Rank inferences for the accelerated failure time models

RANK INFERENCES FOR THE ACCELERATED FAILURE TIME MODELS ZHOU FANG NATIONAL UNIVERSITY OF SINGAPORE 2014 RANK INFERENCES FOR THE ACCELERATED FAILURE TIME MODELS ZHOU FANG (B.Sc. Wuhan University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF STATISTICS AND APPLIED PROBABILITY NATIONAL UNIVERSITY OF SINGAPORE 2014 ii ACKNOWLEDGEMENTS I am so grateful that I have Dr. Xu Jinfeng and Professor Chen Zehua as my supervisors. They are truly great mentors not only in statistics but also in daily life. I would like to thank them for their guidance, encouragement, time, and endless patience. Next, special acknowledgement goes to the faculties and staff of DSAP, especially Associate Professor Li Jialiang and Mr. Zhang Rong. Anytime I encountered difficulties and tried to seek help from them, I was always warmly welcomed. I also thank all my colleges who helped me to make life easier as a graduate student. I wish to express my gratitude to the university and the department for supporting me through NUS Graduate Research Scholarship. Finally, I will thank my family for their love and support. iii CONTENTS Acknowledgements Summary vii List of Tables x List of Figures Chapter Introduction 1.1 1.2 ii xii Introduction to Survival Analysis . . . . . . . . . . . . . . . . . . . 1.1.1 Survival Data and Right Censoring . . . . . . . . . . . . . . 1.1.2 The Cox Proportional Hazards Model . . . . . . . . . . . . . 1.1.3 The Censored Accelerated Failure Time Model . . . . . . . . Semi-parametric Models . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS iv 1.2.1 Partially Linear Model . . . . . . . . . . . . . . . . . . . . . 1.2.2 Varying Coefficients Model . . . . . . . . . . . . . . . . . . . 1.3 Variable Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Objectives and Organization . . . . . . . . . . . . . . . . . . . . . . 12 Chapter Partially Linear Accelerated Failure Time Model 15 2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Partially Linear Model for Uncensored Data . . . . . . . . . . . . . 17 2.3 Existing Rank-based Methods for the Censored Partially Linear Model 19 2.4 Proposed Local Gehan Method . . . . . . . . . . . . . . . . . . . . 22 2.4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.4.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . 26 2.4.3 Optimal Bandwidth . . . . . . . . . . . . . . . . . . . . . . . 30 2.4.4 Estimation of Limiting Covariance Matrix . . . . . . . . . . 31 Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.5.1 Computation Algorithm . . . . . . . . . . . . . . . . . . . . 33 2.5.2 Bandwidth Selection . . . . . . . . . . . . . . . . . . . . . . 34 2.5.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.5 2.6 Chapter Varying-coefficient Accelerated Failure Time Model 46 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2 Extended Local Gehan Procedure . . . . . . . . . . . . . . . . . . . 50 3.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.2.2 Asymptotic Properties . . . . . . . . . . . . . . . . . . . . . 51 CONTENTS 3.3 3.4 v 3.2.3 Optimal Bandwidth . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.4 Estimation of Limiting Covariance Matrix . . . . . . . . . . 56 Numeric Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.3.1 Computation algorithm . . . . . . . . . . . . . . . . . . . . . 