Rebekkah S. Dann. Chapel Hill Approved by: Gary Koch. Amy Herring. Lisa LaVange. Jeanenne Nelson. John Preisser

Transcription

1 MEHODS FOR SRENGHENING HE DESIGN AND ANALYSIS OF LINIAL RIALS O SHOW NON-INFERIORIY OF A NEW REAMEN O A REFERENE REAMEN FOR A BINARY RESPONSE VARIABLE Rebekka S. Da A dissertatio submitted to te faculty of te Uiversity of Nort arolia at ael Hill i artial fulfillmet of te requiremets for te degree of Doctorate of Public Healt i te Scool of Public Healt (Biostatistics ael Hill 006 Aroved by: Gary Koc Amy Herrig Lisa LaVage Jeaee Nelso Jo Preisser

2 ABSRA Rebekka S. Da: Metods for Stregteig te Desig ad Aalysis of liical rials to Sow No-iferiority of a New reatmet to a Referece reatmet for a Biary Resose Variable (Uder te directio of Gary G. Koc No-iferiority cliical trials are icreasigly becomig more romiet i researc ad develomet of ew aramaceuticals. e objective of suc trials is to sow tat te amout by wic a ew treatmet is worse ta a active cotrol is below a secified amout. Metodology secifically for te desig ad aalysis of tese trials is essetial for te assurace of quality trials tat are statistically defesible i te scietific commuity as well as i a regulatory settig, were traditioally focus as bee o sueriority. Stadard metodology must be reviewed ad assessed as to its aroriateess for addressig te o-iferiority yotesis. ategorical data aalysis for a dicotomous rimary edoit may iclude aalysis of a risk ratio or a risk differece wic comares te test ad active cotrol treatmets. e effect of samle size allocatio ad oter arameters of iterest o te erformace of tese metods will be assessed. I additio, aroriate samle size formulas will be develoed ad evaluated to aid i trial laig. I some o-iferiority trials, it is ossible to iclude a lacebo arm as well as a active cotrol arm wic allows o-iferiority to be assessed relative to te ercetage of te differece betwee te cotrol ad lacebo arms tat te test treatmet reserves over lacebo. Metodology for tis assessmet is also of iterest alog wit aroriate samle ii

3 size calculatios. is settig also resets a area of researc for discussio of te oe versus two trials aradigm. Extesios to te metodology for te risk ratio ad risk differece are assessed we stratificatio is ecessary, secifically for large subgrous suc as geder. Metods for stratificatio are a imortat comoet, ad additioally te effects of stratificatio i a o-iferiority settig eed evaluatio. Review, develomet, ad assessmet of tis metodology for categorical data as secifically focused o te o-iferiority settig is a imortat additio to te curret statistical ractice. is researc is a coesive resetatio for eac of te measures of iterest troug assessmet of metodology ad its relatio to aroriate desig comoets suc as samle size calculatio. e imortace of elig statisticias uderstad ad imlemet metods i tese areas is of most cocer. iii

4 AKNOWLEDGEMENS is dissertatio is te roduct of may years of work, surred o by tose I ave bee rivileged to kow. My family as bee a costat source of ecouragemet to ursue educatio ad fiis tis edeavor. My frieds are so imortat to me ad ave walked alogside of me troug te igs ad lows of te ast few years. Lastly, my academic metors ave bee istrumetal i guidig me troug from ig scool to comletig graduate scool, wit muc suort ad affirmatio. Dr. Koc as rovided assistace i teacig me te ractice of biostatistics as well as may oortuities to lauc my career. ak you to all wo ave bee a art of tis rocess. iv

5 ABLE OF ONENS LIS OF ABLES...vii LIS OF FIGURES viii INRODUION. ater I. Review ad Evaluatio of Metods for omutig ofidece Itervals for te Ratio of wo Proortios ad osideratios for No-iferiority liical rials...5 Itroductio..5 Metods 6 Review of Literature.. ofidece Limit omarisos.4 Simulatios 5 Samle Size alculatios...8 oclusios Refereces..3 II. Review of Metods for Oe-sided estig of te Differece betwee Proortios ad Samle Size osideratios 35 Itroductio 35 Review of Metods i te Literature.36 Simulatios for No-iferiority.48 Samle Size osideratios..5 v

6 Discussio 54 Refereces 55 III. Metods for Aalyzig ree-arm rials wit Biomial Proortios as te Primary Edoit...77 Itroductio..77 Metods for Assessig No-iferiority i a ree-arm rial..78 Performace of Metods based o Simulatios for Assessig No-iferiority 8 Results of Simulatios.84 Samle Size Formulas..86 Assessig No-iferiority i a ree-arm rial: vs rials Paradigm 89 Performace of Metods based o Simulatios for Assessig Dual Edoits of Sueriority ad No-iferiority.9 Refereces...95 IV. Metods for Aalyzig Stratified No-iferiority rials wit Biomial Proortios as te Primary Edoit wit iterest i te Risk Ratio or te Risk Differece 9 Itroductio 9 Assessig No-iferiority of a Risk Ratio i a Stratified rial.30 Assessig No-iferiority of a Risk Differece i a Stratified rial.4 Imlicatios for Overall Sigificace wit oditios o Idividual Strata.54 Discussio..58 Refereces..59 V. Discussio ad Future Researc 95 vi

7 LIS OF ABLES able. Software Resources for Metods. 4. Summary of Selected Uer ofidece Limits, aylor Series Exasio Metods..5.3 Summary of Selected Uer ofidece Limits, Quadratic Equatio Metods.6.4 Summary of Selected Oe-Sided P-values, Maximum Likeliood Metods..7. Summary of Selected Lower ofidece Limits at Oe-sided Summary of Selected Oe-sided P-values Summary of Simulated & alculated Power, Samle Size Allocatios :, : Summary of Simulated & alculated Power, Samle Size Allocatios 3:, :, 3: Summary of Simulatio ad Samle Size alculatio Results Summary of Scearios wic maitai Project Level α= from Maca et. al Samle Sizes for ree-arm rial Sceario at 0.85 Power for No-iferiority ad Sueriority Hyotesis, Samle Size Allocatio ::..98 vii

