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

▶
shape constrained nonparametric baseline estimators in
SMART_READER_LITE
LIVE PREVIEW

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

Dec 02, 2022 304 likes •597 views

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

slide-1
SLIDE 1

Leiden University

The Netherlands

Leiden University

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.

slide-2
SLIDE 2

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

λ(x) = lim

∆x↓0

P(x ≤ X < x + ∆x|X ≥ x) ∆x = f (x) S(x)

  • The cumulative hazard function Λ(x) =

x

0 λ(u) du

  • Let C ∼ G denote the censoring time
  • Let Z denote the covariate (age, weight, treatment)
slide-3
SLIDE 3

Leiden University

The Cox proportional hazards model

  • Right-censored data (Ti, ∆i, Zi), for i = 1, . . . , n
  • T = min(X, C) denotes the follow-up time
  • ∆ = {X ≤ C} is the censoring indicator
  • The covariate vector Z ∈ Rp is time invariant
  • X|Z⊥C|Z
  • The Cox model

λ(x|z) = λ0(x)eβ′

0z,

where

  • λ0 is the underlying baseline hazard function
  • β0 ∈ Rp is the vector of the underlying regression coefficients
slide-4
SLIDE 4

Leiden University

Assumptions

  • X ∼ F, C ∼ G, T ∼ H
  • F, G are assumed absolutely continuous.
  • (A.1) Let τF, τG and τH be the end points of the support of

F, G, H. Then τH = τG < τF

  • (A.2) There exists ε > 0 such that

sup

|β−β0|≤ε

❊

  • |Z|2 e2β′Z

< ∞, where | · | denotes the Euclidean norm

slide-5
SLIDE 5

Leiden University

Estimating monotone baseline hazards in the Cox model

  • The NPMLE ˆ

λn 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 λn(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

λn(x; β) by ˆ βn, the maximum partial likelihood estimator

  • We propose ˆ

λn(x) = ˆ λn(x; ˆ βn) as our estimator of λ0

slide-6
SLIDE 6

Leiden University

Estimating monotone baseline hazards in the Cox model

  • Grenander-type estimator
  • 1. Start from the Breslow estimator Λn of the

baseline cumulative hazard Λ0

100 200 300 400 500 600 700 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 survival times baseline cumulative hazards estimates

slide-7
SLIDE 7

Leiden University

Estimating monotone baseline hazards in the Cox model

  • 2. Take its Greatest Convex Minorant (GCM)

Λn

100 200 300 400 500 600 700 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 survival times baseline cumulative hazards estimates

slide-8
SLIDE 8

Leiden University

Estimating monotone baseline hazards in the Cox model

  • 3. The Grenander-type estimator ˜

λn is defined as the left-hand slope of Λn

200 400 600 0.000 0.005 0.010 0.015 0.020 survival times baseline hazards estimates

slide-9
SLIDE 9

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

λC

n (x) → λ0(x) a.s.

  • No limiting distribution available
slide-10
SLIDE 10

Leiden University

Estimating monotone baseline hazards in the Cox model

  • Comparison between the three baseline hazard estimators

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 survival times hazard function estimates

1000 Weibull(3/2,1) observations

LS estimator CC estimator MLE estimator True hazard

slide-11
SLIDE 11

Leiden University

Estimating monotone baseline densities in the Cox model

  • Grenander-type estimator of a monotone baseline density f0
  • Since

F0(x) = 1 − e−Λ0(x)

  • We propose

Fn(x) = 1 − e−Λn(x), where Λn is the Breslow estimator.

  • Define the nonincreasing Grenander-type estimator ˜

fn as the left derivative of the Least Concave Majorant (LCM) of Fn

slide-12
SLIDE 12

Leiden University

Pointwise consistency

  • Theorem 1 (Lopuha¨

a & Nane, 2013a)

Assume that (A.1) and (A.2) hold and that λ0 is nondecreasing on [0, ∞) and f0 is nonincreasing on [0, ∞). Then, for any x0 ∈ (0, τH), λ0(x0−) ≤ lim inf

n→∞

ˆ λn(x0) ≤ lim sup

n→∞

ˆ λn(x0) ≤ λ0(x0+), λ0(x0−) ≤ lim inf

n→∞

˜ λn(x0) ≤ lim sup

n→∞

˜ λn(x0) ≤ λ0(x0+), f0(x0+) ≤ lim inf

n→∞

˜ fn(x0) ≤ lim sup

n→∞

˜ fn(x0) ≤ f0(x0−), with probability one. The values λ0(x0−), f0(x0−) and λ0(x0+), f0(x0+) denote the left (right) limit of the baseline hazard and density function at x0.

