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PROBLEMS RELATED TO EFFICACY MEASUREMENT AND ANALYSES
by Sibabrata Banerjee
In clinical research it is very common to compare two treatments on the basis of an efficacy v r bl M r p f ll f Χ nd Υ d n t th r p n f p t nt n th t tr t nt A nd r p t v l th nt t (Y ( h h n b ll d th pr b b l t nd x f r th Eff t S z f nt r t n l n l t t t h bj t v f th t d t d r v n ff r th t ld p r t tr t nt r
nf r t v l nd bj t v l p r d t th rl r ppr h K rn l d n t
t t n f l n n-p r tr th d th t h n t b n ll t l z d n ppl d
t t t l t l nl d t t p t t n l pl x t h rr nt t d h th t th th d r b t v n nd r rr l t n tr t r th t r d r n th p t t n f
ll p bl d ff r n h rn l th d n b ppl d t th t t n f th OC
( v r Op r t n Ch r t r t rv ll t th pl nt t n f n n - p r tr z r r n f OC h r nd r th OC rv (A C h h x tl
l t th nt t (Y l xpl r d n th d rt t n h th d l d f r th t d t n r l z t th r r f ppl t n O EMS E A E ΤΟ E ICACY MEASUREMENT AND ANALYSES
by Sibabrata Banerjee
A Dissertation Submitted to the Faculty of New Jersey Institute of Technology in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Mathematical Sciences
Department of Mathematical Sciences
May 2007 Copyright © 2007 by Sibabrata Banerjee
ALL RIGHTS RESERVED APPROVAL PAGE
PROBLEMS RELATED TO EFFICACY MEASUREMENT AND ANALYSES
Sibabrata Baneriee
Dr. Sunil.K. War, Dissertation Advisor Date Associate Professor, Mathematical Sciences, NJIT
Dr. Mannish Bhattacharjee, Committee Member Date Professor Mathematical Sciences, NJIT
Dr. Farad Kianifard, Committee Member Date Senior Associate Director, Biometrics, Novartis Parma, NJ
Dr. Sandie Sin}iaray, Committee Member Date Research Scientist R&D, Educational Testing Service, NJ
Dr. Thomas Spencer I, Committee Member 'Date Professor, School of Management, Walden University, NJ
Dr. Kaushik Ghosth Committee Member Dte Assistant Professor, Mathematical Sciences, NJIT BIOGRAPHICAL SKETCH
Author: Sibabrata Banerjee
Degree: Doctor of Philosophy
Date: May 2007
Undergraduate and Graduate Education:
• Doctor of Philosophy in Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 2007
• Master of Science in Statistics, Rniversity of Maryland, Baltimore, MD, 2003
• Master of Statistics, Indian Statistical Institute, Kolkata, India, 1997
• Bachelor of Statistics, Indian Statistical Institute, Kolkata, India, 1995
Major: Mathematical Sciences
Presentations and Publications:
Banerjee S. and Char, Sunil "Problems Related to Efficacy Measurement and Analysis" Joint Statistical Meeting (GSM) Seattle, Washington 2006
Banerjee S. and Boo, Wonsuk "A Comparison Study of Models for the Human Sex Ratio," Joint Statistical Meeting (JSM) Minneapolis, Minnesota, 2005
Banerjee S. and Roy Anindya, "Finite Sample Efficiency of Local linear Estimation in EXPAR Models," Frontiers in Applied and Computational Mathematics (ACM) Conference, Newark, NJ 2004
iv Το b l v d p r nt t l v l f d r t r nd d r p r nt n l
ν ACKNOWLEDGMENT
I ld l t xpr n r r t t d t r S n l Ch r h ntr d d t th pr bl h l v n ntl d nd t n th n ld h v b n p bl th t h n t nt n r nt nd pp rt It ld h v b n p bl t h v d n n th n th t r M nn h h tt h rj n h lp l h d h d r p n I ld l nt h r th n pr bl nd t th l t n r
tr n h nt th t ld l d t th l t n r r d K n f rd d rv p l th n h
n ppr h t rd lv n th pr bl h n n n f h p p r h l v n
n b r f p nt r nd d f r r r h nd h f ll d th r v r
l l Wh t v r 1 h v n d t l rn b t th fl f r r h d nt tl fr h r h Sp n r h d n t d v r l f l b t n th t nd
t t t fr h p r n l ll t n M n r th n t h I ld l l t th n
r S nd S nh r f r h t hn l d n fr t t t n r t
h l nd I v r f rt n t t h v h n tt
r K h Gh t l n r n h l nd n pr f r h h lp d n l th h t b th th t hn l nd n n-t hn l M
f hr r b n t t t n h r lf h lp d th h r np t fr t t t
Al t v r th n th t I h v n d t l rn b t t hn l pr nt t n d t h r
I l th n f l t th p rt nt f M th t l S n nd th n v r t f r pp rt n r nd f r th p t n nd th r f l t v l bl t
vi A E O CO E S
Ch pt r
1 I O C IO 1
1 1 Obj t v 1
1 r nd Inf r t n
O - A AME IC ES S 7
1 f n n th r bl 7
n t E t t n M th d 15
3 S rv v l An l 1
v r Op r t n Ch r t r t C rv 1
3 A AME IC ES S
3 1 Exp n nt l UM UE f r ES
3 M nt C rl S l t n f th E t t r
KE E E SI ES IMA IO 7
1 Appr h 7
E t t n fr p nd nt t 3
3 S l t n 3
5 ECEI E O E A I G C A AC E IS IC C E 51
5 1 n OC 51
5 OC fr n t 51
5 3 E t t n th r v t v f OC nd th A C n th n t 57
5 OC M r f Eff 59
5 5 r l lt 3
vii TABLE OF CONTENTS (Continued) Chapter Page
CO C SIO S A E S IES
A E I SO CE CO ES
E E E CES 77
viii LIST OF TABLES
Table Page
1 tr b t n f l d r r Ch n r l n f r Ant h p rt n v r 9
S r St t t 1
3 C v n l f r ff r nt t f M n f S pl 1
3 1 lt fr M nt C rl S l t n 5
1 Ch f K rn l 31
Eff n f K rn l C p r d t Ep n hn v K rn l 3
3 - C p r n f n t E t t f r ff r nt K rn l n th S nd dth f 1 5 11 nd 11 5 35
C p r n f E t t d M n Int r t d S r d Err r f n t C p t d fr Ind p nd nt nd p nd nt t
5 E t t d r b b l t G v n Al n th th M n S r d Err r
E t t d r b b l t G v n Al n th 95% t tr p C nf d n Int rv l nd E p r l r b b l t 9
5 1 Eff t S z M r fr ff r nt M th d f E t t n
5 C nf d n Int rv l f OC C rv 5
5 3 E t t d Opt l hr h ld l
ix LIST OF FIGURES
Figure Page
1 OC rv 1
1 Un v r t K rn l d n t t t f th ff t f t bl nd d bl d pr r l r n dr v n l n th th rn l t th v l t n p nt n b nd dth 9
Un v r t K rn l d n t t t f th ff t f t bl nd d bl d pr r l r n dr v n l n th th rn l t th v l t n p nt n b nd dth 9
3 n v r t K rn l d n t t t f th ff t f t bl nd d bl d pr r l r n dr v n l n th th Ep n hn v rn l t th v l t n p nt n b nd dth = nd 5 3
n v r t K rn l d n t t t f th ff t f t bl nd d bl d pr r l r n dr v n l n th th Ep n hn v rn l t th v l t n p nt n b nd dth = 9 nd 1 33
5 Un v r t K rn l d n t t t f th ff t f t bl nd d bl d pr r l r n dr v n l n th th Ep n hn v rn l t th v l t n p nt n b nd dth = 15 nd 3
n v r t K rn l d n t t t f th ff t f t bl nd d bl d pr r l r n dr n v r l K rn l nd th b nd dth 3
7 M n nt r t d r d rr r (MISS p r n f th d n t t t p t d fr n nd p nd nt pl nd d p nd nt pl 5
n t t t p r d n t th tr d n t
9 n t t t fr v r l r n b r ( f nd p nd nt b rv t n 7
5 1 nd n th pt l p nt f d r n t n n th OC rv 5
5 Ar nd r th OC rv 57
5 3 h pt l thr h ld v l 59
5 E p r l OC rv nd 95% nf d n b nd 3
χ CHAPTER 1
INTRODUCTION
1.1 Objective
h bj t v f th d rt t n t t d r f efficacy. St rt n th th p f
pr bl f p r n t nt -h p rt n v dr n d bl -bl nd l n l tr l
bootstrap kernel density estimate f th d ff r n f th dr pr p d h
bootstrap confidence intervals f th pr p d d n t r l p t d h t d
d n tr t th t th th d f rl r b t v n nd r dependent tr t r th t r
d r n th p t t n f ll p bl d ff r n n h h f th nf r n
b d A rv f th x t n th d f ff r nt h b n n l d d
h hl ht n th l nt p nt nd l t t n f h th d
h th d r l t l z d t t d rv v l n l r p f ll th
t t f th rv v l f n t n nd th h z rd f n t n A d n t b d l r th t
t t th d r v t v f th r v r p r t n h r t r t rv ntr d d n th
t d An ppl t n f th th d n d r n nt n l xpl r d h r
h d rt t n l t l z l r th h kernel density t t n nearest
neighbor t t n nd ROC regression nd h th t th n b ff t v l d pt d
ppl d t t t l t l n th r f ph r t l t t t
In th nt xt f kernel density t t n bandwidth l t n th d pl
r l r l t r t r r v f h th d r n l d d n r n bl d t l All th
b nd dth l t n th d f ll nd r n r l l f pr bl th t n b b d
nd r t h t pt z t n St h t pt z t n r lt nd pr bl r th b f
l l t t t l th r
1 Ex pl f th pt z t n n l d maximum likelihood t t n likelihood ratio t t Neyman-Pearson Lemma nd pt z t n f th bias nd th variance f n
t t r
.2 r nd Inf r t n
C p r n t tr t nt th r p t t pr r ff v r bl pr bl h h
nl n nt r d n th l n l t d S v r l p r tr nd r p r tr z d
th d r d t f nd l t n t th pr bl r tr ppr h r ft n b d
n n r l t pt n r p r tr z d ppr h r pr r l r n b d t t l th Wilcoxon-Mann-Whitney(WMW) t t
Eff t z (ES pr nt th n t d f th d ff r n b t n t tr t nt
nd r n d r t n In n f th r nt ppr h ES pr nt d th t l
xpr n n t l nd r t d b l n n An d l ES r h t t
p t nd pp l t b th l n n nd t t t n An x pl f h r
v n b th pr b b l t nd x P (Y X) , h r nd d n t th p rf r n
r f t p t n dr ( n f h h b pl b r t d th t th l r r th p rf r n v l th r ff th dr
On f th f r t t d th t rv d th t t n f th nt t P (Y X)
W lf nd (1971 h h v n d r d th t t n nd r th n r l t
pt n ll th t th pt n C nf d n b nd f r P (Y X) nd
r l t d nt t l xpl r d th v r l x pl
