Simple techniques for comparing survival functions with interval-censored data

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1 Simple techniques for comparing survival functions with interval-censored data Jinheum Kim, joint with Chung Mo Nam Department of Applied Statistics University of Suwon Comparing survival functions with interval-censored data p. 1/31

2 Contents Introduction Review Proposed tests Simulation studies Application to real examples Summary Comparing survival functions with interval-censored data p. 2/31

3 Interval censored data Failure time: time to an event of interest, denoted by T > 0 T may not be known exactly, but is known only to lie in a subinterval of the real line, (L,R] T is right-censored if R = ; exactly observed if L = R; and interval-censored if L R Right-censored data: T is either right-censored or exactly observed Comparing survival functions with interval-censored data p. 3/31

4 First example Breast cancer cosmesis data taken from Finkelstein and Wolfe (1985, BCS) A retrospective study with a periodic follow-up to compare early breast cancer patients who had been treated with radiation therapy followed by chemotherapy to those treated with radiotherapy alone Follow-up intervals for those patients who were geographically remote were often longer with increasing time after irradiation therapy Comparing survival functions with interval-censored data p. 4/31

5 Data set of first example Observe time until the appearance of breast retraction (monitored every 4 or 6 months) RT only RT+Chemo Patient # (L i,r i ] Patient # (L i,r i ] 1 (45,_ ] 1 (8,12] 2 (25,37] 2 (0,5].. 46 (46,_ ] 48 (48,_ ] Comparing survival functions with interval-censored data p. 5/31

6 Second example Lung cancer post-operative treatment data taken from Yonsei Cancer Center (Study period: Sep/1990-Sep/1994) A retrospective study to compare the patients who had been treated with radiation therapy followed by chemotherapy to those treated with radiotherapy alone after lung resection Comparing survival functions with interval-censored data p. 6/31

7 Data set of second example Observe time to onset of relapse after resection RT only RT+Chemo Patient # (L i,r i ] Patient # (L i,r i ] 1 (12,14] 1 (48,_ ] 2 (2,4] 2 (0,7].. 30 (20,_ ] 28 (13,16] Comparing survival functions with interval-censored data p. 7/31

8 Goal Let S l (t) = Pr(T > t), l = 1,...,p, denote the survival function of T for the lth group To test H 0 : S 1 (t) = = S p (t), t (0, ) vs. H a : Not all survival functions are equal at t In two examples, focus on comparing treatment effect rates between two groups Comparing survival functions with interval-censored data p. 8/31

9 Notations & assumptions Data: (x i,a i = (L i,r i ]), i = 1,...,n x i = (x i1,...,x i,p 1 ) = (1, 0,..., 0),...,(0,...,0, 1) or (0,...,0) for the individuals who come from the 1st,...,(p 1)th,pth group δ i = 0 or 1 depending on whether the ith individual is right-censored or not The distinct endpoints, L i and R i, of A i are ordered and labelled 0 = s 0 < s 1 < < s m = For i = 1,...,n;j = 1,...,m, α ij = 1 if L i < s j R i ; 0, otherwise Interval-censoring mechanism is independent of the failure time and covariates Comparing survival functions with interval-censored data p. 9/31

10 Tests for treatment comparison Finkelstein & Wolfe (1985, BCS) Finkelstein (1986, BCS) Sun (1996, 1997, StatMed) Pan (2000, StatMed) Sun (2001. LDA) Zhao & Sun (2004, StatMed) Comparing survival functions with interval-censored data p. 10/31

11 Finkelstein (1986, BCS) Likelihood L = i Pr(T i A i x i ) = i {S(L i x i ) S(R i x i )} Introduce the Cox model for S(s x) = {S(s)} exp(x β) L is a function of β and γ = log{ logs(s)} Propose a score test for β = 0, based on U = logl β evaluated at β = 0 and ˆγ(0) Comparing survival functions with interval-censored data p. 11/31

12 Finkelstein (1986, BCS): remarks Resemble the usual logrank test for right-censored data in two sample case, i.e. where U = j (d 1j d j n 1j /n j ), n j = i w rij, d j = i w dij ; n 1j = i x i w rij, d 1j = i x i w dij w rij = k j α ikˆp k / k α ikˆp k, w dij = α ij ˆp j / k α ikˆp k The observed Fisher information matrix corresponding to γ, which need be inverted, could have too many zero off-diagonal entries Comparing survival functions with interval-censored data p. 12/31

13 Sun (1996, StatMed) An interval-censoring version of the usual logrank test for discrete interval-censored data Plug-in d j,n j,d 1j, and n 1j of Finkelstein (1986) into the usual logrank test There was a chance to overestimate these quantities and his test may not reduce to the usual logrank test while right-censored data are available (Zhao & Sun, 2004) Comparing survival functions with interval-censored data p. 13/31

14 Goal Develop a nonparametric test not to require the EM algorithm Develop better estimates of the numbers of failures and of individuals in risk set Always be accessible to estimated null variance regardless of the number of distinct observed times Comparing survival functions with interval-censored data p. 14/31

15 Notations For exact or right-censored observations, re-define L i as the largest among s j s less than L i and R i as L i m i ( 1) : total number of s j s included in A i, i.e. m i = j α ij A i = I i1 I i1 +1 I i1 +m i 1, where I j = (s j 1,s j ] and i 1 {1,...,m} R j : a pseudo risk set of all individuals who have a nonzero probability of being at risk in I j D j : a pseudo failure set of all individuals who have a nonzero probability of failing in I j Comparing survival functions with interval-censored data p. 15/31