slide-13
SLIDE 13

Leiden University

Asymptotic distribution

  • Typical features for isotonic estimators
  • n1/3 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
slide-14
SLIDE 14

Leiden University

Asymptotic distribution

  • For the Grenander-type estimator ˜

λn

  • 1. Inverse process

Un(a) = argmin

x∈[0,T(n)]

{Λn(x) − ax} , for a > 0, where argmin denotes the largest location of the minimum

  • 2. For any a > 0, the following switching relationship holds

Un(a) ≥ x ⇔ ˜ λn(x) ≤ a, with probability one

slide-15
SLIDE 15

Leiden University

Asymptotic distribution

  • For a fixed x0,

P

  • n1/3

˜ λn(x0) − λ0(x0)

  • > a
  • = P
  • n1/3

Un(λ0(x0) + n−1/3a) − x0

  • < 0
  • Moreover

n1/3 Un(λ0(x0) + n−1/3a) − x0

  • = argmin

x∈In(x0)

{❩n(x) − ax}, where In(x0) = [−n1/3x0, n1/3(T(n) − x0)] and for x ∈ In(x0) ❩n(x) = n2/3 Λn(x0 + n−1/3x) − Λ0(x0 + n−1/3x)

  • − [Λn(x0) − Λ0(x0)]

+ Λ0(x0 + n−1/3x) − Λ0(x0) − n−1/3λ0(x0)x

slide-16
SLIDE 16

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(β0, x) = ❊[{T ≥ x}eβ′

0Z]

  • Theorem 2 (Lopuha¨

a & Nane, 2013a)

Assume (A.1) and (A.2) and let x0 ∈ (0, τH). Suppose that λ0 is nondecreasing on [0, ∞) and continuously differentiable in a neighborhood of x0, with λ0(x0) = 0 and λ′

0(x0) > 0. Then,

n1/3

  • Φ(β0, x0)

4λ0(x0)λ′

0(x0)

1/3 ˜ λn(x0) − λ0(x0)

  • →d argmin

t∈❘

{W (t) + t2}, where W is a standard two-sided Brownian motion

  • riginating from zero.
slide-17
SLIDE 17

Leiden University

Asymptotic distribution

  • Theorem 3 (Lopuha¨

a & Nane, 2013a)

Assume (A.1) and (A.2) and let x0 ∈ (0, τH). Suppose that λ0 is nondecreasing on [0, ∞) and continuously differentiable in a neighborhood of x0, with λ0(x0) = 0 and λ′

0(x0) > 0. Then,

n1/3

  • Φ(β0, x0)

4λ0(x0)λ′

0(x0)

1/3 ˆ λn(x0) − λ0(x0)

  • →d argmin

t∈❘

{W (t) + t2}, where W is a standard two-sided Brownian motion originating from zero.

slide-18
SLIDE 18

Leiden University

Asymptotic distribution

  • Theorem 4 (Lopuha¨

a & Nane, 2013a)

Assume (A.1) and (A.2) and let x0 ∈ (0, τH). Suppose that f0 is nonincreasing on [0, ∞) and continuously differentiable in a neighborhood of x0, with f0(x0) = 0 and f ′

0(x0) < 0. Let F0 be the

baseline distribution function. Then, n1/3

  • Φ(β0, x0)

4f0(x0)f ′

0(x0)[1 − F0(x0)]

1/3 ˜ fn(x0) − f0(x0)

  • →d argmin

t∈❘

{W (t) + t2}, where W is a standard two-sided Brownian motion originating from zero.

slide-19
SLIDE 19

Leiden University

Hypothesis testing

  • Likelihood ratio test of H0 : λ0(x0) = θ0 versus

H1 : λ0(x0) = θ0

  • Let Lβ(λ0) the (log)likelihood function
  • For fixed β ∈ ❘p, x0 ∈ (0, τH) and θ0 ∈ (0, ∞) fixed

maximize Lβ(λ0) under H0

  • Propose ˆ

λ0

n(x) = ˆ

λ0

n(x; ˆ

βn) as the constrained NPMLE

slide-20
SLIDE 20

Leiden University

Hypothesis testing

  • Recall ˆ

λn(x) = ˆ λn(x; ˆ βn), the unconstrained NPMLE estimator of a nonincreasing λ0

  • By Theorem 3,

n1/3 ˆ λn(x0) − λ0(x0)

  • →d

4λ0(x0)λ′

0(x0)

