (in the tested group - J or #$); STOP = stop trial and claim futility
!"#$% &' ()* +,-$. )/ .,% 012.-"% "$)#3 2%4#%5.!-( 6%2!"57 89%5.2 .,-. (!5% #3 9%$.!+-((: )++#$ -. .,% 2-;% -5-(:2!27
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
!"# "$%" &' "$# #()%"&!'* %+!,#- Iˆ &* % .%'/!0 ,%.&%+1# 2&"$ ,%1)#* .%'3&'3 4.!0 1 "! J − 15 6!.#!,#."$# )77#. +!)'/ α(0) !4 "$# "87# 9 #..!.- 2$&:$ 2&11 +# #;71%&'#/ &' <77#'/&; <- !4 "$&* 3.!)7 *#()#'"&%1 /#*&3' :%' +# /#:!07!*#/ &'"! "2! 7%."* *&0&1%. "! =>?5 @$# A.*" 7%."- P0 (B#C#:" HJ ), &* +!)'/#/ +8 P0 (ZJℓ ≥ b
4!. ℓ = 1 !. 2) + P0 (ZJℓ < b 4!. ℓ = 1, 2, %'/ ZJ3 ≥ c) ≡ P1b + P1c .
@$# *#:!'/ 7%."- P0 (<::#7" HJ %'/ .#C#:" HIˆ)- &* +!)'/#/ +8 ′ P0 (Z˜Jℓ ≤ ˜b %'/ ZIℓˆ ≥ b
4!. *!0# ℓ ≤ ℓ′ < 3) + P0 (Z˜Jℓ ≤ ˜b,
′
ZIℓˆ < b
4!. ℓ ≤ ℓ′ < 3, %'/ ZI3ˆ ≥ c)
≡P2b + P2c .
<* *$!2' &' <77#'/&; <- D!,=ZJℓ , ZJℓ ? ≈ Nℓ /Nℓ 4!. ℓ ≤ ℓ′ - 2$#.# Nℓ &* "$# "!"%1 ')0+#. !4 7%"&#'"* #'.!11#/ )7 "! "$# ℓ"$ &'"#.&0 %'%18*&*5 E#':#- "$# 7.!+%+&1&"&#* P1b %'/ P1c :%' +# :!07)"#/ +8 "$# .#:).*&,# ')0#.&:%1 &'"#3.%"&!' FG- 75HIJ )*&'3 "$# C!&'" '!.0%1 /#'*&"8 !4 {ZJℓ , 1 ≤ ℓ ≤ 3}5 ′
p
′
G
ACCEPTED MANUSCRIPT !"#$%&'! (#)!$&"*+ &(,!-$*,&.( "*( *+%. /! #%!0 ,. ".)1#,! ,2! 1$./*/&+&,3 P2b =
2 ˆ X
J−1 X
−∞ i=1
P0 (Z˜Jℓ ≤ ˜b, Zjℓ < x
4.$ j ∈/ {i, J}|Ziℓ = x)P0 (Ziℓ
′
≥b
4.$ %.)! ℓ′ ≥ ℓ|Ziℓ = x)φi (x)dx
PT
ℓ=1
b
5*(0 %&)&+*$+3 4.$ P2c 67 8! #%! 2!$! ,2! 9.&(, (.$)*+ 0&%,$&/#,&.( .4 {Ziℓ , ℓ ≤ ℓ′ ≤ 3} ".(0&,&.(*+ .( Ziℓ : ;&,2 ℓ $!1$!%!(,&(- ,2! %,*-! ;2!( HJ &% *""!1,!0: *(0 ,2! 9.&(, (.$)*+ 0&%,$&/#,&.( .4 Z˜Jℓ ≈ ZJℓ − √ Nℓ δ/(2σ) *(0 Zjℓ , j ∈ / {i, J}: ".(0&,&.(*+ .( Ziℓ ;&,2 i 6= J 7 <2! ,2$!%2.+0% b, ˜b < 0 *(0 c *$! 0!,!$)&(!0 /3 %.+'&(- ,2! !=#*,&.(% .$ @) = ǫβ,
P1b + P2b = ǫα, P1c + P2c = (1 − ǫ)α,
NU
PθJ =δ (Z˜Jℓ ≤ ˜b 4.$ ℓ>?
SC
RI
′
5?A6
MA
4.$ ,2! 1$!%"$&/!0 )*B&)#) ,31! C !$$.$ α: 1.;!$ 1 − β *, ,2! *+,!$(*,&'! δ: *(0 ,2! 1$.1.$,&.( 0 < ǫ < 1 .4 ,31! C 5.$ ,31! CC6 !$$.$ %1!(, *, &(,!$&) *(*+3%!%7
D
! "#$%&'() #)*'+, -. $ /$,#-0'1)# 2-,&/-33)# &/'$3 &- &)*& ),#-($*243$/ &5)/$%6
TE
!"! #$%&'()*+,(- '.-)+) )/($%(-% /0.-(12 3&,,&45$6 57(65$6 .'(,+(/5&$ 3&- 5)*0.75* )/-&8.
<2! "+&(&"*+ ,$&*+ )!(,&.(!0 &( ,2! %!".(0 1*$*-$*12 .4 D!",&.( ? &% * $*(0.)&E!0 ,$&*+ 4.$ 1*,&!(,% !B1!$&!("&(- *"#,! &%"2!)&" %,$.F! 5CD6 0#! ,. * 1$.B&)*+ *(,!$&.$ "&$"#+*,&.( +*$-! '!%%!+ .""+#%&.( ;2. *$! ,$!*,!0 ;&,2&( ?@ 2.#$% .4 CD .(%!,7 G*,&!(,% ;&++ #(0!$-. H C %,#0&!% *4,!$ !($.++)!(, /#, /!4.$! $*(0.)&E*,&.(7 <2! .(+3 IJK *11$.'!0 0$#- 4.$ ,2! ,$!*,)!(, .4 *"#,! CD &% CLM,GK: 3!, .(+3 2 −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
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
U
ACCEPTED MANUSCRIPT
TE
D
MA
NU
SC
RI
PT
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ACCEPTED MANUSCRIPT !" #$%&'%(')*"' +),-./.& #$%$)#$)- )# Ziℓ = {Wiℓ − nℓi (nℓi + nℓ0i + 1)/2}/{nℓi nℓ0i (nℓi + nℓ0i + 1)/12}1/2 .
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4& 566"&')/ 78 9" '"():" %& "/%-$ ;.(<=,% ;.( >.:0 Ziℓ , Zjℓ ), i ≤ j ≤ J, 1 ≤ ℓ ≤ ℓ′ ≤ 38 %&' =#" )$ $. '"():" $!" ,)<)$)&? @.)&$ &.(<%, ')#$()A=$).&# .; $!" Ziℓ 8 i ≤ j ≤ J %&' 1 ≤ ℓ ≤ ℓ′ ≤ 38 =&'"( P (Y ≤ X) = 1/2 %&' =&'"( -.&$)?=.=# %,$"(&%$):"# P (Y ≤ X) = 1/2 + θB !" ,)<)$)&? @.)&$ &.(<%, ')#$()A=$).&# #!.9 $!%$ $!" 6(.-"'=("# )& C"-$).&# DBD %&' DBE ;.( #$%&'%(')*"' #%<6," <"%&# 0F3 .; &.(<%,,G ')#$()A=$"' .A#"(:%$).&# -%& A" %66,)"' $. $!" #$%&'%(')*"' +),-./.& #$%$)#$)-# 0123B 4& 6%($)-=,%(8 )& 6,%-" .; $!" #$%$)#$)- 0H3 ;.( $"#$)&? µi ≥ µ0i + δ $. %,,.9 "%(,G #$.66)&? ;.( ;=$),)$G8 9" &.9 '"I&"
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Z˜iℓ = {Wiℓ − nℓi (nℓi + nℓ0i + 1)/2 − nℓi nℓ0i θ}/{nℓi nℓ0i (nℓi + nℓ0i + 1)/12}1/2
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µ = (0, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0.3, 0.3) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0, 0, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.6, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (2, 1.5, 1, 0.5, 0.5, 0.5) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (0.5, 1, 1.5, 2, 2, 2) µ = (0, 0, 0, 0.3, 0.4, 0.5) v = (1, 1, 1, 1, 1, 1)
)*!%+,#p (p1 , ...p6 ) 5 (1.1, 0.7, 0.4, 0.4, 0.2, 2) 89 (0.9, 0.9, 0.6, 0.6, 1, 85) 80.5 (3.8, 4.4, 6.7, 13.6, 2.3, 49.7) 65.7 (14.7, 28.5, 6, 2.2, 0.7, 13.6) 26.5 (0.6, 0.2, 1.4, 7.8, 1.6, 15) 86.8 (60.7, 9, 2.6, 1, 0.4, 13.1) 78 (55.4, 8.6, 2.2, 1.4, 0.6, 10.2) 92.5 (7.4, 13.4, 20, 3, 0.6, 48.1) 83 (5.8, 7.4, 24.9, 3.5, 0.9, 40.5) 72.2 (15.2, 16.2, 13.7, 4.1, 1.3, 21.7) 47.9 (0.2, 0, 0, 0.1, 0, 47.6) Pe
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p (p1 , ...p6 ) 4.8 (0.9, 1, 0.9, 0.4, 0.2, 1.3) 89.8 (0.6, 0.6, 0.8, 1, 0.9, 86) 77.7 (2.1, 3.2, 5.1, 14.3, 2.8, 50.2) 49 (6.5, 20, 5.3, 2.2, 1, 14.1) 25.9 (0.3, 0.2, 1, 7.7, 2.5, 14.3) 72.8 (49.9, 5.9, 2.1, 0.9, 0.3, 13.7) 57.4 (36.9, 5.9, 2.5, 1.3, 0.5, 10.4) 88.7 (5.4, 11.7, 18.3, 2.5, 0.8, 50) 77.1 (3.6, 5, 23, 3.2, 0.9, 41.4) 61.1 (8.9, 11.9, 13, 3.6, 1.7, 21.9) 50.1 (0.1, 0.1, 0, 0, 0.2, 49.7)
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µ = (0, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0.3, 0.3) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0, 0, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.6, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (2, 1.5, 1, 0.5, 0.5, 0.5) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (0.5, 1, 1.5, 2, 2, 2) µ = (0, 0, 0, 0.3, 0.4, 0.5) v = (1, 1, 1, 1, 1, 1)
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p (p1 , ...p6 ) 5.2 (1, 0.8, 0.5, 0.3, 0.2, 2.4) 89 (0.9, 0.7, 0.9, 0.8, 1, 84.8) 83 (3.5, 3.8, 7, 11, 3, 54.6) 64.5 (17.4, 30.7, 2.8, 1.2, 1, 11.3) 33 (0.2, 0.1, 2.7, 7.3, 3.1, 19.6) 90.9 (56, 12.7, 1.6, 0.6, 0.5, 19.6) 82.9 (51.5, 14.1, 2.1, 1, 0.5, 13.7) 94.1 (5.7, 8.6, 13.9, 3.4, 0.8, 61.6) 87.9 (3.2, 3.7, 17.7, 3.3, 1, 58.9) 74.3 (16.1, 14.4, 10.7, 4, 1.8, 27.4) 38 (0.3, 0.1, 0, 0, 0, 37.5)
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p (p1 , ...p6 ) 4.8 (1, 0.8, 0.7, 0.5, 0.4, 1.3) 89.7 (0.5, 0.6, 0.6, 0.6, 0.9, 86.6) 79.8 (1.8, 3.1, 6.3, 10, 3, 55.5) 46.1 (7.9, 20.9, 2.9, 1.5, 0.8, 12.2) 35.1 (0.3, 0.1, 2.6, 8.4, 3.4, 20.2) 81.1 (50, 9.4, 1.4, 0.8, 0.6, 18.9) 67.3 (40.2, 9.4, 2, 1, 0.6, 14) 92.3 (4.6, 8.3, 12.8, 2.5, 1, 63) 84.9 (2.2, 2.9, 17.4, 3.1, 1.2, 58.1) 65.3 (9.1, 11.2, 10.1, 3.9, 2, 29) 37.5 (0.3, 0.2, 0, 0, 0, 37)
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ACCEPTED MANUSCRIPT !!"#$%& ' ()*)$+, "-.%"#./0 .123"$ 4"34%#5 !,%#.%!1"0 )#$ 5,2+! 3"6+"#4%)1 "&4"#3%2#3 !"! #$%$&'( )*+,)-+. /0 1(/1/2)& 1(/+)&'() ,- 3)+4,/- 5!5!
