A joint frailty-copula model between disease progression and death for meta-analysis

Transcription

1 CSA-KSS-JSS Specal Invted Sessons 4 / / 6 A jont fralty-copula model between dsease progresson and death for meta-analyss 3/5/7 Takesh Emura Graduate Insttute of Statstcs Natonal Central Unversty TAIWAN Jont work wth Vrgne Rondeau INSERM Bostatstc Bordeaux Segalen Unversty FRANCE

2 Meta-analyss Survval analyss Part I Revew Jont model & motvatng example Part II Proposed Proposed method -- Copula approach Smulaton Data analyss: -- Meta-analyss of ovaran cancer data

3 Meta-Analyss Synthesze multple ndependent studes N=9 N=57 N=578 Useful to detect small but consstent effect ˇ Treatment effect of chemotherapy on survval n head & neck cancer Pgnon et al. 9 ˇ The effect of CXCL gene on survval n ovaran cancer Ganzfred et al. 3 ˇ The effect of ECRG 4 gene on survval n breast cancer Sabater et al. 3

4 Choce of endponts Tme-to-progresson TTP e.g. recurrence metastass Death OS Two events of nterest Death from any cause Progresson-free survval [ PFS = mn TTP OS ] Meta-analyss analyss on event tmes Head & neck cancer data Pgnon et al. ; 9 Ft separate Cox models on PSF and OS respectvely Ovaran cancer data Ganzfred et al. 3 Ft a Cox model on OS 3 Breast cancer data Sabater et al. Ft separate Cox models on PFS and OS respectvely 4

5 Jont model = a bvarate model for TTP and OS Tme-to-progresson TTP Death OS Bvarate survval models Large lterature In meta-analyss: analyss: Study specfc random effect to explan the heterogenety Bvarate survval analyss Burzykowsk et al. : TTP OS s jontly observed Sem-competng rsks analyss Rondeau et al. : TTP can be dependently censored by OS : TTP s observed f TTP < OS 5

6 Combnng heterogeneous studes usng unobserved random effect called fralty N=9 N=57 N=578 u u u ~ Data structure: f u / / u ug u exp G ndependent studes... G each study contan N subjects j... N Share the same u_ 6

7 Data structure X TTP Recurrence Relapse etc. D OS Death from any cause C Admnstratve censorng e.g. study end Jont fralty model Rondeau et al. r t u ur texp β Z t u u texp β Z hazard for hazard for X D u Study specfc random effect fralty β β Effect on tme - to - progresson Effect on tme - to - death X D

8 Data structure X TTP Recurrence Relapse etc. D OS Death from any cause C Admnstratve censorng e.g.study end Observaton : T T Sem - competng rsks settng : Frst occurng event tme T mn Termnal event tme T Z X mn D... G j... D C C Indcator of progresson I T X Indcator of death I T D N

9 Data structure Entry X Recurrence Fg. Case of D Death C Study end Frst occurng event tme: T mn Termnal event tme : T I T X X D C mn D C D I T D X

10 Meta-analyss for ovaran cancer Table. Data sets n the G 4 ndependent studes of ovaran cancer from Ganzfred et al. 3. Data set The number of observed events GEO accesson Sample sze Relapse Death Censorng number GSE76 N GSE36 N GSE989 N TCGA Total 4 N N

11 4 patterns of event process Relapse Death before censorng Relapse Censorng wthout de Death wthout relapse before censorng Censorng nether relapse nor death

12 Table. Four mutually exclusve cases. Frst occurrng event T T Lkelhood contrbuton Progresson X D Pr X T D T u Progresson X C Pr X T D T u Death D D Pr X T D T u Censorng C C Pr X T D T u

13 log-lkelhood of Rondeau et al. : β G N j β { log logr T exp log T } N N m m u u R T u j j r T f u du where m N j N and m j. Nonparametrc hazard approxmaton va Cubc M-Splne O Sullvan 988; Joly Commenges and Letenneur 998 L r t g r M t L t hm t 3

14 Cubc M-splne bases: Equally spaced knots bases : M t M t M 3 t M 4 t M 5 t 4

15 Baselne hazard va cubc M Baselne hazard va cubc M-splne: splne: t M t M t M t M t M x r

16 Part II: Proposed Methods We generalze the approach of Rondeau to account for the ntra-subject dependence between TTP and OS Copula Nelsen 6 s used as a modelng tool a flexble model for dependence

17 Proposed Idea X TTP Recurrence Relapse etc. D OS Death from any cause Jont fralty model Rondeau et al. r t u ur texp β Z t u u texp β Z X D u tme - to - tme - Stll Independent censorng wthn a cluster Our proposed dea: Relax ths ntra-cluster ndependence assumpton va Copulas progresson to - death D X

18 A Copula Copula approach Proposed C : [] [] [] unquely characterzes the dependence between two contnuous random varables Sklar s Theorem 959: Example : Independence copula: C[ v w] vw Example : Clayton copula Clayton 978 C v w v w / ndependence postvely dependece