S n ff t l (19 h v xpl r d th nt t P (Y X) nd r p r tr z d v r n P (Y — P (X Y n t r l d t n fr t p p l t n In th p p r th h v ntr d d h br d t t r th p r tr ll r p r tr z d 3 pr p rt h t t r nt ll W l x n-M nn-Wh tn (WMW t t t x pt th pr p rt n r l l t d n n r l t pt n h h v l ntr d d n th p p r th t t r f P(Y>X) — ( Y b d n n n-p r tń d n t
t t n pr d r h h v l h n b th th p r l t d nd th r t l j t f t n b v r n t t n nd pt t r lt th t th h br d t t r pseudo-MLE th ll t v r t r lt nd r n r t n t th t th
th d t t r b d n n n-p r tń d n t t t n th d h b n d n
t r l d t h r t d v v d n th t th t t r pt t ll
v l nt t th MW t t t nd b rr t n th d nd b t tr p tr t nt
t v n r ttr t v (b r d d th t t r
M th d b d n b t tr p nf d n nt rv l f th nt t h v b n d d b Ch n nd K n f rd ( It h b n rr tl p nt d t b th th t th r j t n f th Wilcoxon-Mann-Whitney t t ld pl th t th t d tr b t n
nd rl n th b rv t n r n t th h r f r n n ff r nt
n l t n b r n bl nf rr d th t n tr t nt b tt r th n th th r v r th d n t pr v d nt f bl r f ff Al t nt n d n th p p r
nd ll tr t d th n x pl th t t n t r n bl t h ft d l h n th r p n r t r l h Wilcoxon-Mann-Whitney t t ft n d th h ft
d l pt n
C ntr t p r n b t n v r l ES r h v b n d d b A t n
t l ( h h v n d r d th nl d ` r f p r t n b t n t d tr b t n (Coven's d), n f th n n f l ES v r th l t t n f th
'Coven 's d' l n t pt n f h d t t h h ft n n t th h
Odds ratio nd th generalized odds ratio r l xpl r d n th b v nt n d p p r 4
h nt t P (Y r th pr b b l t nd x ntr d d n f th p t t v
ES r n th p p r b A t n t l (
Alth h th pr b b l t nd x n b f r th l r l t n b t n th
ecei e O e a i g C a ac e is ics ( OC rv nd th nt t P (Y h
n r t d r n d nt r t n l n l r r h n p p r b A t n t l ( h
Ke a s τ l d d n th b v p p r b t th nl n n f l f th d t
ll t d n t h d p r r b t l ( h v l d v l p d th d f
nn t n th pr b b l t nd x th th OC nd ntr d d th A ea u e e OC
Cu e (AUC h t nv lv d p rt f th r r n d t n f r v r t
A us e A C n lt rn t v nd x f r th AUC h h p t d n th b f
-p r tr r p n f th v r nt r h h r v n n d ff lt pr t l
t t n l nf nd n ff t AUC eg essio l nt n d n th nt xt
h t p d v l p d n d t l b dd nd p ( 3 nd p ( 3
A pt t ppr h t rd th ff t z r pr bl r n d r d b
b ( l r b d th d f d t r n n ff th pr f th r S v r l p t t v th d r d d nd f n l h r nd d n th b f th p rf r n p r n f th th d n th b v nt n d p p r
OC n pt b rr d fr th n l d t t n th r It n n th
nt xt f d l d n t t t n f r v l t n f th p rf r n f b n r
l f r th nt n tp t
h r f rl xt n v l t r t r n t t n (Y h r Y f ll t n n n d tr b t n nd G r p t v l Wh l th nt r t l n t t n th
nt t (Y p r t v l n t ppr h th pr bl nd dr
nf r n t n d r th nt f th h ft (= Y — f th r nd b rv t n v t r nd Y r t lf r nd v r bl
p r tr d n t t t n n v l th d h h h b n v rl d f r
l n t b l n l t t t n r tr d n t t t n h v r nl
d b t t t n r x pl f n r l n ld nl h v t t t th
n (μ nd th v r n (σ fr th d t Alt rn t l f th d n t f th nt t
G n ld tr t t t th l nd th h p p r t r α nd λ f th
d n t
np r tr d n t t t n n r l th d t d l th n n n d n t h th d r th n l pt n pt n f p f d n t r v n f l f d tr b t n (l t n l xp n nt l f l t l r r d n th th d
h ppr h n b ll d pt d n th nt xt In t d f d l n th th
nd v d l d n t f nd Y n n d l d r tl th nd l t th b v
nt n d nt t th f ll n
In th rr nt t d np r tr d n t t t n th d r ppl d f r
t t n th nt t 00 . Al n th t t t f ( v bt n d f r n thr h ld v l v M r n th ld b v n n th d n t t t n h pt r
Ap rt fr th np r tr d n t t t n p r tr ppr h l h v
r t p t nt l f xpl r d n pr bl p f d l n r x pl xp n nt l ό d bl xp n nt l n f r r n r l z d n f r pt n l d t th
rr p nd n U i o m y Mi imum a ia ce U iase Es ima e (UM UE f PAY
nd nf r n r l t d t t (Al t l 5 CHAPTER
NON-PARAMETRIC RESULTS
In th h pt r th r n l pr bl th t d f t nt h p rt n v dr ll b d r b d n d t l h r l v n f th n n rp r tr d n t t t n th d nd th r ppl t n t th pr bl ll l b p nt d t ff r nt d n t t t n
th d ll b n d r d th th pr bl n th f r r nd S rv v l n l
th d nd r v r p r t n h r t r t rv r v r l l r l t d t
n n rp r tr d n t t t n nl b th l r th th t r f r
t t n th d n t p t nt ll h lp t p th t t f h z rd f n t n
(f l r r t nd r v r p r t n h r t r t rv
2.1 Defining the Problem
h f ll n d t ( bl 1 n d bl rbl nd l n l tr l t p r t
nt h p rt n v dr bt n d fr t l (199 nd Ch n nd K n f rd
( h pr r ff v r bl th h n n bl d pr r fr th b l n
d ff r n f th bl d pr r r d n ft r th tr t nt th th dr fr th
bl d pr r r d n b f r th tr t nt M t f th n l n th nd f pr bl
th r n r l t nd nv lv n unpaired t rt t r th n l pt n
n n rp r tr v r n f th t-t t n l th W l x n rM nn rWh tn ( MW
t t ( l ll d th M nn rWh tn U t t
h MW t t r r n b d t t ll d th W l x n r n - t t nd th
M nn rWh tn U t t ntr d d nd p nd ntl b W l x n (19 5 nd M nn nd
Wh tn (19 7 h p rp f th t t f nd n t f th t r nd v r bl h
th d tr b t n r f n f th r nd v r bl t h t ll l r r th n th 7 8 th r t l d d n f th t t l n th l r r n b d t t r n l d d
n h nn (199 S f th t d b n th t t t th W l x n-M nn-
Wh tn ( MW t t v r th MW t t ft n d h r t p p l t n d tr b t n r d t h v th n r l h p b t n f th h ft d r l t v t th th r b n t nt nt Δ nd r th lt rn t v h p th h
p t t n l p t f th WMW t t t t t h v b n r v d b n r r h r rnh rd t l (19 nd dbr (1995 h v x n d th r f th MW t t n r p t r t t t l p r nn t l ( h v r v d th p rf r n f th MW t t th 11 r l t t t l p
h h v d r l d t t fr ph r l l xp r nt nd f nd th t th
r l t t t l p v v r d ff r nt t fr th W l x n-M nn-
Wh tn t t h p p l r n n th r l t f 11 p n l d SAS 1 S rpl
SASS IMA 3 5 nd SYS A 9
In Ch n nd K n f rd ( th h v n d r d th ff r 00
v n b bl 2. tr b t n f l d r r Ch n fr l n f r Ant h p rt n v r [S r Ch n nd Κ n f rd ( t l (199 ]
h d t t v n n bl 1 ll b r p t dl d n th pr nt t d A
r t t t f th d t t n l d d n th bl t v r h d f th d t th t th r d r ld b pr p r d f r th d ff r nt n l p rf r d t l t r p nt n th t d 0
bl 2.2 S r St t t
It h b n p nt d t n Ch n nd K n f rd A th t PAY
v l nt t th Wi co o Ma u n-W i ey t t t n nt n d t Α oo s a co i e ce i e a f r th b v p r t r Γο h b n p t d n th p p r It h
l b n nt n d th r th t th BMW t t t d n t v nt f bl r
f ff G n r ll p n l n t th r t l v l f th M nn-Wh tn r
t t t r d r v n nf d n nt rv l f th b v nt n d p r t r Γο ld
v th h n d d n ht f th ff It ld l t l r t f th
d n f th t d tr b t n n t n r n f ntl d ff r nt nd r th h ft
d l pt n h nf r t n th t t ld f l t pr v d ld l b v t l r
x pl t ld n t b bl t v nf r t n n th nd rl n n rt nt f th t d tr b t n r nt f bl d ff r n f th n f th t d tr b t n
St d h th t th M nn-Wh tn rU t t t n b nr l bl f th v r n f th t p p l t n r t d ff r nt It l h th dr b f n r n n nf r t n
bt n bl fr th nd rl n d tr b t n ; S S n ff A19 O n t l A19 r
rnb nd M C rt A195 tth ff A19 3 h ntr d d d f d f r f th
W l x n t t t t t t br d r f r f n ll h p th th t nv lv n l v r n
nd b n f n r l t h h nl r f rr d t th hr n r h r pr bl
Co e s n n rp r tr r f ff t z nl d n p h tr nd th r r f ppl d t t t nd p rt l rl n pr bl th t h v
l r tr t r th n t d b v It v n b th f ll n f r l
h r Y nd r th r p t v n nd Sp th p l d t nd rd
d v t n Sl nd S2 r th t nd rd d v t n f th t pl f z i & 2 r p t v l h dv nt f r n th th r th t t n t b d n d tr b t n l pt n nd n t fr t r f r n n r t t t n t
nd r t nd th ff t z j t b l n t th pr r b d l t f r x pl f r
ll ff t z 5 f r d r t nd f r h h ff t z (A t n t l h l t nn t b t t t ll n n f l
v r t t h th t th n l v r n `C v n d n t
n n f l t t r f ff t z It t p t th r nd h v C h n d v l ll r th n nd t h v t p r t d p r f p p l t n r x pl
n th f ll n r t t t th `C v n d ll r th n th pr r b d b t th n r f r p rt 2
bl 2. `C v n d l f r ff r nt t f M n f S pl
h `C v n d v l v n b 19 ( nd t n n r d l th t th ff t z h ld b f r r th n d r t G v n n bl 3 r th `C v n d v l f r d ff r nt r t f th n p n th r n n r l t n th pr nt x pl
t v n ll ff t z th n f n pl h t b t l t b t 5 t b r th n th th r h t nd rd d v t n f n pl b n 3 t b r th n th th r pl r l n xpl n n th l v l f th ff t z
C n d r th pr r ff v r bl f th t dr n t n In th t
th h n n bl d-pr r ft r d n t r n th dr t th t th p r f v r bl fr t d ff r nt d tr b t n d f n d n th d n b t p bl th d ff r nt v r n h p nd tr t r h t th pr b b l t d n t f b th th v r bl ld h v ll th p t v n th b t f th r l l n
Add t n ll th r nd p nd nt f n n th r h t t f l t f n nd r th n r l pt n f n l v r n t nv lv d h n n r l t n t
d; th r f rr d t th BevrensrFisveru Problem.
