Shape Constrained Nonparametric Baseline Estimators in the Cox Model - PowerPoint PPT Presentation

Shape Constrained Nonparametric Baseline Estimators in the Cox Model Joint work with Rik Lopuha¨ a (TU Delft) Tina Nane, Center for Science and Technology Studies. Leiden University The Netherlands Leiden University

Basic concepts in survival analysis • Events of interest - death, onset (relapse) of a disease, etc • Let X ∼ F denote the survival time, with density f • Functions that characterize the distribution of X • The survival function S ( x ) = P ( X > x ) • The hazard function P ( x ≤ X < x + ∆ x | X ≥ x ) = f ( x ) λ ( x ) = lim ∆ x S ( x ) ∆ x ↓ 0 � x • The cumulative hazard function Λ( x ) = 0 λ ( u ) d u • Let C ∼ G denote the censoring time • Let Z denote the covariate (age, weight, treatment) Leiden University

The Cox proportional hazards model • Right-censored data ( T i , ∆ i , Z i ) , for i = 1 , . . . , n • T = min ( X , C ) denotes the follow-up time • ∆ = { X ≤ C } is the censoring indicator • The covariate vector Z ∈ R p is time invariant • X | Z ⊥ C | Z • The Cox model λ ( x | z ) = λ 0 ( x ) e β ′ 0 z , where • λ 0 is the underlying baseline hazard function • β 0 ∈ R p is the vector of the underlying regression coefficients Leiden University

Assumptions • X ∼ F , C ∼ G , T ∼ H • F , G are assumed absolutely continuous. • (A.1) Let τ F , τ G and τ H be the end points of the support of F , G , H . Then τ H = τ G < τ F • (A.2) There exists ε > 0 such that � | Z | 2 e 2 β ′ Z � sup < ∞ , ❊ | β − β 0 |≤ ε where | · | denotes the Euclidean norm Leiden University

Estimating monotone baseline hazards in the Cox model • The NPMLE ˆ λ n of a nondecreasing baseline hazard • Let T ( 1 ) ≤ · · · ≤ T ( n ) denote the ordered follow-up times • For β fixed, maximize the (log)likelihood function over all nondecreasing baseline hazards and obtain ˆ λ n ( x ; β ) • zero, for x < T ( 1 ) • constant on [ T ( i ) , T ( i + 1 ) ) , for i = 1 , 2 , . . . , n − 1 • ∞ , for x ≥ T ( n ) • Replace β in ˆ λ n ( x ; β ) by ˆ β n , the maximum partial likelihood estimator • We propose ˆ λ n ( x ) = ˆ λ n ( x ; ˆ β n ) as our estimator of λ 0 Leiden University

Estimating monotone baseline hazards in the Cox model • Grenander-type estimator • 1. Start from the Breslow estimator Λ n of the baseline cumulative hazard Λ 0 3.5 3.0 baseline cumulative hazards estimates 2.5 2.0 1.5 1.0 0.5 0.0 100 200 300 400 500 600 700 survival times Leiden University

Estimating monotone baseline hazards in the Cox model • 2. Take its Greatest Convex Minorant (GCM) � Λ n 3.5 3.0 baseline cumulative hazards estimates 2.5 2.0 1.5 1.0 0.5 0.0 100 200 300 400 500 600 700 survival times Leiden University

Estimating monotone baseline hazards in the Cox model • 3. The Grenander-type estimator ˜ λ n is defined as the left-hand slope of � Λ n 0.020 baseline hazards estimates 0.015 0.010 0.005 0.000 0 200 400 600 survival times Leiden University

Estimating monotone baseline hazards in the Cox model • Another estimator of a nondecreasing baseline hazard was proposed by Chung and Chang (1994) • Consistency: ˆ λ C n ( x ) → λ 0 ( x ) a . s . • No limiting distribution available Leiden University

Estimating monotone baseline hazards in the Cox model • Comparison between the three baseline hazard estimators 1000 Weibull(3/2,1) observations 3.0 LS estimator CC estimator MLE estimator True hazard 2.5 hazard function estimates 2.0 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 survival times Leiden University