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!"#$$ %&#% θj = µj − µ0j #'( θ = (θ1 , ..., θJ )) *+, θ ∈/ Θ0 , θj > 0 -+, #$$ 1 ≤ j ≤ J . #'( (!/'! √ j(θ) = #,01#21≤j≤J pj θj . 3)!). j(θ) "+,,!45+'(4 %+ %&! 5#%3!'% 4670,+65 j 83%& %&! $#,0!4% 9: 3'-+,1#%3+' '617!, Ij . 8&3"& (!5!'(4 +' θ #'( 8&3"& 83$$ 7! (!'+%!( 7; Ij (θ)) <+%! %&#% Ij (θ) 34 $3'!#, 3' n #'( %&#% %&! %;5! == !,,+, β(θ) (!/'!( 3' >?@ 4#%34/!4 >AB@
lim n−1 log β(θ) = − lim n−1 log Ij(θ) (θ) = max pj (µj − µ0j )2 /(4σ 2 ), n→∞
n→∞
1≤j≤J
MA
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3' C3!8 +- %&! #4;15%+%3" 7!&#C3+, +- D#6443#' %#3$ 5,+7#73$3%3!4) =' -#"%. Ij(θ) (θ) 34 #$4+ %&! 9: (3C!,0!'"! In (θ, θ 0 ). 8&!,! θ 0 ∈ Θ0 34 %&! 13'313E!, +Q(λ|θ) = (θ − λ)(3#0(p1 , ..., pJ )(θ − λ)′ / (2σ 2 )
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+C!, λ ∈ Θ0 G %&! !25!"%!( C#$6! 6'(!, θ +- %&! $+0#,3%&1 +- %&! $3H!$3&++( ,#%3+ +- θ %+ λ 7#4!( +' # 4#15$! +- 43E! n/2 34 n2 Q(λ|θ)) I#&#(6, JK. AFL 4&+8' %&#% exp {−(1 + o(1)In (θ, θ0 )} 34 %&! #4;15%+%3"#$$; 13'31#$ C#$6! >#4 n → ∞@ -+, %&! %;5! == !,,+,. #% #' #$%!,'#%3C! θ ∈/ Θ0 . +- #'; $!C!$Mα %!4% +- H : θ ∈ Θ0 7#4!( +' # 4#15$! +- 43E! n/2) N $!C!$Mα %!4% 8&+4! %;5! == !,,+, #%%#3'4 %&34 #4;15%+%3"#$$; 13'31#$ ,#%! #% !C!,; #$%!,'#%3C! θ 34 4#3( %+ 7! #$%$&'( )*+,)-4) O&34 8#; +- "+15#,3'0 %!4%4 C3# $#,0! (!C3#%3+' #55,+231#%3+'4 #55$3!4 %+ 0!'!,#$ 5#,#1!%,3" -#13$3!4) PC!' 3' "#4!4 8&!,! $#,0! (!C3#%3+' #55,+231#%3+'4 %+ %;5! == !,,+,4 #,! (3Q"6$% %+ (!,3C!. I#&#(6, JK. AFL 4600!4%4 %+ /2 %&! %;5! == !,,+, #% #' #$%!,'#%3C! θ #'( %+ %,; (!,3C3'0 $#,0! (!C3#%3+' #55,+231#%3+'4 -+, %&! %;5! = !,,+, 3'4%!#(G 45!"3/"#$$;. -+, supλ∈Θ α(λ)) R! #$4+ 4600!4%4 %&! "$+4!$; ,!$#%!( 3(!# +- (!,3C3'0 $#,0! (!C3#%3+' #55,+231#%3+'4 -+, %&! 5MC#$6! +- %&! %!4% #% θ ∈/ Θ0 ) O&!4! $#,0! (!C3#%3+' #55,+231#%3+'4 $!#( %+ # $313%3'0 S6#'%3%; inf λ∈Θ q(λ|θ). 8&3"& 34 "#$$!( %&! I#&#(6, !2#"% 4$+5! #'( 34 1#2313E!( 8&!,! q = Q 03C!' 7; >AF@)
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S1551-7144(14)00131-1 doi: 10.1016/j.cct.2014.09.001 CONCLI 1071
To appear in:
Contemporary Clinical Trials
Received date: Revised date: Accepted date:
24 June 2014 30 August 2014 1 September 2014
Please cite this article as: Lai Tze Leung, Lavori Philip W., Liao Olivia Yueh-Wen, Adaptive Choice of Patient Subgroup for Comparing Two Treatments, Contemporary Clinical Trials (2014), doi: 10.1016/j.cct.2014.09.001
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
20330)$* 1'&% 40! /09'#3.$& 203.#!"'0) 04 B#);") '20!&' "' D"$20L0)M' !#); '13+ >0 #50"% 06'21!")( -& 3#") "%&#' 1)%&!$*")( 01! 3& -0%0$0(* 6* &2-)"2#$ #!(13&) ' ")50$5")( )0).#!#3& !"2 ' # "' "2'8 /& ?!' 20)'"%&! ") <&2 "0) N -& .#!#3& !"2 2#'& 04 )0!3#$$* %"' !"61 &% !&# 3&) 01 203&' /" - 20330) ;)0/) 5#!"#)2& #)% 3&#)
µj
E0!
µ0j G
40! .# "&) '16(!01.
j
#)% -& )&/ E0! 20) !0$G !&# 3&) + D& %&5&$0. -&
6#'"2 -&0!* 40! #%#. "5& 2-0"2& 04 .# "&) '16(!01. 0 203.#!& -& /0 !&# 3&) ' ") -"' '&
")(8 ?!' 40!
%&'"()' /" - ?L&% '#3.$& '"H& #)% -&) 40! # -!&&9' #(& (!01. '&O1&) "#$ %&'"() -# "' 1'&% 0 #..!0L"3# & #) #%#. "5& %&'"() /" - 3"%9201!'& %& &!3")# "0) 04
-& .# "&) '16(!01. #)% '#3.$& '"H& !&9&' "3# "0)+
<&2 "0) P ("5&' 41! -&! %"'21''"0)8 &L &)'"0)' #)% 20)2$1%")( !&3#!;'+
! "#$%& '()*+, %- .+*'*',.%&#/ -*+0#/ $)''%-1 !"! #$%&'()(*+ (,-).% /*0 123340+567-*43-. *89).&0(*)8 08: '.-/03-8+<1..0'&
n
.# "&) ' #!& !#)%03"H&% 0 -& )&/ #)% 20) !0$ !&# 3&) ' #)% -& !&'.0)'&' #!& )0!3#$$*
%"' !"61 &%8 /" - 3&#) .!&9%&?)&% '16(!01.'
µj
Πj
40! 40!
-& )&/
!&# 3&) #)%
j = 1, ..., J +
µ0j
D& #''13&
N
40!
-#
-& 20) !0$
!&# 3&) "4
-& .# "&) 4#$$' ")
-& !&'.0)'&' -#5& 20330) ;)0/) 5#!"#)2&
ACCEPTED MANUSCRIPT σ2
!"#$%&'(%& #( )%*%(+% ,%- .(&&'/01 )%$2),0 3,4',%5$& +'00 /$ -'&5)&&$- '% 6$5#'(% 7
8$#
ΠJ
-$%(#$&
#9$ $%#'4$ .,#'$%# .(.)0,#'(% (% +9'59 , #4,-'#'(%,0 4,%-(:';$- 5(%#4(00$- #4',0 <=>?@ 5(:.,4'%A #9$ #+(
x+
-$%(#$&
?9$ C)00/,5*D8$'/0$4
max(x, 0)E
ΠJ
'&
n 4 (µJ
IJ =
− µ0J )2+ /σ 2 E
%(#'%A #9,# #9$ 5(:.,4'&(% '%3(03$& #$&#'%A #9$ (%$D&'-$- %)00 91.(#9$&'&
PT
#4$,#:$%#& B(5)&$&
+9$4$
µJ ≤ µ0J .
?9$ C8 '%B(4:,#'(% %):/$4E (4 4$0,#'3$ $%#4(.1E 2),%#'F$& #9$ ,:()%# (B '%B(4:,#'(% '% #9$ &,:.0$ #( B4(: #9$ 5(%#4(0 :$,%
#9$(41 (B $G5'$%# .,4,:$#4'5 #$&#&H &$$ IJKE ILK
µ0J E ,%- .0,1& ,% ':.(4#,%# 4(0$ '% #9$ ,&1:.#(#'5
M% .,4#'5)0,4E 'B
RI
µJ
IJ
'& %(# 0,4A$ $%()A9E #9$% #9$ =>? :,1
SC
-'&#'%A)'&9 #9$ #4$,#:$%# :$,%
%(# 9,3$ &)G5'$%# .(+$4E ' $ E .4(/,/'0'#1 (B &9(+'%A , &'A%'F5,%# :$,% -'N$4$%5$ /$#+$$% #9$ #4$,#:$%# ,%- #9$ 5(%#4(0
Πj
/$ #9$ .4$3,0$%5$ (B .,#'$%# &)/A4().