8 LIS OF FIGURES Figure. Summary of Simulated ye I Error, aylor Series Exasio Metods...8. Summary of Simulated ye I Error, Quadratic Equatio Metods Summary of Simulated ye I Error, Maximum Likeliood Metods Summary of Simulated ye I Error, Overview of Better Metods 9.5 omariso of Simulated Power, Deviace ad aylor Series Metods 30.6 omariso of Simulated Power, Deviace ad Adated Agresti Metods omariso of Simulated Power, Deviace ad Bailey Metods omariso of Simulated ad alculated Power, aylor Series Metod omariso of Simulated ad alculated Power, aylor Series Adjusted Ala Metod.3.0 omariso of Simulated ad alculated Power, Farrigto-Maig Metod...3. omariso of Simulated ad alculated Power, Farrigto-Maig Metod omariso of Simulated ad alculated Power, Farrigto-Maig 3 Metod omariso of Simulated ad alculated Power, Deviace Simulated ad aylor Series alculated Metod omariso of Simulated ad alculated Power, Deviace Simulated ad F-M alculated Metod..34. Summary of Simulated ye I Error, By Samle Size Allocatio..66. Summary of Simulated ye I Error, By Samle Size Allocatio Summary of Simulated ye I Error, By Samle Size Allocatio Summary of Simulated ye I Error, By Samle Size Allocatio..67 viii

9 .5 Summary of Simulated ye I Error, By No-iferiority Margi Summary of Simulated ye I Error, By Pi.68.7 Summary of Simulated Power, By Samle Size Allocatio, Ala=0.05, Delta=-No-if Margi/ Summary of Simulated Power, By Samle Size Allocatio, Ala=0.05, Delta= Summary of Simulated Power, By Samle Size Allocatio, Ala=0.05, Delta= Summary of Simulated Power, By Samle Size Allocatio, Ala=0.05, Delta= omariso of Simulated Power, Newcombe Hybrid Score ad Pearso Metods..7. omariso of Simulated Power, Farrigto-Maig 3 ad Pearso Metods..7.3 omariso of Simulated Power, Farrigto-Maig 3 ad Newcombe Hybrid Score Metods omariso of Simulated Power, Agresti & affo ad Deviace Metods.7.5 omariso of Simulated Power, Wald ad Deviace Metods omariso of Simulated Power, Wald ad Agresti & affo Metods 73.7 omariso of Simulated ad alculated Power, F-M 3 Metod.74.8 omariso of Simulated ad alculated Power, Newcombe Hybrid Score Simulated Metod & F-M 3 alculated Metod omariso of Simulated ad alculated Power, Pearso Simulated & F-M 3 alculated metod.75.0 omariso of Simulated ad alculated Power, Wald Metod..75. omariso of Simulated ad alculated Power, Deviace Simulated Metod & Wald alculated Metod omariso of Simulated ad alculated Power, Agresti & affo Simulated Metod ad Wald alculated Metod...76 ix

10 3. Summary of Simulated ye I Error, By Samle Size Allocatio Summary of Simulated ye I Error, By No-iferiority Margi Summary of Simulated ye I Error, By Pi Summary of Simulated ye I Error, By Pi P Summary of Simulated ye I Error, By (Pi Pi P Summary of Simulated ye I Error, By otal Samle Size omariso of Simulated Power, RMLE & Pearso Simulated Power omariso of Simulated Power, RMLE & Deviace Simulated Power omariso of Simulated Power, RMLE & Wald Simulated Power omariso of Simulated Power, RMLE & Agresti & affo Simulated Power omariso of Simulated Power, RMLE & WLS Simulated Power omariso of Simulated Power RMLE Metod, By otal Samle Size & Samle Size Allocatio omariso of Simulated ad alculated Power, RMLE Metod, By Samle Size Allocatio omariso of Simulated ad alculated Power, RMLE Metod, By No-iferiority Margi omariso of Simulated ad alculated Power, RMLE Metod, By Pi omariso of Simulated ad alculated Power, RMLE Metod, By Pi P omariso of Simulated ad alculated Power, RMLE Metod, By (Pi Pi P omariso of Simulated ad alculated Power, RMLE Metod, By otal Samle Size omariso of Simulated ad alculated Power, Wald Metod, By Samle Size Allocatio, Ala=0.05, Lambda=..09 x

11 3.0 omariso of Simulated ad alculated Power, Wald Metod, By Samle Size Allocatio, Ala=0.05, Lambda=, Pi <=0.7, Pi P>=0.4, Allocatio=::, 3:: omariso of Simulated ad alculated Power, WLS Metod, By Samle Size Allocatio, Ala=0.05, Lambda=.0 3. omariso of Simulated ad alculated Power, WLS Metod, By Samle Size Allocatio, Ala=0.05, Lambda=, Pi <=0.7, Pi P>=0.4, Allocatio=::, 3:3:, ::, 3:: omariso of Simulated ad alculated Power, Agresti & affo Simulated Metod & RMLE alculated Metod, By Samle Size Allocatio, Ala=0.05, Lambda= omariso of Simulated ad alculated Power, Agresti & affo Simulated Metod & RMLE alculated Metod, By Samle Size Allocatio, Ala=0.05, Lambda=, Pi <=0.7, Pi P>= omariso of Simulated ad alculated Power, Deviace Simulated & RMLE alculated Metod, By Samle Size Allocatio, Ala=0.05, Lambda=. 3.6 Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Wald Metod, By Samle Size Allocatio Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Agresti & affo Metod, By Samle Size Allocatio Summary of Simulated ye I Error, No-iferiority i wo Searate rials, RMLE Metod, By Samle Size Allocatio Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Wald Metod, By No-iferiority Margi Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Agresti & affo Metod, By No-iferiority Margi Summary of Simulated ye I Error, No-iferiority i wo Searate rials, RMLE Metod, By No-iferiority Margi Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Wald Metod, By Pi Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Agresti & affo Metod, By Pi 6 xi

12 3.34 Summary of Simulated ye I Error, No-iferiority i wo Searate rials, RMLE Metod, By Pi Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Wald Metod, By Pi P Summary of Simulated ye I Error, No-iferiority i wo Searate rials, Agresti & affo Metod, By Pi P Summary of Simulated ye I Error, No-iferiority i wo Searate rials, RMLE Metod, By Pi P Summary of Simulated Power, No-iferiority i wo Searate rials, Wald Metod, By Samle Size Allocatio Summary of Simulated Power, No-iferiority i wo Searate rials, Agresti & affo Metod, By Samle Size Allocatio Summary of Simulated Power, No-iferiority i wo Searate rials, RMLE Metod, By Samle Size Allocatio Summary of Simulated Power, No-iferiority i wo Searate rials, Wald Metod, By No-iferiority Margi Summary of Simulated Power, No-iferiority i wo Searate rials, Agresti & affo Metod, By No-iferiority Margi Summary of Simulated Power, No-iferiority i wo Searate rials, RMLE Metod, By No-iferiority Margi Summary of Simulated Power, No-iferiority i wo Searate rials, Wald Metod, By Pi Summary of Simulated Power, No-iferiority i wo Searate rials, Agresti & affo Metod, By Pi 3.46 Summary of Simulated Power, No-iferiority i wo Searate rials, RMLE Metod, By Pi Summary of Simulated Power, No-iferiority i wo Searate rials, Wald Metod, By Pi P Summary of Simulated Power, No-iferiority i wo Searate rials, Agresti & affo Metod, By Pi P.3 xii

