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Introduction Adaptive smooth tests Simulations, real example Summary Adaptive Neymans smooth tests of homogeneity of two samples of survival data David Kraus Institute of Information Theory and Automation, Prague & Charles University

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Introduction Adaptive smooth tests Simulations, real example Summary

Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

David Kraus

Institute of Information Theory and Automation, Prague & Charles University in Prague, Dept of Statistics http://www.davidkraus.net/wnar2007/

WNAR/IMS 2007 Irvine, 24–27 June 2007

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary

Outline

1

Introduction Motivation Existing directional and versatile tests

2

Adaptive smooth tests Construction of Neyman’s test Adaptive tests, selection rules Asymptotic behaviour

3

Simulations, real example Simulation study Illustration

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Motivation

Two-sample problem

two groups of survival data (T1,1, δ1,1), . . . , (T1,n1, δ1,n1) (T2,1, δ2,1), . . . , (T2,n2, δ2,n2) Tj,i survival times (possibly right-censored) δj,i event indicators αj(t) hazard functions H0: α1 = α2

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Motivation

Motivation and aim

directional tests

designed for specific alternatives may fail against other alternatives

existing omnibus tests

consistent against arbitrary alternatives sometimes weak in small samples

aim: test with robust power

should not fail against a wide range of alternatives should not lose much to directional tests

I propose a new test: Neyman-type smooth test

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Existing directional and versatile tests

Weighted logrank test

test statistic τ K(t)dU0(t) = τ K(t) ¯ Y1(t) ¯ Y2(t) ¯ Y(t) d ¯ N2(t) ¯ Y2(t) − d ¯ N1(t) ¯ Y1(t)

compares Nelson–Aalen estimators in the two groups

Gρ,γ weights K(t) = [ˆ S(t−)]ρ[1 − ˆ S(t−)]γ (ρ, γ ≥ 0) good (often optimal) choices of ρ, γ for proportional hazards, early, middle or late differences of hazards not good for detection of crossing hazards

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Existing directional and versatile tests

Combinations of Gρ,γ tests

consider a ‘cluster’ {Z1, . . . , Zk} of Gρ,γ statistics, e.g., {G0,0, G2,0, G2,2, G0,2} combine them

T max = max{|Z1|, . . . , |Zk|} T sum = |Z1| + · · · + |Zk|

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Existing directional and versatile tests

Kolmogorov–Smirnov and similar tests

use the whole path of the logrank process U0(t) (asymptotically a Brownian motion in transformed time) Kolmogorov–Smirnov supt∈[0,τ] |U0(t)|/

σ0(τ) Cramér–von Mises Anderson–Darling

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Construction of Neyman’s test

Neyman’s embedding idea

null model α1 = α2 embedded in α2(t) = α1(t) exp{θTψ(t)} functions ψ1(t), . . . , ψd(t) model the hazard ratio ψj(t) basis functions in transformed (standardised) time, e.g., Legendre polynomials, cosines, interval indicators instead of H0: α1 = α2 versus A: α1 = α2 we have H0: θ = 0 versus Hd: θ = 0

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Construction of Neyman’s test

Cox model formulation

model α2(t) = α1(t) exp{θTψ(t)} can be viewed as a Cox model with time-varying covariates set Zj,i = 1[j=2] (second group indicator) then we have a Cox model λj,i(t) = Yj,i(t)α(t) exp{θTψ(t)Zj,i} with d artificial covariates ψ1(t)Zj,i, . . . , ψd(t)Zj,i their significance is to be tested

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Construction of Neyman’s test

Score test statistic

partial likelihood score vector U(τ) = τ ψ(t) ¯ Y1(t) ¯ Y2(t) ¯ Y(t) d ¯ N2(t) ¯ Y2(t) − d ¯ N1(t) ¯ Y1(t)

is a vector of weighted logrank processes

asymptotically zero-mean Gaussian under H0 score statistic for θ = 0 Td = U(τ)Tˆ σ(τ)−1U(τ) → χ2

d

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Adaptive tests, selection rules

Data-driven test

how to choose basis functions? which and how many? idea of adaptive tests

1 select the most likely alternative:

choose a subset S ⊂ {1, . . . , d}

2 test against that alternative:

smooth test with selected functions TS = US(τ)Tˆ σSS(τ)−1US(τ)

Schwarz’s selection criterion (BIC) S = arg max

C∈S

{TC − |C| log n} (S is a class of nonempty subsets of {1, . . . , d})

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Adaptive tests, selection rules

Classes of subsets

S ⊂ 2{1,...,d} \ ∅ (∅ must be excluded because of the heavy penalty) Ledwina (1994, JASA): nested subsets S = Snested = {{1}, {1, 2}, . . . , {1, . . . , d}} Claeskens & Hjort (2004, Scand. J. Statist.): all subsets S = Sall = 2{1,...,d} \ ∅ Janssen (2003, Statist. Decis.): always include some d0 directions of high priority S = {C ∪ C0 : C ∈ S′} (wlog C0 = {1, . . . , d0})

