Survival Prediction Under Dependent Censoring: A Copula-based Approach

Transcription

1 Survival Prediction Under Dependent Censoring: A Copula-based Approach Yi-Hau Chen Institute of Statistical Science, Academia Sinica 2013 AMMS, National Sun Yat-Sen University December Joint work with Takeshi Emura (National Central University)

2 Survival Analysis survival data for the onset time of some event of interest (disease, death, cancer recurrence...) are commonly collected in studies in medicine and many other fields of science owing to the limitation of observation period and other factors such as dropout of the study subjects, data on survival time T is usually censored by some censoring time U such as the study termination time or the dropout time

3 subject 1 T = T,δ = 1 subject 2 T = U,δ = 0 T > U (censored) time origin

4 the actual survival data we observe takes the form ( T,δ) T = min(t,u), δ = I(T U) one of the aims of survival analysis is to identify factors that may explain and predict survival time T well based on the censored survival data ( T,δ)

5 Censoring Mechanism conventional survival analysis is based on independent censoring assumption, assuming that the censoring time is independent of the survival time, conditional on the covariates may be easily violated in practice the independent censoring assumption is even more stringent in the univariate analysis than in the multivariate analysis since T U X 1,X 2 T U X 1

6 T X 1 dependence induced by X 2 U X 2

7 even if the independent censoring assumption is satisfied with multivariate X, it may not hold for some univariate covariate X

8 Effects of Dependent Censoring estimation of survival rates/regression parameters with wrongly assumed independent censoring is subject to serious bias (Zheng and Klein 1995 Biometrika; Huang and Zhang 2008 Biometrics; Chen 2010 JRSSB) adverse effects on variable (gene) selection

9 Hazard Rates the net hazard rate (given covariate X) for death is defined as h(t X) = Pr( t T t+dt T t,x )/dt the apparent hazard rate in the presence of censoring is h (t X) = Pr( t T t+dt,t U T t,u t,x )/dt which is the hazard rate we can directly estimate from the censored survival data

10 Apparent vs. Net Hazards when T U X, h( X) = h ( X) h( X) h ( X) generally under dependent censoring dependent censoring generally leads to biased estimation of the net hazard rate

11 A Copula Framework of Dependent Censoring by Sklar s theorem, the joint survival function of T and U (conditional on covariate X) can be written as Pr( T > t,u > u X ) = C( Pr(T > t X), Pr(U > u X) ) where C : [0,1] 2 [0,1] is called copula that describes explicitly marginal survivals and dependence structure between T and U

12 Some Examples Clayton copula C α (u,v) = ( u α +v α 1 ) 1/α,(u,v) [0,1] 2,α > 0 Frank copula { C α (u,v) = log α 1+ (αu 1)(α v } 1),(u,v) [0,1] 2, α > 0,α 1 α 1 Gumbel copula C α (u,v) = exp { {( logu) α +( logv) α } 1/α},(u,v) [0,1] 2,α > 1

13 α: dependence parameter (1-1 correspondence to Kendall s τ)

14 Bias Assessment under copula model Pr( T > t,u > u X ) = C α ( S T (t X), S U (u X) ) S T (t X) = Pr(T > t X), S U (u X) = Pr(U > u X) h (t X) = r α (t X) h(t X) r α (t X) = C(1,0) α (S T (t X), S U (u X)) S T (t X) C α (S T (t X), S U (u X)), C (1,0) α (u,v) = C α(u,v) u

15 the apparent effect of X on survival β α(t) = log h (t X = 1) h (t X = 0) = β(t)+logr α(t X = 1) r α (t X = 0) where β(t) = log h(t X=1) h(t X=0) is the net effect of X when C α (u,v) = uv, i.e., T and U are independent, r α ( X) = 1, hence the apparent effect β coincides with the net effect β in general cases the bias arises

16 Clayton Copula Net effect = 1 Net effect = 1 β = Apparent effect % censored Not censored 50% censored 60% censored β = Apparent effect % censored 50% censored 40% censored Not censored α =Association parameter α =Association parameter

17 Frank Copula Net effect = 1 Net effect = 1 β = Apparent effect % censored 50% censored 40% censored Not censored β = Apparent effect % censored Not censored 50% censored 40% censored α =Association parameter α =Association parameter

18 Survival Prediction with High Dimensional Data a recent focus of research due to abundance of high-throughput genomic/genetic data such more detailed personalized data provide potentially useful predictors for survival analysis hope to achieve more accurate prognosis and develop personalized treatment strategies

19 a challenge in statistical analysis due to the p n nature of the data

20 Examples van de Vijver et al. (2002 New England Journal of Medicine) utilized expression profiles from 24,885 genes for 295 breast cancer patients to identify patients who would benefit from adjuvant therapy, leading to a new criterion which reduces patients risk over traditional guidelines based only on histological and clinical characteristics Chen at al. (2007 New England Journal of Medicine) examined expression profiles over 672 genes for 125 non-small-cell lung cancer patients to identify a gene signature closely associated with survival outcome in patients with non-small-cell lung cancer