Estimating monotone baseline densities in the Cox model • Grenander-type estimator of a monotone baseline density f 0 • Since F 0 ( x ) = 1 − e − Λ 0 ( x ) • We propose F n ( x ) = 1 − e − Λ n ( x ) , where Λ n is the Breslow estimator. • Define the nonincreasing Grenander-type estimator ˜ f n as the left derivative of the Least Concave Majorant (LCM) of F n Leiden University

Pointwise consistency • Theorem 1 (Lopuha¨ a & Nane, 2013a) Assume that (A.1) and (A.2) hold and that λ 0 is nondecreasing on [ 0 , ∞ ) and f 0 is nonincreasing on [ 0 , ∞ ) . Then, for any x 0 ∈ ( 0 , τ H ) , ˆ ˆ λ 0 ( x 0 − ) ≤ lim inf λ n ( x 0 ) ≤ lim sup λ n ( x 0 ) ≤ λ 0 ( x 0 +) , n →∞ n →∞ ˜ ˜ λ 0 ( x 0 − ) ≤ lim inf λ n ( x 0 ) ≤ lim sup λ n ( x 0 ) ≤ λ 0 ( x 0 +) , n →∞ n →∞ ˜ ˜ f 0 ( x 0 +) ≤ lim inf f n ( x 0 ) ≤ lim sup f n ( x 0 ) ≤ f 0 ( x 0 − ) , n →∞ n →∞ with probability one. The values λ 0 ( x 0 − ) , f 0 ( x 0 − ) and λ 0 ( x 0 +) , f 0 ( x 0 +) denote the left (right) limit of the baseline hazard and density function at x 0 . Leiden University

Asymptotic distribution • Typical features for isotonic estimators • n 1 / 3 rate of convergence • non-normal limiting distribution • Groeneboom (1985) recipe 1. Define an inverse process 2. Use the switching relationship 3. Use the Hungarian embedding (KMT construction) to derive the limiting distribution of the inverse process 4. Obtain the limiting distribution of the monotone estimator Leiden University

Asymptotic distribution • For the Grenander-type estimator ˜ λ n 1. Inverse process U n ( a ) = argmin { Λ n ( x ) − ax } , x ∈ [ 0 , T ( n ) ] for a > 0 , where argmin denotes the largest location of the minimum 2. For any a > 0 , the following switching relationship holds U n ( a ) ≥ x ⇔ ˜ λ n ( x ) ≤ a , with probability one Leiden University

Asymptotic distribution • For a fixed x 0 , � n 1 / 3 � � � ˜ P λ n ( x 0 ) − λ 0 ( x 0 ) > a � n 1 / 3 � � � U n ( λ 0 ( x 0 ) + n − 1 / 3 a ) − x 0 = P < 0 • Moreover n 1 / 3 � � U n ( λ 0 ( x 0 ) + n − 1 / 3 a ) − x 0 = argmin { ❩ n ( x ) − ax } , x ∈ I n ( x 0 ) where I n ( x 0 ) = [ − n 1 / 3 x 0 , n 1 / 3 ( T ( n ) − x 0 )] and for x ∈ I n ( x 0 ) ❩ n ( x ) = n 2 / 3 � � � Λ n ( x 0 + n − 1 / 3 x ) − Λ 0 ( x 0 + n − 1 / 3 x ) − [Λ n ( x 0 ) − Λ 0 ( x 0 )] � + Λ 0 ( x 0 + n − 1 / 3 x ) − Λ 0 ( x 0 ) − n − 1 / 3 λ 0 ( x 0 ) x Leiden University