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Asymptotic behaviour

Asymptotics under H0

|S| → d∗ in probability where d∗ = min{|C| : C ∈ S} (minimum dimension) TS → max{VC(τ)TσCC(τ)−1VC(τ) : C ∈ S, |C| = d∗} in distribution specifically

for S = Snested with d0 = 0: TS → χ2

for S = Sall with d0 = 0: TS → max of dependent χ2

for S = Snested with d0 > 0: TS → χ2

for S = Sall with d0 > 0: TS → χ2

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Asymptotic behaviour

Consistency

fixed alternative α1(t) = α2(t) smooth tests (both fixed and data-driven) are consistent if τ ψ∗(t) ¯ y∗

1(t)¯

y∗

2(t)

¯ y∗(t) (α2(t)−α1(t))dt = 0 (at least 1 comp.) (¯ y∗

1, ¯

y∗

2, ψ∗ limits of n−1 ¯

Y1, n−1 ¯ Y2, ψ under the alternative) meaning of the condition: ψj’s are not ‘completely wrong’, some of ψj’s contribute to approximation of hazard ratio (θ = 0 doesn’t solve the limiting estimating equation)

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Asymptotic behaviour

Behaviour under local alternatives

local alternatives α2(t) = α1(t) exp{n−1/2η(t)} |S| → d∗ in probability TS → max of noncentral χ2

d∗

with S = Snested, d0 = 0: TS behaves asymptotically like logrank under loc. alt. (the reason for taking d0 > 0 directions of primary interest) with S = Sall, d0 = 0: TS behaves like T max with d0 > 0: TS behaves like Td0

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Simulation study

Level of asymptotic tests

α1(t) = α2(t) = 1, light censoring, balanced sample sizes d = 4, d0 = 0 d = 7, d0 = 4 Td T nested

S

T all

S

T nested

S

T all

S

n (χ2

4)

(χ2

1)

(max χ2

1)

(χ2

4)

(χ2

4)

50 0.0664 0.1265 0.0701 0.0945 0.1167 100 0.0608 0.0960 0.0600 0.0860 0.1084 400 0.0537 0.0656 0.0516 0.0662 0.0848 two sources of problems

1 insufficient concentration of S in smallest sets 2 inaccuracy of normal approximation for the score

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Simulation study

Convergence of selection criteria under H0

Selection probabilities of two smallest dimensions d = 4, d0 = 0 d = 7, d0 = 4 n |S| = 1 |S| = 2 |S| = 4 |S| = 5 Nested 50 0.9366 0.0518 0.9482 0.0444 subsets 100 0.9607 0.0325 0.9662 0.0294 400 0.9848 0.0134 0.9844 0.0145 All 50 0.9804 0.0176 0.8857 0.1101 subsets 100 0.9891 0.0099 0.9184 0.0796 400 0.9967 0.0030 0.9594 0.0400

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Simulation study

Permutation approximation

test statistics are exchangeable (permutation invariant) under H0 use permutation principle:

1 randomly assign observations to groups

(sampling without replacement)

2 compute the test statistic 3 repeat 1, 2 many times (∼ 2000) 4 p-value: how extreme is the observed statistic?

(based on the sample of permutations)

alternatively: bootstrap (sampling with replacement)

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Simulation study

Comparison with power of other tests

Situations from Fleming, Harrington & O’Sullivan (1987, JASA)

Prop. haz.

Late diff. Middle/early Early Middle G0,0 0.792 0.340 0.232 0.134 0.306 G2,0 0.655 0.056 0.357 0.562 0.173 G0,2 0.517 0.876 0.070 0.097 0.121 G2,2 0.676 0.302 0.241 0.135 0.588 T max 0.734 0.796 0.319 0.474 0.457 KS 0.772 0.274 0.556 0.558 0.468 CM 0.705 0.058 0.478 0.511 0.319 T nested

(d0 = 0) 0.677 0.803 0.541 0.733 0.419 T all

S (d0 = 0)

0.547 0.687 0.751 0.788 0.609 T nested

(d0 = 4) 0.563 0.826 0.769 0.797 0.638 T all

S (d0 = 4)

0.516 0.793 0.766 0.785 0.619

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary Illustration

Example: survival with gastric cancer

500 1000 1500 2000 2500 3000 0.0 0.2 0.4 0.6 0.8 1.0 Chemotherapy Chemotherapy plus radiotherapy

Data from Stablein & Koutrouvelis (1985, Biometrics) T p S G0,0 0.47 0.637 G2,0 2.59 0.009 G0,2 1.99 0.053 G2,2 0.41 0.684 T max 2.59 0.021 KS 2.20 0.047 T8 17.55 0.023 T nested

(d0 = 4)13.59

0.018 {1, 2, 3, 4} T all

(d0 = 4)

13.59 0.03 {1, 2, 3, 4} T nested

(d0 = 0)13.45

0.005 {1, 2} T all

(d0 = 0)

13.32 0.01 {2}

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data

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Introduction Adaptive smooth tests Simulations, real example Summary

Summary

I have proposed

smooth test for α1 = α2 using Neyman’s embedding idea strategies for data-driven choice of basis functions

advantage of the proposed approach

power appears to be more stable than power of other versatile procedures

this presentation and a tech. report available from http://www.davidkraus.net/wnar2007/

David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data