B(4 , #4',0 +'#9 , #(#,0 &,:.0$ &';$ %):/$4 B(4
Πj
'&
Ij =
(µj − µ0j )2+ /(2σ 2 )
n 4 pj (µj
n
− µ0j )2+ /σ 2 E
'& #9$ $".$5#$- %):/$4 (B &)/ O$5#& '%
+9'59 '& #9$ .4(-)5# (B #9$ .4$3,0$%5$
(B &)59 .,'4&
Πj
Πj
?9$ C8 '%B(4:,#'(%
pj E
#9$ C8 '%B(4:,#'(%
4$5$'3'%A #9$ %$+ #4$,#:$%# ,%- 5(%#4(0E 4$&.$5#'3$01E ,%-
P(# (%01 -($& #9'& &9(+ #9$ #4,-$D(N /$#+$$% #9$ .4$3,0$%5$ (B #9$
D
n/2
npj
#9,# 4,%-(:';$& .,#'$%#& #( #9$ #+( #4$,#:$%#&
B4(: , .,'4 (B .,#'$%#& '%
#9$ $".$5#$- %):/$4
?9$4$B(4$
NU
pj
MA
8$#
.,#'$%# &)/A4(). ,%- #9$ :,A%'#)-$ (B #9$ -'N$4$%5$
µj − µ0j
'% 59((&'%A #9$ .,#'$%# &)/A4(). #( 5(:.,4$
(B
Ij
m/pj
(3$4
1≤j≤J
TE
#9$ #+( #4$,#:$%#&E /)# '# ,0&( &)AA$&#& #9,# ,% ,&1:.#(#'5,001 (.#':,0 59('5$ (B &)/A4(). '& #9$ :,"':';$4 Q$ 5,% ,0#$4%,#'3$01 4$A,4- , &,:.0$ &';$ (B
B4(: #9$ $%#'4$ .(.)0,#'(%
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
m
B4(:
Πj
,& ,% $N$5#'3$ &,:.0$ &';$ (B
! ! "# $%&'$#( )**+,)&- .,+ /0$1 2)3*4$ 2'5$ (+')42 6'%5$ #9$4$ '& #1.'5,001 0'##0$ '%B(4:,#'(% B4(: .4$3'()& &#)-'$& ,/()# #9$ &)/A4(). $N$5# &';$ B(4
j 6= J E
+$ /$A'% +'#9 , &#,%-,4- =>? #( 5(:.,4$ #9$ %$+ #4$,#:$%# +'#9 #9$ 5(%#4(0 (3$4 #9$ $%#'4$
.(.)0,#'(% #$&#'%A
R$#$4:'%,#'(% (B #9$ &,:.0$ &';$
HJ : µJ ≤ µ0J E
4$5,00'%A #9,#
µJ
,%-
n
µ0J
4$&.$5#'3$01E (3$4 #9$ $%#'4$ .,#'$%# .(.)0,#'(%
'& /,&$- (% .(+$4 ,# ,% ,&&):$- $N$5# &';$
Hi : µi ≤ µ0i 'B
HIˆ
+'#9
B(4
?9$ :,'% '%%(3,#'(% (B #9$ .4(.(&$- #4',0 -$&'A% '& #9,#
Iˆ
'% #9$ $3$%#
HJ
'& %(# 4$O$5#$-E #( 5(%#'%)$ #$&#'%A
i = IˆE ,%- 5,% 50,': #9$ %$+ #4$,#:$%# #( /$ /$##$4 #9,% 5(%#4(0 B(4 #9$ .,#'$%# &)/A4().
'& 4$O$5#$-
θ = (θ1 , ..., θJ )E
µJ − µ0J
,4$ #9$ :$,% 4$&.(%&$& (B #9$ %$+ ,%- 5(%#4(0 #4$,#:$%#&E
'# ,00(+& ,-,.#'3$ 59('5$ (B #9$ .,#'$%# &)/A4().
Iˆ
µj − µ0j
Q$ 5,% '%#$4.4$#
Iˆ
,& ,% $&#':,#$ (B
I =
,4A:,"i6=J
Ij
8$##'%A
θj = µj − µ0j
,%-
#9$ .4(/,/'0'#1 (B , B,0&$ 50,': '& #9$ #1.$ M $44(4
α(θ) =
Pθ (=$O$5# HJ ) + Pθ (θIˆ ≤ 0, Pθ (θIˆ ≤ 0,
,55$.#
HJ
,55$.#
,%- 4$O$5#
S
HJ
HIˆ)
,%- 4$O$5#
HIˆ)
'B
θJ ≤ 0
'B
θJ > 0,
ACCEPTED MANUSCRIPT θ ∈ Θ0 #
+%&"& *) (!
$%&"&
Θ0
'!()*)+) !
,-)& '-,*2 *
HJ
,-- .(/--0 1,",2&+&" 3&'+!")
*) "&5&'+&6 $%&(
θJ > 0#
θ
θj ≤ 0
!" )!2&
α(θ) ≤ α#
2/-+*1-& +&)+*(8 =>?# =@? +! ),+*) ; +%*) '!()+",*(+9
!" ,--
J,%,6/" &A'*&('; =K?9
<( L11&(6*B L $& 8*3& , 1"&'*)& )+,+&2&(+ !
j ≤ J}
,(6 +%&"& !"& +%& +;1& << &""!" ,+
β(θ) =
J−1 X
Pθ (Iˆ = i,
i=1
$%*'% *) +%& 1"!7,7*-*+; !
,''&1+
HJ
Hi ).
MNO
!" +%& (&$ +"&,+2&(+9 :&
α9 n
1,+*&(+) +! +%& &B1&"*2&(+,- +"&,+2&(+ ,(6 +%& '!(+"!-9 :& "&5&'+
ni n0i (ˆ µi − µ ˆ0i )2+ /σ 2 ≥ cα ni + n0i
µ0i ) *) +%& 2&,( "&)1!()& ! i = J # $%&"& µ ˆi (ˆ
1,+*&(+) *(
Πi
,(6 "&5&'+
HIˆ
*
RHS ˆ
I
≥ cα 9
F%& +%"&)%!-6
cα
HJ
*
M>O
"!2 +%& +"&,+2&(+ M'!(+"!-O ,"2 ,(6
Iˆ 6= J
+%& 8&(&",-*C&6 -*Q&-*%!!6 ",+*! )+,+*)+*' RHS i # $%*'% *) +%& -& +E%,(6 )*6& !
i = 1, ..., J − 1#
!'/) %&"& !(
+%& 1"!'&6/"& ,(6 *+) *21-&2&(+,+*!( +! 2,*(+,*( , 1"&)'"*7&6 +;1& < &""!" '!()+",*(+ !(
*) +%& +%& '!""&)1!(6*(8 ),21-& )*C&9 P+%&"$*)& $& '%!!)& +%& 1,+*&(+ )/78"!/1 !
*)
*)
,(6
,-)&-; &(6*(8 /1 $*+% (! 1!)*+*3& '-,*2
F%& +"*,- ",(6!2-; ,))*8()
!"
θ∈ / Θ0
Θ0
D
-&3&-
+%& ,);21+!+*'
TE
+%& 6&)'"*1+*!( !
<()+&,6 $& +";
+%& 1"!1!)&6 1"!'&6/"& +!8&+%&" $*+% *+) 6&"*3,+*!(9 4!+& +%,+ +%& '!21-&2&(+ !
NU
{θ : θj > 0
D*('& +%& (/--
F%& GH *( !"2,+*!( (/27&") *( +%& 1"&'&6*(8 )/7)&'+*!( I+ (*'&-;
MA
&A'*&('; "&)/-+ !
α +&)+ *) (!+ &B1&'+&6 +! &B*)+9
SC
$*+% +%& '!('&1+ !
θ∈ / Θ0 9
RI
%;1!+%&)*) *) %*8%-; '!21!)*+&# , /(* !"2-; 2!)+ 1!$&" /- -&3&-E
n → ∞9
4!+& +%,+
,(6 $*-- 6&)'"*7& , 1"!'&6/"& +%,+
:& ,-)! $,(+ +%& 1"!'&6/"& +! 7& &A'*&(+ *( +%& )&()& ! 2,B*2*C*(8 +%& 1!$&" ,+
+! ,++,*( ,);21+!+*' &A'*&('; ,)
j ≤ J.
7&',/)& *( +%,+ ',)& $& $!/-6 (!+ 8! !( +! +&)+ ,
)/71!1/-,+*!(9 :& $,(+ +! 2,*(+,*( , +;1& < &""!" '!()+",*(+ /)&) +%& '-!)&6 +&)+*(8 1"*('*1-& *( +%& +%&!"; !
)/'% +%,+
PT
!"
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ni (n0i )
$*+% +%& -,"8&)+ 3,-/&
M>O# ,2!(8 ,-- )/78"!/1)
*) '%!)&( )/'% +%,+
α(θ) ≤ α
!" ,--
θ ∈ Θ0 #
,(6 $& (&B+ 6&)'"*7& *+) '!21/+,+*!(9 <+ $*-- 7& )%!$( *( L11&(6*B L +%,+
sup α(θ) = α(0),
M@O
θ∈Θ0 7; 2,Q*(8 /)& !
ni ≈ n0i
+%& '-!)&6 +&)+*(8 1"*('*1-& =>?# =@?9
F! '!21/+&
α(0)#
$& I")+ /)& +%& ,11"!B*2,+*!()
M)*('& )+/6; )/75&'+) ,"& &T/,--; -*Q&-; +! "&'&*3& +%& (&$ +"&,+2&(+ !" '!(+"!-O ,(6
+%&"&7; ,11"!B*2,+*(8 +%& ",(6!2 3,"*,7-&)
ni
,(6
n0i
7; +%& '!()+,(+
ni + n0i ≈ npi #
npi /2 ,(6 ni n0i /(ni +n0i ) 7; npi /49
α(0) ',( +%&( 7& '!21/+&6 ,) , )/2 ! *(+&8",(6)# !3&" '&"+,*( )&+)# ! +%& 2/-+*3,"*,+& √ µi − µ ˆ0i )/2σ 9 F%& '!3,"*,('& 2,+"*B ! +%*) (Z1 , ..., ZJ ) /(6&" θ = 0# $%&"& Zj = npi (ˆ
F%& &""!" 1"!7,7*-*+; (!"2,- 6&()*+; !
2/-+*3,"*,+& (!"2,- 6*)+"*7/+*!( *) 1,"+*'/-,"-; )*21-& *( +%& ',)&
@
Π1 ⊂ · · · ⊂ ΠJ #
$%*'% $& ,))/2& *( +%&
ACCEPTED MANUSCRIPT !"#!$ %!&'# ! () *+! ,(*-.'*-/0 &$-/-&'$ *1-'$ 2! &1-%!2 -/ *+! !&(/2 3'1'01'3+ () 4!&*-(/ 56 7/ *+- /! *!2
*+! !"#'*-(/
p pi /pj
)(1
i ≤ j6
α = α(0) = P0 (ZJ ≥ cα ) + φi (x)
J−1 Xˆ ∞ i=1
P0 (ZJ < cα , Zj < x
&'/ %! 2!*!1,-/!2 %= ($.-/0
j∈ / {i, J}|Zi = x)φi (x)dx,
)(1
cα
- *+! 2!/ -*= )#/&*-(/ () *+! *'/2'12 /(1,'$
cα
Zi 6
SC
?+!1!