Ev n f t n n th t th n r th d n f n tr t nt b r th n th
th r n ld t ll b n r f th n r t th t n "h h b r ?" h
t n th f f r th r f ff t z C n d r t pl fr t d ff r nt p p l t n h b rv t n fr th t p p l t n r nd p nd nt f
n n th r b th th n nd b t n th pl If th t d n t h d p r n
n d f n t b th v t r f p r d ff r n h t t n n d l th p r
f nd p nd nt pl nd dd t n ll th n t b f th z lth h n th x pl th pl z r th If n ld l t th v t r f ll p bl d ff r n th r dr fr th d tr b t n f th tr d ff r n t th t th r d nt ll d tr b t d th th r t l d tr b t n f th d ff r n b t th r n t nd p nd nt th h rt f th d ff r n v l ld b rr l t d th
h th r h d n t p t d fr th d ff r n n th v t r ld n t nl pr v d th h p f th d ff r n t ld l pr v d t t f th median nd
th r quintiles, mean nd th variance nd n b t l z d t t th t t f th d r v t v f th d n t It ld l v l r d f th nt f p r t n f th t d tr b t n Appl t n f bootstrap pr d r h lp n th r v l f b At th p nt t h ld b n t d th t MW t t t t lf p t d n th t
ntr d th h rt f d p nd n nd t th r lt nb d t n b
h n th t t b l n t l r r l f tr t t t n l th t t t
G v n p r f v t r f b rv d v l fr th t d tr b t n n n l t th d ff r n fr ll p bl b n t n h th d ff r n d n t d b AY - Χ h h th bbr v t n f r nx1 = A 1 mη h r Υ nd X r
t v t r f d n n n x 1 nd x 1 r p t v l nd = Υ — Χ 4 . Ma - W i ey U n nb d t t r f r th p r t r Θ h h d f n d
r nt n d tr b t n t r d t
t th t th d f n t n f f r th n n- nt d tr b t n nv lv th
dd t n l t r 1 AY = h t r ntr d d t dj t f r t n th pl
v r th dd d t r 1 AY = n r th t th nf r t n f t d nd
n t nt th th xpr n f n r nt l pr b b l t v n b th f ll n
xpr n
h ecei e O e a i g C a ac e is ic rv ppl d n th nt xt f ff
r nt l n th r pp l n r f t d r b f th r l t n h p
f th r nd r th rv nd th f ll n nt t P AY th pr b b l t
r f ff t z
2.2 n t E t t n M th d
A n th t th d tr b t n h nt n d r v t v d n t t t n pr d r
n b ppl d th n l pt n t th d n t f D x t h d n t
t t n pr d r h lp n v l z n th p r t n b t n th t tr t nt d tr b t n h t t pt r th h p f th tr d n t 15 n t n b p t d n v r n n-p r tr th d h vistogram, kernel density estimation, maximum penalized likelivood estimation, th nearest neigvbor metvod t l K rn l th d n b b d n v r l d ff r nt rn l
th ntr t n pr p rt f th nd d t rn l Ad pt v nd v r bl rn l
th d r p p l r n d ff r nt t t t l ppl t n
Ap rt fr th h f rn l th r r v r l th d f h f b nd dth n kernel density estimation. v r d ff r nt b n t n f h f b nd dth nd rn l l d t t t d d n t th d ff r nt h p nd pr p rt
ft n nh r t d fr th rn l d f r t t n In dd t n d p nd n p n th
h f p n lt f n t n l t n f rn l nd b nd dth n b ff t d
rt n t l th t t n pr d r r l t v l n n t v t th h f rn l b t
n t v t th h f b nd dth
h t p p l r th d f t t n d n t th h t r th d
Alth h th th d nt t v nd t f ll nd p rh p th t d l
pt d n h v r t ff r fr r dr b h t p rt nt
n th h p f th d n t h h n d p nd h v l n th h n bin-widtv nd th n b r f b n Al th h t r j t ll t n f b x t d v r n
n th r r pr nt n th fr n r th r l t v fr n f th b n v r th t r t d n t b b l t l nt n nd h v d r v t v f n r ll
rd r h h t r d n t t t n t pt r f nd nt l p t f th nd rl n tr t r h v r h ft d h t r AAS d r t f t f th pr bl n nt r d b th rd n r h t r ppr h S S lv r n A19 r S tt
A199 f r d t l d d n
h rn l d n t t t n th th r h nd d p nd n th h f rn l
nd th b nd dth h nd rl n h p f th d n t t t d n t h n h 6 th r n bl h f b nd dth v r th d r th b nd dth th th r
ll b th h p f th d n t ll b nd dth h n v rl r h rv nd t d b nd dth n th t th p rt nt d t l fr th h p f th d n t
3 S rv v l An l
Hazard rate o failure rate nl d n rv v l n l It d f n d
h r fAt) is th d n t f n t n nd FAt) is th l t v d tr b t n f n t n t th t f th r nd v r bl Y nd h v pr p rt n l vazard rates, nd th n t nt f pr p rt n l t th t f th f ll n r l t n h ld
h n n rp r tr pr p rt v r l r pr p rt h ld nd r r
nd t n h h d n t th d n t
n n nf r n b d n th pr p rt n l t n t nt n f t d r tl
nt rpr t bl nf r n n P AY Ex t n r l b l t t t l l -r n t t h v b n d v l p d t dr nf r n n th n t nt f pr p rt n l t
2.4 v r Op r t n Ch r t r t C rv
v r Op r t n Ch r t r t A OC rv b rr d fr th n l d t t n th r v r l l r l t d t th ES Α p nt n th OC rv n b xpr d
Wh r Y c nd t l f t n nt th l f
r ° nd ι d n t d d nd h lth p p l t n r p t v l S dd nd
p A 3 f r pr h n v d n n th h rv n tr t d b j n n d ff r nt p nt b d n d r n thr h ld c, lth h th x t v l f c n t r pr nt d n th rv d f n t n th r n f th rv [ 1] x [ 1] r th 1 p nt f v f OC rv n th nt xt f ES ld d n t PAYE c 1
Ρ(Χ c) h t b rv t n r n ll fr th l r d n t d Χ
h p r l ROC rv n b bt n d fr p r f d t v t r pl b pl tt n th p r l rv v l f n t n f th t r nd v r bl n th
p r l ROC rv t d ff r nt h p b d n d ff r nt pl b rv t n
Ex pl f ROC rv th ll tr t n r n l d d n th l t n t n G v n
n r 1 pl t f OC rv f t p p l t n d d nd n r l h r th h r z nt l x r pr nt th n r l p p l t n
r 1 OC rv
M th d l r t th n d n n n-p r tr d n t t t n n b
ppl d t t t th th r t l h p f th ROC rv Alth h th r n
n r n p p l r t f th f th ROC rv n f ld l r d l nd l n l
t t t n t n r r h r h v xpl r d th d n t t t n ppr h n th
nt xt
Area under tve ROC curve (AUROC), th n t th nt t
bt n d b r n th r nd r th ROC rv It l h d th t AUROC th r t ll l t th ES. n n nf r n b d n AUROC x tl v l nt t th t b d n ES.