19 Fralty Jont fralty-copula model Proposed model Rondeau et al. r t u ur texp β Z t u u texp β Z Copula Pr X model: x D y u C [ exp{ R tme - x to - tme - u progresson to - death } exp{ D y u X } ] where C s a copula Nelsen 6 and R x x u r v u dv y u y v u dv

20 Dependence parameter Copula parameter Related to ntra-class Kendall s tau X D u e.g. Clayton copula: X D u /

21 Why such an elaborate copula model s necessary? Death mmedately after progresson. Strong dependence between TTP and OS Kendall s tau >.5 Only a few covarates consstently measured across studes n meta-analyss Resdual dependence between TTP and OS unadjusted by covarates

22 How ndependent censorng volate? Survval tme Dependency nduced by x x gene ; gven Censorng tme x gene ; not gven Fgure Emura and Chen 4 SMMR Survval TTP and censorng OS tmes usually cannot be condtonally ndependent gven only x regardng as unobserved covarate x

23 The data observed T T for... G and j... N. Frst occurrng event T T Lkelhood contrbuton Progresson X D Pr X T D T u Progresson X C Pr X T D T u Death D D Pr X T D T u Censorng C C Pr X T D T u 3

24 β G N j { logr log β Log-lkelhood proposed r N j T [ R T T u ] [ R T log T T } u where R t R texp βz t texp β Z D Intra-class dependence structure ] D [ R [ T R T T T u ] u s t u C [ exp us exp u t ] ] f u du Heterogenety [ ] D / D [] D / D D D D D [] [ ] [] / where [ ] [] [] D D / s D D / t and D D / st. 4

25 Log-lkelhood proposed Independent copula C v w vw reduces to the log-lkelhood of Rondeau et al. : β G N j β { logr log T exp log T } N N m m u u R T u j j r T f u du where m N j N and m j. 5

26 Penalzed lkelhood wth cubc M-splne Drectly follow Rondeau et al. β β r t dt t dt L L L L M k t M t dt t dt hkh M k t M t k k r r r t dt gk g Optmal smoothng parameters routne Lkelhood cross-valdaton LCV score ˆ ˆ LCV tr{ ˆ H PL H } dt ˆ s the log-lkelhood evaluated at the penalzed MLE ˆ H s the converged Hessan matrx of the penalzed ML estmaton PL 6 Ĥ s the converged Hessan matrx of the un-penalzed ML estmaton

27 Standard error SE 95% confdence nterval for ˆ.96SE ˆ.96 { H ˆ } PL 95% confdence nterval for the baselne hazard r s t ˆ rˆ x.96se{ rˆ x } M xˆ g.96 M x{ HPL } gm x where M t M x M x. L r 7

28 Smulatons: G=5 N= or Smulaton settngs: Fralty: u ~ Gamma / where.5 Covarate: Z ~ Unf Jont fralty-copula model Pr X x D y u [ exp{ R x u } exp{ y u } ] / at X D u. 5. Margnals: R x u u r x exp Z y u u y exp Z where r = and = C ~ Unf 5 5~3% censored subjects 8

29 Table 3. Smulatons wth 5 runs G 5 N N Mean SD SE CP% Mean SD SE CP% CEN =5% = = = = Mean SD SE CP% Mean SD SE CP% CEN =3% = = = = Mean SD SE CP% Mean SD SE CP% CEN =6% = = = = Mean SD SE CP% Mean SD SE CP% CEN =34% = = = = CEN represents the percentage that censorng tme precedes both death and progresson tmes

30 Smulatons: runs G=5 Settng b: =- =- r x and y 3

31 Meta-analyss for ovaran cancer Table. Data sets n the G 4 ndependent studes of ovaran cancer from Ganzfred et al. 3. Data set The number of observed events GEO accesson Sample sze Relapse Death Censorng number GSE76 N GSE36 N GSE989 N TCGA Total 4 N N

32 Synthesze 4 ndependent genomc studes of ovaran cancer N= N=58 N=78 N=557 A recently dscovered CXCL gene expresson Dgnfcant bomarker of OS n ovaran cancer Popple et al. ; Ganzfred et al. 3 Our goal: Confrm ther fndng by our proposed jont model on TTP and OS: r t u ur texp t u u texp CXCX CXCX hazard for hazard for TTP OS 3

33 Result of fttng the jont model Proposed Rondeau et al. exp exp RR for TTP 95%CI RR for OS 95%CI Heterogenety Var u SE Copula parameter SE fxed Kendall s tau / fxed Penalzed log-lkelhood Ganzfred et al. 3 reported RR= for OS based on 4 studes 33

34 Result of fttng the jont model Ganzfred et al. 3 34

35 Summary We proposed a jont fralty-copula model for dependence between TTP and OS Extend the jont fralty model of Rondeau et al. allow ntra-cluster dependence va copulas Data analyss: Proposed method vs. method of Rondeau et al. produces smlar result even for the sgnfcant ntra-cluster dependence Future work Future work Is the method of Rondeau et al. robust aganst the amount of ntra-cluster dependence? We wth Dr. Nakatoch and Murotan are searchng the meta-analyss data of Sabater et al. PLoS ONE Thank you!