58 3.3.2 Bandwidth selection . . . . . . . . . . . . . . . . . . . . . . 58 3.3.3 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Chapter Variable Selection in the Partially Linear Accelerated Failure Time Model 74 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.2.1 Penalized global Gehan estimator . . . . . . . . . . . . . . . 77 Numerical Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.1 Tuning parameter selection . . . . . . . . . . . . . . . . . . 78 4.3.2 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 4.4 Chapter Conclusion and Discussion 86 Chapter A Theoretical Proof of Chapter 89 A.1 Lemma A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.2 Lemma A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 A.3 Lemma A.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 A.4 Proof of Lemma 2.4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 94 A.5 Proof of Theorem 2.4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 94 CONTENTS vi A.6 Proof of asymptotic normality of α ˆ . . . . . . . . . . . . . . . . . . 101 A.7 Lemma A.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 A.8 Proof of Theorem 2.4.3 . . . . . . . . . . . . . . . . . . . . . . . . . 103 Chapter B Theoretical Proof of Chapter 107 B.1 Lemma B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.2 Lemma B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 B.3 Lemma B.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 B.4 Proof of Lemma 3.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 112 B.5 Proof of Theorem 3.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 113 B.6 Lemma B.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 B.7 Proof of Theorem 3.2.3 . . . . . . . . . . . . . . . . . . . . . . . . . 118 Bibliography 123 vii SUMMARY In many biomedical studies, the response of interest is the time until some event to occur. Such data is usually referred to as lifetime data, failure time data or survival data. By linearly relating the logarithm of survival time to the covariates, a semiparametric accelerated failure time model is often used to examine the covariate effect, providing an easy and direct interpretation. In some applications, the assumption that the covariate effects are linear and constant may be too restrictive. Hence it is desirable to develop more flexible models incorporating nonlinear or varying covariate effects. We consider the partially linear and varying coefficients accelerated failure time models for the analysis of right censored survival data. Rank-based inferential procedures along with the kernel smoothing method are proposed. Firstly, in the censored partially linear accelerated failure time model, we propose a local Summary viii Gehan loss function-based estimation procedure using the kernel smoothing method. The estimation can be obtained through standard ’quantreg packages’ available in R. Under mild regularity conditions, we establish the asymptotical normality of the local Gehan estimator. A resampling procedure is also developed to estimate the limiting covariance matrix. We then extend the local Gehan estimator to two global Gehan estimators. One is obtained by averaging the local ones at all observed points. Another is the minimizer of profile Gehan loss function. Without considering local