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## Statistical Principles of Clinical Trials and Epidemiology

by: Jordane Kemmer

23

0

36

# Statistical Principles of Clinical Trials and Epidemiology ST 520

Marketplace > North Carolina State University > Statistics > ST 520 > Statistical Principles of Clinical Trials and Epidemiology
Jordane Kemmer
NCS
GPA 3.79

Daowen Zhang

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COURSE
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Daowen Zhang
TYPE
Class Notes
PAGES
36
WORDS
KARMA
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## Popular in Statistics

This 36 page Class Notes was uploaded by Jordane Kemmer on Thursday October 15, 2015. The Class Notes belongs to ST 520 at North Carolina State University taught by Daowen Zhang in Fall. Since its upload, it has received 23 views. For similar materials see /class/223935/st-520-north-carolina-state-university in Statistics at North Carolina State University.

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Date Created: 10/15/15
Power and Sample Size Calculation for Logrank Test under a Nonproportional Hazards Model Daowen Zhang Department of Statistics North Carolina State University zhang statncsuedu http www4 stat ncsu eduNdzhang2 Joint work with Hui Quan Department of Biostatistics amp Programming Sano Aventis GUIROOM OUTLINE Motivating example Rimonabant trial on cardiovascular risk Review of the logrank test statistic Distributions of the logrank test statistic Detailed power calculation Example and simulation Results Summary 1 Motivating example Rimonabant trial Rimonabant trial Assess the bene t of Rimonabant on reducing cardiovascular risk Placebo controlled Primary endpoint time to cardiovascular event event rates expected to be low in each group Log rank test was proposed to assess the treatment effect Power and sample size consideration should also be based on the log rank test It is straightforward if treatment effect is characterized by 1t es 0 t 7 1t hazard of cardiovascular event for treatment 0t hazard of cardiovascular event for placebo If 0 and censoring process independent of treatment group log rank test statistic T has distribution Schoenfeld 1981 Biometrika T 3 W 61 e121 6 Allocation probability to the treatment D Expected total of deaths under Ha from both groups 0 Can be used to calculate power and sample size if the treatment effect model PH model is reasonable 0 However o Other issues 1 censoring information cannot be retrieved 2 drop out information can be retrieved during the study 0 How to handle drop out 1 treat it as censoring assumption 2 conduct ITT analysis efficiency loss 0 Problem how to calculate power and sample size for each strategy Which is better 0 Need to investigate the distribution of the log rank test statistic for our problem 2 Review of the logrank test statistic I o The standard log rank test statistic Where 0 Under H0 2 31 I So ltgt H0 2 1t 0t T 5 Mo 1 So reject H0 if T Z zag 0 Under Ha 1t 7E 0t but 1t 0t Schoenfeld 1981 Biometrika TNlt 1 Where me log 1tA0lttgtrlttgt1 WWW U000 77t1 7TtVtdt12 Where Vt describes process of observing deaths 7Tt gt 6 if Q5 censoring process is the same in both groups 0 Special case PH alternative 1t 0 t Ha e mO then T 1 W 61 6w 1 0 Can be used to calculate the power for PH alterative 10 3 Distribution of the logrank test statistic l o It is reasonable to assume the alternative for our problem 1t 1 t E 0t0 39Aot e 0 t e t0oo 1t hazard of treated group 0t hazard of untreated group 0 Distributions of the log rank test statistic under Ha for two strategies 1 Strategy 1 Treat drop out as censoring 2 Strategy 2 Conduct ITT analysis 11 Distribution for Strategy 1 0 Direct use of the result of Schoenfeld 1981 Biometrika gt TNlt 1 W5 ft Wm mch U000 tHl 7TtVtdt12 x61 6 X D total expected of deaths from two groups in the study D total expected of deaths from two groups after to 0 Power PZ gt W zag 22 o Concern approximation good enough better one 12 o The use of a series of double expectation theorem leads to 1 e f1 e 1f0 5 D1 total of deaths from treated group after to D0 total of deaths from untreated group after to qj m 61 6gtlt o Assumption drop out independent of the unerlying survival time had the patient not dropped out the same in both groups 0 Let D1 total expected of deaths from treated group D0 total expected of deaths from untreated group DI total expected of deaths from treated group before to D3 total expected of deaths from placebo group before to DZDO l Dl D1ZD1 Dl DOZDOD 13 Distribution for Strategy 2 o Lakatos 1988 Biometrics derived an approx dist of the log rank test under any Ha 1 t 7E AER t At o ASH hazard of the group randomized to placebo 1 t hazard of the group randomized to treatment 0 Partition patient time 0 L A l F Um n1 With equal Width A ti ti1 F L A F A accrual