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Njit-Etd2007-041 Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment of the order would involve violation of copyright law. Please Note: The author retains the copyright while the New Jersey Institute of Technology reserves the right to distribute this thesis or dissertation Printing note: If you do not wish to print this page, then select “Pages from: first page # to: last page #” on the print dialog screen The Van Houten library has removed some of the personal information and all signatures from the approval page and biographical sketches of theses and dissertations in order to protect the identity of NJIT graduates and faculty. ABSTRACT 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 vrbl Mr pfll f Χ nd Υ dnt th rpn f ptnt n th t trtnt A nd rptvl th ntt (Y (hh n b lld th prbblt ndx fr th Efft Sz f ntrt n lnl ttt h bjtv f th td t drv n ff r tht ld pr t trtnt r nfrtvl nd bjtvl prd t th rlr pprh Krnl dnt ttn fl nn-prtr thd tht h nt bn ll tlzd n ppld tttl tl nl d t t pttnl plxt h rrnt td h tht th thd rbt vn ndr rrltn trtr tht r drn th pttn f ll pbl dffrn h rnl thd n b ppld t th ttn f th OC (vr Oprtn Chrtrt rv ll t th plnttn f nn- prtrz rrn f OC h r ndr th OC rv (AC hh xtl l t th ntt (Y l xplrd n th drttn h thdl d fr th td t nrlz t thr r f ppltn OEMS EAE ΤΟ EICACY 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 blvd prnt t lvl f dr tr nd dr prnt n l ν ACKNOWLEDGMENT I ld l t xpr nr rttd t r Snl Chr h ntrdd t th prbl h l vn ntl d nd tn thn ld hv bn pbl tht h ntnt nrnt nd pprt It ld hv bn pbl t hv dn nthn tht r Mnnh htthrj n hlp l hd h dr pn I ld l nt h r th n prbl nd t th ltn r trn hnt tht ld ld t th ltn r rd Knfrd drv pl thn h n pprh trd lvn th prbl hn n n f h ppr h l vn nbr f pntr nd d fr rrh nd h flld th r vr ll Whtvr 1 hv nd t lrn bt th fl f rrh dnt tl fr h r h Spnr h dntd vrl fl b t n tht nd ttt fr h prnl lltn M nr thn t h I ld l l t thn r Snd Snhr fr h thnl dn fr t t t nr t hl nd I vr frtnt t hv h n tt r Kh Ght l nr n hl nd n prfr h hlpd nl th h t bth th thnl nd nn-thnl M f hrr bn tttn hrlf hlpd th hr npt fr t t t Alt vrthn tht I hv nd t lrn bt thnl prnttn d t hr I l thnfl t th prtnt f Mthtl Sn nd th nvrt fr pprtn r nd fr th ptn nd thr flt vlbl t vi AE O COES Chptr Page 1 IOCIO 1 11 Objtv 1 1 rnd Infrtn 2 O-AAMEIC ESS 7 1 fnn th rbl 7 nt Ettn Mthd 15 3 Srvvl Anl 1 vr Oprtn Chrtrt Crv 1 3 AAMEIC ESS 31 Expnntl UMUE fr ES 20 3 Mnt Crl Sltn f th Ettr .. 26 KEE ESI ESIMAIO 7 1 Apprh 7 Ettn fr pndnt t 3 3 Sltn 3 5 ECEIE OEAIG CAACEISIC CE 51 51 n OC 51 5 OC fr nt .. 51 53 Ettn th rvtv f OC nd th AC n th nt 57 5 OC Mr f Eff 59 55 rl lt .. 3 vii TABLE OF CONTENTS (Continued) Chapter Page COCSIOS A E SIES .. 66 AEI SOCE COES 68 EEECES 77 viii LIST OF TABLES Table Page 1 trbtn f ld rr Chn r ln fr Anthprtnv r 9 2.2 Sr Sttt 1 3 Cvn l fr ffrnt t f Mn f Spl 1 31 lt fr Mnt Crl Sltn 5 1 Ch f Krnl 31 4.2 Effn f Krnl Cprd t Epnhnv Krnl 3 3- Cprn f nt Ett fr ffrnt Krnl n th S nddth f 15 11 nd 115 35 4.4 Cprn f Ettd Mn Intrtd Srd Errr f nt Cptd fr Indpndnt nd pndnt t 44 5 Ettd rbblt Gvn Aln th th Mn Srd Errr 48 4.6 Ettd rbblt Gvn Aln th 95% ttrp Cnfdn Intrvl nd Eprl rbblt 9 51 Efft Sz Mr fr ffrnt Mthd f Ettn 64 5 Cnfdn Intrvl f OC Crv 5 53 Ettd Optl hrhld l 66 ix LIST OF FIGURES Figure Page 1 OC rv 1 1 Unvrt Krnl dnt tt f th fft f t blndd bld prr lrn dr vn ln th th rnl t th vltn pnt n bnddth 9 4.2 Unvrt