1、1Survival was the primary endpoint of this study;the time to recurrence was also determined.A minimum of 150 eligible patients per treatment arm was planned for the trial.This ensured that the trial would have a power of 0.90 to detect a ratio of the control-group hazard to the combination-therapy-g
2、roup hazard of 2.0.This assigned a pairwise comparison of treatments by a one-sided log-rank test in which a value of 0.05 indicated statistical significance.2生存時間解析例数設計特徴検定検定？検定，症例数設計可能？検定例数設計原理Freedman式Shoenfeld式関係？SAS例数設計複雑問題計算3次情報量最大？）10万人0.01年（3.5日）10人100年間）人5年間生存時間解析数情報量期間考慮必要4222112:222212121
3、21割合群併率群最終時点生存率群最終時点生存割合群併ddN222)1(2)1(HRHRzzd22)(log(2HRzzdFreedmanShoenfeld群比群群2:HR2:1:12215）SASV9POWERGLMPOWER）nQuery登録期間，途中脱落，区分指数）SWOGstatisticaltool登録期間，途中脱落，非劣性，交互作用，層，例数不均等，一標本問題http:/www.swogstat.org/stat/public/default.html6 t tests for means equivalence tests for means confidence interval
4、s for means tests of a binomial proportion multiple regression one-way analysis of variance multi-way analysis of variance contrast78Log-rank test for equality of survival curvesTest based on exponential survival,accrual period Test based on exponential survival,accrual period,drop outsLog-rank test
5、,user specified survival rates,accrual period,drop outs(simulation)9 Test significance level,0.050 1 or 2 sided test?2 Group 1 proportion 1 at time t 0.700 Group 2 proportion 2 at time t 0.800 Hazard ratio,h=ln(1)/ln(2)1.598 Power(%)80 n per group 296QueryFreedman式採用10S Sa am mp pl le e S Si iz ze e
6、 p pe er r G Gr ro ou up p1 10 00 01 15 50 02 20 00 02 25 50 03 30 00 03 35 50 04 40 00 04 45 50 0P Po ow we er r5 50 06 60 07 70 08 80 09 90 01 10 00 0 L Lo og g-r ra an nk k t te es st t o of f s su ur rv vi iv va al l i in n t tw wo o g gr ro ou up ps s f fo ol ll lo ow we ed d f fo or r f fi ix
7、xe ed d t ti im me e,c co on ns st ta an nt t h ha az za ar rd d r ra at ti io o=0.050(2)伊=0.700 衣=0.800 11When the sample size in each group is 296,a 0.050 level two-sided log-rank test for equality of survival curves will have 80%power to detect the difference between a Group 1 proportion 1 at tim
8、e t of 0.700 and a Group 2 proportion 2 at time t of 0.800(a constant hazard ratio of 1.598);this assumes no dropouts before time t.1213CRABCRAB統計統計(例数設計）例数設計）One Arm Binomial One Arm Survival One Arm Normal Two Stage Two Arm Binomial Two Arm Survival Two Arm Normal Binomial Interaction Survival Int
9、eraction SurvivalEquivalence Expected Deaths 1415a)生存時間分布指数分布仮定b)薬剤比例的効果（時点一様死亡低下）c)解析方法検定用d)群試験行等16 検定有意水準検定有意水準(通常通常5 5)有意差見逃確率有意差見逃確率(通常通常20%20%)SDSD個体間大個体間大D D()予想群間平均値差予想群間平均値差(生物学的検出差生物学的検出差)222)(2DSDzzN群平均値差例数設計群平均値差例数設計17 50%（0.500)考慮 NSDNSDNSDNNyyt2222212DDzNSDtD2zzNSDtD2222)(2DSDzzN18OBS P