Asymptotic distribution • 3. No embedding available for the Breslow estimator • 3’. Linearization result of the Breslow estimator (Lopuha¨ a & Nane, 2013b) • Let Φ( β 0 , x ) = ❊ [ { T ≥ x } e β ′ 0 Z ] • Theorem 2 (Lopuha¨ a & Nane, 2013a) Assume ( A . 1 ) and ( A . 2 ) and let x 0 ∈ ( 0 , τ H ) . Suppose that λ 0 is nondecreasing on [ 0 , ∞ ) and continuously differentiable in a neighborhood of x 0 , with λ 0 ( x 0 ) � = 0 and λ ′ 0 ( x 0 ) > 0 . Then, � � 1 / 3 � � Φ( β 0 , x 0 ) ˜ n 1 / 3 { W ( t ) + t 2 } , λ n ( x 0 ) − λ 0 ( x 0 ) → d argmin 4 λ 0 ( x 0 ) λ ′ 0 ( x 0 ) t ∈ ❘ where W is a standard two-sided Brownian motion originating from zero. Leiden University

Asymptotic distribution • Theorem 3 (Lopuha¨ a & Nane, 2013a) Assume ( A . 1 ) and ( A . 2 ) and let x 0 ∈ ( 0 , τ H ) . Suppose that λ 0 is nondecreasing on [ 0 , ∞ ) and continuously differentiable in a neighborhood of x 0 , with λ 0 ( x 0 ) � = 0 and λ ′ 0 ( x 0 ) > 0 . Then, � � 1 / 3 � � Φ( β 0 , x 0 ) ˆ n 1 / 3 { W ( t ) + t 2 } , λ n ( x 0 ) − λ 0 ( x 0 ) → d argmin 4 λ 0 ( x 0 ) λ ′ 0 ( x 0 ) t ∈ ❘ where W is a standard two-sided Brownian motion originating from zero. Leiden University

Asymptotic distribution • Theorem 4 (Lopuha¨ a & Nane, 2013a) Assume ( A . 1 ) and ( A . 2 ) and let x 0 ∈ ( 0 , τ H ) . Suppose that f 0 is nonincreasing on [ 0 , ∞ ) and continuously differentiable in a neighborhood of x 0 , with f 0 ( x 0 ) � = 0 and f ′ 0 ( x 0 ) < 0 . Let F 0 be the baseline distribution function. Then, � � 1 / 3 � � Φ( β 0 , x 0 ) n 1 / 3 ˜ f n ( x 0 ) − f 0 ( x 0 ) 4 f 0 ( x 0 ) f ′ 0 ( x 0 )[ 1 − F 0 ( x 0 )] { W ( t ) + t 2 } , → d argmin t ∈ ❘ where W is a standard two-sided Brownian motion originating from zero. Leiden University

Hypothesis testing • Likelihood ratio test of H 0 : λ 0 ( x 0 ) = θ 0 versus H 1 : λ 0 ( x 0 ) � = θ 0 • Let L β ( λ 0 ) the (log)likelihood function • For fixed β ∈ ❘ p , x 0 ∈ ( 0 , τ H ) and θ 0 ∈ ( 0 , ∞ ) fixed maximize L β ( λ 0 ) under H 0 • Propose ˆ n ( x ) = ˆ n ( x ; ˆ λ 0 λ 0 β n ) as the constrained NPMLE Leiden University

Hypothesis testing • Recall ˆ λ n ( x ) = ˆ λ n ( x ; ˆ β n ) , the unconstrained NPMLE estimator of a nonincreasing λ 0 • By Theorem 3 , � 4 λ 0 ( x 0 ) λ ′ � 1 / 3 n 1 / 3 � ˆ � 0 ( x 0 ) { W ( t ) + t 2 } λ n ( x 0 ) − λ 0 ( x 0 ) → d argmin Φ( β 0 , x 0 ) t ∈ ❘ { W ( t ) + t 2 } ≡ C ( x 0 ) argmin t ∈ ❘ ≡ C ( x 0 ) g ( 0 ) , 2 where g(x) is the slope at x of the GCM of { ❲ ( t ) + t 2 } Leiden University

Hypothesis testing • Similarly, it can be shown that n 1 / 3 � ˆ � C ( x 0 ) λ 0 g 0 ( 0 ) , n ( x 0 ) − λ 0 ( x 0 ) → d 2 where g 0 is the constrained slope process of the GCM of { ❲ ( t ) + t 2 } Leiden University

Hypothesis testing • Banerjee & Wellner (2001) Leiden University

Hypothesis testing • Banerjee & Wellner (2001) Leiden University