smoothing, a global Gehan estimator based on piecewise linear approximation to the nonparametric term is also proposed. Simulation results suggest their favorable performance in terms of bias and variance. Compared with the existing methods such as the stratified method and the spline method, the proposed methods exhibit certain advantages. Real data applications are also conducted to illustrate the practical utilities of the proposed methods. Secondly, we extend the local Gehan procedure to the censored varying coefficients accelerated failure time model. The varying coefficients model allows extra dynamics of covariate effects and includes many aforementioned models as special cases. Under mild regularity conditions, we prove that the local Gehan loss-based estimator continues to enjoy good properties. Theoretical properties are established and numerical examples are given for the illustration. In computation, a censored version cross validation method is also proposed to choose the smoothing parameter. In parallel with the partially linear model, a resampling method by random perturbation is proposed for inferential purposes. Finally, we study the problem of variable selection in the censored partially linear accelerated failure time model. Combined with the penalty, the global Gehan loss function with piecewise linear approximation offers a convenient tool for simultaneous estimation and variable selection. Extensive simulation studies were conducted to investigate the properties of the proposed variable selection procedures in terms of both the reduced Summary model error and the probability of identifying the correct model. ix B.5 Proof of Theorem 3.2.2 B.5 113 Proof of Theorem 3.2.2 By Lemma 3.2.1, γn−1 Q∗n (v1 , v2 ) = Bn (v1 , v2 ) + rn (v1 , v2 ). where rn (v1 , v2 ) →p ∗ )T , uniformly over any bounded set. Note that γn−1 Q∗n (v1 , v2 ) is minimized by (θˆn∗T , α ˆ 2n and Bn (v1 , v2 ) is minimized by ∗ T (θ˜n∗T , α ˜ 2n ) = −γn−2 A−1 (SnT1 (0, 0), Sn2 (0, 0))T . By similar argument as in the proof of Theorem 2.4.2, we can obtain ∗ ∗ T (θˆn∗T , α ˆ 2n ) = (θ˜n∗T , α ˜ 2n ) + op (1). This implies the asymptotic representation (3.9). We next show the asymptotic normality ˆ(z0 ). From (3.9), we have of a √ nh(ˆ a(z0 ) − a(z0 )) = −γn−2 [4τ υf (z0 )pΣ(z0 )]−1 Sn11 (0, 0) + op (1), (B.1) where Sn11 (0, 0) = 2γn [n(n − 1)]−1 ∆i I(εi + δi (z0 ) ≤ εj + δj (z0 ))(Xi − Xj )Kh (Zi − z0 ) i=j Kh (Zj − z0 ) + ∆j I(εj + δj (z0 ) ≤ εi + δi (z0 ))(Xj − Xi )Kh (Zj − z0 )Kh (Zi − z0 ) Thus, we can write −γn−2 Sn11 (0, 0) = Sna1 (0, 0) + Sna2 (0, 0), where Sna1 (0, 0) = 2γn−1 [n(n − 1)]−1 ∆i [I(εi ≤ εj )](Xj − Xi )Kh (Zi − z0 )Kh (Zj − z0 ) i=j +∆j [I(εj ≤ εi )](Xi − Xj )Kh (Zj − z0 )Kh (Zi − z0 ) B.5 Proof of Theorem 3.2.2 114 Sna2 (0, 0) = 2γn−1 [n(n − 1)]−1 ∆i [I(εi + δi (z0 ) ≤ εj + δj (z0 )) − I(εi ≤ εj )] i=j (Xj − Xi )Kh (Zi − z0 )Kh (Zj − z0 ) + ∆j [I(εj + δj (z0 ) ≤ εi + δi (z0 )) −I(εj ≤ εi )](Xi − Xj )Kh (Zj − z0 )Kh (Zi − z0 ) We next prove that Sna1 (0, 0) → N (0, υν0 f (z0 )Σ(z0 )) (B.2) where Σ(z0 ) = E(Xi XTi |Zi = z0 ). Note that we can write √ Sna1 (0, 0) = n[n(n − 1)]−1 hn (Di , Dj ) i=j where hn (Di , Dj ) = wn (Di , Dj ) + wn (Dj , Di ) is symmetric with wn (Di , Dj ) = h−3/2 ∆i [I(εi ≤ εj )](Xj − Xi )K( Z j − z0 Z i − z0 )K( ) h h Similarly to the arguments in the proof