13 3.49 Summary of Simulated Power, No-iferiority i wo Searate rials, RMLE Metod, By Pi P Summary of ye I Error, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By Samle Size Allocatio Summary of ye I Error, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By No-iferiority Margi Summary of ye I Error, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By Pi Summary of ye I Error, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By Pi P Summary of Simulated Power, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By Samle Size Allocatio Summary of Simulated Power, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By No-iferiority Margi Summary of Simulated Power, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By Pi Summary of Simulated Power, No-iferiority: ombie Data usig RMLE Metod, Sueriority: wo Searate rials usig F-M 3 Metod, By Pi P.8 4. Summary of Simulated ye I Error for Stratified Risk Ratio, By reatmet Allocatio Summary of Simulated ye I Error for Stratified Risk Ratio, By Null Hyotesis Risk Ratio Summary of Simulated ye I Error for Stratified Risk Ratio, By Pi Summary of Simulated ye I Error for Stratified Risk Ratio, By Pi Summary of Simulated ye I Error for Stratified Risk Ratio, By (Pi Pi...63 xiii

14 4.6 Summary of Simulated ye I Error for Stratified Risk Ratio, By otal Samle Size omariso of Simulated Power for Stratified Risk Ratio, Gart-S & Deviace Simulated Power, By reatmet Allocatio omariso of Simulated Power for Stratified Risk Ratio, Gart-S & YH Simulated Power, By reatmet Allocatio omariso of Simulated Power for Stratified Risk Ratio, Gart-S & Gart Simulated Power, By reatmet Allocatio omariso of Simulated Power for Stratified Risk Ratio, ML Logit & MH Simulated Power, By Pi omariso of Simulated Power for Stratified Risk Ratio, Logit & Agresti Simulated Power, By Pi omariso of Simulated Power for Stratified Risk Ratio, MH & Logit Simulated Power, By Pi omariso of Simulated Power for Stratified Risk Ratio, MH & Wald Simulated Poewr, By Pi omariso of Simulated Power for Stratified Risk Ratio, Gart-S & Wald Simulated Power, By Pi omariso of Simulated ye I Error for Stratified Risk Ratio, By Strata Allocatio, Ala=0.05, reatmet Allocatio=: omariso of Simulated ye I Error for Stratified Risk Ratio, By Strata Allocatio, Ala=0.05, reatmet Allocatio=: omariso of Simulated ye I Error for Stratified Risk Ratio, By Strata Allocatio, Ala=0.05, reatmet Allocatio=: Summary of Simulated Power for Stratified Risk Ratio, By Strata Allocatio, Ala=0.05, reatmet Allocatio=: Summary of Simulated Power for Stratified Risk Ratio, By Strata Allocatio, Ala=0.05, reatmet Allocatio=: Summary of Simulated Power for Stratified Risk Ratio, By Strata Allocatio, Ala=0.05, reatmet Allocatio=:..70 xiv

15 4. omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart-S Simulated & Nam alculated Power, By Strata Allocatio, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart-S Simulated & Nam alculated Power, By Strata Allocatio, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart-S Simulated & Nam alculated Power, By Strata Allocatio, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart Simulated & Nam alculated Power, By Strata Allocatio, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart Simulated & Nam alculated Power, By Strata Allocatio, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart Simulated & Nam alculated Power, By Strata Allocatio, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart-S Simulated & S alculated Power, By (Pi - Pi, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart-S Simulated & S alculated Power, By (Pi - Pi, Ala=0.05, eta=, reatmet Allocatio=: omariso of Simulated & alculated Power for Stratified Risk Ratio, Gart-S Simulated & S alculated Power, By (Pi - Pi, Ala=0.05, eta=, reatmet Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece, By reatmet Allocatio, Ala=0.05, Strata Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece, By Null Hyotesis Risk Differece, Ala=0.05, Strata Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece, By Pi, Ala=0.05, Strata Allocatio=: 76 xv

16 4.33 Summary of Simulated ye I Error for Stratified Risk Differece, By Pi, Ala=0.05, Strata Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece, By (Pi Pi, Ala=0.05, Strata Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece, By otal Samle Size, Ala=0.05, Strata Allocatio=: omariso of Simulated Power for Stratified Risk Differece, Gart & Nam ad Gart & Nam-S Simulated Power, By reatmet Allocatio, Ala=0.05, Strata Allocatio=:, Delta= omariso of Simulated Power for Stratified Risk Differece, Gart & Nam ad YH Simulated Power, By Pi, Ala=0.05, Strata Allocatio=:, Delta= omariso of Simulated Power for Stratified Risk Differece, Gart & Nam ad Deviace Simulated Power, By reatmet Allocatio, Ala=0.05, Strata Allocatio=:, Delta= omariso of Simulated Power for Stratified Risk Differece, Gart & Nam-S ad Deviace Simulated Power, By reatmet Allocatio, Ala=0.05, Strata Allocatio=:, Delta= omariso of Simulated Power for Stratified Risk Differece, Gart & Nam ad MH Simulated Power, By Pi, Ala=0.05, Strata Allocatio=:, Delta=0, reatmet Allocatio=:, : omariso of Simulated Power for Stratified Risk Differece, Gart & Nam ad WLS Simulated Power, By Pi, Ala=0.05, Strata Allocatio=:, Delta=0, reatmet Allocatio=: omariso of Simulated Power for Stratified Risk Differece, Gart & Nam ad Wald Simulated Power, By Pi, Ala=0.05, Strata Allocatio=:, Delta=0, reatmet Allocatoi=: omariso of Simulated Power for Stratified Risk Differece, WLS ad Wald Simulated Power, By Pi, Ala=0.05, Strata Allocatio=:, Delta=0, reatmet Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece, By Strata Allocatio, Ala=0.05, reatmet Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece, By Strata Allocatio, Ala=0.05, reatmet Allocatio=:..83 xvi