<+!1!)(1!8 -/ *+- &' ! *+! *+1! +($2
PT
≈
:>;
RI
&' !8 9(.:Zi , Zj ;
!"! # $%&'( )*+'*,-./0 1*).$, 2.-3 /1/(-.4* (/-.*,- )'5$%&'( )*0*6-.&, -A!
µJ − µ0J
#/2!1$=-/0 *+!
',3$!
-A! &'$&#$'*-(/ -/ ' B9< -
*=3-&'$$= %' !2 (/
NU
<+! !@!&*
(,!
1!$'*!2 *#2-! '/2 '$ ( (/ &(/ *1'-/* (/ )#/2-/0 '/2 *#2= 2#1'*-(/8 ?+-&+ $!'2 *( *+! /(*-(/ () C-,3$-!2
*+! &(#1 ! () *+! *1-'$6
<+-
+'
MA
'$*!1/'*-.!D -/ EF8 36G5H6 <+! (% !1.!2 !@!&* -A! ,'= 2-@!1 #% *'/*-'$$= )1(, *+! ' $!2 *( '2'3*-.! 2! -0/
9+'3*!1 G () EKH6 4-/&! (#1 2! -0/ '$ ( '$$(?
?-*+ ,-2I&(#1 !
*1!'*,!/* &(,3'1- (/
(.!1
#,!2 !@!&* -A! 2#1-/0
',3$!
-A! 1!I! *-,'*-(/J
,'$$!1 3'*-!/*
#%01(#3
!! '/2
*#2-! '%(#* *+! !@!&* -A!
D
*+!1! - # #'$$= $-**$! -/)(1,'*-(/ '* *+! %!0-//-/0 () *+! *1-'$ )1(, 31!.-(#
TE
-/ *+! #%01(#3 8 '/ '2'3*-.! 2! -0/ *+'* &'/ %(*+ &+(( ! *+! 3'*-!/* #%01(#3 '/2 1!I! *-,'*! *+! ',3$! -A! - 3'1*-&#$'1$= '**1'&*-.!6
L'1*1(@ '/2 M'- EGH8 ENH +'.! 2!.!$(3!2 ' *+!(1= () !O&-!/* '2'3*-.! 2! -0/ )(1 ',3$! -A! 1!I! *-,'*-(/
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
*+'* -/.($.! PI *'0! QMB *! * 6 R* *+! S1 * -/*!1-, '/'$= - 8 *+! ',3$! -A! )(1 *+! !&(/2 *'0! - ! *-,'*!26 7) *+! QMB *! * 1!T!&* *+! /#$$ +=3(*+! - (1 *(3 !'1$= )(1 )#*-$-*= '* *+! !&(/2 -/*!1-, '/'$= - 8 *+! *1-'$ *(3 6 U*+!1?- ! *+! *1-'$ &(/*-/#! *( *+! *+-12 *'0! ?+-&+ &(11! 3(/2 *( *+! ,'V-,#, ',3$! -A! () *+! *1-'$6 <+! ! '2'3*-.! 2! -0/
&'/ %! '331(V-,'*!2 %=
*'/2'12 01(#3
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! *-,'*! *+! ',3$! -A! )(1 *+! !&(/2 *'0!J !! EGH8 E5WH8 E55H6 X! /!V* !V*!/2 *+! ! '331(V-,'*-(/ () *+! '2'3*-.! 2! -0/ *( ' PI *'0! 01(#3 !"#!/*-'$ 2! -0/ -/ ?+-&+ *+! $' * *'0! &(11! 3(/2 *( *+! ,'V-,#, ',3$!
-A! '/2 *+!
',3$!
-A! #3 *( *+!
!&(/2
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/!'1 *+! ,-2I3(-/* () *+! S1 *I *'0! '/2 S/'$
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HJ 8
(1 (,! -/Y'*-(/ *+!1!()8 2!*!1,-/!2 %= *+! 3(?!1 '* (,! !@!&* -A!
3(3#$'*-(/6 R 3(-/*!2 (#* -/ EFH8 *+- ,'V-,#, ',3$! -A! +' (12!1 ?! S1 * *! *
HJ
'/2 &'/ *+!/ 2- &(/*-/#! *! *-/0
HJ
8δ 2 | log α|/σ 2 6
δ
)(1 *+! !/*-1!
R* '/ -/*!1-, '/'$= - 8
!'1$= )(1 !O&'&= (1 )#*-$-*=6 7) !'1$= *(33-/0 )(1 !O&'&=
(& 8 ?! *!1,-/'*! *+! *1-'$ '/2 &$'-, *+'* *+! /!? *1!'*,!/* - %!**!1 *+'/ *+! &(/*1($ (/ '.!1'0! (.!1 *+! !/*-1! 3(3#$'*-(/6 7) *(33-/0 (& )(1 )#*-$-*= () *! *-/0
HJ 8
?-*+ *+! ,( * 31(,- -/0 3'*-!/* #%01(#38 *+'* - 8 *+! #%01(#3
>
*+!/ ?! '&&!3*
i 6= J
HJ
'/2 &(/*-/#! *+! *1-'$
*+'* ,'V-,-A! QMB i 2!S/!2 %= :P;8
ACCEPTED MANUSCRIPT !" #$"% ni &'( n0i )*+,&-*( . "%* -/))*0+/'($'1 0&2+,* 0$3*0 &" "%* "$2* /4 $'"*)$2 &'&,.0$05 64 "%* "*0" 4/) HJ (/*0 '/" 0"/+7 "%*' #* -/'"$'!* "/ "%* '*8" 0"&1* /4 "%* 9:0"&1* (*0$1' &'( )*+*&" "%$0 +)/-*(!)*5
nℓi nℓ0i (ˆ µℓ − µ ˆℓ0i ) nℓi + nℓ0i i
,
σ
>@?
RI
Ziℓ =
s
PT
;/ * 2/)* 0+*-$<-7 0$2$,&),. "/ "%* 0=!&)* )//" /4 "%* ,*4":%&'( 0$(* /4 >9?7 ,*"
HJ 7 "*)2$'&"*07 &'( -,&$20 *C-&-. /4 "%* '*# ")*&"2*'" $4
NU
ZJℓ ≥ b.
SC
(*'/"* "%* "*0" 0"&"$0"$- 4/) +&"$*'" 0! 1)/!+ Πi &" "%* ℓ"% $'"*)$2 &'&,.0$05 A" "%$0 &'&,.0$07 "%* ")$&, )*B*-"0
>D?
MA
6" &--*+"0 HJ &'( +)/-**(0 "/ "%* 0! 1)/!+ 0*,*-"$/' $4
>E?
Z˜Jℓ ≤ ˜b
TE
D
#%*)* Z˜iℓ $0 "%* "*0" 0"&"$0"$- /4 "%* &,"*)'&"$F* Ki : µi ≥ µ0i + δ 7 "%&" $07 Z˜iℓ =
s
nℓi nℓ0i (ˆ µℓ − µ ˆℓ0i − δ) ℓ ni + nℓ0i i
,
σ.
>G?
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
H'-* HJ $0 &--*+"*( &" 0"&1* ℓ &'( & 0! 1)/!+ Iˆ7 #$"% "%* ,&)1*0" F&,!* /4 Ziℓ 4/) i 6= J 7 $0 -%/0*'7 "%* 4!"!)* *')/,,2*'" /4 "%* ")$&, #$,, $'-,!(* +&"$*'"0 /4 "%$0 0! 1)/!+ /',.7 #%$,* "%* 2&8$2!2 "/"&, 0&2+,* 0$3* N
)*2&$'0 "%* 0&2*5 I$2$,&) "/ "*0"$'1 HJ 7 "%* ")$&, )*B*-"0 HIˆ &" 0"&1* ℓ′ &'( "*)2$'&"*0 #$"% &' *C-&-. -,&$2 /4 "%* '*# ")*&"2*'" $' "%$0 0! 1)/!+ $4 ′
ZIℓˆ ≥ b.
>JK?
′ Z˜Iℓˆ ≤ ˜b.
>JJ?
6" 2&. &,0/ "*)2$'&"* 4/) 4!"$,$". $4
64 '*$"%*) >D? '/) >E? *F*) /--!)07 "%* ")$&, +)/-**(0 "/ "%* <'&, 0"&1*7 $' #%$-% -&0* HJ $0 )*B*-"*( $4 ZJ3 ≥ c
>JL?
&'( $0 &--*+"*( /"%*)#$0*5 M$1!)* J 1$F*0 & 0-%*2&"$- 0!22&). /4 "%* 0*=!*'-* /4 (*-$0$/'0 $' "%$0 1)/!+ 0*=!*'"$&, ")$&,5 @
ACCEPTED MANUSCRIPT
PT
Final Analysis
SC
RI
Start 1st Interim Analysis 2nd Interim Analysis Accrue !! Test "! If eff, stop(J) If fut, select subgroup #$, test "%$ If eff, stop(#$) If fut, STOP Else accrue !%$& only Test "%$& If eff, stop(#$) If fut, STOP Else accrue !%$& only
Test "! If eff, stop(J) If fut, select subgroup#$, test "%$ If eff, stop(#$) If fut, STOP Else accrue !%$& only Test "%$& If eff, stop(#$) Else, STOP
MA
NU
Else accrue !!
Test "%$& If eff, stop(#$) Else, STOP
Test "! If eff, stop(J) Else, STOP
D
Else accrue !!
TE
!! =entire population; !%$& =selected subgroup; eff=efficacy boundary crossed; fut=futility boundary crossed; stop(·) = stop trial for efficacy (in the tested group - J or #$); STOP = stop trial and claim futility
!"#$% &' ()* +,-$. )/ .,% 012.-"% "$)#3 2%4#%5.!-( 6%2!"57 89%5.2 .,-. (!5% #3 9%$.!+-((: )++#$ -. .,% 2-;% -5-(:2!27
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
!"# "$%" &' "$# #()%"&!'* %+!,#- Iˆ &* % .%'/!0 ,%.&%+1# 2&"$ ,%1)#* .%'3&'3 4.!0 1 "! J − 15 6!.#!,#."$# )77#. +!)'/ α(0) !4 "$# "87# 9 #..!.- 2$&:$ 2&11 +# #;71%&'#/ &' <77#'/&; <- !4 "$&* 3.!)7 *#()#'"&%1 /#*&3' :%' +# /#:!07!*#/ &'"! "2! 7%."* *&0&1%. "! =>?5 @$# A.*" 7%."- P0 (B#C#:" HJ ), &* +!)'/#/ +8 P0 (ZJℓ ≥ b
4!. ℓ = 1 !. 2) + P0 (ZJℓ < b 4!. ℓ = 1, 2, %'/ ZJ3 ≥ c) ≡ P1b + P1c .
@$# *#:!'/ 7%."- P0 (<::#7" HJ %'/ .#C#:" HIˆ)- &* +!)'/#/ +8 ′ P0 (Z˜Jℓ ≤ ˜b %'/ ZIℓˆ ≥ b
4!. *!0# ℓ ≤ ℓ′ < 3) + P0 (Z˜Jℓ ≤ ˜b,
′
ZIℓˆ < b
4!. ℓ ≤ ℓ′ < 3, %'/ ZI3ˆ ≥ c)
≡P2b + P2c .