10、 ZOBS P Z 1 0.010 -2.32635 1 0.010 -2.32635 2 0.025 -1.95996 2 0.025 -1.95996 3 0.050 -1.64485 3 0.050 -1.64485 4 0.100 -1.28155 4 0.100 -1.28155 5 0.200 -0.84162 5 0.200 -0.84162 6 0.500 -0.00000 6 0.500 -0.00000 7 0.800 7 0.800 0.841620.84162 8 0.900 8 0.900 1.281551.28155 9 0.950 1.64485 9 0.950
11、1.6448510 0.975 10 0.975 1.959961.9599611 0.990 2.3263511 0.990 2.32635日本Z米国Z世界共通Z 019NSDNSDNSDdifVdifEHNSDNSDNSDdifVdifEHyydif22212220122,:2,0:D20H0H1棄却限界値0NSDz2NSDz2DDNSDdifSD2)(21222)(22)(2)(22DDDDSDzzNSDzzNNSDzzNSDzNSDz22232425）SDSD 比例比例 例）倍必要倍例）倍必要倍 10101212見積，見積，1.441.44倍倍 ）反比例反比例 例）効果倍必要例）効果倍
12、必要 12121010見積，見積，1.441.44倍倍）検定有意水準厳必要増大）検定有意水準厳必要増大 例）有意水準例）有意水準0.050.050.010.01約約1.51.5倍倍）小必要増大小必要増大 例）例）0.200.200.100.10約約1.31.3倍倍222)(2DSDzzN26 22)()1()1()1(2cEccEEPPPPPPzPPzN22222/)(22DSDzSDzN22)(2DSDzzN2)1()1()1(:SDPPPPPPPPPPPccEEcEcE群割合対照群割合試験群割合，27 標準薬改善率PC：.40新薬改善率PE ：.45，.50，.60，.70(=0.05,0
13、.20）(0.05,0.10,0.20,0.30)0.05 0.10 0.20 0.30 N 1552 388 97 4228 群 1 2 3 4 5 6 実 時 間 4 5 6 9 10 11 人年法比Wald検定29指数分布(exponentialdistribution)確率密度関数：生存関数：関数：時点一定(瞬間死亡率表)exp()(ttS)exp()(ttf)(th30確率密度関数確率密度関数f生存関数生存関数関数関数h)exp()(ttf)exp()(ttS)(th31 itdhdhhV2（死亡数逆数比例）(人年法)MLE：分散：観察人年死亡数:,:itd320041.0,0093
14、.0667.00.1670.111HR111.0111063167.09543222212111222221111dhhVdhhVhhtdhtdhii3323.03167.03111.0)167.0111.0()()(222121222212122122dhdhhhhhVhh34 計算2)(2)(2121dddhhhzzZ121222121212dhdhhhhhVhhZdhhhhdhhhdhdhhhdhdhhhZ22212121222121212221235HR1/2：比dHRHRdhhhhzzdhhhhZ2)1()1(2)/1()/1()(2)()(2222122122122122222)1
15、(2)1(HRHRzzd36群生存時間曲線差指数分布仮定：1)0exp(:0loglogloglog:loglog:12012120120120HRHHRHHShoenfeldHFreedman 370logloglog:120HRHddhhhVhhhV112loglog221212121211)log()log()log()log()log()log(ddhhhhVhhZdHRdhhddhhZ2)log(2)log()log(11)log()log(121222)(log(2HRzzd38Freedman：Shoenfeld：展開（2次）222)1(2)1(HRHRzzd22)(log(2H
16、Rzzd2)()()()(2axagaxagagxg2)log(11)log(2)1(2)1(1HRwhen 222HRHRHRHRHRHR39log(HR)HR1周展開（2次）(HR)=(HR-1)(HR+1)HR1周展開4)1(21112HRHRHRHR)1()1(2)log(HRHRHR4)1(2)1(4)1(2)1(2)1log(2)log(22HRHRHRHRHR40FreedmanShoenfeldTaylorShoenfeld：log(HR)Freedman：2*(HR-1)/(HR+1)Taylor：(HR-1)-(HR-1)2/2;41結局，Freedman式Shoenfel
17、d式，指数分布仮定例数設計？何故，検定例数設計？42時点 群群 群 群群 群 生存死亡計 群 1 2 3 4 5 6実時間 4 5 6 9101143群群生存死亡集合全体群期待死亡数E群観測死亡数O3/6005.05.012112112111212ppnnnEOV442群群生存死亡集合全体5群期待死亡数E群観測死亡数O3/50016.04.022212222212222ppnnnEOV451群群生存死亡集合全体4群期待死亡数群観測死亡数3/410112461群群生存死亡2集合全体3群期待死亡数群観測死亡数2/301002470群群生存死亡2集合全体2群期待死亡数群観測死亡数2/21010148
18、0群群生存死亡1集合全体1群期待死亡数群観測死亡数1/1100014956.28997.0517.1)()()0()0(111221320431530630)(2222222262625252424232322222121222iiiiiiEOVEOIUEOEOEOEOEOEOEO50 HRethethHRethgroupthgroupgroupgroupzzththlog,)()()(:2)(:1 2:1z,1:0)exp()()(00000比例51MLE：-1.688山頂上山頂上傾傾52)()0()()0()0()0()0()0()0()0(log)(22220iiiiEOVIEOUIUI