17 4.46 Summary of Simulated ye I Error for Stratified Risk Differece, By Strata Allocatio, Ala=0.05, reatmet Allocatio=: Summary of Simulated Power for Stratified Risk Differece, By Strata Allocatio, Ala=0.05, reatmet Allocatio=:, Delta= Summary of Simulated Power for Stratified Risk Differece, By Strata Allocatio, Ala=0.05, reatmet Allocatio=:, Delta= Summary of Simulated Power for Stratified Risk Differece, By Strata Allocatio, Ala=0.05, reatmet Allocatio=:, Delta= omariso of Simulated & alculated Power for Stratified Risk Differece, Gart & Nam Simulated ad Nam alculated Power, By reatmet Allocatio, Ala=0.05, Delta= omariso of Simulated & alculated Power for Stratified Risk Differece, Gart & Nam Simulated ad Nam alculated Power, By Pi, Ala=0.05, Delta= omariso of Simulated & alculated Power for Stratified Risk Differece, Deviace Simulated ad Nam alculated Power, By reatmet Allocatio, Ala=0.05, Delta= omariso of Simulated & alculated Power for Stratified Risk Differece, Deviace Simulated ad Nam alculated Power, By Pi, Ala=0.05, Delta= omariso of Simulated & alculated Power for Stratified Risk Differece, Wald Simulated ad Nam alculated Power, By reatmet Allocatio, Ala=0.05, Delta= omariso of Simulated & alculated Power for Stratified Risk Differece, Gart & Nam Simulated ad Wald alculated Power, By Pi, Ala=0.05, Delta= omariso of Simulated & alculated Power for Stratified Risk Differece, Wald Simulated ad Wald alculated Power, By Pi, Ala=0.05, Delta= Summary of Simulated ye I Error for Stratified Risk Ratio wit Side oditios o Idividual Strata, Gart-S Metod, reatmet Allocatio=:.89 xvii

18 4.58 Summary of Simulated ye I Error for Stratified Risk Ratio wit Side oditios o Idividual Strata, Gart-S Metod, reatmet Allocatio=: Summary of Simulated ye I Error for Stratified Risk Ratio wit Side oditios o Idividual Strata, Gart-S Metod, reatmet Allocatio=: omariso of Simulated Power for Stratified Risk Ratio wit Side oditios o Idividual Strata, Gart-S Metod, By Strata Allocatio & (Pi Pi, eta=, Overall Strata Ala=0.05, Idividual Strata Ala= omariso of Simulated Power for Stratified Risk Ratio wit Side oditios o Idividual Strata, Gart-S Metod, By Strata Allocatio & (Pi Pi, eta=, Overall Strata Ala=0.05, Idividual Strata Ala= omariso of Simulated Power for Stratified Risk Ratio wit Side oditios o Idividual Strata, Gart-S Metod, By Strata Allocatio & (Pi Pi, eta=, Overall Strata Ala=0.05, Idividual Strata Ala= Summary of Simulated ye I Error for Stratified Risk Differece wit Side oditios o Idividual Strata, Gart & Nam-S Metod, reatmet Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece wit Side oditios o Idividual Strata, Gart & Nam-S Metod, reatmet Allocatio=: Summary of Simulated ye I Error for Stratified Risk Differece wit Side oditios o Idividual Strata, Gart & Nam-S Metod, reatmet Allocatio=: omariso of Simulated Power for Stratified Risk Differece wit Side oditios o Idividual Strata, Gart & Nam-S Metod, By Strata Allocatio & (Pi Pi, Delta=0, Overall Strata Ala=0.05, Idividual Strata Ala= omariso of Simulated Power for Stratified Risk Differece wit Side oditios o Idividual Strata, Gart & Nam-S Metod, By Strata Allocatio & (Pi Pi, Delta=0, Overall Strata Ala=0.05, Idividual Strata Ala= xviii

19 4.68 omariso of Simulated Power for Stratified Risk Differece wit Side oditios o Idividual Strata, Gart & Nam-S Metod, By Strata Allocatio & (Pi Pi, Delta=0, Overall Strata Ala=0.05, Idividual Strata Ala= xix

20 INRODUION No-iferiority cliical trials are icreasigly becomig more romiet i te researc ad develomet of ew aramaceuticals. Metodology secifically for te desig ad aalysis of tese trials is a essetial comoet for te assurace of quality trials tat are statistically defesible i te scietific commuity as well as i a regulatory settig, were traditioally te focus as bee o sueriority. e mai goal of o-iferiority trials is to sow tat te ew exerimetal medicatio (test treatmet is ot uaccetably worse ta te curret stadard of care (active cotrol treatmet by a secified amout, but te test treatmet may ave oter desirable asects suc as a better safety rofile or roerties wic make atiet comliace better. is is a reversal from te goals of a sueriority trial wic geerally icludes te test treatmet i comariso to a lacebo were te goal is to sow tat tis ew treatmet is more effective ta lacebo. I certai disease areas suc as ifectios, te use of a lacebo cotrol arm is uetical due to widesread use of te active cotrol for treatmet of te disease. ere are secific guidace documets wic discuss te issues surroudig te desig ad imlemetatio of o-iferiority trials. e IH-E0 guidace o te oice of otrol Grou ad Related Issues i liical rials rovides te ratioale for use of a active-cotrol treatmet i a o-iferiority settig. I additio, te trial must address assay sesitivity troug istorical evidece of efficacy, ad te coduct of te

21 trial must make efforts to icrease comliace ad miimize droouts, sice oor trial coduct ca bias tese trials toward o-iferiority. Additioally, te coice of a margi for testig te iferiority yotesis must be establised by cliical ad statistical judgmet.ese issues are all very imortat asects i te desig of a o-iferiority trial. However, te curret discussio will assume tat tese issues are aroriately addressed ad te focus will iclude statistical issues related to samle size calculatio, samle size allocatio, ad aalysis i o-iferiority trials. ese issues are essetial for statisticias wo eed to kow ow to better desig ad aalyze tese trials, wit secific emasis o metods related to dicotomous categorical data. Stadard metodology must be reviewed ad assessed as to its aroriateess for addressig te o-iferiority yotesis. ategorical data aalysis for a dicotomous rimary edoit may iclude aalysis of a risk ratio or a risk differece wic comares te test ad active cotrol treatmets. is assessmet of o-iferiority is erformed by comutig a cofidece iterval ad determiig if te alicable limit is below (or similarly above te re-secified o-iferiority margi. A test statistic ca also be used for tis assessmet were rejectio of te ull yotesis of iferiority would require a -value less ta te re-secified ala level. Metods for comutig eiter te cofidece iterval or te corresodig test statistic will be assessed accordig to teir erformace wit resect to tye I error ad ower troug simulatios for relevat scearios. e effect of samle size allocatio o te erformace of tese metods will also be assessed. I additio, aroriate samle size formulas will be develoed to aid i trial laig. I some o-iferiority trials it is ossible to iclude a lacebo arm as well as a active cotrol arm. is lacebo arm ca address issues related to assay sesitivity ad 3