<* *$!2' &' <77#'/&; <- D!,=ZJℓ , ZJℓ ? ≈ Nℓ /Nℓ 4!. ℓ ≤ ℓ′ - 2$#.# Nℓ &* "$# "!"%1 ')0+#. !4 7%"&#'"* #'.!11#/ )7 "! "$# ℓ"$ &'"#.&0 %'%18*&*5 E#':#- "$# 7.!+%+&1&"&#* P1b %'/ P1c :%' +# :!07)"#/ +8 "$# .#:).*&,# ')0#.&:%1 &'"#3.%"&!' FG- 75HIJ )*&'3 "$# C!&'" '!.0%1 /#'*&"8 !4 {ZJℓ , 1 ≤ ℓ ≤ 3}5 ′
p
′
G
ACCEPTED MANUSCRIPT !"#$%&'! (#)!$&"*+ &(,!-$*,&.( "*( *+%. /! #%!0 ,. ".)1#,! ,2! 1$./*/&+&,3 P2b =
2 ˆ X
J−1 X
−∞ i=1
P0 (Z˜Jℓ ≤ ˜b, Zjℓ < x
4.$ j ∈/ {i, J}|Ziℓ = x)P0 (Ziℓ
′
≥b
4.$ %.)! ℓ′ ≥ ℓ|Ziℓ = x)φi (x)dx
PT
ℓ=1
b
5*(0 %&)&+*$+3 4.$ P2c 67 8! #%! 2!$! ,2! 9.&(, (.$)*+ 0&%,$&/#,&.( .4 {Ziℓ , ℓ ≤ ℓ′ ≤ 3} ".(0&,&.(*+ .( Ziℓ : ;&,2 ℓ $!1$!%!(,&(- ,2! %,*-! ;2!( HJ &% *""!1,!0: *(0 ,2! 9.&(, (.$)*+ 0&%,$&/#,&.( .4 Z˜Jℓ ≈ ZJℓ − √ Nℓ δ/(2σ) *(0 Zjℓ , j ∈ / {i, J}: ".(0&,&.(*+ .( Ziℓ ;&,2 i 6= J 7 <2! ,2$!%2.+0% b, ˜b < 0 *(0 c *$! 0!,!$)&(!0 /3 %.+'&(- ,2! !=#*,&.(% .$ @) = ǫβ,
P1b + P2b = ǫα, P1c + P2c = (1 − ǫ)α,
NU
PθJ =δ (Z˜Jℓ ≤ ˜b 4.$ ℓ>?
SC
RI
′
5?A6
MA
4.$ ,2! 1$!%"$&/!0 )*B&)#) ,31! C !$$.$ α: 1.;!$ 1 − β *, ,2! *+,!$(*,&'! δ: *(0 ,2! 1$.1.$,&.( 0 < ǫ < 1 .4 ,31! C 5.$ ,31! CC6 !$$.$ %1!(, *, &(,!$&) *(*+3%!%7
D
! "#$%&'() #)*'+, -. $ /$,#-0'1)# 2-,&/-33)# &/'$3 &- &)*& ),#-($*243$/ &5)/$%6
TE
!"! #$%&'()*+,(- '.-)+) )/($%(-% /0.-(12 3&,,&45$6 57(65$6 .'(,+(/5&$ 3&- 5)*0.75* )/-&8.
<2! "+&(&"*+ ,$&*+ )!(,&.(!0 &( ,2! %!".(0 1*$*-$*12 .4 D!",&.( ? &% * $*(0.)&E!0 ,$&*+ 4.$ 1*,&!(,% !B1!$&!("&(- *"#,! &%"2!)&" %,$.F! 5CD6 0#! ,. * 1$.B&)*+ *(,!$&.$ "&$"#+*,&.( +*$-! '!%%!+ .""+#%&.( ;2. *$! ,$!*,!0 ;&,2&( ?@ 2.#$% .4 CD .(%!,7 G*,&!(,% ;&++ #(0!$-. H C %,#0&!% *4,!$ !($.++)!(, /#, /!4.$! $*(0.)&E*,&.(7 <2! .(+3 IJK *11$.'!0 0$#- 4.$ ,2! ,$!*,)!(, .4 *"#,! CD &% CLM,GK: 3!, .(+3 2 −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
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
U
ACCEPTED MANUSCRIPT
TE
D
MA
NU
SC
RI
PT
!"#$!%#"& '!(%) *+, '- .#// 01 (22!#"$1) 3!% $41 5$6)78 9!:2%#5#"& !3 %12%151"$($#;15 3%!: :6/$#2/1 )#59#2/#"15 #"9/6)#"& "16%!/!&7 (") 0#!5$($#5$#958 (") #")121")1"$ !3 $41 #";15$#&($#!"(/ 5#$15< =41 :(>#:6: 5(:2/1 5#?1 9(/96/($#!" #5 0(51) !" ( "!"@()(2$#;1 $%#(/ $! )1$19$ 2%!A19$1) )#B1%1"91 01$.11" 1")!;(596/(% $%1($:1"$ (") 5$(")(%) :1)#9(/ :("(&1:1"$ 3!% $4151 2($#1"$58 65#"& C#/9!>!"D5 %("E 56: !3 $41 F("E#" 59!%15 (5 $41 $15$ 5$($#5$#9 $! $15$ $41 "6// 472!$415#5 P (Y ≤ X) ≤ 1/2 !3 "! #:2%!;1:1"$ #" $41 F("E#" 59!%1 X 07 $41 1")!;(596/(% $%1($:1"$ 9!:2(%1) $! $41 59!%1 Y 3%!: $41 5$(")(%) :1)#9(/ $%1($:1"$ (%: #" $41 1"$#%1 2!26/($#!"< C1 3!//!. $41 (22%!(94 #" GHI8 2
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
ij
11
J
J
i,j ij
J
1
2
1
12
3
2
21
4
3
22
ℓ i
i
K
5
4
31
11
ACCEPTED MANUSCRIPT !" #$%&'%(')*"' +),-./.& #$%$)#$)- )# Ziℓ = {Wiℓ − nℓi (nℓi + nℓ0i + 1)/2}/{nℓi nℓ0i (nℓi + nℓ0i + 1)/12}1/2 .
0123
PT
4& 566"&')/ 78 9" '"():" %& "/%-$ ;.(<=,% ;.( >.:0 Ziℓ , Zjℓ ), i ≤ j ≤ J, 1 ≤ ℓ ≤ ℓ′ ≤ 38 %&' =#" )$ $. '"():" $!" ,)<)$)&? @.)&$ &.(<%, ')#$()A=$).&# .; $!" Ziℓ 8 i ≤ j ≤ J %&' 1 ≤ ℓ ≤ ℓ′ ≤ 38 =&'"( P (Y ≤ X) = 1/2 %&' =&'"( -.&$)?=.=# %,$"(&%$):"# P (Y ≤ X) = 1/2 + θB !" ,)<)$)&? @.)&$ &.(<%, ')#$()A=$).&# #!.9 $!%$ $!" 6(.-"'=("# )& C"-$).&# DBD %&' DBE ;.( #$%&'%(')*"' #%<6," <"%&# 0F3 .; &.(<%,,G ')#$()A=$"' .A#"(:%$).&# -%& A" %66,)"' $. $!" #$%&'%(')*"' +),-./.& #$%$)#$)-# 0123B 4& 6%($)-=,%(8 )& 6,%-" .; $!" #$%$)#$)- 0H3 ;.( $"#$)&? µi ≥ µ0i + δ $. %,,.9 "%(,G #$.66)&? ;.( ;=$),)$G8 9" &.9 '"I&"
NU
SC
RI
′
01J3
MA
Z˜iℓ = {Wiℓ − nℓi (nℓi + nℓ0i + 1)/2 − nℓi nℓ0i θ}/{nℓi nℓ0i (nℓi + nℓ0i + 1)/12}1/2
TE
D
;.( $"#$)&? Ki : P (Y ≤ X | Πi ) = 1/2 + θB +)$! Ziℓ %&' Z˜iℓ '"I&"' AG 0123 %&' 01J38 9" -%& %66,G $!" I/"' #%<6," #)*" '"#)?& )& C"-$).& DBD %&' $!" ?(.=6 #"K="&$)%, '"#)?& )& C"-$).& DBE 9)$! σ = 1B !" $!("#!.,'# cα )& 0E3 %&' b, ˜b, c )& 0L3M01D3 ("<%)& $!" #%<" A"-%=#" .; $!" %;.("<"&$).&"' ,)<)$)&? <=,$):%()%$" &.(<%, ')#$()A=$).& ;.( $!" #$%&'%(')*"' +)-./.& #$%$)#$)-#B !"! #$%&'()*+ ,-'&',(%&).(),. /0 '1'$()2% ,3)*),'3 (&)'3 1%.)+*.
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
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× )<" #$%$=#8 "/-"6$ )& #-"&%(). CS 9!"(" $!" :%()%&-" )# QBJ8 %&' )& #-"&%(). CH 9!"(" $!" :%()%&-" )# DB !" N%&R)& #-.("# .; 6%$)"&$# )& $!" "&M '.:%#-=,%( $("%$<"&$ %(< %(" %##=<"' $. A" &.(<%,,G ')#$()A=$"' 9)$! <"%& µij %&' :%()%&-" vij 9!"& $!" O+4× )<" #$%$=# ;%,,# )& $!" -",, cij 8 9!)-! !%# % 6(":%,"&-" qij B !" µij %(" ?):"& )& % :"-$.( µ -.&#)#$)&? .; µ11 , µ12 , µ21 , µ22 , µ31 %&' µ32 0)& $!%$ .('"(3 %&' $!" vij %(" ?):"& )& % #)<),%( :"-$.( ;.( "%-! #-"&%(). 1Q
ACCEPTED MANUSCRIPT *#$
qij
%+$ , -$!
"#$ %-$+%,$ (%.5'$ ( 6$
03 "#$ "+ %'4 "#$ 5+0&%& ' "7
%!2 "#$ 0-$+%'' 5+0&%& ' "7
("055 !, %" %!7
30+ (0.$
8+$50+"$2
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1# '$ =$( ,! F <($(
pf
03 $%+'7
˜b = −1.46, b = 2.39, c = 2.31
cα = 2.16)
*0 $G%. !$ "#$
/0++$(50!2 !, "0
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α = 0.05, β = 0.2
( +$50+"$2
! *%&'$ I4 =$( ,! > <($(
=$( ,! > %((<.$( "#$ $H$/" ( 6$
θ = 0.064
! /$''
˜b = −1.46, b = 2.37, c = 2.28,
cij )
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! "#$ ($/0!2 ( .<'%" 0! ("<27 #%-$ 2 H$+$!" 5+$-%'$!/$ +%"$(4 1 "# .0(" 5%" $!"( 3%'' !,
cα = 2.14)
%!2
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MA
>'' "#+$$ 2$( ,!( #%-$ "75$ C $++0+ /0!"+0''$2 %" A9 %!2 $%+'7 ("055 !, 30+ 3<" ' "7 /%! +$2
(%.$ %( "#%" 03 =$( ,! F:4 &<" "#$+$
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"#$ $H$/" ( 6$( %,+$$( 1 "# "#$ %((<.5" 0!( #%( "#$ # ,#$(" 501$+4 %!2 %'(0 +$(<'"(
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
! % (%.5'$ ( 6$ "#%"
( 0! %-$+%,$ %" '$%("
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)4
).