Φ(β0, x0)

1/3

argmin

t∈❘

{W (t) + t2} ≡ C(x0)argmin

t∈❘

{W (t) + t2} ≡ C(x0) 2 g(0), where g(x) is the slope at x of the GCM of {❲(t) + t2}

slide-21
SLIDE 21

Leiden University

Hypothesis testing

  • Similarly, it can be shown that

n1/3 ˆ λ0

n(x0) − λ0(x0)

  • →d

C(x0) 2 g0(0), where g0 is the constrained slope process of the GCM of {❲(t) + t2}

slide-22
SLIDE 22

Leiden University

Hypothesis testing

  • Banerjee & Wellner (2001)
slide-23
SLIDE 23

Leiden University

Hypothesis testing

  • Banerjee & Wellner (2001)
slide-24
SLIDE 24

Leiden University

Hypothesis testing

  • Replacing β by ˆ

βn in Lβ(λ0) gives 2 log ξn(θ0) = 2Lˆ

βn(ˆ

λn) − 2Lˆ

βn(ˆ

λ0

n)

  • Theorem 5 (Nane, 2013)

Suppose that (A.1) and (A.2) hold and let x0 ∈ (0, τH). Assume that λ0 is nondecreasing on [0, ∞) and continuously differentiable in a neighborhood of x0, with λ0(x0) = 0 and λ′

0(x0) > 0. Then, under the null hypothesis,

2 log ξn(θ0) →d ❉, where ❉ =

[(g(u))2 − (g0(u))2]du.

slide-25
SLIDE 25

Leiden University

Interval estimation

  • Pointwise confidence intervals for λ0(x0)
  • Likelihood ratio method

{θ : 2 log ξn(θ) ≤ q(❉, 1 − α) where q(❉, 1 − α) is the (1 − α)th quantile of ❉ (Banerjee & Wellner, 2005)

  • Asymptotic distribution

[ˆ λn(x0)−n−1/3 ˆ Cn(x0)q(❩, 1−α/2), ˆ λn(x0)+n−1/3 ˆ Cn(x0)q(❩, 1−α/2)], where ❩ = argmin{❲(t) + t2} and q(❩, 1 − α/2) is the (1 − α/2)th quantile of the distribution ❩ (Groeneboom & Wellner, 2001)

slide-26
SLIDE 26

Leiden University

Current research

  • Citation analysis
  • Event of interest - time to first citation, 5th citation, etc
  • Time frame - first five years after publication (field specific)
  • Censored data
  • Covariates - document type, collaboration type, number of

authors, number of pages, etc

  • Nondecreasing baseline hazard
slide-27
SLIDE 27

Leiden University

References

  • BANERJEE, M. and WELLNER, J.A. 2001. Likelihood ratio tests for monotone functions.

The Annals of Statistics. Vol 29, No. 6, 1699 − 1731.

  • BANERJEE, M. and WELLNER, J.A. 2005. Score statistics for current status data:

comparisons with likelihood ratio and Wald statistics. International Journal od

  • Biostatistics. Vol 1, Art. 1, 29 pp. (electronic).
  • CHUNG, D. and CHANG, M.N. 1994. An isotonic estimator of the baseline hazard

function in Cox’s regression model under order restriction. Statistics & Probability letters.

  • No. 21, 223 − 228.
  • GREONEBOOM, P. 1985. Estimating a monotone density, In proceedings of the Berkeley

conference in honor of Jerzy Neyman and Jack Kiefer , 2, 539 − 555.

  • GREONEBOOM, P. and WELLNER, J.A. 2001. Computing Chernoff’s distribution.

Journal of Computational and Graphical Statistics. No. 10, 388 − 400.

  • KOML ´

OS,J.,MAJOR,P. and TUSN´ ADY, G. 1975. An approximation of partial sums of independent r.v.’s and the sample d.f. Z.Wahrsch. verw. Gebiete, No. 32, 111 − 131.

  • LOPUHA¨

A, H.P. and NANE, G.F. 2013a. Shape constrained nonparametric estimators of the baseline distribution in Cox proportional hazards model. Scandinavian Journal of

  • Statistics. Vol. 40, No. 3, 619 − 646.
  • LOPUHA¨

A, H.P. and NANE, G.F. 2013b. An asymptotic linear representation for the Breslow estimator. Communications in Statistics - Theory and Methods. No. 42, 1314–1324.

  • NANE, G.F. 2013. A likelihood ratio test for monotone baseline hazard functions in the

Cox model. Submitted. arxiv:1304.1295v1.