period F follow up time L study length 14 0 Under Ha Xf t 7E A3t t A3t TNlt 1 51191 Pi N 2 DZ 151pi 1pi N p 12 2 Di u om 1 Di n1tiXti QUOAMWHA total expected of deaths in tn n1 2 52 ATti3ti 3 pi n1t n0tz 15 4 n0t n1t number of patients at risk can be calculated iteratively nkt 1 ti lt F nkti1 tiA A tigtF L ti 7 n1 Assume constant accrual rate in 0 A 0 Need to know the hazard function for each randomized group 16 0 Assume 0t 0 gt A3 0 0 Assume drop out process has no effect on untreated group Z N mph 0 Then it is reasonable to assume 1tZ as 1 Case 1 Z S to 0 2 Case 2 Z gt to 0 t E 0t0 MWZ A1 t E toZ 1 t E ZOO Where 1 E MAO eg 1 UJ1 1 17 o The survival function for group randomized to treatment 51 t EIT2t EEIT2tZ mama 0 Case 1 Z lt to SltZ 0t 0 Case 2 Z Z to e W t e 0t0 e AOtO A1t t0 t E e Aoto A1Z t0 1tz t E ZOO 18 0 Can calculate Sft and fft and hence I mt SW 0 Then can calculate the nc q in N q 1 for the log rank test Xi t o For better numerical accuracy A needs to be small say 1 1000 if unit year 19 4 Detailed power calculation for strategy 1 l 0 Some assumptions 1 Other than drop out end of study is the only other censoring can be relaxed 2 0 A is the accrual period a accrual rate can be at 3 F follow up period L A l F total study length 4 F 2 to 5 0t 0 20 0 Consider tt l dt in 0A t tdt A L 0 Average of patients entering into study in t t l dt Qadt treatment group 1 6adt placebo group 21 o The probability that a patient entering at t is observed to die in the study ie dies before L is PT S rninL t Z c The probability that a patient entering at t is observed to die before to is PT S rnint0Z 22 o For placebo group PT g minL t 2 EEIT g minL t The inner expectation can be shown to be 1 e 0Lt Z gt L t EIT g minL t zmz 1 6amp02 Z lt L t gt A0 PT g minL t 2 m A0 Te ltAorgtltL t 23 o The total expected of deaths in the study for placebo group A Do a1 6PT S minL t Zdt 0 MA eltxorm1 0T oT o The total expected of deaths for placebo group before to A DE a1 6PT mint0Zdt 0 aA1 60 1 A0Tt0 0 T e 24 o For treatment group 739 739 0 T lt 39 L Z AoTto P m1n t AOT A1T O7 1 e AO Al o AIWLTXL t 1 739 o The total expected of deaths in the study for treatment group D1 a6 KA A1T2 1 A0A1t0 A1TL A1TA 1 o The total expected of deaths for treatment group before to Di A a6PT mint0Zdt 0 CLAer 0 739 1 e 07t0 25 5 Example and simulation results I Expect new treatment takes effect after 1 year gt to 1 Rate to have cardiovascular risk 003 per year A0 003 Expect 25 reduction when new treatments takes its full effect 1 00225 Accrual rate a 1000 patients month Study length L 50 months Expect 10 per year drop out rate Signi cance level 04 005 targeted power 09 How long should the accrual period A be And sample size 26 095 090 Power 085 accrual rate 12000 patientsyear study length 417 years Accrual period in years 27 Power 095 090 085 080 075 9000 10000 11000 12000 13000 14000 Accrual rate number of patients per year 28 Power 095 090 085 080 075 accrual rate 12000 patientsyear accrual period 142 years 36 38 40 42 44 Study length in years 29 Power 092 090 088 086 accrual rate 12000 patientsyear accrual period 142 years study length 417 years Weight w 30 Power 075 080 085 090 095 070 accrual rate 12000 patientsyear study length 417 years 14 1 6 1 8 Accrual period in years 20 31 Power Lo 03 039 0 quot x a m f Lo g 1quot xx 00 f f x 039 r x 1 O I I 00 f I I O r I f I Lo 39 I accrual period 17 months 0 study length 50 months 0 l 039 z 8000 9000 10000 11000 12000 13000 14000 Accrual Rate number of patients per year 32 Power 075 080 085 090 095 070 accrual rate 12000 patientsyear accrual period 17 months 4 O 42 4 4 Study Length in Year 33 software S plus function logmnkpowe alpha005 lambda0 lambdalz t00 wt05 tau0 acrate acperiodz slengthz theta05 nsub1000 ittF signifance level of the logrank test hazard for placebo hazard for treatment to used in the formula weight for residual treatment effect dropout rate accrual rate accrual period study lenght slength acperiodgtt0 allocation prob number of subintervals for ITT analysis ag for ITT analysis 34 6 Discussion I Delayed treatment effect l drop outs present challenge to statisticians Proposed two strategies 1 Treat drop outs as censored observations a Assumption drop out process independent of underlying true time to event b Drop out processes almost the same in both groups c Calculation straightforward d Don t need to specify the hazard for untreated group 2 Conduct ITT analysis a May be what regulatory agencies want b May have enough power only if residual treatment effect is 35 relatively large 70 in our example c Can be computationally intensive small A d Have to specify the hazard for untreated group 0 Derived formula easy to use con rmed by simulation to have good statistical properties 0 Can include other censoring 36

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