E t t f r th ES l b bt n d b th d r t ppr h h n t b n
f th pr r f f th ppr h f th t d C A E
A AME IC ESU S
np r tr d h lp h n n p r tr pt n t b t f t r r
v n v l d t n th pr n f n n d tr b t n th rt n n n n p r t r
t p bl t f nd UM U f th ES h pr bl lv d x tl th
n ld lv UM U pr bl n th p r tr t p Al t l ( 5 h v
tt pt d t lv th pr bl nd r th pt n f G n r l z d Un f r d tr b t n nd f ll f nd UM UE t t f 8= P (Y nd l f
= (Y } h r Γ ll d th r l b l t r h h v d d
x pl f t t t l t l r n n th nt xt f d t r f b r n nd h ft
G n r l z d Un f r ld b d h n th nt xt b t n l n l
ff nd r l t d ppl t n th xp n nt l f l f d tr b t n p rt l rl th
xp n nt l d tr b t n t b r ppr pr t A n n r l l f d tr b t n h th xp n nt l f l ld l d t np r tr d
h l p l l n r l d tr b t n th n n v r n nd xp n nt l d tr b t n r th d tr b t n th th n n l p r t r l pr n
. Exp n nt l UMVU f r ES
E t t n f th nt t (Y n th xp n nt l d tr b t n h b n v r d b
n r r h r E t t n f (Y h r b th nd Y h v d bl xp n nt l d tr b t n v r d n l t l ( 5 In th pr d n p p r d bl xp n nt l d tr b t n th n n l p r t r nd n n n l t n p r t r h b n
n d r d h xp n nt l d tr b t n v r ttr t v h b f th
20 2 r l pr p rt nd v r l th r nt r t n pr p rt h th h r t r z t n b d n b l t d ff r n ( r nd b n 197 S th nd Sh h (19 1 h v
xpl r d th l r b nd f th v r n f th t t r f P (Y X ), h r Y nd X
r d tr b t d xp n nt ll th p r t r λ nd μ h h v l xpl r d th b nd f r th M n S r d Err r f th M x l h d E t t r h n n f th t p r t r λ nd μ n n n
h r r v r l th r t d n th bj t th t t v d nt th t
t t n P (Y X) n th xp n nt l t p nd d p p l r pr bl Ex pl f th t d r Iv h n (199 Cr r nd C p (1997 Al t l ( nd M C l
(1991 h f ll n r lt d r v d n th xp n nt l d tr b t n f r t r nd v r bl h th r t l v l f ES d n t d b
nb d t t r f , nd t n d b th complete sufficient t t t ld b th 22 UM U estimator. We will show that the following unbiased estimator of (Y > X ) would serve the purpose: I (Y > X 1 ) where
Hence, this estimator would be given by the following expression
Note that Al and X 1 are conditionally independent of each other. In other words,
Al 1 s it is independent of X 1 s , which would be clear from the following derivation h t n l b v d th n f fr t n nv lv n b n l r nd v r bl Note that both forms of the UMVU estimator are valid for m >_ 2. t d th f n t pl pr p rt f th UM U t t r l n th th MM E
t t r I nt C rl l t n r p rf r d th r n f v l f λ 1 nd λ
h z f th pl dr n fr th t xp n nt l p p l t n r t n t b 3
nd v th nd I nt C rl l t n r d n n h h t t r
v n l n th th n r d rr r (MMSE n bl 3 1 h MSE v l f r th 26 MA E t t r p r bl t th ISE f th E UE S n th I E r t
p t nd th UM U t t r d n t v n dd d dv nt v r th E E f r th ISE n rn d t pr f r bl t th pl r t t r CHAPTER 4
KERNEL DENSITY ESTIMATION
4.1 Approaches
C rn l d n t t t n n b r l z d th l t n pr f th a e age s i e
is og am (AS ppr h rt ll ll n np r tń d n t t t n l r th n b h n t b pt t ll rn l th d (S tt 199 h Ge e a Ke e
eo em d t rr ll nd S tt (199 t bl h th
G v n d tr b t n f n t n th d n t n b r tt n
nd t r f n t n n th t n b v n n rn l f n t n K ( nd rr v
t th C rn l d n t t t r
h rn l t t f th d n t f n t n n b r tt n 7 h th b d f rn l d n t t t n
h kernel density estimator nt n d b v l ll d th rz n r r
nbl tt t t r th h n kernel function K (.) d f n d (A) d r b d
b v
r n th n b r f b rv t n i thr h n Alth h th r
pp d t b nd p nd nt b rv t n fr th d tr b t n th rn l th d
ft n d h n th pt n v l t d r v ll d th bandwidtv r th smootving operator. It nt ll th h lf l n th f th nd n h h th
th n t pl
r d t l d d r pt n f d n t t t n n n l t n f th f ll n r r S tt (199 ndl t l ( S lv r n (19 Azz l n
(19 1 S h lt nd Q r b rr (1971 nd n n (19
h rn l d n t t t r b n t n fr th rn l ppl d t h
v l t n p nt h th t t r nh r t th pr p rt f th rn l h ph n n n r ph ll xpr d n th d r ( r 1 h d n t v n th r r p t d f r th d t n nt h p rt n v dr th x pl ntr d d n
Ch pt r h rn l f n t n r nl r d b l f t n t nh n th v l
ff t Τ • .. • • , • • , 0 th n b r f b rv t n f ll n nt th t b n nt d E h f th b rv t n
v n l ht nd th t ld th r t n l r h t r In rn l d n t
t t n l r ppr h t n nd n nt rv l f th l n th t h th b nd dth t n r nd t p l v l t n p nt h b rv t n f ll n nt th t
nt rv l r t l z d t n tr t th rn l t t nd ht n d t th p nt
n h th t th p nt l t t th nt r t th h h t ht nd th p nt f rth t fr th nt r t th l t ht h ph n n n xpl n d
r ph ll n th d r ( r
S f th p p l r h f rn l r v n n bl 1 Wh l v l t n th p rf r n f th rn l d n t t t r t p rt nt t h n ppr pr t l f n t n th t ll b bl t p r th f tt d d n t v r th h l pp rt An
xp t d l f n t n n n th r n b l l t d b v r n v r ll p bl l v l M n Int r t d S r Err r (MISS n h h nd th
p r d t n rb tr r rn l f n t n (S lv r n 19 h b l Keg r pr nt th Ep n hn v rn l It h b n ll t bl h d th t nd r th pt n f
nd p nd n th Ep n hn v rn l tp rf r t f th tr rn l f r
ff n n d r d S S lv r n (19 f r r nf r t n 31 bl 4. Ch f C rn l
Alth h th r r r l t v l f t d n th ff n f rn l f r d p nd nt d t th nd p nd n pt n ft n v l t d n r l l f pr bl nd
t t f th pr p rt f th d n t t t r t ll h ld ll t l (1995 h v
h n th t v n n tr n l d p nd nt d t n pt t ll pt l b nd dth f r nd p nd nt d t d h l n r l r t nd t n
( h x t n f th xth d r v t v f th tr d n t r t f d In th p p r th h v l h n th t th MISS xpr n f r th d p nd nt pl h
l r f r th MISS xpr n f r th d n t t t fr n nd p nd nt
pl A t p l h rt f th ff n f th r rn l p r d t th Ep n hn v
rn l v n n bl 2 bl 4.2 Eff n f C rn l C p r d t Ep n hn v K rn l (S r S lv r n 19
r 4. UUn v r t rn l d n t t t f th ff t f t bl d pr r l r n dr l n th th Ep n hn v rn l t th v l t n p nt n b nd dth = nd 5
G v n n r 3 nd 5 r th pl t f UUn v r t d n t t t f
th bl d pr r v l n d ff r nt b nd dth nd th Ep n hn v rn l h
b nd dth th thn p r t r h l r r th b nd dth th th r th
rv If th h n b nd dth t ll t ht n r t p r r hn nt th
d n t On th th r h nd f h n b nd dth t l r t ld th t th
rv b nd n pt l nt h r b t ld ntr d b nt th d n t t t n pr d r h th n r t r n n b ll tr t d n l r r b nd dth
th th t f d t p nt
r 4.4 Un v r t rn l d n t t t f th ff t f bl d pr r l r n dr l n th th Ep n hn v rn l t th v l t n p nt n b nd dth =9 nd 1
C n d r th x pl n Ch pt r h t f ll p bl d ff r n fr th
d t n bl d rpr r dr 3 1 ( h h r d Ev n th th h v l
f d t t n b n th t th ll b nd dth r h p t ld rf r th
x pl f d n t r p t d p r t l th ph n n n ll rf
h ll tr t n f th v n n th r 3 nd 5 h h n b nd dth n
th r l n th f r t nd nd r r ( , (9 1 (15 r p t v l
h t th f r t r n th r 3 rr p nd t b nd dth f nd th nd
r rr p nd t b nd dth f . h n r l n t n l r fr th d r th
d n t t t b r h th d nd v r r th d th th f l r r nd
l r r b nd dth 4
r 4.6 Un v r t rn l d n t t t f th ff t f t bl d pr r l r n dr n v r l rn l nd th b nd dth
In th r th d n t pl t rr p nd n t th rn l r l r- d d f r p r n Wh n d ff r nt rn l r h n p n th b nd dth f x d th
t t d n t v r t h h pl t ld h t ll tr t th r v n rn l r d f r th th n p r t n nd th t t d d n t f th d ff r n pl tt d n th f r
nv t t th d n t t t th v r n h f rn l n
xp r nt nd t d th th d t t n l th v r bl d n t t t n r lt n d t nd d ff r nt rn l r v n b l h f rth r ll tr t th f t th t h f n f th f n t n rn l f n t n l ttl d ff r n n th
v r ll t t n th d
bl 4. C p r n f n t E t t f r ff r nt C rn l U n th S nd dth f 1 5 6 h kernel f n t n ll h n t f th f ll n pr p rt
h pt t d tr b t n f th rn l t t r r rdl f th h n rn l
v n b ( ( + βλ σ2 h r f (x) th tr d n t nd th th r n t nt
r v n b
t th t th bias factor fax d p nd n th h n kernel ll th b nd dth
(v) nd th nd d r v t v ( f ( f th tr d n t v l t d t th p nt x
h r f r n z t n f bias ld nv lv ppr pr t h f th kernel nd th bandwidtv.
t th h f th kernel nd th bandwidtv ld ff t th v l f σ
n th l ld b t f nd pt l kernel nd bandwidtv t n z b th bias
ac o β nd v r n σ h nv nt n t n z th Integrated Squared Error
('SE), mean integrated square error (MISS) r th ppr x t f r l f r th MISS
(AMISS).
h r r th r f r f l f n t n th t ld l d t d ff r nt t p f o ima a wi On h x pl th Ku ack- ei e i o ma io v n b
h r K th rn l f n t n nd μ th nd rd r nt th r p t t th rn l f n t n nd th tr d n t ff r nt b n t n f oss u c io s ke e u c io s ld v d ff r nt pt l h f a wi nd h n d ff r nt
e si y es ima es
l r (19 9 h xpl r d th pt l h f smoo i g a ame e
( a wi n oo s a In t d f p t n th nt r l f r th d d l f n t n h d th oo s a mea nd oo s a a o ima io s f r nt r l h
h n th t th pt l h f b nd dth f r th th d n d th th pt l
h n ke e e si y es ima io
h nd r z ( h v p r d v r l b nd dth l t n th d n ke e smoo i g f OC cu es h r p r l t d h th t th A ma smoo i g me o p rf r b t n t th n p r d 38 4.2 Estimation from Dependent Data
K rn l d n t t t n th d ppl d n d t th t r d t b nd p nd nt
p nd nt d t t pp r n t r ll nd th r f r r ft n d t t t th d n t
A r lt t t th t r nf r r n t r f l r r variance r bt n d If th n t r f d p nd n n n th n n pr v nt n b xp t d n t r f l r v r n r t l t b tt r b nd f th p bl b nd v r n n b bt n d th th xtr n l d f th d p nd nt tr t r n th d f r r d n th b r th v r n n b d pt d t f rth r pr v t h t l p rt nt nd f l t n f th d t t d p nd nt nd f t h t th n t r f h d p nd n
Wh n d n t t t r p t d fr ll p bl d ff r n f th t
ff v r bl fr nd p nd nt pl nl th v r n f th t t d t r r t nd th b r n n ff t d h pt t pr p rt f th t t
h normality, r d t n f bias (slow) witv increased sample size, r n t b
ttr t v b f r If th x t xpr n f th v r n b t n th rn l r
h d v r bl r n n th n n n p th v n tr n r r lt h
r z d nd pr v d b l
New Result 4.2.1 t 2 m nd Al A2 An b nd p nd nt pl fr t n n n d tr b t n E nd Y n nt n d n t fa nd
fl r p t v l t Al A2 WM b th l t f ll p bl d ff r n f X nd A 39
generalized Hoeffding's theorem, or the generalized Rurstatistics. For more detailed discussions on the generalized Rurstatistics see Lehman (1963).