19、UddUUdLdU53U(0)-1.517I(0)=0.8997One step 推定値 -1.68576857.10.89971.517-)0()0(1IUb)0()0(IU54DATA WORK;INPUT GROUP TIME CENSOR;CARDS;1 4 1 1 9 11 4 1 1 9 11 5 1 2 10 11 5 1 2 10 12 6 1 2 11 12 6 1 2 11 1PROC phreg DATA=WORK;model TIME*CENSOR(0)=GROUP/ITPRINT;55 Maximum Likelihood Iteration HistoryMaxim
20、um Likelihood Iteration History IterIter Ridge Log Likelihood GROUP Ridge Log Likelihood GROUP 0 0 -6.579251212010 0.000000000 0 0 -6.579251212010 0.000000000 1 0 -5.343241803607 1 0 -5.343241803607 -1.685705465-1.685705465 2 0 -5.343240134138 -1.687845205 2 0 -5.343240134138 -1.687845205Last Evalua
21、tion of the GradientLast Evaluation of the Gradient GROUP GROUP-7.852452E-7-7.852452E-7-1.68570.8997221.5167)0()0(1IUb56検定対数比推定値：対数比0)0()0()0()0()0()0(1,)0()0()()0(,)()0()0()0()()(222121211122222222222IUIIUbVbIbVIUbbEOVIEOUIUEOVEOiiiiiiii5722222222122111)(log)(21)(log)(4)(4/4:45.05.0)()0()0(1HRzzdHR
22、zzzzezzeeennnEOVIzzIbbVbiiiii群当死亡数：数群合58 FreedmanShoenfeld式展開行，次式近似等 Freedman方必要N少大 検定：Cox回帰反復測定回行対数比=0検定近似的Shoenfeld式等591)時点2群生存率S(t)2)生存時間(M)基3)人年法推定指数分布:21MMHR分布:21MMHR)(log)(log12tStSHR 総観察時間総数 60例題生存時間分布指数分布仮定，次条件例数設計行.0.05（両側），0.20手術単独群(cont)5年生存率：0.65補助化学療法群(drug)5年生存率：0.80610862.05)65.0log(:
23、drug0446.05)8.0log(:cont5)5(log)5exp()5()(,)exp()(,)(dchhSStStttSth62）各群計算）比計算）1.96，0.84）計算52.00862.00446.0cdhhHR5.14165.080.0229.38229.38)152.0(2)152.0()84.096.1()1(2)1()(21222222dNHRHRzzd63data samplesize;alpha=0.05;beta=0.20;t=5;pc=0.65;pd=0.80;h2=-log(pd)/t;h1=-log(pc)/t;hr=h2/h1;za=probit(1-alp
24、ha/2);zb=probit(1-beta);ef=(za+zb)*2*(hr+1)*2/(2*(hr-1)*2);nf=2*ef/(2-pd-pc);es=2*(za+zb)*2/(log(hr)*2);ns=2*es/(2-pd-pc);64T PC PD H2 H1 ALPHA BETA5 0.65 0.8 0.044629 0.086157 0.05 0.2 HR ZA ZB EF NFNF ES NSNS0.518 1.96 0.84 38.9 141.5141.5 36.3 131.9131.9NF：Freedman公式NS：Shoenfeld公式割合差例数設計行N=15165F
25、reedman式 Shoenfeld式222)1(2)1(HRHRzzd22)(log(2HRzzdzHRHRdz112zHRdz)log(266data samplesize;alpha=0.05;beta=0.20;t=5;pc=0.65;pd=0.80;n1=150;n2=150;h2=-log(pd)/t;h1=-log(pc)/t;hr=h2/h1;za=probit(1-alpha/2);e=(n1*(1-pc)+n2*(1-pd)/2;zbf=(e*2)*.5*abs(hr-1)/(hr+1)-za;zbs=(e/2)*.5*abs(log(hr)-za;pf=probnorm(
26、zbf);ps=probnorm(zbs);67alpha beta t pc pd n1 n2 h2 h1 0.05 0.2 5 0.65 0.8 150 150 0.04 0.09 hr za e zbf zbs pf ps0.52 1.96 41.25 0.92 1.03 0.82 0.85PF：Freedman公式PS：Shoenfeld公式681)2）登録期間，登録3）症例追加4）症例脱落5)期間69登録期間R年期間期間T年年S(T+R)時間累積登録累積登録人数人数生存率時間一様分布指数分布累積発症率S(T)S(T+)70割合nperyearR期待数dtRtTRSdtRteventS
27、dtRteventPRRR1)(11)(11)(000割合dtRtTRR1)(exp10割合RRTR1exp)(exp1割合R：登録期間，nperyear：年当登録数71登録期間：年,期間：最大年年間登録例数：例対照群年生存率：65%N150薬剤群年生存率：%N=150群併割合：32%検出力：87%（）72data power;r=2;nperyear=150;alpha=0.05;pc=0.65;pd=0.80;t=5;za=probit(1-alpha/2);lambda=-log(pd+pc)/2)/t;h2=-log(pd)/t;h1=-log(pc)/t;hr=h2/h1;do t=