µ = (0, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0.3, 0.3) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0, 0, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.6, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (2, 1.5, 1, 0.5, 0.5, 0.5) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (0.5, 1, 1.5, 2, 2, 2) µ = (0, 0, 0, 0.3, 0.4, 0.5) v = (1, 1, 1, 1, 1, 1)
)*!%+,#p (p1 , ...p6 ) 5 (1.1, 0.7, 0.4, 0.4, 0.2, 2) 89 (0.9, 0.9, 0.6, 0.6, 1, 85) 80.5 (3.8, 4.4, 6.7, 13.6, 2.3, 49.7) 65.7 (14.7, 28.5, 6, 2.2, 0.7, 13.6) 26.5 (0.6, 0.2, 1.4, 7.8, 1.6, 15) 86.8 (60.7, 9, 2.6, 1, 0.4, 13.1) 78 (55.4, 8.6, 2.2, 1.4, 0.6, 10.2) 92.5 (7.4, 13.4, 20, 3, 0.6, 48.1) 83 (5.8, 7.4, 24.9, 3.5, 0.9, 40.5) 72.2 (15.2, 16.2, 13.7, 4.1, 1.3, 21.7) 47.9 (0.2, 0, 0, 0.1, 0, 47.6) Pe
Pf
.10
/03 /8
/05 52
4013
/24 5518 010
/58 /.1/ 414
015
//5 //
/57 /313 .17
N
PT
RI
SC
NU
p (p1 , ...p6 ) 4.8 (0.9, 1, 0.9, 0.4, 0.2, 1.3) 89.8 (0.6, 0.6, 0.8, 1, 0.9, 86) 77.7 (2.1, 3.2, 5.1, 14.3, 2.8, 50.2) 49 (6.5, 20, 5.3, 2.2, 1, 14.1) 25.9 (0.3, 0.2, 1, 7.7, 2.5, 14.3) 72.8 (49.9, 5.9, 2.1, 0.9, 0.3, 13.7) 57.4 (36.9, 5.9, 2.5, 1.3, 0.5, 10.4) 88.7 (5.4, 11.7, 18.3, 2.5, 0.8, 50) 77.1 (3.6, 5, 23, 3.2, 0.9, 41.4) 61.1 (8.9, 11.9, 13, 3.6, 1.7, 21.9) 50.1 (0.1, 0.1, 0, 0, 0.2, 49.7)
!"#$ %& '($)!*+,- ./!)!.*$)+0*+.0 +, */$ .!0$ qij = 1/6 12) !## i, j
!"#$% '
69.2
39.8
61.3
68.9
22.3
28.1
28.9
28.8
69.9
94.5
4.8
p
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
AC !"#$% & CE /0/ 010 2/13 PT 564 6513 .13 E/50D/010 4 /23 5.17 513 M /24 4413 0418 A
ACCEPTED MANUSCRIPT
56
)5.
)2
)8
)7
)4
)3
)/
)6
)0
)5
).
µ = (0, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0.3, 0.3) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.3, 0.3, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0, 0, 0.3, 0.3, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.6, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0, 0, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (1, 1, 1, 1, 1, 1) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (2, 1.5, 1, 0.5, 0.5, 0.5) µ = (0.5, 0.4, 0.3, 0, 0, 0) v = (0.5, 1, 1.5, 2, 2, 2) µ = (0, 0, 0, 0.3, 0.4, 0.5) v = (1, 1, 1, 1, 1, 1)
)*!%+,#-
AC !"#$% & CE /0. 012 PT 67. 7614 E/03D/314 /3/ 6512
p (p1 , ...p6 ) 5.2 (1, 0.8, 0.5, 0.3, 0.2, 2.4) 89 (0.9, 0.7, 0.9, 0.8, 1, 84.8) 83 (3.5, 3.8, 7, 11, 3, 54.6) 64.5 (17.4, 30.7, 2.8, 1.2, 1, 11.3) 33 (0.2, 0.1, 2.7, 7.3, 3.1, 19.6) 90.9 (56, 12.7, 1.6, 0.6, 0.5, 19.6) 82.9 (51.5, 14.1, 2.1, 1, 0.5, 13.7) 94.1 (5.7, 8.6, 13.9, 3.4, 0.8, 61.6) 87.9 (3.2, 3.7, 17.7, 3.3, 1, 58.9) 74.3 (16.1, 14.4, 10.7, 4, 1.8, 27.4) 38 (0.3, 0.1, 0, 0, 0, 37.5)
.14
.13
3314
Pf
/6. 0418 5810
//7 6312 010
/58 /212 .18
/56 3612 .10
//5 //12 512
PT
RI
SC
NU
/13 M 53 5713A
Pe
/68 /410 .1/
/35
N
p (p1 , ...p6 ) 4.8 (1, 0.8, 0.7, 0.5, 0.4, 1.3) 89.7 (0.5, 0.6, 0.6, 0.6, 0.9, 86.6) 79.8 (1.8, 3.1, 6.3, 10, 3, 55.5) 46.1 (7.9, 20.9, 2.9, 1.5, 0.8, 12.2) 35.1 (0.3, 0.1, 2.6, 8.4, 3.4, 20.2) 81.1 (50, 9.4, 1.4, 0.8, 0.6, 18.9) 67.3 (40.2, 9.4, 2, 1, 0.6, 14) 92.3 (4.6, 8.3, 12.8, 2.5, 1, 63) 84.9 (2.2, 2.9, 17.4, 3.1, 1.2, 58.1) 65.3 (9.1, 11.2, 10.1, 3.9, 2, 29) 37.5 (0.3, 0.2, 0, 0, 0, 37)
!"#$% '
57.2
47.2
75.5
79
28.4
35.9
36.5
25.6
72.9
94.5
4.8
p
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
!"#$ %& '($)!*+,- ./!)!.*$)+0*+.0 +, */$ .!0$ q = (0.2, 0.1, 0.3, 0.1, 0.1, 0.2)
ACCEPTED MANUSCRIPT
ACCEPTED MANUSCRIPT !"#$" %&!' !()'# %&)*$ +,%& -.. / )" ()"$0 1&$ !22")!3& 4*$5 67 89/: ,* %) 3)'543% (4 %,2 $ %$*%,'# ); %&$ !<$"!#$ %"$!%($'% $=$3% ;)" %&$ )<$"! $!5,'# %) %&$ &72)%&$*$*
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PT
2)24 !%,)'*0 .%")'# 3)'%") ); BCDE ,(2)*$* (!'7 3)'*%"!,'%* )' %&$ 2!"!($%",3 *2!3$ +&$' %&$ "$*2)'*$* 7 5,*%",64%$5@ !'5 89/: !55"$**$* %&$ 2")6 $( ); 3)'*%"43%,'# (4 %,2 $ %$*%,'# 2")>
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θj = µj − µ0j
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ACCEPTED MANUSCRIPT !"#$% &'(& ())*+" ( (,&#-! .'*#.! */ ,(&#!%& "01$2*0, #% &'! 3%() (%()4"#" 56!"#$% 7 #% &'! &(1)!"89 &'!% 6!"#$% 7 :(4 1! ,2!/!22! &* &'! $2*0, "!;0!%&#() 6!"#$% < !",#&! &'! "(:,)! "#=! "(-#%$"> ?#:*% (% ?#:*% @ABC '(-! ,2*,*"! ( .)("" */ ( (,&#-! !%2#.':!%& !"#$%"9 ;0#&! #D!2!%& (% .*%"# !2(1)4 :*2!
PT
2!"&2#.&#-! &'(% &'(& ,2*,*"! '!2!#%9 +'#.' ())*+ &'! !)#$#1#)#&4 .2#&!2#( &* 1! ( (,&#-!)4 0, (&! (& #%&!2#: (%()4"!" */ ( &2#()> E'!4 ()"* ,*#%& *0& &'(& F&'! :*2! *%! 2!"&2#.&" &'! !)#$#1#)#&4 .2#&!2#(9 &'! )*%$!2 ,(&#!%&
RI
(..20() +#)) &(G!H (% &'(& &'!#2 ( (,&#-! !%2#.':!%& !"#$% F#" :*"& ,*+!2/0) 2!)(&#-! &* &'! "&(% (2 %*%I ( (,&#-! (,,2*(.' +'!% *%)4 ( ":()) "01"!& */ ,(&#!%&" 1!%!3&9 '*+!-!29 &'#" #" !J(.&)4 +'!% &'! (..20()
SC
2(&! #" :*"& !.2!("! >H K*&! &'(& *02 (,,2*(.' *!" %*& "!!G &* 3% &'! ,(&#!%& "01$2*0, /*2 +'#.' &'! %!+ &2!(&:!%& #D!2" &'! :*"& /2*: &'! .*%&2*)> L%"&!( 9 #& "!!G" &* 3% &'! ,(&#!%& "01$2*0, &'(& '(" &'!
NU
)(2$!"& M0))1(.GIN!#1)!2 #%/*2:(&#*% %0:1!2 Ij 5+'#.' #%-*)-!" 1*&' &'! ,2!-()!%.! (% &'! :!(% &2!(&:!%& #D!2!%.!8 /*2 j 6= J 9 (/&!2 (..!,&#%$ &'! %0)) '4,*&'!"#" HJ : µJ ≤ µ0J /*2 &'! !%! ,*,0)(&#*% ΠJ >
MA
O!(&&" !& ()> @APC 2!.!%&)4 2!,*2&! ( 2(% *:#=! .)#%#.() &2#() .*:,(2#%$ LQI&R< /*))*+! 14 !% *-(".0)(2
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TE
'!&!2*$!%!#&4 */ &'! "&0 4 ,*,0)(&#*% (% &'! ,*""#1#)#&4 */ "#$%#3.(%& #:,2*-!:!%&" */ &'! !% *-(".0)(2 (,,2*(.' #% "*:! ,(&#!%& "01$2*0,"9 !3%! 14 "!-!2#&4 */ &'! "&2*G!> Y*+!-!29 &'!"! "01$2*0," +!2! %*& ,2!I",!.#3! (% %* "&(&#"&#.())4 "#$%#3.(%& &2!(&:!%& #D!2!%.! 1!&+!!% &'! &+* ,*"&I'*. "!-!2#&4I1("!