22 aroriate trial coduct. is also allows o-iferiority to be assessed relative to te lacebo arm, usig te ercetage of effect tat te test treatmet as over lacebo comared to te effect tat te cotrol treatmet reserves over lacebo. Metodology for tis assessmet is also of iterest alog wit aroriate samle size calculatios. is settig also resets a area of researc for discussio of te oe versus two trials aradigm. Frequetly, regulatory agecies require two cofirmatory trials. However, if tese trials are ru i a idetical maer wit similar rotocols, it may be beeficial to ru oe large trial. e imlicatios of tese scearios are assessed related to tye I error cotrol ad te resultig ower for rejectig te ull yotesis of iferiority. Extesios to te metodology for te risk ratio ad risk differece are assessed we stratificatio is ecessary, secifically for large subgrous suc as geder. Metods for stratificatio are a imortat comoet, ad additioally te effects of stratificatio i a o-iferiority settig. Review, develomet, ad assessmet of tis metodology for dicotomous data secifically focused o te o-iferiority settig is a imortat additio to te curret statistical ractice. is researc is a coesive resetatio for eac of te measures of iterest troug assessmet of metodology ad its relatio to aroriate desig comoets suc as samle size calculatio. e imortace of elig statisticias uderstad ad imlemet metods i tese areas is of most cocer. 4

23 Refereces. U. S. Deartmet of Healt ad Huma Services, Food ad Drug Admiistratio. Guidace for Idustry, E0 oice of otrol Grou ad Related Issues i liical rials. Rockville, MD 00.. Hug, H. M. J., Wag, S. J., sog Y., Lawrece, J., ad O Neill, R.. Some Fudametal Issues wit No-Iferiority estig i Active otrolled rials. Statistics i Medicie 003,,

24 ater Review ad Evaluatio of Metods for omutig ofidece Itervals for te Ratio of wo Proortios ad osideratios for No-iferiority liical rials I. Itroductio Ratios of roortios are ofte called risk ratios i a cliical trials settig. ese ratios are used to comare two ideedet grous, usually o two differet treatmets. A o-iferiority cliical trial ca comare a active cotrol grou to a grou takig a ew treatmet for a efficacy outcome (or a lacebo grou to a grou takig a ew treatmet for a safety outcome. e goal is to sow tat te ew treatmet is ot uaccetably worse ta te active cotrol (or lacebo treatmet. e ew treatmet may ave oter beeficial asects suc as a reductio i severity of side effects, easier use, or lower cost. Assessig o-iferiority is ofte doe troug a cofidece iterval for te risk ratio of te two grous, articularly if cotrol failure rates are small (e.g., 0.0 or cotrol success rates are large (e.g., If failure rates are very small (e.g., < 0.05 te te odds ratio ca be coservatively used to aroximate te risk ratio (we defied so as to ave te larger exected rate i te umerator ad te smaller exected rate i te deomiator. For situatios were failure rates are larger (e.g., > 0.0, te te differece i rates is tyically

25 emasized,3,4. I some cases, if te ew treatmet grou as a risk tat is ot more ta twice tat of te cotrol grou for a failure outcome troug a uer cofidece limit of or less, te te ew treatmet will be judged o-iferior. is o-iferiority limit ca be set at a variety of re-determied levels deoted θ 0. Accordigly, a corresodig test of o-iferiority as te ull yotesis as Ho: θ = π / π θ 0 ad te alterative yotesis as HA: θ = π / π < θ 0 were θ = π / π is te oulatio risk ratio for te test grou versus te cotrol grou wit π as te oulatio roortio of evets i te test grou ad π as te oulatio roortio of evets i te cotrol grou, ad θ 0 = was te reviously metioed examle. ere are may metods i existece for comutig a cofidece iterval for a risk ratio. Several of te metods for formig cofidece itervals for ratios of two ideedet biomial roortios will be reviewed ad evaluated for teir statistical erformace. ese metods iclude use of a aylor Series exasio to estimate variace, solutios to a quadratic equatio, ad maximum likeliood metods. Simulatios were used to idetify te better metods for cotrollig te tye I error rate wile maitaiig ower. Alicatios of tese fidigs iclude samle size calculatios wic arise i radomized cliical trials coducted to sow o-iferiority. II. Metods A. aylor Series Exasio Metods 7

26 e literature cotais may metods for formig cofidece itervals for risk ratios. e first grou of tese uses a variace formed troug a aylor Series exasio. e followig metod see i (., ereafter called te aylor Series metod, is te simlest i tis grou discussed by Katz, Batista, Aze ad Pike 5 ad used by SAS i te FREQ rocedure 6 ad by Equivest 7 to form a 00(-α% cofidece iterval for a risk ratio: / y / ex loge ± z α (. y / y y were y is te umber of evets ad is te total samle size i te treatmet grou, y is te umber of evets ad is te total samle size i te cotrol grou, ad z α is te 00(- α ercetile from a stadard ormal distributio. I 988, Gart ad Nam 8 revised tis origial metod so tat te cofidece iterval would be defied if y or y were equal to zero. e formula see i (. is tis modified cofidece iterval used by StatXact 9 for risk ratios. ( y exlog e ( y 0.5 /( 0.5 /( / 0.5 ± zα (. 0.5 y 0.5 y is Modified aylor Series metod adds a alf to te evet cout for eac grou as well as te total samle size for eac grou. e last metod i tis grou of aylor Series exasio metods is adated from a cofidece iterval for a sigle biomial roortio roosed by Agresti ad oull 0. For a cofidece iterval for a sigle biomial roortio, Agresti ad oull suggested addig alf te squared z-value (at te corresodig ala level to eac outcome for eac grou to roduce a more coservative iterval. is strategy was adated for a test of o-iferiority were te ull yotesis is ot oe of equality. e additioal z α couts must be 8