of Lemma B.3, it can be shown that E[ hn (Di , Dj ) ] = o(n) By Lemma B.1, this implies that hn (Di , Dj ) = E[hn (Di , Dj )|Di ] + op (1), Sna1 (0, 0) = √ n(2n−1 n i=1 rn (Di ) + op (1)) since it is easy to check that r¯n = 0. We have rn (Di ) = E[hn (Di , Dj )|Di ] = E(wn (Di , Dj )|Di ) + E(wn (Dj , Di )|Di ) Zj − z Zi − z )∆i E{(Xj − Xi )K( )|Xi , Zi , Ui∗ , εi } h h Zj − z Z i − z0 +h−3/2 G(εi )K( )E{∆j (Xi − Xj )K( )|Xi , Zi , Ui∗ , εi } h h Z i − z0 = h−1/2 (G(εi ) − 1)∆i K( )[( K(t)f (z0 + th)dt)Xi h = h−3/2 (1 − G(εi ))K( B.5 Proof of Theorem 3.2.2 − E(Xj |Zj = z0 + th)K(t)f (z0 + th)dt] h−1/2 G(εi ) − 115 G(u)h(u)duK( Zi − z )[( h K(t)f (z0 + th)dt)Xi E(Xj |Zj = z0 + th)K(t)f (z0 + th)dt] Under condition (C3), E(Xi |Zi = z0 ) = 0, rn (Di ) → h−1/2 (G(εi ) − 1)K( +h−1/2 G(εi ) Zi − z )[( h G(u)h(u)duK( K(t)f (z0 + th)dt)Xi ∆i ] Z i − z0 )[( h K(t)f (z0 + th)dt)Xi ] Furthermore, E[rn (Di )rn (Di )T ] → E{h−1 (G(εi ) − 1)2 K ( Zi − z )[( h K(t)f (z0 + th)dt)Xi ∆i ] [( K(t)f (z0 + th)dt)Xi ∆i ]T + h−1 G2 (εi )( [( K(t)f (z0 + th)dt)Xi ][( K 2( Zi − z )[( h Z i − z0 )[( h K(t)f (z0 + th)dt)Xi ]T } K(t)f (z0 + th)dt)Xi ∆i ] [( K(t)f (z0 + th)dt)Xi ∆i ]T } + h−1 EG2 (εi )E( [( K(t)f (z0 + th)dt)Xi ][( E{K ( Zi − z )[( h Z i − z0 ) h K(t)f (z0 + th)dt)Xi ]T + 2h−1 G(εi )(G(εi ) − 1) K(t)f (z0 + th)dt)Xi ∆i ][( → h−1 E(G(εi ) − 1)2 E{K ( G(u)h(u)du)2 K ( G(u)h(u)du)2 E{K ( Z i − z0 ) h K(t)f (z0 + th)dt)Xi ]T } + 2h−1 E[G(εi )(G(εi ) − 1)] K(t)f (z0 + th)dt)Xi ∆i ][( K(t)f (z0 + th)dt)Xi ]T } B.5 Proof of Theorem 3.2.2 116 Since G(εi ) ∼ U (0, 1), EG(εi ) = 12 , E(G(εi ))2 = 13 , var(G(εi )) = 12 , E(G(εi ) − 1)2 = EG2 (εi ) − 2EG(εi ) + = 1/3, then f (z0 ) E(∆2i Xi XTi |Zi = z0 + th)f (z0 + th)K (t)dt + υf (z0 ) E(Xi XTi |Zi = z0 + th)f (z0 + th)K (t)dt − f (z0 ) E(∆i Xi XTi |Zi = z0 + th)f (z0 + th)K (t)dt 3 → f (z0 )E(∆2i Xi XTi |Zi = z0 )ν0 + υf (z0 )E(Xi XTi |Zi = z0 )ν0 − f (z0 )E(∆i Xi XTi |Zi = z0 + th)ν0 = υν0 f (z0 )Σ(z0 ). E(rn (Di )rnT (Di )) → So, Sna1 (0, 0) → N (0, υν0 f (z0 )Σ(z0 )) To prove the asymptotic normality of Sna1 (0, 0), it is sufficient to check the LindebergFeller condition: ∀ε > 0, n−1 n T i=1 E{rn (Di )rn (Di ) I( √ rn (Di ) > ε n)} → 0. This can be easily verified by applying the dominated convergence theorem. However, the asymptotic representation of Sna2 (0, 0) is slightly different with that in the proof of Theorem 2.4.2. Next we show that Sna2 (0, 0) = 2h2 [τ υf (z0 )µ2 Σ(z0 )a (z0 ) + o(1)] + op (1). γn We may write Sna2 (0, 0) = 2[n(n − 1)]−1 h∗n (Di , Dj ) i=j (B.3) B.5 Proof of Theorem 3.2.2 117 where h∗n (Di , Dj ) = wn∗ (Di , Dj ) + wn∗ (Dj , Di ) is symmetric with wn∗ (Di , Dj ) = nh−1 γn [∆i (I(εi +δi (z0 ) ≤ εj +δj (z0 ))−I(εi ≤ εj ))](Xj −Xi )K( Z j − z0 Z i − z0 )K( ). h h By applying Lemma B.1, it can be shown that Sna2 (0, 0) = E[h∗n (Di , Dj )] + op (1). Note that δj (z0 ) − δi (z0 ) = 1 [(Zj − z0 )2 XTj − (Zi − z0 )2 XTi ]a (z0 ) + [(Zj − z0 )2 − (Zi − z0 )2 ]φ (z0 ) 2 = o((Zj − z0 )2 ) + o((Zj − z0 )2 ). It follows by using the same arguments in the proof of Lemma B.1 that E[h∗n (Di , Dj )] = 2nh−1 γn E{ (Xj − Xi )K( [G(ε + δj (z0 ) − δi (z0 )) − G(ε)]g(ε)dε Z j − z0 Zi − z )K( )} h h = 2nh−1 γn [τ υ + O(h)]E[(δj (z0 ) − δi (z0 ))(Xj − Xi )K( = G(u)h(u)du Z j − z0 Z i − z0 )K( )](1 + o(1)) h h 2h2 [τ υf (z0 )µ2 Σ(z0 )a (z0 ) + o(1)]. γn This proves (B.3). By combining (B.2) and (B.3) and using the approximation given in (B.1), we obtain (3.10). B.6 Lemma B.4 B.6 118 Lemma B.4 If E[ Hn (Di , Dj ) ] = O(h−2 ), then √ ˆn ) = o(1) almost surely and Un = n(Un − U r¯n + o(1) a.s. Proof : The proof of Powell, Stock and Stoker (1989) for Lemma A.1 suggests that ˆn ] = O(n−2 h−2 ). By theorem 1.3.5 of Serfling (1980), E[ Un − U n i=1 E[ ˆn ] = Un − U ˆn = o(1) almost surely. The second result O(n−1 h−2 ) < ∞. This implies that Un − U ˆn . follows by an application of the strong law of large numbers to U B.7 Proof of Theorem 3.2.3 Let θ∗ and α∗ be defined the same as before. We introduce the reparametrized objective function n n ¯ ∗ (θ∗ ; α∗ ) = 2/[n(n − 1)] Q n ∆i [(εi − γn α2∗ (Zi − z0 )/h − γn θ∗T Ui + δi (z0 )) i=1 j=1 −(εj − γn α2∗ (Zj − z0 )/h − γn θ∗T Uj + δj (z0 ))]− Kh (Zi − z0 )Kh (Zj − z0 )(Wi + Wj ). +∆j [(εj − γn α2∗ (Zj − z0 )/h − γn θ∗T Uj + δj (z0 )) −(εi − γn α2∗ (Zi − z0 )/h − γn θ∗T Ui + δi (z0 ))]− Kh (Zj − z0 )Kh (Zi − z0 )(Wj + Wi ). T (θ ∗ , α∗ ), S ¯n2 (θ∗ , α∗ ))T =( Let S¯n (θ∗ , α2∗ ) = (S¯n1 2 ¯ ∗T ∗ ∗ θ∗ Qn (θ , α2 ), ¯∗ ∗ ∗ T α∗2 Qn (θ , α2 )) , we can show that S¯n (θ∗ , α2∗ ) has a similar local linear approximation as stated in Lemma B.3. B.7 Proof of Theorem 3.2.3 119 T (θ ∗ , α∗ ), where To make the proof concise, we prove this for S¯n1 T S¯n1 (θ∗ , α2∗ ) n n −1 ∆i I[(εi − γn α2∗ (Zi − z0 )/h − γn θ∗T Ui + δi (z0 )) = 2γn [n(n − 1)] i=1 j=1 ≤ (εj − γn α2∗ (Zj − z0 )/h − γn θ∗T Uj + δj (z0 ))](Ui − Uj )Kh (Zi − z0 )Kh (Zj − z0 ) (Wi + Wj ) + ∆j I[(εj − γn α2∗ (Zj − z0 )/h − γn θ∗T Uj + δj (z0 )) ≤ (εi − γn α2∗ (Zi − z0 )/h − γn θ∗T Ui + δi (z0 ))](Uj − Ui ) Kh (Zj − z0 )Kh (Zi − z0 )(Wj + Wi ) Let Un = γn−1 [S¯n1 (θ∗ , α2∗ )−S¯n1 (0, 0)] = 2[n(n−1)]−1 i=j (Wi +Wj )Mn (Di , Dj , θ ∗ , α∗ ), where Mn (Di , Dj , θ∗ , α2∗ ) = [mn (Di , Dj , θ∗ , α2∗ ) + mn (Dj , Di , θ∗ , α2∗ )] and mn (Di , Dj , θ∗ , α2∗ ) = ∆i [I(εi − γn α2∗ (Zi − z0 )/h − γn θ∗T Ui + δi (z0 ) ≤ εj − γn α2∗ (Zj − z0 )/h − γn θ∗T Uj + δj (z0 )) −I(εi + δi (z0 ) ≤ εj + δj (z0 ))] (Ui − Uj )Kh (Zi − z0 )Kh (Zj − z0 ) Note that Un = 4n−1 n i=1 Wi [(n n ∗ ∗ j=1,j=i Mn (Di , Dj , θ , α2 )], − 1)−1 conditional on {Di }ni=1 , this is a weighted average of Wi . Note that n E(Un |{Di }ni=1 ) = 4n−1 E( n [(n − 1)−1 i=1 j=1,j=i n n = 4n−1 Mn (Di , Dj , θ∗ , α2∗ )]Wi |{Di }ni=1 ) [(n − 1)−1 i=1 Mn (Di , Dj , θ∗ , α2∗ )] j=1,j=i = 2[n(n − 1)]−1 Mn (Di , Dj , θ∗ , α2∗ ). i=j n Var(Un |{Di }ni=1 ) = 2n−2 n [(n − 1)−1 i=1 Mn (Di , Dj , θ∗ , α2∗ )]2 . j=1,j=i B.7 Proof of Theorem 3.2.3 120 By Lemma B.4, it can be shown that 2[n(n − 1)]−1 Mn (Di , Dj , θ∗ , α2∗ ) = E(Mn (Di , Dj , θ∗ , α2∗ )) + o(1) = γn A∗ θ∗ + o(1) i=j almost surely, where A∗ = 4τ υf (z0 )diag(Ip , µ2 Ip ) It is also easy to check that 2n−2 n i=1 [(n − 1)−1 Σ(z0 ). n ∗ ∗ j=1,j=i Mn (Di , Dj , θ , α2 )] = o(1) almost surely. Thus for almost surely every sequence {Di }ni=1 , Un = γn A∗ θ∗ + op (1), where op (1) is in the probability space generated by {Wi }ni=1 . Similar to the proofs of Lemma 3.2.1 and the asymptotic representation in Theorem 3.2.2, we can show that for almost surely every sequence {Di }ni=1 , √ ˆ ¯(z0 ) − a(z0 )) = −γn−2 [4τ υf (z0 )Σ(z0 )]−1 S¯n11 (0, 0) + op (1), nh(a (B.4) where op (1) is in the probability space generated by {Wi }ni=1 , and S¯n11 (0, 0) = 2γn [n(n − 1)]−1 (Wi + Wj )[(∆i I(εi + δi (z0 ) ≤ εj + δj (z0 )))(Xi − Xj ) i=j +(∆j I(εj + δj (z0 ) ≤ εi + δi (z0 )))(Xj − Xi )]Kh (Zj − z0 )Kh (Zi − z0 ) The approximation (B.1) can be strengthened to almost surely convergence, i.e., √ nh(ˆ a(z0 ) − a(z0 )) = −γn−2 [4τ υf (z0 )Σ(z0 )]−1 Sn11 (0, 0) + op (1), a.s. (B.5) Combining (B.4) and (B.5), we have that for almost surely every sequence {Di }ni=1 , √ ˆ ¯(z0 ) − a ˆ(z0 )) = −γn−2 [4τ υf (z0 )Σ(z0 )]−1 [S¯n11 (0, 0) − Sn11 (0, 0)] + op (1). nh(a Note that γn−2 [S¯n11 (0, 0) − Sn11 (0, 0)] B.7 Proof of Theorem 3.2.3 = 2γn−1 [n(n − 1)]−1 i=j 121 1 [(Wi − ) + (Wj − )][(∆i I(εi + δi (z0 ) ≤ εj + δj (z0 )))(Xi − Xj ) 2 +(∆j I(εj + δj (z0 ) ≤ εi + δi (z0 )))(Xj − Xi )]Kh (Zi − z0 )Kh (Zj − z0 ) n = 4γn−1 n−1 i=1 (Wi − ){(n − 1)−1 [∆i (I(εi + δi (z0 ) ≤ εj + δj (z0 )))(Xi − Xj ) j=1,j=i +(∆j I(εj + δj (z0 ) ≤ εi + δi (z0 )))(Xj − Xi )]Kh (Zi − z0 )Kh (Zj − z0 ). And E{γn−2 [S¯n11 (0, 0) − Sn11 (0, 0)]|{Di }ni=1 } = 0. We have Var{γn−2 [S¯n11 (0, 0) − Sn11 (0, 0)]|{Di }ni=1 } n = 16γn−2 n−2 (n − 1)−2 [∆i I(εi + δi (z0 ) ≤ εj + δj (z0 ))(Xi − Xj ) { i=1 j=1,j=i +∆j I(εj + δj (z0 ) ≤ εi + δi (z0 ))(Xj − Xi )]Kh (Zi − z0 )Kh (Zj − z0 )}2 = V1 + V2 + V3 . where n V1 = 16γn−2 n−2 (n − 1)−2 h−4 [∆i [I(εi + δi (z0 ) ≤ εj + δj (z0 ))]2 i=1 j=1,j=i T (Xi − Xj )(Xi − Xj ) K ((Zi − z0 )/h)K ((Zj − z0 )/h), n V2 = 16γn−2 n−2 (n −2 −4 − 1) [∆j [I(εj + δj (z0 ) ≤ εi + δi (z0 ))]2 h i=1 j=1,j=i T (Xj − Xi )(Xj − Xi ) K ((Zj − z0 )/h)K ((Zi − z0 )/h), n V3 = 16γn−2 n−2 (n − 1)−2 h−4 i=1 j1 =i j2 =i [∆i I(εi + δi (z0 ) ≤ εj1 + δj1 (z0 ))(Xi − Xj1 ) +∆j1 I(εj1 + δj1 (z0 ) ≤ εi + δi (z0 ))(Xj1 − Xi )] [∆i I(εi + δi (z0 ) ≤ εj2 + δj2 (z0 ))(Xi − Xj2 ) B.7 Proof of Theorem 3.2.3 +∆j2 I(εj2 + δj2 (z0 ) ≤ εi + δi (z0 ))(Xj2 − Xi )] K ((Zi − z0 )/h)K((Zj1 − z0 )/h)K((Zj2 − z0 )/h). 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Journal of the American statistical association, 101, 476, 1418-1429. 128 [...]... Analysis 5 and requires weaker assumptions for censored regression than the former, it provides a more powerful tool for the study of the model (1.5) in practice Therefore, our research mainly focuses on the rank- based inferences for the censored accelerated failure time model On the basis of Jin et al (2003), for the accelerated failure time model (1.5) with the ˜ observed censored data (1.1), we can... of non-parametric terms for the partially linear accelerated failure time model The remainder of this thesis is organized in three main chapters In chapter 2, we study the estimation problem for the accelerated failure time partially linear model We discuss the merits and drawbacks of existing rank- based methods for the model fitting Noticing these drawbacks, we propose a local rank procedure along with... are still surviving at the time when the study is terminated and their true survival times were not recorded This was the case with Stanford heart transplant data, analyzed by Miller and Halpern (1982), etc What was observed for each of the subjects was the censoring indicator taking value 1 for death and 0 for surviving, and the minimum of the survival time and the censoring time See, Fan and Gijbels... CHAPTER 2 Partially Linear Accelerated Failure Time Model 2.1 Motivation Censored data can be analyzed under the accelerated failure