Result 4.2.2 In the above set up, let f,,, be the true density of the random variable
the bandwidth used and μ ιι 2 K(
The expectation is exactly equal to the expectation of the density estimate obtained using independent data. The reason of equality is the well known fact that even in the case of correlated random variables the sum of expectation is the expectation of the sum. ( n th f t th t th M r nd v r bl lth h d p nd nt r rt nl d nt ll d tr b t d h n b r tt n th f ll n xpr n
l t x v B = —t, h n B = x + vt nd dB = vdt. h n th b v
xpr n b th f ll n nt t ( t th t ( nt n d n t fa , h h b n t t d
ll th t th d n t t t p t d fr th d p nd nt d t p rf r
l t ll It h ll th n r t rn pt t pr p rt nd n r l t h b f n t n l l t th n bt n d n th f nd p nd nt d t h n r l b l f th t th b nd dth l t n th d th t r f r nd p nd nt d t r n t
t ppl bl n th f d p nd nt d t v r th r r l v nt r b ll
t l (1995 ddr n th th d f b nd dth l t n n th nt xt f d n t 4 t t n fr d p nd nt d t In th p p r r f rr d b v th h v h n th t v n
n tr n l d p nd nt d t n th pt t ll pt l b nd dth p t d
th th nd p nd n pt n d h A tt r f f t th nd r d p nd n nd d b th rn l f n t n ppl d n th d p nd nt d t tr t r h
t tr n r nfl n th n th r n l d p nd n rd n t th p p r
h tr t r f d p nd n n th n r d r b d b v n b f r l t d
th th v r n tr x v n n th f ll n d pl t th v r n tr x f th m x n d t p nt bt n d fr ll p bl p r d ff r n b d n t d b Σ
r j r 1 — Χ ; ( = nd j =1 n nd th th r
d nt ll d tr b t d lth h th r n t n r l nd p nd nt rth r d n t n
th tr x Σ Χ n b r tt n th f ll n xpr n pl n p l t n th b v tr x Σ n b r tt n
r O d n t th KKr n r pr d t f tr I d n t th d nt t tr x f rd r m x m J d n t th x tr x th ll th ntr l t n t Ρ 1 nd 2 d n t th tr v n b l
If th t t f th nt t σα σα nd σ2 r v l bl th n n n
l t t Σ
Α l d f r xpr n f th nv r f th v r n tr x n b bt n d f ll n th r lt f M ll r (19 1 h xpr n n t rn n b d t p t th th r t t d d nd b f th f ll n t r n W th t
pt n b t l n r b n t n f th tr d n t f ll n nt th f l f d n t th ppr h nn t b d ff t v l In dd t n n nn t xp t
n rr t d pl t b nd p nd nt Onl x pt n f r b th th r tr t n h n th d t r r nd pl fr th n r l d tr b t n
4. S l t n
An xp r nt th n n d n t nd t d n th nt xt f d n t t t n fr
nd p nd nt d t v n b l h l f th xp r nt t h th d n t
t t p t d fr d p nd nt nd nd p nd nt d t p r th h th r
d n th th r t l r lt d d n th pr d n t n n n xp t
p r bl r lt fr th t nd f t t h t t fr pl t l r nd pl pt b n h r t j d th p rf r n f th t t fr d p nd nt pl h r th d p nd n rt f ll ntr d d p n n nd th
t d f t dr d d n h pt r Wh r v r v l bl th tr d n t d n th xp r nt
h nd p nd nt d t r pl d fr Ν( 2 p p l t n th d p nd nt d t
r bt n d fr t nd p nd nt pl f N(0,1) nd N(1, 1) p p l t n In th
f th d ff r n f th t n r l p p l t n N(0,1) nd N(1,1) l th 44 r p r th p rf r n f th t d n t t t r th tr d n t l d
n th xp r nt
bl 4.4 C p r n f E t t d M n Int r t d S r d Err r f n t C p t d fr Ind p nd nt nd p nd nt t
h MISS l n th th pl z d n th d n t t t n l
v n In th t t n th Ep n hn v rn l d nd b d r v l d t n
th d d f r th l t n f pt t ll pt l b nd dth h MISS
l l t d n th f ll n xpr n
h th d f MISS d r b d n S n t l (199 r ](x) nd fax)
r th t t d d n t nd th tr nd rl n d n t r p t v l h n b r n th n b r f v l t n p nt f th d n t h h h n t b th r 51 r 1
h r n f th x v l h n f r th d n t t t r b t n - nd t n r th n l n f ll th pr b bl pl fr h f th tr nd rl n d n t
h nd p nd nt d n t t t r l l t d n n b r f d t p nt
h r v r fr t 3 On th th r h nd th d p nd nt d n t t t r 4 l l t d n ll p bl d ff r n f t r l pl h f z 1 t 15
h n b r r x tl h lf f th pl z t n f r th nd p nd nt d n t
N t t h th r r n b r f d p nd nt d t p nt nd j t l n th pr v N v r fr t 3 Int r t n l t b rv d th t th d n t
t t rr p nd n t th d p nd nt d t r b f r l t b tt r th n th r
nd p nd nt nt rp rt h bl ll tr t th f nd n
MISS fr d p & End d t
r 4. M n nt r t d r d rr r (MISS p r n f th d n t t t p t d fr n nd p nd nt pl nd d p nd nt pl
h MISS v l p t d fr d n t t t b d n nd p nd nt d t
l n th th r nt rp rt (d n t p t d fr d p nd nt d t r v n n r
7 p th n th l th MISS fr d p nd nt d t lt pl d b f t r f t th p r r MISS v l f r pl z thr h 5 t b p r bl 46 n f th t f pl n n t nt th n th l t n l rl h th t th d n t t t n fr th d p nd nt d t t p rf r b tt r th n th nd p nd nt
N t b n b th th pl th n b r f v r bl nd x
In r n l n t n f th d n t t t f r N = 5 v n l n
th th tr d n t nd th d p nd nt d n t p t d fr th p r f n r l pl
h f z 1 5
d n t t t p r d th th r l d n t
Figure 4.8 n t t t p r d th th tr d n t
v r n n rr tl p nt t th t th t t p t d n th f
N nd p nd nt d t t r b d n Ν d t p nt h r th r r x = d t p nt ( f r l r r n b r d n th t t n fr th d p nd nt t n
v r ll pr n n th d p nd nt ht b l r t v b t th `p r n t pr n n t b p r bl Int r t n l v n th f ll n xp r nt h t t f t r r lt 4
th th t t bt n d fr th nd p nd nt pl nd th t l d n t N(1,2).
r 4. n t t t fr v r l r n b r ( f nd p nd nt b rv t n
In r 9 th r p rt d d n t t t fr th d p nd nt d t p t d fr p r f nd p nd nt pl f z h th rt f ll n r t d pl h d t p nt
z h x tl l t th z f th nd p nd nt pl dr n fr th
N(1,2) p p l t n t th t b th th d n t t t p t d fr d p nd nt nd
nd p nd nt d t p rf r v r d
ll th t th d n t f (d ff r n b t n t dr th r p t t th pr r ff v r bl h b n p t d fr th d p nd nt d t It h l b n
nt n d th t th d n t h th p nt- r t d n t n nd b th t 48 unaffected by the particular structure of this dependence. However, the ultimate goal is to
utilize this as an effect size measure.
bl 4. Estimated Arobabilities Along with the Mean Squared Errors
Given in Table 4.5 are the different possible values of the difference ( and the estimated probabilities for the data set in Table 2.1 in Chapter 2. The mean squared errors based on betstrap are also given along with these values.
fix ideas, note that the estimated probability of the difference being larger than
0 is given by 0.56744 (5.931e-05), where the quantity given in parenthesis is the bootstrap based mean squared error of this probability. For a randomly chosen subject, the probability of observing a favorable response under the application of treatment relative to treatment A is 0.56744, where the similar quantity based on Mann-Whitney statistics is 0.57541 (Chen and Kianifard, 2000). 4 bl 4.6 E t t d Ar b b l t G v n Al n th 95% t tr p C nf d n Int rv l nd E p r l Ar b b l t
S l rl n n l t v r l p bl v l f th d ff r n nd th
rr p nd n pr b b l t l n th t n r d rr r h r f r th ppr h
v r l b r t nd r t nd n f th d ff r n nd th rr p nd n b t tr p b d 95% nf d n nt rv l v n n bl 5
C p r d t Ch n nd KK n f rd ( th ppr h f ff t z r h
f nd nt l d ff r n In th p p r nt n d b v th bl d rpr r v r bl
d t b d r t b t n th pr nt t d t d t b nt n
( th r th x t n f th d n t nn t b d lth h th b rv t n r d r t
h p nt- 95% b t tr p nf d n nt rv l f th pr b b l t f th d ff r n v l r t r th n z r h rp r th n th nf d n nt rv l v n n Ch n
nd KK n f rd ( Al b d n th t d th rr p nd n nf d n nt rv l f 0 th n r nt l pr b b l t f b rv n r f v r bl r p n fr tr t nt r l t v t tr t nt A th t P(Y — Ρ( Y ( 1 519 1 h h d n t n l d h r b t r th t th d ff r n n f nt t th 5% l v l CHAPTER 5
RECEIVER OPERATING CHARACTERISTIC (ROC) CURVE
5.1 Using ROC
Receiver Operating Characteristic (ROC) curve is a plot of `sensitivity' (correct assertion of disease) versus 'A — specificity' (incorrect positive) of a test. In other words, the probability of a diseased subject being correctly identified is `sensitivity' and the probability of a subject without the disease being correctly identified as healthy is
`specificity'. This curve is plotted with respect to the thresholds of discrimination of the test under discussion, where `sensitivity' is represented along the vertical axis and `Aur specificity' is represented along the horizontal axis. The concept of ROC has originated in the signal detection theory but is rapidly gaining acceptance in medical applications, several branches of engineering and psychometric applications.