16 Uniform assumption and weights Under H 0, the true failure time of the ith individual, given A i, is uniformly distributed over {s i1,s i1 +1,...,s i1 +m i 1}, i.e. Pr(T i = s k A i ) = 1/m i, k = i 1,i 1 + 1,...,i 1 + m i 1 Under the model, w rij = Pr(T i s j A i, i = 1,...,n) = k j α ik / k α ik : conditional prob. of individual i being at risk in I j w dij = Pr(T i I j A i, i = 1,...,n) = δ i α ij / k α ik : conditional prob. of individual i failing in I j Comparing survival functions with interval-censored data p. 16/31

17 Illustration: artificial data Data Interval-censored:(1,2], (3,5], (4,6] Right-censored: (3, ) (2,3] Exact: (4,4] (3,4] s j s: s 1 =0,1,2,3,4,5,6,s 7 = Covering Interval-censored:(1,2]=I 2, (3,5]=I 4 I 5, (4,6]=I 5 I 6 Right-censored: (3, )=I 3 Exact: (4,4]=I 4 Comparing survival functions with interval-censored data p. 17/31

18 Illustration: interval-censored i m i 1 α ij w rij w dij α ij w rij w dij α ij w rij w dij s j Comparing survival functions with interval-censored data p. 18/31

19 Illustration: right-censored & exact i m i 4 α ij w rij w dij α ij w rij w dij n j d j s j Comparing survival functions with interval-censored data p. 19/31

20 Risk set and death set As in Sun (1996) nonparametric test, plug-in d j,n j,d jl, and n jl into the usual logrank test n j = n i=1 w rij : counts of R j in I j d j = n i=1 w dij : counts of D j in I j n jl = l i w rij : pesudo-counts of individuals at risk from population l in I j d jl = l i w rij : pesudo-counts of failures from population l in I j Remark: For right-censored data, d j,n j,d jl, and n jl reduce to corresponding values in the usual logrank test Comparing survival functions with interval-censored data p. 20/31

21 Proposed test Propose a logrank-type statistic U = (U 1,...,U p 1 ), where U l = m j=1 (d jl d j n jl /n j ), l = 1,...,p 1 To test H 0, propose a logrank-type test based on U, P = U ˆΣ 1 U, where ˆΣ is an estimated covariance of U Use P χ 2 (p 1) approximately under H 0 Comparing survival functions with interval-censored data p. 21/31

22 Alternatives for ˆΣ An ad-hoc version of right-censored data by plugging d j,n j,d jl, and n jl into covariance matrix of usual logrank test statistic Applying multiple imputation method to impute T for a subject with δ = 1 (Pan, 2000; Sun, 2001) Comparing survival functions with interval-censored data p. 22/31

23 Simulation studies: design pars T : generated from Weibull(β,λ) for Group 1; Weibull(β,r h λ) for Group 2 with β=0.5, 1.0, 2.0 and r h =1.0, 1.8, 3.0 λ : determined to satisfy mean failure time of 12 Interval-censored data: L = max(0,t u 1 ),R = T + u 2 with u 1,u 2 : discrete uniform over {1,...,D};D=2, 3, 5 C : censoring indicator from Bernoulli(c p );c p =0,.3,.5 If C=0, set R= Sample size: n=50, 100 Replications: 3000 (with SE=0.004), M =50 Comparing survival functions with interval-censored data p. 23/31

24 Results: level when n=50 D 2 5 β c p LR U P na P mi LR U P na P mi Comparing survival functions with interval-censored data p. 24/31

25 Results: power when n=50 D 2 5 β c p r h LR U P na P mi LR U P na P mi Comparing survival functions with interval-censored data p. 25/31

26 First example: revisit Breast cosmesis data Treatment n Right Cen.(%) RT only RT+Chemo Total Test Statistic p value U P na P mi Comparing survival functions with interval-censored data p. 26/31

27 First example: survival curves Survival Curves by Treatment Group RT+Chemo RT Only 0.8 Estimated Survival Probability Survival Times (in Months) Comparing survival functions with interval-censored data p. 27/31

28 Second example: revisit Lung cancer post-operative data Treatment n Right Cen.(%) RT only RT+Chemo Total Test Statistic p value Score P na P mi Comparing survival functions with interval-censored data p. 28/31

29 Second example: survival curves Survival Curves by Treatment Group RT+Chemo RT Only 0.8 Estimated Survival Probability Survival Times (in Months) Comparing survival functions with interval-censored data p. 29/31

30 Concluding remarks Proposed a nonparametric test for comparing survival functions with interval-censored data The size of proposed test is well controlled; its power is comparable to that of efficient logrank test under proportional hazards Extend to more complicated censoring scheme, for example, truncated and doubly interval-censored (e.g., incubation time in AIDS study) Comparing survival functions with interval-censored data p. 30/31

31 Thank you Comparing survival functions with interval-censored data p. 31/31

Part III. Hypothesis Testing. III.1. Log-rank Test for Right-censored Failure Time Data

Part III. Hypothesis Testing. III.1. Log-rank Test for Right-censored Failure Time Data 1 Part III. Hypothesis Testing III.1. Log-rank Test for Right-censored Failure Time Data Consider a survival study consisting of n independent subjects from p different populations with survival functions