Krnl dnt tt f th fft f t blndd bld prr lrn dr vn ln th th rnl t th vltn pnt n bnddth 9 3 nvrt Krnl dnt tt f th fft f t blndd bld prr lrn dr vn ln th th Epnhnv rnl t th vltn pnt n bnddth = nd 5 3 4.4 nvrt Krnl dnt tt f th fft f t blndd bld prr lrn dr vn ln th th Epnhnv rnl t th vltn pnt n bnddth = 9 nd 1 33 5 Unvrt Krnl dnt tt f th fft f t blndd bld prr lrn dr vn ln th th Epnhnv rnl t th vltn pnt n bnddth = 15 nd 3 nvrt Krnl dnt tt f th fft f t blndd bld prr lrn dr n vrl Krnl nd th bnddth 3 7 Mn ntrtd rd rrr (MISS prn f th dnt tt ptd fr n ndpndnt pl nd dpndnt pl 5 4.8 nt tt prd nt th tr dnt 46 9 nt tt fr vr lr nbr ( f ndpndnt brvtn 7 51 ndn th ptl pnt f drntn n th OC rv 5 5 Ar ndr th OC rv 57 53 h ptl thrhld vl .. 59 5 Eprl OC rv nd 95% nfdn bnd 3 χ CHAPTER 1 INTRODUCTION 1.1 Objective h bjtv f th drttn t td r f efficacy. Strtn th th pf prbl f prn t nt-hprtnv dr n dbl-blnd lnl trl bootstrap kernel density estimate f th dffrn f th dr prpd h bootstrap confidence intervals f th prpd dnt r l ptd h td dntrt tht th thd frl rbt vn ndr dependent trtr tht r drn th pttn f ll pbl dffrn n hh f th nfrn bd A rv f th xtn thd f ff rnt h bn nldd hhlhtn th lnt pnt nd lttn f h thd h thd r l tlzd t td rvvl nl r pfll th tt f th rvvl fntn nd th hzrd fntn A dnt bd lrth t tt th drvtv f th rvr prtn hrtrt rv ntrdd n th td An ppltn f th thd n drnnt nl xplrd hr h drttn l tlz lrth h kernel density ttn nearest neighbor ttn nd ROC regression nd h tht th n b fftvl dptd ppld tttl tl n th r f phrtl ttt In th ntxt f kernel density ttn bandwidth ltn thd pl rl rl trtr rv f h thd r nldd n rnbl dtl All th bnddth ltn thd fll ndr nrl l f prbl tht n b bd ndr tht ptztn Stht ptztn rlt nd prbl r th b f ll tttl thr 1 2 Expl f th ptztn nld maximum likelihood ttn likelihood ratio tt Neyman-Pearson Lemma nd ptztn f th bias nd th variance f n ttr .2 rnd Infrtn Cprn t trtnt th rpt t prr ff vrbl prbl hh nl nntrd n th lnl td Svrl prtr nd rprtrzd thd r d t fnd ltn t th prbl rtr pprh r ftn bd n nrlt ptn rprtrzd pprh r prrl rn bd tt l th Wilcoxon-Mann-Whitney(WMW) tt Efft z (ES prnt th ntd f th dffrn btn t trtnt ndr ndrtn In n f th rnt pprh ES prntd thtl xprn nt l ndrtd b lnn An dl ES r ht t pt nd ppl t bth lnn nd tttn An xpl f h r vn b th prbblt ndx P (Y X) , hr nd dnt th prfrn r f t ptn dr (n f hh b plb r t d tht th lrr th prfrn vl th r ff th dr On f th frt td tht rvd th ttn f th ntt P (Y X) is Wlf nd (1971 h hv ndrd th ttn ndr th nrlt ptn ll tht th ptn Cnfdn bnd fr P (Y X) nd rltd ntt l xplrd th vrl xpl Snff t l (19 hv xplrd th ntt P (Y X) nd rprtrzd vrn P (Y — P (X Y n trl dt n fr t ppltn In th ppr th hv ntrdd hbrd ttr th prtr ll rprtrzd 3 prprt h ttr ntll Wlxn-Mnn-Whtn (WMW ttt xpt th prprtn r lltd n nrlt ptn h hv l ntrdd n th ppr th ttr f P(Y>X) — (Y bd n nn-prtń dnt ttn prdr h hv l hn bth th prl td nd thrtl jtftn b vrn ttn nd ptt rlt tht th hbrd ttr pseudo-MLE th ll t v rt rlt ndr n rtn t tht th thd ttr bd n nn-prtń dnt ttn thd h bn d n trl dt hr td v vdn tht th ttr pttll vlnt t th MW ttt nd b rrtn thd nd bttrp trtnt t vn r ttrtv (b rdd th ttr Mthd bd n bttrp nfdn ntrvl f th ntt hv bn dd b Chn nd Knfrd ( It h bn rrtl pntd t b th tht th rjtn f th Wilcoxon-Mann-Whitney tt ld pl tht th t dtrbtn ndrln th brvtn r nt th hrfr n n ff rnt nl t n b rnbl nfrrd tht n trtnt bttr thn th thr vr th d nt prvd ntfbl r f ff Al t ntnd n th ppr nd lltrtd th n xpl tht t nt rnbl t
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