References to ROC can be found even in classical regression problem and discriminant analysis, machine learning and data mining. The following section discusses some estimation problems in this area.
5.2 ROC from Density
Recall that the ROC curve is the plot of `sensitivity' versus `A — specificity' with respect to several thresholds of discrimination. If the observations from the diseased group is denoted as Y and the observations from the healthy group denoted by then the standard
way of interpreting the ROC curve is a = Y ( (A — where the distribution of A and are FY and respectively. The point is the survival probability of at a specified point. Also the survival function of Y, i.e. (A— F) is denoted by i and the
51 2
nv r f th d tr b t n f n t n f X d n t d A n th t th l t v d tr b t n f n t n tr tl n t n th v l f n Oth r f th
tr t n t n pr p rt r l x d th n r pr nt th n r l z d nv r f th
l t v d tr b t n f n t n h f n t n l f r f th n r l z d nv r v n b th f ll n xpr n
Obt n n th nv r f d tr b t n f f n t n ft n t pl t d fr th r t l p nt f v nd xp n v fr p t t n l p nt f v
An th r f r pr nt n th OC rv b n th f ll n n t t n
E 1 (A = c n A— E (c) = VE (c) = h th OC rv pl tt d
( n t n b n ( = E ( A (A — = Y (c v r F (c) In th r
rd th b v r p r tr z t n h lp d nt f th OC rv th pl t f th
rv v l f n t n f th v r bl nd A h r th f r r r pr nt th h lth r p
f bj t nd th l tt r r pr nt th d d r p f bj t
An p rt nt f OC rv t v l t p rf r n f t t th t h v b n r d n p nd r b d n nt n tp t If th OC rv b d n
`t t A n r r t th ( A p nt p r d t OC curve b d n n th r t t
`t t th n `t t A ld b b tt r t t p r d t `t t Al n th l n
ld t nd rd t t ld h v OC rv n thr h p nt th t v r n r t th
( A p nt h h ld b d n t d th i ea oi fr n n h OC rv d pl th p bl tr d - ff b t n th t nd f rr r ( p A997 S n th r
p rt nt f th OC rv t d t r n th pt l thr h ld p nt n rd r t
x z b th ` n t v t nd ` p f t h x z n ` n t v t 53 nd n z n A — p f t
t th t th ( A p nt rr p nd t A % n t v t nd A % p f t
p ( 3 h d t l d d n n th t t t l pr p rt f OC rv v r
f th r t t th t ld r nd l l f th bj t nt d d nd h lth
r p th n th OC rv b d n th t t t ld b th l n f l t r pr nt d
h h f th pt l thr h ld v l th t h ld b d ld d p nd n
n f t r h th t d t pr v l n f th d nd n
( p A997 In th b n f th t nf r t n n t h th pt l p nt
t t l t t th i ea oi (v n ll
An lt rn t n pt l thr h ld v l n b h n b h n th p nt f r p bl fr th l n f l t h h th OC f th d-h t t th t t th t r nd l l f d d bj t fr th h lth n In th f ll n
t n n t t n th d f th pt l thr h ld p nt ld b d v l p d n th d n t t t n ppr h h r lt 5 A nd 5 ld b d f r th
t t n l r th h b v r lt t t d th t pr f n l d (
h pl b t f l r lt n b t l z d t bt n th d r v t v f th OC
rv t d ff r nt thr h ld p nt A d r t f th b v r lt t f nd th pt l thr h ld p nt b d n th x l v rt l d t n fr th d-h t t r 5 A
n v r pl f d p t r f OC rv nd th p nt n th OC rv th p nt f rth t fr th l n j n n ( nd (A A
r f: A n r l d r f r pr bl v n n r 5 A An p nt n th OC rv v n b R (p). t th p rp nd l r d t n f n rb tr r p nt n th OC rv fr th l n f l t b Hap).
r . nd n th pt l p nt f d r n t n n th OC rv
nd n th pt l p nt f r d r n t n th r p t t th t nd f p bl rr r n pr bl Ιι l t r r n t n t r
l r pr bl r x pl l t l t r r n t n b v n th f ll n 6
Alth h th π v l n b t t d b f tt n th l t r r n rv th r nt ll th n f th r n l , . S t r n t pr d t r t t th p bl , v l fr th bt n d t t f th π ; v l h v r t d r n t n pr bl nd lth h h n n th f ll n th nv nt n th n t n r l th b t h f π ο
h f n t n l v l f th nt t p Ramp) — p • h r f r th r f th
triangle drawn by the points (0,0), R and (A,A) is given by p ρ n th t t l
r f th tr n l nd th tr n l f r d b ( (A A nd (A ld b
Ρ +A ] , [ ίρ — h h tr v ll l r b nd f r th r nd r th OC rv
t th t th OC rv d t b n v nd n pr t l t t n th
n t b th In OC th t h t ll l r r v r bl ( (Y c _ A d
ll r pr nt d l n th v rt l x v r n v OC n b fl pp d f
th t h t ll l r r v r bl r pr nt d l n th h r z nt l x In th r n l
f r f th OC rv th r pr nt d t t h t b t l t d th d-h t t
th r f r th rv t b v th l n f l t
. E t t n th r v t v f OC nd th A C U n th n t
h f ll n l r th ld b f l t t t th v l f c th pt l
thr h ld v l n f th r pl f x d b rv t n fr nd g th n th r
l f t n nt n r p r th th r n b d n f n nd v d l b rv t n f ll n
th r ht d r th l ft d f c
St p Ch n nt rv l f thr h ld p nt C h th t th t x l d th p nt
h r th r f th d n t v l b z r Cl rl th pt l thr h ld p nt f
x t ld b l n t th t C Step 5 nd th pt l v l c, h h rr p nd t r =A
Step 6 S n th rr p nd n p r lr d n n t ld b t t b t th r n l ( Y v l
f nd r n bl rr r b nd f r th pt l c *, n n ε ε A h th t C th t nt n n f r h h th r v l b t n [A— ε A + ε] r 5 3
ld r l rl ll tr t th pt l pr p rt f th thr h ld v l nd l th f t th t th r t f th d n t l t n t th thr h ld p nt
r . h pt l thr h ld v l
v r th l r th d r b d n th t n r nt th x t n f th
pt l p nt th n x pt n f th tr v l f pl t l p r t d d n t r l t pl t l d nt l d n t h n n n b h n f r b d l d tr b t n th n v OC rv If th OC rv h v th n th pt l p nt n b h n fr v r l nd d t p nt n h h n l t h n
l b l xtr fr th nd d t l l xtr
.4 OC M r f Eff
A r p r tr z d v r n f th OC rv th pl t f th t rv v l f n t n f th
d r n thr h ld v l h v r n f th rv n b d n th t d f ff t
z ll n r b t l ( n n r pl th nt n t f th d d p p l t n (Y I ° nd th h lth p p l t n (Y I ° b A nd X r p t v l
h r A nd X d n t v l f th pr r ff v r bl f r th dr nd r t d
h OC rv n th nt xt ld pp r t b l r t th ideal point f th t
f th r ff dr r pr nt d l n th v rt l x n th t h h r v l f th t ld rr p nd t b tt r ff
S n th rv v l f n t n f th t nd rl n v r bl r p t d t
v r l p bl thr h ld n n p r th rr p nd n pr b b l t t th thr h ld v l On th th r h nd n n r z th h l OC f n t n nt
n l v l n l th r nd r th rv (AUC h h n b h n t b l t
P(Y X) . S th OC rv v th p r n f th v l P( A c)
nd Ρ (X > c) f r d ff r nt v l f nd th r nd r t n t t f th
nt t PAY X) .
A f n t n l l r l t d t th r p r tr z d v r n f th OC rv th
v rt l h ft f n t n (A v n b h r (A Ι ° h th d tr b t n E nd (Y Ι ° h th d tr b t n r nd
d n t th t nd rl n pr p rt f th p p l t n b n n d r d r
x pl ld b th pl b r p nd th r p h r th dr nd r t d h b n d n t r d h r l t n b t n th OC rv nd th v rt l h ft f n t n d f n d b v v n n Gh t nd rd ( 7
r R(p) th OC rv nd th f n t n A( d n t th v rt l h ft f n t n
r rl d v l p nt f th h ft f n t n n th nt xt f t pl pr bl
(A97
n th h ft f n t n n l b t l z d n th ff t z t t n r
x pl th n h lp n th t t n f th r nd r th OC rv nd r d ff r nt thr h ld v l h n t rn ld b f l t d nt f th p r r dr h n th r r t dr nd r t d
h f ll n t r lt 5 A nd 5 r p l f th r lt b Gh h
nd ń ( 7 h r r lt b d n Generalized Rank Set Samples aGRSS) nd n th pr nt nt xt pl f d f r th f Simple Random Sample aSRS). h t r lt v n b l n b ff t v l d f r th t t n f th p nt- v r n
f th OC rv nd th n t l z d t r t nf d n b nd f r th OC rv h
n l b xt nd d n f nd n nf d n nt rv l f r th ES bt n d th r
nd r th OC rv
lt .4. If t Simple Random Samples (S S f z m nd n r ll t d fr t p p l t n h r th p p l t n OC rv d n t d b th n th
p r l OC rv v n by h th f ll n pt t pr p rt Here q lies in the open interval (0,A) , q denotes the minimum of p and q f and f are the densities corresponding to the distributions E and F .
Therefore if we call the expression in (5.A) as K( then a point-wise confidence band of the ROC curve can be constructed inverting the above relations. This confidence interval is given in the following equation
t nd rd n r l p r nt l th r t th r ht l t α 2.