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
"01$2*0," +(" *1"!2-! > L& #" ,*#%&! *0& #% @AP9 ,>AZA[C &'(& #/ #& 1!.(:! .)!(2 02#%$ &'! "&0 4 &'(& &'! (""0:,&#*% */ !;0() &2!(&:!%& #D!2!%.! *-!2 &'! &+* "01$2*0," +(" #%-()# 9 &'! &2#() ,2*&*.*) :#$'& 1! (:!% ! &* (..*0%& /*2 &'#" 0%!J,!.&! (% !\.#!%& :!&'* &* (
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ACCEPTED MANUSCRIPT !!"#$%& ' ()*)$+, "-.%"#./0 .123"$ 4"34%#5 !,%#.%!1"0 )#$ 5,2+! 3"6+"#4%)1 "&4"#3%2#3 !"! #$%$&'( )*+,)-+. /0 1(/1/2)& 1(/+)&'() ,- 3)+4,/- 5!5!
SC
RI
PT
!"#$$ %&#% θj = µj − µ0j #'( θ = (θ1 , ..., θJ )) *+, θ ∈/ Θ0 , θj > 0 -+, #$$ 1 ≤ j ≤ J . #'( (!/'! √ j(θ) = #,01#21≤j≤J pj θj . 3)!). j(θ) "+,,!45+'(4 %+ %&! 5#%3!'% 4670,+65 j 83%& %&! $#,0!4% 9: 3'-+,1#%3+' '617!, Ij . 8&3"& (!5!'(4 +' θ #'( 8&3"& 83$$ 7! (!'+%!( 7; Ij (θ)) <+%! %&#% Ij (θ) 34 $3'!#, 3' n #'( %&#% %&! %;5! == !,,+, β(θ) (!/'!( 3' >?@ 4#%34/!4 >AB@
lim n−1 log β(θ) = − lim n−1 log Ij(θ) (θ) = max pj (µj − µ0j )2 /(4σ 2 ), n→∞
n→∞
1≤j≤J
MA
NU
3' C3!8 +- %&! #4;15%+%3" 7!&#C3+, +- D#6443#' %#3$ 5,+7#73$3%3!4) =' -#"%. Ij(θ) (θ) 34 #$4+ %&! 9: (3C!,0!'"! In (θ, θ 0 ). 8&!,! θ 0 ∈ Θ0 34 %&! 13'313E!, +Q(λ|θ) = (θ − λ)(3#0(p1 , ..., pJ )(θ − λ)′ / (2σ 2 )
>AF@
TE
D
+C!, λ ∈ Θ0 G %&! !25!"%!( C#$6! 6'(!, θ +- %&! $+0#,3%&1 +- %&! $3H!$3&++( ,#%3+ +- θ %+ λ 7#4!( +' # 4#15$! +- 43E! n/2 34 n2 Q(λ|θ)) I#&#(6, JK. AFL 4&+8' %&#% exp {−(1 + o(1)In (θ, θ0 )} 34 %&! #4;15%+%3"#$$; 13'31#$ C#$6! >#4 n → ∞@ -+, %&! %;5! == !,,+,. #% #' #$%!,'#%3C! θ ∈/ Θ0 . +- #'; $!C!$Mα %!4% +- H : θ ∈ Θ0 7#4!( +' # 4#15$! +- 43E! n/2) N $!C!$Mα %!4% 8&+4! %;5! == !,,+, #%%#3'4 %&34 #4;15%+%3"#$$; 13'31#$ ,#%! #% !C!,; #$%!,'#%3C! θ 34 4#3( %+ 7! #$%$&'( )*+,)-4) O&34 8#; +- "+15#,3'0 %!4%4 C3# $#,0! (!C3#%3+' #55,+231#%3+'4 #55$3!4 %+ 0!'!,#$ 5#,#1!%,3" -#13$3!4) PC!' 3' "#4!4 8&!,! $#,0! (!C3#%3+' #55,+231#%3+'4 %+ %;5! == !,,+,4 #,! (3Q"6$% %+ (!,3C!. I#&#(6, JK. AFL 4600!4%4 %+ /2 %&! %;5! == !,,+, #% #' #$%!,'#%3C! θ #'( %+ %,; (!,3C3'0 $#,0! (!C3#%3+' #55,+231#%3+'4 -+, %&! %;5! = !,,+, 3'4%!#(G 45!"3/"#$$;. -+, supλ∈Θ α(λ)) R! #$4+ 4600!4%4 %&! "$+4!$; ,!$#%!( 3(!# +- (!,3C3'0 $#,0! (!C3#%3+' #55,+231#%3+'4 -+, %&! 5MC#$6! +- %&! %!4% #% θ ∈/ Θ0 ) O&!4! $#,0! (!C3#%3+' #55,+231#%3+'4 $!#( %+ # $313%3'0 S6#'%3%; inf λ∈Θ q(λ|θ). 8&3"& 34 "#$$!( %&! I#&#(6, !2#"% 4$+5! #'( 34 1#2313E!( 8&!,! q = Q 03C!' 7; >AF@)
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
0
0
!5! 67/2)& 4)24,-8 1(,-+,17) $-& 1(//0 /0 9:; O+ 5,+C! >T@. ,!"#$$ %&#% Hj : θj ≤ 0 #'( Θ0 = {θ : Hj 34 %,6! -+, 4+1! j}) *+, θ ∈ Θ0 . $!% J(θ) (!'+%! %&! 4!% +- %&! %,6! &;5+%&!4!4) O&! -#13$;M834! !,,+, ,#%! >*UP @ 34 (!/'!( #4 Pθ { !V!"% Hj -+, 4+1! j ∈ J(θ)}. #'( +'! "#' 64! %&! "$+4!( %!4%3'0 5,3'"35$! >WOX@ %+ "+'%,+$ *UP JY. TL) O&34 5,3'"35$! #4461!4 %&#% %&!,! 34 # $!C!$Mα %!4% +- !C!,; 3'%!,4!"%3+' &;5+%&!434 ∩i∈I Hi . I ⊂ {1, ..., J}. #'( 64!4 3% %+ %!4% %&! 3'%!,4!"%3+' &;5+%&!434 3- #'( +'$; 3- #$$ 3'%!,4!"%3+' &;5+%&!4!4 ∩i∈I Hi 83%& I $ I ∗ #,! %!4%!( #'( ,!V!"%!() ∗
AB
ACCEPTED MANUSCRIPT !"#$ % &'($ ) $**+* !, #+--!&&$. !/ %". +"0' !/ ∩i∈J(θ) Hi !, &$,&$. %". *$1$#&$. 2".$* 3456 !& /+00+7, &8%&
9:;<≤ α=
:$ "+7 2,$ % >%*!%"& +/ 345 &+ (*+>$ ?@A= 9!*,& "+&$ &8%& &8$ (*+#$.2*$ !" $#&!+" B=B !, &%"&%-+2"& &+
PT
?%A C*,& &$,&!"D HJ E' 2,!"D FG
!, %##$(&$.H &8$ #*!&!#%0 >%02$ *$-%!", &8$ ,%-$ /+* ?%A %". ?EA= )/ 7$ 2,$ 345 %". %((0' maxi∈I FG
RI
&8$ !"&$*,$#&!+" 8'(+&8$,!, ∩i∈I Hi 7!&8 I ⊂ {1, ..., J − 1}6 &8!, D!>$, &8$ 9:;< /+* &$,&!"D Hi , 1 ≤ i ≤ J − 1
SC
%/&$* %##$(&!"D HJ = I+&$ &8%&
NU
P (FG
i∈I∪{J} /
i∈I
?JKA
√
MA
%". &8%& &8$ ,L2%*$ *++& +/ &8$ FG< ,&%&!,&!#, %*$ 1+!"&0' "+*-%0 *%".+- >%*!%E0$,6 7!&8 -$%", ≤ 0 $M#$(& /+* FG$ !"&$*,$#&!+" 8'(+&8$,$,= )& &8$" /+00+7, &8%&
9:;<≤ α(0)6 78!#8 !, &8$ &'($ ) $**+* +/ &$,&!"D &8$ D0+E%0 "200 8'(+&8$,!, ∩Ji=1 Hi 2".$* &8$ (%*%-$&$*
TE
D
>%02$ 0= 48!, ,8+7, &8%& α(θ) ≤ α(0) /+* θ ∈ Θ0 =
!"! #$%&'()*' *+ ,-. %* /0*12 (&31&'%)45 6&()/' 345 !, %0,+ %((0!#%E0$ &+ D*+2( ,$L2$"&!%0 .$,!D" NJKO6 NJPO= )" (%*&!#20%*6 /+* &8$ QR,&%D$ .$,!D" !" $#&!+"
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
B=Q6 ,2ED*+2( ,$0$#&!+" !--$.!%&$0' /+00+7, %##$(&%"#$ +/ HJ %& %" !"&$*!- %"%0',!, %". &8$" (*+#$$., 7!&8 &8$ ,$0$#&$. ,2ED*+2(6 2,!"D &8$ -%M!-2- +/ FG< i +>$* i 6= J %, &8$ ,$0$#&!+" #*!&$*!+" %& &8$ !"&$*!%"%0',!,= S0&8+2D8 &8$*$ ,$$-, &+ E$ %..!&!+"%0 #+-(0!#%&!+" E$#%2,$ &8$ QR,&%D$ .$,!D" %00+7, $%*0' ,&+((!"D
/+* /2&!0!&'6 &8!, #%" E$ $%,!0' !"#+*(+*%&$. !"&+ &8$ %*D2-$"& &+ ,8+7 &8%& %" %"%0+D +/ ?JKA ,&!00 8+0., /+* &8$ D*+2( ,$L2$"&!%0 .$,!D"H ,$$ % ,!-!0%* %*D2-$"& !" S(($".!M T +/ NBUO /+* % #0+,$0' *$0%&$. (*+E0$-=
!!"#$%& '( )*%#+ $%,+-%./+%*# *0 ,+1#$1-$%2"$ 3%45*&*# ,+1+%,+%5, *6"- ,/.7-*/!, 1#$ +%8" 7!8! 9*0:154 +*0 '155 ;*<40)4';&( *+ (%4'6406)=&6 >)5;*$*' (%4%)(%);( 3+",!.$* &8$ ,&%".%*.!V$. :!0#+M+" ,&%&!,&!#, Ziℓ D!>$" E' ?J@A /+* (%&!$"& ,2ED*+2( i %& &8$ ℓ&8 !"&$*!%"%0',!,6 2".$* &8$ "200 8'(+&8$,!, +/ "+ &*$%&-$"& .!W$*$"#$ E$&7$$" &8$ "$7 %". &8$ ,&%".%*. &*$%&-$"&,= :$ "+7 .$*!>$ %" $M(0!#!& /+*-20% /+* 3+> (Ziℓ , Zjℓ )6 7!&8 1 ≤ i ≤ j ≤ J %". 1 ≤ ℓ ≤ ℓ′ = 4+ ,!-(0!/' &8$ ′
"+&%&!+"6 0$& W .$"+&$ &8$ :!0#+M+" ,&%&!,&!# E%,$. +" % ,%-(0$ X1 , ..., Xm , Y1 , ..., Yn %". 0$& W ′ .$"+&$ &8%& E%,$. +" % 0%*D$* ,%-(0$ X1 , ..., Xm′ , Y1 , ..., Yn′ 7!&8 m ≤ m′ %". n ≤ n′ = 48$ "200 8'(+&8$,!, +/ "+ &*$%&-$"& .!W$*$"#$ &*%",0%&$, !"&+ &8$ ,%-$ .!,&*!E2&!+" +/ Xi %". Yj =
JX
!"#$ &8$ Xi %". Yj %*$ !=!=.=6 %
ACCEPTED MANUSCRIPT E(W ) = m(m + n + 1)/2
!"#$"%$ &'()*#"!'%*"+ "%,-(.#! ,*/.