27 distributed to eac grou accordig to te ull yotesis (θ 0 ad te allocatio of samle size to eac grou (R= / as see i (.3 ( y exlog e ( y γ /( γ /( γ γ ± z y ot, α ot, γ y γ γ ot, γ ot, / (.3 were R γ z ot, = α *, R γ z ot, = α *, R γ = γ ot, θ0 *, ad θ 0 γ = γ ot, *. θ 0 For examle, at a α=0.05 level a additioal z α = (.96 8 couts must be added. For a settig wit twice as may atiets allocated to te test grou ta te cotrol grou, R=, ad a ull yotesis of θ 0 =, tere are a total of γ =3.56 evets added to te test grou, γ =0.89 evets added to te cotrol grou wit a total of γ ot, =5.33 added to te overall umber of atiets i te test grou ad γ ot, =.67 atiets added to te cotrol grou. e aylor Series Adjusted Ala metod was added so as to correct iflatio of tye I error by te aylor Series metod see i iitial simulatios. is metod is te aylor Series metod wit a ala level tat is less ta te ala level for te α=0.05 sceario. For examle, tis metod would use a ala level of = 0.05 we α=0.05 was secified. e coice of was motivated by fidigs from te simulatios for te secific scearios reseted were tis modificatio was eeded to offset te small iflatio i tye I error of te aylor Series metod. is adjustmet of te ala level is deedet o te alicatio at ad ad simulatios ca be used to determie te aroriate adjustmet for ay sceario. is adjustmet to te ala level is a way to 9

28 address studies wit fiite samles rater ta ifiite (or very large samles by icreasig te z-criterio for sigificace sligtly (i.e., for ala=0.05 te z-criterio would icrease from.96 to.00. B. Solutio to Quadratic Equatio Metods e ext grou of metods is sligtly more comlicated because te cofidece limits are te solutios to a quadratic equatio. After algebraic maiulatios, a quadratic form of te equatios rovided below are te solved for θ. e uer ad lower cofidece limits are te smaller ad larger of te two solutios, resectively. However, tese metods may roduce comlex-valued results (we square roots of egative umbers are ivolved. Fieller first reseted te most basic of tese metods i 944 as see i (.4, ereafter called te Quadratic metod were ˆ = y / ad ˆ = y /. ˆ ( ˆ (ˆ θ ˆ θ ˆ ( ˆ = z α (.4 e secod of tis grou of metods i (.5 was roosed by Bailey i 987 wic is a modificatio of te Quadratic metod to roduce limits wit more desirable roerties as will be discussed i more detail i te literature review sectio. ˆ 9 / 3 / 3 (ˆ ( ˆ θ θ / 3 / 3 ˆ / 3 / 3 ˆ ( ˆ = z α (.5 0

29 e last of tis grou of metods was roosed by Farrigto ad Maig i wit tree ossible variatios o te equatio i (.6. ~ π (ˆ θ ˆ ( ~ π ~ ~ π ( π θ = z α (.6 Eac variatio suggests comutig ~ π ad ~ π i a differet maer. e first of tese, F-M, uses te observed values ad sets ~ π = ˆ ad ~ π = ˆ. e secod variatio, F-M, uses fixed margial totals to comute ~ π θ = 0 ( ˆ ˆ ( θ 0 ad ~ ( ˆ ˆ π =. ( θ 0 e tird variatio, F-M 3, uses maximum likeliood estimatio uder te ull yotesis to obtai ~ π ad ~ π wit details foud i Farrigto ad Maig s aer 4 ad solutios for ~ π ad ~ π below: ~ b b 4ac π = ad ~ π ~ = π / R a were a =, b = θ 0 ˆ ˆ, ad c = θ0 ˆ ˆ. I additio, Gart ad Nam 8 summarize a iterval attributed to Noeter were te equatio i (.7 is solved for θ to yield uer ad lower cofidece limits, θ L ad θ U. θ ( ˆ ˆ ( ˆ / ˆ θ = z θ ( θˆ ( θˆ α (.7. Maximum Likeliood Metods

30 e tird grou of cofidece iterval metods icludes tose tat use maximum likeliood estimators for te roortio of evets i te treatmet ad cotrol grous based o te joit distributio of te evets as te roduct of two ideedet biomial distributios for te treatmet ad cotrol grous. e first of tese metods calculates a deviace statistic as see i (.8 Deviace = *[log L(ˆ π, πˆ log L( θ 0 πˆ *, πˆ * ] (.8 were πˆ ad πˆ are te maximum likeliood estimators of π ad π uder te alterative * yotesis ad ( θ0πˆ ad * ˆπ are te corresodig maximum likeliood estimators uder te ull yotesis θ = θ 0. e secod of tese maximum likeliood metods is based o a Pearso statistic i te form of [(observed exected / exected] i (.9. { y * θ ˆ 0π } * θ ˆ π 0 {( * y ˆ ( θ 0π } * ( θ ˆ π 0 { y * ˆ π } * ˆ π {( * y ˆ ( π } * ( ˆ π (.9 usig * 0 ˆπ θ ad ˆ π *, te maximum likeliood estimators of π ad π uder te ull yotesis θ = θ 0. Kooma 3 roosed tis metod i 984, ad StatXact 9 is a software ackage tat rovides tese cofidece itervals. I additio, Bedrick 4 discusses a set of metods termed te ower divergece metods see i (.0 were various values of λ ca be used, wit tis discussio focusig o λ=-0.5, 0.5, 0.67,.0, ad.5.

31 λ I = λ( λ (.0 ˆ ˆ ~ λ ( ˆ ˆ ~ λ ˆ ˆ ~ λ ( ˆ ˆ ~ λ e Deviace, Pearso, ad Power Divergece metods roduce test statistics for wic -values ca be obtaied usig te ci-square distributio uder oe degree of freedom. e aroriate cofidece limits ca be foud troug a iterative rocess. e yotesized ratio θ of π to π is modified util te desired -value (e.g., 0.05 or 0.05 is obtaied. is rocess idetifies te largest θ 0 tat would ot be rejected as H 0 : θ θ 0. e ratio tat roduces te desired -value is te te uer cofidece limit. is iterative rocess requires cagig te maximum likeliood estimator ertaiig to te ull yotesis as θ 0 cages. is grou of metods is more comlicated ta te oters due to te iterative ature of fidig te cofidece itervals as all yoteses ot rejected, tus requirig itesive comuter resources. A summary of available software resources for te comutatio of te metods described ca be foud i able.. III. Review of Literature Differet combiatios of te metods described above ave bee comared i te literature. I 978, Katz et al. 5 comared te aylor Series metod ad te Quadratic metod usig simulatios ad calculatig coverage robabilities. Katz et al. suggested tat te Quadratic metod could be erratic ad may ot roduce cofidece limits at all; te aylor Series metod was recommeded for use istead of te Quadratic metod. 3