time model, if the relationship between the logarithm of failure time and the covariates is assumed to be linear a priori While this has the advantage of producing good model estimates when the true relationship is consistent with the linear assumption, the resulting estimates... alternative to the proportional hazard model, the accelerated failure time model (Kalbfleisch and Prentice, 1980, p3234; Cox and Oakes, 1983, p.64-65), has become more appealing in handling the censored failure time data See also, Wei (1992) 1.1 Introduction to Survival Analysis 1.1.3 4 The Censored Accelerated Failure Time Model In parallel with the parametric proportional hazards model, the censored accelerated. .. nonlinear effect on the time to prostate cancer recurrence (Qi et al., 2011) In these studies, the uncertain effect is treated as the nonparametric component whenever the accelerated failure time partially linear model is employed If the predictor with the uncertain effect is independent of other predictors, the regression coefficients can still be estimated in a pseudo linear model When the independence... (1.3) Thus, the failure time variable T admits the linear regression form log T = β T X + ε, (1.5) with ε = log T0 It is called as accelerated failure time model or AFT model for short This ordinary regression model form (1.5) is easier to interpret the estimates of regression coefficients and requires no proportional hazards assumption as compared to model (1.2) These are important reasons for its increasing... resulting estimates may not be good when the dependence of the response on one of the covariates is uncertain To solve this, the accelerated failure time partially linear model is often used, incorporating 2.1 Motivation a nonparametric component into the accelerated failure time model for more flexibility This model can also be viewed as partially linear model for censored data Deviations from an assumed... popularity The estimation methods and their theoretical properties for the censored accelerated failure time model have been studied extensively, for example, the least square based approach ( Ritov, 1990; Lai and Ying, 1991a) as in Buckley and James (1979) and the rank based approach (Tsiatis, 1990; Lai and Ying, 1991b; Ying, 1993; Jin et al., 2003) proposed initially by Prentice (1978) Since the latter... In the absence of local smoothing, a global Gehan estimator based on piecewise linear approximation of the nonparametric term is also proposed In Chapter 3, we extended the local rank procedure along with kernel smoothing to the varying coefficients model for failure time data This is a pioneer work which allows one to capture the nonlinear interaction effects of covariates on the logarithm of failure time . RANK INFERENCES FOR THE ACCELERATED FAILURE TIME MODELS ZHOU FANG NATIONAL UNIVERSITY OF SINGAPORE 2014 RANK INFERENCES FOR THE ACCELERATED FAILURE TIME MODELS ZHOU FANG (B.Sc assumptions for censored regression than the former, it provides a more powerful tool for the study of the model (1.5) in practice. Therefore, our research mainly focuses on the rank- based inferences for. the rank- based inferences for the censored accelerated failure time model. On the basis of Jin et al. (2003), for the accelerated failure time model (1.5) with the observed censored data (1.1),