21 Compound-Covariate (CC) Method for Prediction (Tukey, 1993 Controlled Clinical Trials) genes are pre-selected or screened, one-by one, by univariate (Cox) regression analysis risk score formed by linear combination of pre-selected genes, with the weight of each gene given by univariate regression coefficient estimate an easy way to tackle high-dimensional covariates

22 has been widely adopted in real applications (Beer et al., 2002 Nature Medicine; Chen et al., 2007 New England Journal of Medicine; Matsui et al Clinical Cancer Research)

23 Comparative Performances of Existing Methods comparative studies by Wessels et al. (2002 Bioinformatics), Lai et al. (2006 BMC Bioinfomatics), Lecocke et al. (2006 Cancer Informatics), Sun and Li (2012 Bometrics) concluded that CC often yields consistently better results on microarray datasets than more sophisticated multivariate approaches (PCA, Ridge, Lasso,...)

24 Theoretical Justifications: Shrinkage Method (Emura et al PLoS ONE) under independent censoring and all covariates being independent, we can show that the univariate Cox regression estimator (i) has a limiting value 0, when the true coefficient is 0 (ii) has a limiting value lying between the true coefficient value and 0, when the true coefficient is not 0 univariate estimates are shrinkage of multivariate parameters towards zero avoids over-fitting

25 Theoretical Justifications: Model Averaging (Buckland et al Biometrics) each building model is given by a simple univariate model each model is given an equal weight

26 Gene Selection Accommodating Dependent Censoring a common copula model for each gene: Pr(T > t,u > u X j ) = C α (S T (t X j ),S U (u X j )) a single dependence parameter α proportional hazards model for marginal survival functions of T and U given each covariate: (Λ 0j, Γ 0j : baseline cumulative hazards) S T (t X j ) = exp { Λ 0j (t)exp(β j X j ) } S U (u X j ) = exp { Γ 0j (u)exp(γ j X j ) }

27 Parameter Estimation semiparametric maximum likelihood estimation (Chen 2010 JRSSB): for fixed α, maximizing with respect to Ω j = (β j,γ j,λ 0j,Γ 0j ) the log likelihood il i (Ω j ), where l i (Ω j ) = δ i [ βj X ij +logη 1ij ( T i ;Ω j )+logdλ 0j ( T i ) ] +(1 δ i ) [ γ j X ij +logη 2ij ( T i ;Ω j )+logdγ 0j ( T i ) ] Φ α { exp ( Λ0j ( T i )e β jx j ),exp ( Γ0j ( T i )e γ jx j )}

28 η 1ij (t;ω j ) = Φ (1,0) { α exp( Λ0j ( T i )e β jx j ),exp( Γ 0j ( T i )e γ jx j ) } exp ( Λ 0j ( T i )e β ) jx j η 2ij (t;ω j ) = Φ (0,1) { α exp( Λ0j ( T i )e β jx j ),exp( Γ 0j ( T i )e γ jx j ) } exp ( Γ 0j ( T i )e γ ) jx j Φ α = logc α

29 Prognosis Index ˆβ j (α): the SMLE of β j for fixed α standard error for ˆβ j (α) (by the inverse of observed information matrix) (Chen 2010 JRSSB) gene selection based on the significance test for β j

30 risk score, or prognosis index (PI) for survival prediction for subject i: PI = K j=1 ˆβ j (α)x ij where K is the number of genes selected for prediction when α is chosen as the value leading to independence copula, C α (u,v) = uv, the copula method reduces to that the traditional compound covariate method

31 Determination of the Value of α due to the non-identifiability of α with the censored survival data (Tsiatis 1975 PNAS), the likelihood may provide little information on α a practical approach is to choose α maximizing prediction power we adopt the predictive power measure given by Harrell s concordance measure (c-index) (Harrell et al Statistics in Medicine) and M-fold cross-validation

32 M-fold Cross-validated c-index divide the whole sample into M subsamples of about equal size each time, remove a subsample, and use the remaining subsamples to estimate parameters with a chosen α obtain PI(α) for subjects in the subsample removed, and calculate the c-index (i,k) δ ii( T i < T k )I{PI i (α) > PI k (α)}+δ k I( T k < T i )I{PI k (α) > PI i (α)} (i,k) δ ii( T i < T j )+δ k I( T k < T i )

33 sum the c-index values over M subsamples and obtain CV(α) = M m=1 c m (α) the value α maximizing CV(α) is chosen M = 5 is sufficient for good performance in our experience

34 Simulation n = 100, p = dim(x) = 100 (T,U) follows Pr(T > t,u > u) = ( ) e tαexp(β X) +e uαexp(γ 1/α X) α = 0.5,2,8 (Kendall s τ = 0.2, 0.5, 0.8, respectively) γ = β; 50% censoring

35 the first q = 5, 10, 20 genes have non-zero coefficients (informative genes); the remaining p q genes have zero coefficients (noninformative genes) the analysis is based on Clayton copula and 5-fold cross-validated c-index