Note that the empirical ROC curve ( R ) mentioned above is computed from the raw data just like the empirical distribution is computed. The empirical ROC of the example of the double-blind drugs introduced in chapter 2 is given in the following 62 r 5 h h r z nt l x r pr nt th dr A nd th v rt l x r pr nt th dr
E p rE l OC rv nd C nf d n nd
r .4 E p r l OC rv nd 95% nf d n b nd
d n th th d d r b d n th b nn n f th h pt r ( r 5
n t t f th r nd r th p r l OC rv p t d h v l f th pr b b l t rr p nd n t th pt l p nt f p r t n v n b 779 A th n t t n f r th p nt p *. h rr p nd n OC rv v l Rap *) v n b
9 9A5 5 h r f r th t t d r nd r th OC rv v n b
p r n f th t t fr d ff r nt th d h x pl f th d bl - bl nd dr d n th ff t z r nt d n th th d d n t t t (S Ch pt r d n pt l p nt f p r t n (S rl r th h pt r t d n n r l nn-St ltj nt r t n ° d n E p r l d tr b t n f n t n
iema i eg a io d l th f nd n th r nd r rv ( n b nd d
nt rv l [a ] h r th nt rv l d v d d nt f n t n b r f b nt rv l e
h th r p t d b th n f r t n l
A r n r l f th nn nt r t n th iema -S ie es i eg a io ( Ap tl A99 h nv lv t f n t n In th pr nt d n th t f n t n r (c nd ( (c r p t v l h ld d l th th
f th f r Σ (t Δp ( ll th t th r n l v r bl c th thr h ld v l n t k= r pr nt d n th OC rv t n t t f th ES th n f th pp r
nd th l r f th nt r l p t d n n r l nn-St ltj
nt r l
5.5 Numerical Results
h nf d n nt rv l f th OC rv b d n th xpr n v n n th t n
(5 v n b l ( bl 5 h r p rt d v l f nd ( r th p r l v l p t d fr th d t t v n n t bl A h nf d n nt rv l th S S b d r lt pl f d fr th n r l r lt f r G SS Gh t nd r ( 7 64 h C nf d n nt rv l f th p r l OC rv r n bl h rp t th t t t ll p nt nf d n b nd b d n th t nd rd d v t n f th
OC rv v l t d t r d f p nt n th lt 5
bl .2 C nf d n Int rv l f OC C rv
h t t n l r th f r th pt l thr h ld ppl d t th b n t n f n r l v r bl ld th f ll n r lt S pl f z h r d n t r
r dr n fr th Y nd d n t nd C p nt r t t d fr th f tt d d n t ( t n 5 3 n th rr nt h pt r p rt d v l r th d n
t t bt n d fr r pl t n f th pr G v n n p r nth r th 6 d n b l t d v t n f th t t
bl . E t t d Opt l hr h ld l
h n t rth b rv t n h r th t th t t r r n bl t bl v n
h n d n t r t t d n f r n b r f d t p nt lth h th n
r d rr r r r l t v l h h r f r th ll pl z CHAPTER 6
CONCLUSIONS AND FUTURE STUDIES
h j r f nd n f th t d th f t th t nf r t n fr d t t th h v d p nd n n b xtr t d th th f t l pr v d d Al d n t t t n
th d nd p rt l rl rn l th n n b d n ff r nt h
t t d d n t h th n r t d n t n nd ld ll r MISS p r d t th d n t t t p t d fr nd p nd nt d t th h v th f r f th b f n t n
C nf d n nt rv l b d n b t tr p nd d bl b t tr p th d r t
ff t v n th nt xt f f n t r nf n t d n n l pr bl l d n t t t n
G v n th t n th r l rld n h t r th ll r d t t h h r ft n
p rf t nd d p nd nt th t d n x pl h r h v p t n n rt nl
v r f th h rt n n n ff t v Eff r nt pr bl th t n t ntl nd r f n l n l t d In th rr nt f dv n d
p t n b th th pr bl nd th ppr h t rd t l t n n b ddr d thr h d v l p n ff nt l r th nd pr r
h r r n p bl th rr nt t d n b xt nd d n d ff r nt d r t n
(1 h r t l nd n r pl nt t n nd d v l p nt f d bl b t tr p n d p nd nt d t
( t f nd th r t l t t f th d n t h n th d t th th
v r n v r n tr x = I O P + (J — I O Ρ th r n n r t bl
(3 d p nd nt d n t t t n n b t l z d n n n-p r tń r r n ( th n th d n d p nd nt t p
66 67
( G n r l z l t n t d t n l d n r l xt r d n t
(5 Appl th n np r tr n th d t t t th OC rv n I xt r f r h t Ar (MM pr r h n b d n b pr d n n Er nl t l (
( S pt t r lt n th n r l f r f th rn l v r nt r x pl f th nd rl n d tr b t n n r l l d f r xpr n n b bt n d f r th v r n ft r th ppl t n f th rn l f n t n S l rl n n n nt n d tr b t n f th nd rl n d n t ld v p r n f th r lt APPENDIX
SOURCE CODES
Some selected source codes written in R are included in this section. R is the free version of the S/Splus software (Ihaka and Gentleman, A996 ). The R packages along with relevant packages are available for download for free from the comprehensive R network, http://cran.r-project.org/.
The R version 2.3.A is used for the present study along with the editor Tinn-R version
A.A7.2.4, available for download from the following website http://www.sciviews.org/Tinn-Ri.
Each of these codes is linked with one or more chapters. There are several lines of comments included in each code to clearly explain the objective.
The following program is used to compute the UMVU estimator in the exponential case. See Chapter 3 for the related derivations. 69
} r t rn ( b
} END OF PROGRAM 1
h f ll n d d t d th M nt C rl l t n f th t t r b r p t dl ll n th b v r t n h tp t r r rd d n f l nd v d f r p t pr n h f ll n pr r l l t th b t tr p d n t t t r fr th d t t
ntr d d n Ch pt r nd d r p t dl t ll tr t n f th th d th t d v l p d n th rr nt t d 71 = e si y( w=" c " ke e ="e a ec iko " om=- o = =1 ; = o m( y$ mea =1 s =sg ( og = A SE
a =seg(1 1 y= ; o ( [ a ] ce = 75 1 y= y e=" " a =" " y a =" e si y" a =" a ues" mai a e si y es ima es com a e wi e ea e si y" y im=c( co =" ue" c = w =
c i ( a y$ [ a ] a is (1 a =c (1 1 3 39 9 a e s=c (- -5 - 1 7 as= oi s( $y[ a ] co a e " c =15 ce = 75 1 y= y e a " w = oi s( y$y[ a ] ca a e" c = ce = 75 1 y=3 y e" " w = eg =c(" ea e si y" " e s-es i e " " e s-es ee " ; ege ( 3 eg co = c(" ack" " e " " ue" c =c( 15 y=c(1 3 w =c(
ea e o igi a a a = ea cs (" /cou ses/ esea c /c i ica / a a_ ρ_a_ cs " ea e = UE Ca cu a e e i e e ce = i ( [ 1] ρ[ ] Β = 5 ss= ;
= ma o ec(51 (Β+1 ec a e e ace o i g a ia e i e as co um wi e a ues [ ( +1 ]= e si y( w=" " ke e ="e a ec iko " =51 om=- o = $χ
ow s a e oo s a esem i g
o (i i 1
{ = sam e e ace= = e si y w a " ke e ="e a ec iko " =51 om=- o = e si y as ee e a ua e a e i e a ues [ i]=ss$y Sa e us e y }
= seg(- y=5 = a o ec (17 o (i i 1 { [ i]=a y(as a ay( 1 y C [ + ] [ i] } I = a y(as a ay( mea SS = a y(as a ay( 1 a ISE = a y (( -I 1 mea e o a i i ies o e a ues gi e wi e MMSE s c i ( ou (1-I igi s=5 ou (MSE igi s=
oo s a 95 CI o e es ima e o a i i ies a y(as a ay( qua i e ρ o =c( 5 975
E O OG AM 72 The following program is self explanatory. The ultimate goal of this program is to utilize the density estimate and find the efficacy measure following the methods described in Chapter 4. i e = as e(" / A A/ y C/ OO e si y_ " 1 " " see =" ome i e = as e{"C / A A/ΒΟΟΤ e si y_ " Β " " see =" I
o (i i 1
= sam e( y e ace= Y = e si y( w a c " ke e ae a ec iko " om= ow y o= i y = Y=c i ( $ $y Sa e us e a yes w i e a e( Y i e= i e[i] Use w i e a e o w i e i e i es }
EA I OW a= ma ο ec(51 ec a e e ace o i g a ia e
o (i i 1
{ = ea a e( i e[i] ea e a es i is em a [ i]= $ Sa e o y e e a ua e e si y }
m( ow us e e e a e ai a
is is ou e si y i wou a e e a ues me ia a ues/ 5 a 97 5 a ues & a ; 5 co um s
ou _ e = ma o ec(51 5 ou _ e [ 1 ] = Y [ 1 ] ou _ e ] = a y(a me ia ; ou _ e ] = a y( a ; ou _ e [ c( 5 ] = c i (a y(a me ia a y a o (i i 1 51
{ ou _ e [i 3] = gua i e(a[i ] o e= 5 ; ou _ e [i ] = gua i e( [i ] o = 975 ;
}
a (ce = 7 ce a is=1 ce a =1 ce mai = o (ou _ e { ] c =e+e y im=c( 35 oi s (ou _ e [ 3] co a e " c a " oi s(ou _ e [ ] co = ue" c a "
oi s(ou _ e [ ]+ ou _ e [ 5] co =g ey " c =" " y e a " ρoi s(ou _ e [ ]- ou _ e [ 5] co ak aki" c a " y e a " CC = seq(- y=5 y C (CC [ 1 3 ] ou _ e [ 1] a[ 1 ] sa y( (CC y C = ou _ e [ 1] =a[ 1] my u - u c io ( y {y[w ic mi (a s( -y ]} a y(as a ay( my u y = ou _ e [ I] m( Y i e a ou _ e