28、0 to 6 by.1;n=nperyear*2;pevent=1-(1/r)*exp(-lambda*(r+t)*(exp(lambda*r)-1)/lambda;e=n*pevent/2;zbf=(e*2)*.5*abs(hr-1)/(hr+1)-za;pf=probnorm(zbf);output;end;73図期間検出力変化図期間検出力変化検出力数74）生存時間解析，症例数設計，比基，同様原理行）複雑問題（非劣性，登録期間，脱落）等，nQuery，SWOGHP対応）近将来，POWER生存時間解析機能加期待75優越性非劣性劣与勝zzdhdhhhZ2212zzdhdhhhZ221276da
29、ta inferiority;alpha=0.05;beta=0.20;t=5;pc=0.65;pd=0.80;h2=-log(pd)/t;h1=-log(pc)/t;hr=h2/h1;za=probit(1-alpha/2);zb=probit(1-beta);do delta=0 to-0.3 by-0.05;ef=(za+zb)*2*(hr+delta+1)*2/(2*(hr+delta-1)*2);nf=2*ef/(2-pd-pc);output;end;proc print;run;77alpha beta t pc pd h2 h1 hr alpha beta t pc pd h2
30、 h1 hr zaza zbzb delta delta efef nfnf0.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84 0.00 38.92 141.540.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84 0.00 38.92 141.540.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.05 29.88 108.660.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.05 29.88 108.660.05 0.2 5 0.6
31、5 0.8 0.04 0.09 0.52 1.96 0.84-0.10 23.30 84.710.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.10 23.30 84.710.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.15 18.39 66.860.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.15 18.39 66.860.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.20 14.66 53.300.05 0.2
32、5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.20 14.66 53.300.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.25 11.78 42.820.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.25 11.78 42.820.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.30 9.52 34.620.05 0.2 5 0.65 0.8 0.04 0.09 0.52 1.96 0.84-0.30 9.52 34.6278必要状況
33、）解析検定代一般化検定用）胃代表生存時間分布分布指数分布以外分布想定）群例数（割合減）群以上）比例的効果7980datadata data;r1=-log(0.6500.650)/5 5;r2=-log(0.8000.800)/5 5;do n=142142;do i=1 1 to 10001000;dose=0 0;do j=1 1 to n;t=ranexp(49894989)/r1;censor=2 2;if t gt 5 5 then do t=5 5;censor=0 0;end;output;end;dose=1 1;do j=1 1 to n;t=ranexp(49894989)
34、/r2;censor=2 2;if t gt 5 5 then do t=5 5;censor=0 0;end;output;end;end;end;81ods listing close;procproc lifetestlifetest data=data;time T*censor(0 0);strata dose;by n i;ods output HomTests=out;runrun;ods listing;datadata out;set out;if 0 0Probchisq0.050.05 then sign=1 1;else sign=0 0;procproc freqfr
35、eq;tables test*sign/nopercent nocol;82 表表 :Test:Test*sign sign度数度数|列列|-2Log(LR|Log-Rank|Wilcoxon|-2Log(LR|Log-Rank|Wilcoxon|合計合計|)|)|-+-+-+-+-+-+-+-+0|186|190|194|5700|186|190|194|570|18.60|19.00|19.40|18.60|19.00|19.40|-+-+-+-+-+-+-+-+1|814|810|806|2430 1|814|810|806|2430|81.40|81.00|80.60|81.40|81.00|80.60|-+-+-+-+-+-+-+-+合計合計 1000 1000 10001000 10001000 3000 300083