32 Agai i 984, Kooma 3 used simulatios ad coverage robabilities to comare te aylor Series metod ad te Pearso maximum likeliood metod. Fidigs suggested tat te Pearso metod maitais a coverage robability closer to te ( - α level, ad i additio, te oe-sided robabilities of exceedig te uer limit or beig lower ta te lower limit are muc closer to α. erefore, Kooma recommeded use of te Pearso metod. I 987, Bailey exteded te Quadratic metod to roduce Bailey s metod, wic sould reduce te skewess of te cofidece iterval as well as maitai te omial coverage robability better ta te Quadratic metod. is ew metod is also comared to te aylor series metod ad te Pearso metod. Bailey cocluded tat is metod results i cofidece limits tat are closer to te omial level ta te aylor Series metod. I additio, Bailey s metod more ofte maitais te omial coverage robability better ta te Pearso metod. Gart ad Nam 8 roduced a comreesive comariso of te metods reseted revious to 988. ey idicated tat te Quadratic metod ad Bailey s metod ted to roduce cofidece limits tat are eiter above or below te omial coverage robability, wereas te Modified aylor Series metod ad te Pearso metod acieve coverage robabilities close to te omial level, wit te Pearso metod sligtly better. ey also discuss a skewess-corrected score metod, wic is iterative i ature, tat is sligtly better ta te Power Divergece metod (λ=0.5 of Bedrick 4. I 990, Farrigto ad Maig 4 reseted results o te tree variatios of quadratic metods for roducig cofidece limits for risk ratios. eir recommedatio was 4

33 te tird of tese metods, F-M 3, based o maximum likeliood estimatio for te roortios. IV. ofidece Limit omarisos A iitial comariso of te metods icludes comutig te uer cofidece limits for selected cases. At a oe-sided ala level of 0.05, te uer cofidece limits are reseted for eac of te metods roducig cofidece limits ad -values for te metods roducig a test statistic (wit cofidece limits comuted troug a iterative rocess. e metods are groued by te tree metod tyes: te aylor Series variace exasio metods (able., te quadratic metods (able.3, ad te maximum likeliood metods (able.4. Witi te aylor Series variace exasio metods, te aylor Series metod ad te aylor Series Adjusted Ala metod roduce iger cofidece limits for te : allocatio wereas te Adated Agresti metod as iger cofidece limits for te allocatios tat lace more samle size i te test treatmet for te 3:, :, ad 3: allocatios. e Quadratic metod ad Farrigto-Maig metod roduce very similar uer cofidece limits due to teir similarity i comutatio. Noeter s metod roduces iger cofidece limits for all samle size allocatios. Farrigto-Maig metods ad 3 also roduce similar uer cofidece limits for te selected cases reseted. e Deviace ad Pearso metods roduce similar -values. e grou of Power Divergece metods yields decreasig -values for icreasig coices of λ. 5

34 V. Simulatios Data were geerated from kow distributios to comare te beavior of te metods wit resect to ower ad tye I error. Scearios icluded varyig te followig arameters:. π, te oulatio roortio of evets i te cotrol grou: 0.0, 0.5, 0.0, 0.5. θ= π /π, te oulatio risk ratio: 0.667, 0.800,.000,.50,.500,.000, π, te oulatio roortio of evets i te test grou: π = θπ 4. θ 0, te ull yotesis risk ratio:.5,.0,.5 5. α, te oe-sided ala level: 0.005, 0.05, , te samle size i te test grou is calculated to ave 85% ower to cotradict te ull yotesis θ 0, give a risk ratio of for test versus cotrol wit =R : = ( z z α β Rπ { l( / θ } 0 π 7. Samle size allocatio for test:cotrol as :, :, 3:, :, 3: For eac combiatio of te arameters, 00,000 simulatios were geerated usig a radom samle from te two biomial distributios of y ~ bi(, π ad y ~ bi(, π. For eac combiatio of y ad y, uer cofidece limits or test statistics wit corresodig -values for all metods were calculated. If y or y were equal to zero or te metod failed to roduce a valid result, te te exact cofidece limit for te odds ratio was te default. is modificatio usig te odds ratio is coservative because it emloys exact metodology ad because te odds ratio exceeds te risk ratio we bot exceed oe. As a 6

35 ote, if y =0 te te uer cofidece limit for te odds ratio is essetially ifiite, ad so it was set to 00 ad te ull yotesis of iferiority was ot rejected. is modificatio, were te uer cofidece limit was set to 00, is also ecessary i cases were te grou of quadratic metods lead to square roots of egative umbers (i.e., comlex solutios or were te Deviace or Pearso metods fail to roduce iterretable results because of comutatioal sigularities. No modificatios were ecessary for te Modified aylor Series or te Adated Agresti metods. For eac metod, a idicator variable was created for eac simulatio tat takes te value of if te uer cofidece limit roduced was less ta θ 0 ad 0 oterwise or similarly if te -value was less ta ala te idicator takes te value ad 0 oterwise. is idicator was te averaged across all 00,000 simulatios to roduce a robability. For θ < θ 0, tis robability is te ower for te test of o-iferiority ad ca be writte i te followig maer: ower = r(reject H O : θ=π /π θ 0 H A : θ < θ 0 true. For θ = θ 0, tis robability is te tye I error rate for te test of o-iferiority, ad ca be writte i te followig maer: α = tye I error = r(reject H O : θ=π /π θ 0 H O : θ θ 0 true. A summary of te tye I error of te metods geerated from te 00,000 simulatios is dislayed i Figure. for te aylor Series metods, Figure. for te quadratic metods, ad Figure.3 for te maximum likeliood metods. Farrigto-Maig metod is droed from summaries due to its similarities to te Quadratic metod. Dislays iclude oly te α=0.05 level wit similar atters see for te oter ala levels. e erformace of te metods wit resect to te tye I error varies i relatio to te samle size allocatio of treatmet to cotrol. All of te aylor series exasio metods ave aroximately omial tye I error rates for te : allocatio. However, as more 7