"#$ 0"%(W )
= mn(m + n + 1)/121
.. 2345
67 89 "#$ 667 ::3;:::<7 =>. "(. "%,-(.#! &"# ). - .$ !' >'? !>"!
)=
mn mn(n′ − 1) mn(m′ − 1) mn(m′ + n′ + 1) + + = . 4 12 12 12
D.!
Z = {W − m(m + n + 1)/2}/{mn(m + n + 1)/12}1/2
Z′
*(*+"%+I ?*!>
@'/(Z, Z
′
)
W, m, n7
*# 6+"&. 'J
). !>. !"#$"%$*E.$ F*+&'G'# !"!* !*& "#$ $.H#.
=>.# *! J'++'? J%'( A4BC !>"!
SC
W ′ , m′ , n ′
h i {mn(m′ + n′ + 1)}/ {mn(m + n + 1)}1/2 {m′ n′ (m′ + n′ + 1)/12}1/2 1/2 , 1/2 mn m′ n ′ . m+n+1 m′ + n ′ + 1
=
NU
=
¯ m − Y¯n )/(m−1 + n−1 )1/2 Z = (X
¯m X
!>"! >"/. &'(('# /"%*"#&.
'J *7*7$7 ') .%/"!*'#
mn/(m + n)
X1 , ..., Xm , Y1 , ..., Yn
* !>. L*#J'%("!*'# &+'&MN 'J
?>.%.
mn m+n
B(t), t ≥ 0
*#J'%("!*'# !*(.
*
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¯ m − Y¯n X
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T%'?#*"# ('!*'#7
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mn/(m + n)5
(m−1 + n−1 )−1 =
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J'% !>. "(6+. (."#
" H% ! ') .%/.$ )I O'))*# "#$ P*.,(-#$ 233<7
TE
!"! #$%$&$'( ')** +$,&-$.)&$/' /0 (Ziℓ :
¯ m − Y¯n 5 X
17
A39C
D
Y¯n
MA
=>. 6%.&.$*#, J'%(-+" * !>. "(. " !>"! *# !>. &" . "#$
A4BC
PT
′
RI
@'/(W, W
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
B
mn m+n
,
A34C
*#$.6.#$.#! *#&%.(.#!
*# !>.
"#$ !>* *#$.6.#$.#! *#&%.(.#! 6%'6.%!I * - .$ *# !>. %.&-% */. *#!.,%"!*'#
"+,'%*!>( *# P.&!*'# 37:7 U'% !>. !"#$"%$*E.$ F*+&'G'# !"!* !*& 5 A39C >'? !>. "(. &'/"%*"#&. !%-&!-%. " !>"! J'% !>. !"#$"%$*E.$ "(6+. (."# "
min(m, n) → ∞W
¯ m − Y¯n )/(m−1 +n−1 )1/2 7 (X
B(mn/(m + n)) Wm,n − m(m + n + 1)/2 ≈ . 1/2 {mn(m + n + 1)/12} {mn/(m + n)}1/2
V# A33C5 ?. $.#'!. F*+&'G'#X %"#M -( )I
≈
V# J"&!5 A34C !*++ >'+$ " I(6!'!*&"++I
!' $.#'!. >"/*#, !>.
Wm,n
A33C
!' .(6>" *E. !>. $.6.#$.#&. '# !>. "(6+. *E. 5 "#$ - .
"(. +*(*!*#, S'*#! $* !%*)-!*'#7 Q
A33C '/.% " 6%. &%*).$ .! 'J *#J'%("!*'# !*(. 'J !>. J'%(
*# A34C5 ?. &'# *$.% !>. S'*#! $* !%*)-!*'# *#
mk nk /(mk + nk ), k = 1, ..., K 7
=' 6%'/. A33C5 ?.
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"# " I(6!'!*&"++I #.,+*,*)+. %.("*#$.% !.%(5 "#$ *!
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√
Nθ
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ACCEPTED MANUSCRIPT
nℓi nℓ0i ℓ ni + nℓ0i
1/2
θ∼
1 (pi Nℓ )1/2 θ, 2
PT
!"# $ %! &'( %#)(* +,( %+!-.!/.#)(. 0#'12"2- %+!+#%+#1% Ziℓ #- 3456 %+#'' ,!7( +,( %! ( '# #+#-8 127!/#!-1( %+/$1+$/( !% #- +,( -$'' 1!%( 9$+ #+% (!- #% 2: +,( 2/.(/
NU
SC
RI
/(1!''#-8 +,!+ &!+#(-+% !/( /!-.2 #)(. +2 +/(!+ (-+ !-. 12-+/2' !-. +,!+ +,( +2+!' -$ 9(/ 2: &!+#(-+% #%$98/2$& i (-/2''(. $& +2 +,( ℓ+, #-+(/# !-!';%#% #% 2: +,( 2/.(/ pi Nℓ < =(-1( +,( '# #+#-8 >2#-+ .#%+/#9$+#22: (Ziℓ : 1 ≤ i ≤ J, 1 ≤ ℓ ≤ L) #% +,!+ 2: ! $'+#7!/#!+( -2/ !' !% #- +,( -$'' 1!%( 9$+ ?#+, (!-% 1 1/2 µ< 2 (pi Nℓ /N )
!"!#!$%!&
MA
@4A B,(/-2C =< D (!%$/( 2: !%; &+2+#1 (E1#(-1; :2/ +(%+% 2: ! ,;&2+,(%#% 9!%(. 2- +,( %$ 2: 29%(/7!+#2-%< F,( D--!'% 2: G!+,( !+#1!' H+!+#%+#1%< 4IJKLKM356N5IMOJPQ<
TE
D
@KA B,(/-2C =< H(R$(-+#!' D-!';%#% !-. S&+# !' T(%#8-< H21#(+; :2/ U-.$%+/#!' !-. D&&'#(. G!+,( !+#1%L 4IQK< @MA G!/1$% V* W(/#+) X* Y!9/#(' ZV< S- 1'2%(. +(%+#-8 &/21(.$/(% ?#+, %&(1#!' /(:(/(-1( +2 2/.(/(. !-!';%#% 2: 7!/#!-1(< [#2 (+/#\!< 4IQ]L]M3M6N]JJO]]P<
AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
@5A =21,9(/8 ^* F! ,!-( DB< G$'+#&'( B2 &!/#%2- W/21(.$/(%< _2,- 0#'(; ` H2-%* U-1< a(? ^2/\L 4IbQ< @JA [!,!.$/ VV< H+21,!%+#1 12 &!/#%2- 2: +(%+%< D--!'% 2: G!+,( !+#1!' H+!+#%+#1%< 4I]PLM43K6NKQ]OKIJ< @]A c!# Fc* H,#, GB< W2?(/* %! &'( %#)( !-. !.!&+!+#2- 12-%#.(/!+#2-% #- +,( .(%#8- 2: 8/2$& %(R$(-+#!' 1'#-#1!' +/#!'%< [#2 (+/#\!< KPP5LI43M6NJPQOJKb< @QA [!/+/2C _* c!# Fc* H,#, GB< H(R$(-+#!' X"&(/# (-+!+#2- #- B'#-#1!' F/#!'%< H&/#-8(/ a(? ^2/\L KP4M< @bA [!/+/2C _* c!# Fc< XE1#(-+ !.!&+#7( .(%#8-% ?#+, #.d12$/%( %! &'( %#)( !.>$%+ (-+ #- 1'#-#1!' +/#!'%< H+!+#%+#1% #- G(.#1#-(< KPPbLKQ34P6N4JIMO4]44< @IA [!/+/2C _* c!# Fc< Y(-(/!'#)(. '#\('#,22. /!+#2 %+!+#%+#1% !-. $-1(/+!#-+; !.>$%+ (-+% #- (E1#(-+ !.!&+#7( .(%#8- 2: 1'#-#1!' +/#!'%< H(R$(-+#!' D-!';%#%< KPPbLKQ3M6NKJ5OKQ]< @4PA _(--#%2- B* F$/-9$'' [< G#.d12$/%( %! &'( %#)( 2.#e1!+#2- #- 1'#-#1!' +/#!'% 9!%(. 2- +,( 29%(/7(. +/(!+ (-+ (C(1+< H+!+#%+#1% #- G(.#1#-(< KPPMLKK3]6NIQ4OIIM< @44A _(--#%2- B* F$/-9$'' [< D.!&+#7( !-. -2-!.!&+#7( 8/2$& %(R$(-+#!' +/#!'%< [#2 (+/#\!< KPP]LIM346N4OK4< 4I
ACCEPTED MANUSCRIPT !"# $% &' ()* +(' (*), -./ 012*3*24 52,66*78 *7 93*7*9)3 2:*)35/ ;2)2*52*95 )7< =25 =72%:>)9%/ "?!"@ABCDEC!AFC"G/
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RI
!C# TN%:7,M $' ;)O)8% H/ -7 TN%:7,MU;)O)8% 52)2*52*95 )7< 5%V1%72*)3 :)7W 2%525/ S77)35 ,> 2N% ;2)2*52*95/
SC
!XYA@GBCDEZ"AFZCA/
!A# ;*J,7 [' ;*J,7 H/ S<)62*O% %7:*9NJ%72 <%5*875 >,: 93*7*9)3 2:*)35/ \*,52)2*52*95/ "?!G@!CBCDE]!GF]"A/
TN)33%78%5 ,> <%9*5*,7 J)W*78
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MA
"?!C@CABADE!C?ZF!C!C/
!Y# \)N)<1: HH/ H)2%5 ,> 9,7O%:8%79% ,> %52*J)2%5 )7< 2%52 52)2*52*95/ S77)35 ,> K)2N%J)2*9)3 ;2)2*52*95/
D
!X]Y@GZB"DEG?GFG"C/
TE
!Z# R%77*5,7 T' +1:7I133 \/ T,7`:J)2,:4 &N)5% ==a=== 93*7*9)3 2:*)35 P*2N N46,2N%5%5 5%3%92*,7 )2 *72%:*JE -66,:217*2*%5 )7< 3*J*2)2*,75/ \*,J%2:*9)3 R,1:7)3/ "??]@CZBCDE]A?F]AA/
!X# +)78 b' c%33%: [/
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AC CE P
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65
\*,U
J%2:*95/ !XXX@AABCDE!!ZZF!!X"/
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"!# (%NJ)77 L/
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$,3<%7Ub)4' =79/ ;)7 0:)79*59,@
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