36 Predictor Structure scenario 1 (tag genes): each of informative genes is positively correlated to non-informative genes: We have several sets of correlated genes. In each set, there is only one tag gene associated with the survival, while other genes are not associated with the survival given the tag gene

37 X 1 X q T

38 scenario 2 (gene pathway): the informative genes are positively correlated: We have a set of correlated genes jointly associated with the survival

39 X 1 X q T

40 Evaluation Criteria for Gene Selection sensitivity: sensitivity = p j=1 I(P j P (q),β j 0) p j=1 I(β j 0) 100% P j : p value of the Wald s test for H 0 : β j = 0 P (j) : the jth smallest value from {P 1,...,P p } Specificity: specifiity = p j=1 I(P j > P (q),β j = 0) p j=1 I(β 100% j = 0)

41 higher sensitivity (specificity) better ability to identify informative (noninformative) genes

42 Simulation Results (tag gene structure: q = 5, p = 100) β = (0.8,...,0.8,0,...,0) } {{ } 5 }{{} 95 method Kendall s τ sensitivity specificity independence copula copula method improves sensitivity by 9 12% ˆα s identified by CV are

43 Simulation Results (tag gene structure: q = 10, p = 100) β = (0.4,...,0.4,0,...,0) } {{ } 10 }{{} 90 method Kendall s τ sensitivity specificity independence copula copula method improves sensitivity by 10 11% ˆα s identified by CV are

44 Simulation Results (tag gene structure: q = 20, p = 100) β = (0.2,...,0.2, 0.2,..., 0.2,0,...,0) }{{}}{{}}{{} method Kendall s τ sensitivity specificity independence copula copula method improves sensitivity by 5 6% ˆα s identified by CV are

45 Simulation Results (pathway structure: q = 5, p = 100) β = (0.4,...,0.4,0,...,0) } {{ } 5 }{{} 95 method Kendall s τ sensitivity specificity independence copula both methods perform equally well ˆα s identified by CV are

46 Simulation Results (pathway structure: q = 10, p = 100) β = (0.2,...,0.2, 0.2,..., 0.2,0,...,0) }{{}}{{}}{{} method Kendall s τ sensitivity specificity independence copula copula method improves sensitivity by 16 17% ˆα s identified by CV are

47 Simulation Results (pathway structure: q = 20, p = 100) β = (0.1,...,0.1, 0.1,..., 0.1,0,...,0) }{{}}{{}}{{} method Kendall s τ sensitivity specificity independence copula copula method improves sensitivity by 12% ˆα s identified by CV are

48 Summary of Simulation Results univariate gene selection based on the copula framework improves the ability of identifying informative genes, compared with the conventional selection procedure under independent censoring the improvement is more significant in Scenario 2 (correlated informative genes) than in Scenario 1 (independent informative genes) similar conclusions hold for p = 500 similar conclusions hold for the analysis based on Frank copula

49 Non-Small-Cell Lung Cancer Data (Chen at al NEJM) data contains expression values from 672 genes for n = 125 patients (38 died and others censored; 70% censoring) the patients are divided into 63:62 training/test datasets in the same way as Chen et al. p = 485 genes with CV > 3% are included for analysis Chen et al. reported 16 genes most predictive for the survival of NSCLC patients

50 Results: Gene Selection top 16 genes selected by Chen et al. (independent censoring) and by the copula method (6 genes appear in both lists)

51 Independence Copula No. Gene β p value Gene β p value 1 ANXA ZNF DLG MMP ZNF HGF DUSP HCK CPEB NF LCK ERBB STAT NR2F RNF AXL IRF CDC STAT DLG HGF IGF ERBB RBBP NF COX FRAP DUSP MMD ENG HMMR CKMT1A

52 Results: Survival Prediction Kaplan-Meier estimates of survival functions for the good and poor prognosis groups in the test data, classified by the PI values (cutpoint=median) smaller p value for the test (of survival functions) means better survival prediction based on grid search for K (# genes selected for prediction) {10, 20,..., 100}, both the compound covariate and copula methods attain minimum p value at K = 80

53 Results: Kaplan-Meier Plots of Good vs. Poor Prognosis Groups based on 80 genes

54 Survival probability Univariate Cox P value = Months Survival probability Proposed method P value = Months

55 Results: Kaplan-Meier Plots of Good vs. Poor Prognosis Groups based on 16 genes

56 Survival probability Univariate Cox P value = Months Survival probability Proposed method P value = Months

57 Summary univariate gene selection is a convenient and effective tool when p n issue of dependent censoring is more prominent in such univariate analysis dependent censoring may lead to substantial bias for regression coefficient estimation; the bias can be analytically assessed under a copula framework

58 univariate gene selection procedure accommodating dependent censoring can be performed under a copula framework, together with a predictive performance measure for identifying dependence level between survival and censoring times such a method has greater power to identify informative genes and achieves better survival prediction performance, compared with conventional methods based on independent censoring assumption Thank You!!