36 samle size is laced i te test grou, te tye I error rates become iflated iger ta te omial level. e Adated Agresti metod yields tye I error rates closest to te omial level, but tis metod still sows iflatio for te 3:, :, ad 3: allocatios. Out of te grou of quadratic metods, te Quadratic metod ad Noeter s metod ave tye I error rates tat are cosistetly below te omial level for all allocatios. However, Bailey s, F-M, ad F-M 3 ave aroriate tye I error rates for te : ad : allocatios wit iger ta omial tye I error rates for te 3:, :, ad 3: scearios. e grou of maximum likeliood metods erform similarly for te : allocatio, wit tye I errors aroximately omial or just sligtly iger ta omial. e Deviace metod erforms adequately for all samle size allocatio scearios wit tye I errors close to te omial level. e Pearso metod as sligtly iflated tye I errors for all oter scearios. e grou of ower divergece metods yields iger tye I errors as λ icreases wit λ=-0.5 yieldig lower ta omial tye I errors ad λ=.5 yieldig iger ta omial tye I errors. Figure.4 rovides a graical summary of te metods wit better tye I error erformace icludig te aylor series metod, Adated Agresti metod, Bailey s metod, ad Deviace metod. Discussios of ower will be limited to tese metods for scearios were te simulated tye I error is aroriately cotrolled. e Deviace metod seems to erform aroriately for all samle size allocatio scearios, wit sligtly iger tye I errors for te : allocatio. Figure.5 comares te aylor Series ower to te Deviace ower for te : allocatio, for te ull yotesis θ 0 =. ese metods ted to erform similarly i tis settig. Figure.6 is a comariso of te Adated Agresti ad Deviace simulated owers for te allocatios icludig : ad 8

37 9 :. e Deviace metod roduces similar or sligtly iger simulated owers i tese scearios. Figure.7 dislays Bailey s metod comared to te Deviace metod for te : allocatio settig, also sowig similar simulated owers betwee te two metods. ese fidigs suggest tat i te : or : allocatio settigs, te simler aylor Series or Adated Agresti metods erform similarly to te comuter itesive Deviace metod wit resect to ower. However, te Deviace metod may be te referred metod for allocatios wit more samle size i te test grou i order to maitai te omial tye I error level. VI. Samle Size alculatios A immediate alicatio of tese results arises i te desig of o-iferiority cliical trials. e aylor Series metod rovides a fairly straigtforward form from wic to obtai samle size calculatios. A coservative form of te variace is see i (.. * / / log var v R R y y y y e = < = θ π θ π (. Motivatio for obtaiig a samle size formula begis wit formulatio of a z-statistic i (. were θ 0 is te value of θ uder te ull yotesis. v * log log z 0 e e θ θ = (.

38 0 e equatio i (.3 results from squarig equatio (. ad writig z i terms of te tye I ad tye II errors wic roduces equatio (.4 after algebraic maiulatios. v * log z (z 0 e θ θ = β α (.3 log R z (z 0 e = θ θ θ π β α (.4 is form is te solved for te samle size,, ad ca be writte as i (.5 0 e log R z (z θ θ θ π = β α (.5 wic deeds oly o a re-secified oe-sided tye I error (α, ower (-β, evet rate i te cotrol grou (π, te samle size allocatio (R= /, ad a yotesized ratio of evets i te treatmet versus te cotrol grou (θ wit θ 0, te ull yotesis, secified. is formula is useful i ractice due to ease of comutatio. o evaluate weter formula (.5 roduces samle sizes tat maitai te resecified ower, results were comared to tose obtaied from simulatios. ese results were based o 00,000 simulatios. e samle size formula (.5 was writte i terms of ower as see i equatio (.6 α β θ π θ θ = z R log z 0 e (.6

39 were ower = Φ(z β ad Φ(. is te stadard ormal robability. I additio, tis samle size formula ad ower calculatio i (.5 ad (.6 ca be modified for te aylor Series Adjusted Ala metod wic cotrols tye I error better ta te aylor Series metod (altoug i allocatios wit more subjects i te test grou, te tye I error is still above te omial level. is adjustmet uses a ala level of lower ta tat secified. For examle, at a secified α=0.05 te critical value would be calculated at =0.05. I additio Farrigto ad Maig 4 reset samle size formula (.7 ad ower formula (.8 based o teir metods. z β = = { z ~ ~ ~ ~ α π( π Rθ0π ( π zβ π( π Rθ0π ( π } ( π θ π { ~ ~ ( z ( R ~ ( ~ π θ0π α π π θ0π π } π ( π 0 Rθ π ( π 0 (.7 (.8 were ~ π ad ~ π are secified differetly for eac of te tree metods. e first metod reseted by Farrigto ad Maig use ~ π ~ π π = ad π =. e secod of tese metods uses te followig values: ~ π θ = 0 ( π π ( θ 0 ad ~ π ( π = ( θ 0 π Farrigto-Maig metod 3 relaces ~ b b 4ac π = ad ~ π ~ = π / R a ~ π ad ~ π usig te followig equatios: were a =, b = θ0 π π, ad c = θ0 π π.

40 For combiatios of α, π, θ, R, ad geerated i te 00,000 simulatios, a ower based o te samle size formula was calculated usig formula (.6 ad te aylor Series Adjusted Ala formula. Power was also calculated usig te samle size formulas reseted by Farrigto ad Maig for F-M metod, F-M metod, ad F-M metod 3. is calculated ower was te comared to te ower obtaied from te simulatios for eac metod. Figures.8.4 graically dislay te comariso betwee te calculated ad simulated ower for te aylor Series, aylor Series Adjusted Ala, F-M, F-M, ad F- M 3 metods, at a ala level of 0.05 for a ull yotesis θ 0 =. For most cases, te simulated ower is similar to or larger ta te calculated ower; terefore te samle size formulas are somewat coservative wic is beeficial we determiig samle size for cliical trials. e Deviace metod does ot ave a corresodig samle size formula, terefore te simulated ower from tis metod is comared to te calculated aylor series ower i Figure.3f ad te calculated F-M 3 ower i Figure.3g. e simulated Deviace ower is bot larger ad smaller ta te aylor Series calculated ower for secific scearios. However, te calculated F-M 3 ower seems to agree cosistetly wit te Deviace ower for scearios wit ower iger ta For ower values lower ta 0.80, te F-M 3 calculated ower yields sligtly iger values. However, we laig a trial it is usually ecessary to ave at least 0.80 ower ad i tese cases te F-M 3 calculatios would be aroriate.