E O OG AM 4 74 h f ll n pr r ll b d t p t th v r n rn l f th
G n pr nd r p f p nt h r l v nt th r n b f nd n Ch pt r 5
t th t th thr t p tr n t n n fr GRASSERS to SRS, th n fr th
h ft f n t n t th OC th n b th th v l t n p nt f r th rn l t n t b th
In th r rd KRSRS(p, q) t n t b KR,sRS (P, p). S t n 5 n
Ch pt r 5 f r r nf r t n # END OF PROGRAM 5 #
h f ll n f n t n ll p t th p nt nf d n nt rv l f r th
p r l OC rv b d n th v r n rn l xpl n d b v h n th
v r pr b b l t nl th v l f th nt z f n t n fr th t nd rd n r l d tr b t n n d t b h n d h r lt j t l th r lt d n th b v pr r r v n n t n 5 f Ch pt r 5 6 REFERENCES
A t n t r n pl S nd Arndt S ( Ar b b l t Ind x n Int t v np r tr Appr h t M r n th S z f r t nt Eff t S a is ics i Me ici e 25 59A-
A r t A ( Ca ego ica a a A a ysis W l S r n Ar b b l t nd St t t
Al Μ Μ l M nd W J. ( 5 Inf r n n AYE n G n r l z d Un f r tr b t n Ca cu a S a is ica Associa io u e i 57, 35-
Al Μ M W nd l Μ ( Inf r n n l b l t (Y n - p r t r Exp n nt l tr b t n I e a io a ou a o S a is ica Scie ces 3, AA9-A 5
Ap tl Τ Μ (A99 Ma ema ica A a ysis Seco E i io Add n-W l
Azz l n A (A9 A A t n th E t t n f tr b t n n t n nd Q nt l b C rn l M th d iome ics 68 A 3 -3
r nn dbr nd Sp n M W ( ff r nt O t f th W l x n-M nn-Wh tn t fr ff r nt St t t e Ame ica S a is icia 54 A 7 -77
rnh rd G All n M rb l Μ nd M r W (A9 Inv t t n n th l b l t f S El nt r np r tr M th d n St t t l An l S t S a is ica So wa e e s e e 4, A9-
rnb W M C rt C (A95 A tr b t n- r pp r C nf d n nd f r Ar Y } d n Ind p nd nt S pl f nd Υ e A a s o Ma ema ica S a is ics 29 55 -5
n A/W nd Azz l n A (A997 A ie Smoo i g ec iques o a a A a ysis e Ke e A oac i S- us I us a io s Oxf rd St t t l S n S r Oxf rd n v r t Ar
r b C p MKS nd Al nz Τ A ( U n th OC C rv f r G n r t nt Eff t n Cl n l r l S a is ics i Me ici e 25 575-59
Ch n Μ nd KK n f rd ( A n np r tr pr d r t d th l n ll n n f l ff r S a is ics 1, 93- 9
Ch n S (A99 E p r l l h d C nf d n IInt rv l f r np r tr n t E t t n iome ics 83 3 9-3 A
77 8 Cr r E nd C p U (A997 A t n UM UE f r( Y n th Exp n nt l
C Commu ica io s i S a is ics — eo y a Me o s 26, A 5A - A 55
C A (A97 E p r l Ar b b l t Al t nd St t t l Inf r n f r nl n r M d l n th -S pl C e A a s o S a is ics 2, 7- 77
dd E nd p M S ( 3 S p r tń r n f r th Ar Und r th v r Op r t n Ch r t r t C rv ou a o e Ame ica S a is ica
Associa io 8, 9 - A7
Efr n nd b h r n (A993 A I o uc io o e oo s a Ch p n ll/C C
Er nl A M n S C t ll E nd Adr n A ( n S p r tr OC An l S a is ics i Me ici e 2 , 39 5-39
rr nd r ( E t t n f th Ar Und r th OC C rv S a is ics i Me ici e 2 3 93-31
Gh t K nd rd C ( 7 E p r l Ar Appr h t S S pl Ar bl d n n d S t S pl A I se S a Ma ( n pr
ll G nd nd n ( 3 I pr v d M th d f r nd dth S l t n Wh n E t t n OC C rv S a is ics & o a i i y e e s 64 A 1-A 9
ll M h r SA nd r n Y C (A995 On nd dth Ch f r n t E t t n W th p nd nt t e A a s o S a is ics 2 A- 3
nl A nd M l (A9 h M n n nd U f th Ar Und r v r Op r t n Ch r t r t ( OC C rv a io ogy 4 , 9-3
ndl W M ll r M S p rr h S nd W r tz A ( S p r tr nd S p r tń M d l S i ge Se ies i S a is ics
b r S nd YY h (A99 o us a a A a ysis I S a is ica Me o o ogy i e a maceu ica Scie ces d MA rr r M r l- r 357-3
h nd G ntl n (A99 A n f r t An l nd Gr ph
ou a o Com u a io a a G a ica S a is ics 3 99 - 3A
Iv h n (A99 Unb d E t t r f Ar (Χ Y nd h r r n n th C f n f r nd -p r t r Exp n nt l tr b t n ou a o
Ma ema ica Scie ce 8 , 79 - 793
n n A (A9 v d W r n t E t t n f r St t t nd t An l b W S lv r n ou a o e Ame ica S a is ica Associa io 8 A 9- 7 79
n n A (A99A nt v l p nt n np r nl tr n t E t t n ou a o e Ame ica S a is ica Associa io 86 A3 5-
K r A Ρ nd Er ϋn AC ( 5 E p r l b np r nl tr K rn l n t E t t n S a is ics & o a i i y e e s 75 3A5-3
n A nd W ll (A99 Ar d A C -St d n Env r n nt l Ep d l ou a o e oya S a is ica Socie y Se ies A 157 5- 9
M n r A ( 1 h Av r Ar Und r C rr l t d v r Op r t n Ch r t r t C rv Α np r tń Appr h d n G n r l z d -S pl W l x n St t t A ie S a is ics 50 3 337-3
h n E (A9 3 b t E t t n n An l f r n e A a s o Ma ema ica S a is ics 34 3 957-9
h nn E (A99 o a a e ics: S a is ica Me o s ase o a ks a e ise s e Upp r S ddl v r Ar nt ll
l d C ( E t t n f C nv x OC rv S a is ics & o a i i y e e s 59 A 99-AAA
l d C nd h Y (A999 K rn l E t t r f th OC C rv r tt r th n E p r l S a is ics & o a i i y e e s 44 3 A-
l d C (A99 U n S th d v r Op r t n Ch r t r t C rv t S r z nd C p r n t S t ou a o e Ame ica S a is ica Associa io 93 A35 -A3
dbr (A995 M r p t r St t t f r d l S nt t C i ica a E e ime a a maco ogy a ysio ogy 22, 97 -9
M nn nd Wh tn (A9 7 On t f Wh th r n f nd r bl St h t ll r r th n th Oth r e A a s o Ma ema ica S a is ics 18 A 5 -
M ll r K S (A9 A On th Inv r f th S f M tr ( n t M th t M z n 54 7-7
M C l I (A99A Inf r n n A Y n th W b ll C Commu ica io s i S a is ics — Simu a io a Com u a io 20 A A 9-A
b G ( C nf d n Int rv l f r n Eff t S z M r d n th M nn-Wh tn St t t rt A G n r l I nd l- r -b d M th d S a is ics i Me ici e 25 5 3-557 80 b G ( b C nf d n Int rv l f r n Eff t S z M r d n th M nn-Wh tn St t t rt A pt t I th d nd Ev l t n Statistics in Medicine 25 559-573
O n B., r ll K nd n n (A9 np r tr pp r C nf d n nd f r Ar Y } nd C nf d n t f r Ar Y } Wh n Χ nd Υ r r l Journal of tve American Statistical Association 59 3 7 9 -9
l M Al I I nd W ( 5 E t t n nd t n f (Y n -p r t r Exp n nt l tr b t n Statistics 39 5 A5-
p MKS ( 3 Tve Statistical Evaluation of Medical Tests for Classification and Prediction. Oxf rd Oxf rd Un v r t Ar
p MKS (A997 A r n M d l n r r f r v r Op r t n Ch r t r t C rv n M d l n t t n Biometrics 84 3 595-
tth ff (A9 3 U f th W l x n St t t f r G n r l z d hr n - h r r bl Tve Annals of Matvematical Statistics 34 A59 -A599
r M nd b n (A97 A Ch r t r z t n d n th Ab l t ff r n f d nd r bl Annals of Matvematical Statistics 41, AA3- A
nbl tt M (A95 r n S np r tr E t t f n t n t n Annals of Matvematical Statistics 27, 3 - 37
S th Y S nd Sh h S (A9 A E t t n ( Y f r th Exp n nt l tr b t n Commune. Start. Tveory Metv. A10 A 39- 7
S G (A9 A np r tr C p r n f S pl On f Wh h C n r d Biometrics 53 3- 599-
S tt W (A99 M lt v r t n t E t t n h r r t nd l z t n Wile y-Interscience.
S n E nd ll A (A99 An Ev l t n f th A r f K rn l n t E t t r f r n An l Ecology 77 7 75- 5
S h lt A nd Q nb rr C (A97A On Unb d E t t n f n t n t n Tve Annals of Matvematical Statistics 42 A 3 -A 3
S lv r n W (A9 Density Estimation for Statistics and Data Analysis. nd n Ch p n ll
S n ff S hb r Υ nd r (A9 Alt rn t v E t t n Ar d r f r r( Y n C t r z d t Biometrics 42 95-9 7 81 l r C C (A9 9 t tr p Ch f th S th n r t r n K rn l n t E t t n Biometrics 6 7 5-7A
rr ll G nd S tt W (A99 r bl K rn l n t E t t n Annals of Statistics 20 3 A 3 -A 5
tt ńn t n M nd M ll G M (A9 3 K rn l- d n t E t t fr In pl t t Journal of tve Royal Statistical Society. Series 45 5 -
n ll G ( Statistical Rules of Tvumb. W l S r n Ar b b l t nd St t t
n tr n E S nd C (A999 Ar p rt f np r tr t f r E rl C p r n f r t nt n Cl n l r l n th Ar n f S rr t Endp nt Biometrics 55 AA7A-AA7
W lt r S ( Ar p rt f th S r v r Op r t n Ch r t r t ( OC C rv f r n t t t Statistics in Medicine 19 1 37-A 5
W l x n (A9 5 Ind v d l C p r n b n n M th d Biometrics Bulletin 1 - 3
W lf A nd (A97A On C n tr t n St t t nd p rt n t Tve American Statistician 2 7-3
h nd r z ( C p r n f nd dth S l t n M th d f r K rn l S th n f OC C rv Statistics in Medicine 21 A 5- 55