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
Introduction Adaptive smooth tests Simulations, real example Summary Outline Introduction 1 Motivation Existing directional and versatile tests Adaptive smooth tests 2 Construction of Neyman’s test Adaptive tests, selection rules Asymptotic behaviour Simulations, real example 3 Simulation study Illustration David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data
Introduction Adaptive smooth tests Simulations, real example Summary Motivation Two-sample problem two groups of survival data ( T 1 , 1 , δ 1 , 1 ) , . . . , ( T 1 , n 1 , δ 1 , n 1 ) ( T 2 , 1 , δ 2 , 1 ) , . . . , ( T 2 , n 2 , δ 2 , n 2 ) T j , i survival times (possibly right-censored) δ j , i event indicators α j ( t ) hazard functions H 0 : α 1 = α 2 David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data
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
Introduction Adaptive smooth tests Simulations, real example Summary Existing directional and versatile tests Weighted logrank test test statistic � τ � τ Y 1 ( t ) ¯ ¯ � d ¯ Y 2 ( t ) − d ¯ � Y 2 ( t ) N 2 ( t ) N 1 ( t ) K ( t ) dU 0 ( t ) = K ( t ) ¯ ¯ ¯ Y ( t ) Y 1 ( t ) 0 0 compares Nelson–Aalen estimators in the two groups G ρ,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
Introduction Adaptive smooth tests Simulations, real example Summary Existing directional and versatile tests Combinations of G ρ,γ tests consider a ‘cluster’ { Z 1 , . . . , Z k } of G ρ,γ statistics, e.g., { G 0 , 0 , G 2 , 0 , G 2 , 2 , G 0 , 2 } combine them T max = max {| Z 1 | , . . . , | Z k |} T sum = | Z 1 | + · · · + | Z k | David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data
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 U 0 ( t ) (asymptotically a Brownian motion in transformed time) � Kolmogorov–Smirnov sup t ∈ [ 0 ,τ ] | U 0 ( t ) | / σ 0 ( τ ) ˆ 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
Introduction Adaptive smooth tests Simulations, real example Summary Construction of Neyman’s test Neyman’s embedding idea null model α 1 = α 2 embedded in α 2 ( t ) = α 1 ( t ) exp { θ T ψ ( t ) } functions ψ 1 ( t ) , . . . , ψ d ( t ) model the hazard ratio ψ j ( t ) basis functions in transformed (standardised) time, e.g., Legendre polynomials, cosines, interval indicators instead of H 0 : α 1 = α 2 versus A : α 1 � = α 2 we have H 0 : θ = 0 versus H d : θ � = 0 David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data
Introduction Adaptive smooth tests Simulations, real example Summary Construction of Neyman’s test Cox model formulation model α 2 ( t ) = α 1 ( t ) exp { θ T ψ ( t ) } can be viewed as a Cox model with time-varying covariates set Z j , i = 1 [ j = 2 ] (second group indicator) then we have a Cox model λ j , i ( t ) = Y j , i ( t ) α ( t ) exp { θ T ψ ( t ) Z j , i } with d artificial covariates ψ 1 ( t ) Z j , i , . . . , ψ d ( t ) Z j , 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
Introduction Adaptive smooth tests Simulations, real example Summary Construction of Neyman’s test Score test statistic partial likelihood score vector � τ Y 1 ( t ) ¯ ¯ � d ¯ Y 2 ( t ) − d ¯ Y 2 ( t ) N 2 ( t ) N 1 ( t ) � U ( τ ) = ψ ( t ) ¯ ¯ ¯ Y ( t ) Y 1 ( t ) 0 is a vector of weighted logrank processes asymptotically zero-mean Gaussian under H 0 score statistic for θ = 0 T d = U ( τ ) T ˆ σ ( τ ) − 1 U ( τ ) → χ 2 d David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data
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 T S = U S ( τ ) T ˆ σ SS ( τ ) − 1 U S ( τ ) Schwarz’s selection criterion (BIC) S = arg max { T C − | C | log n } C ∈S ( 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
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 = S nested = {{ 1 } , { 1 , 2 } , . . . , { 1 , . . . , d }} Claeskens & Hjort (2004, Scand. J. Statist. ): all subsets S = S all = 2 { 1 ,..., d } \ ∅ Janssen (2003, Statist. Decis. ): always include some d 0 directions of high priority S = { C ∪ C 0 : C ∈ S ′ } (wlog C 0 = { 1 , . . . , d 0 } ) David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data
Introduction Adaptive smooth tests Simulations, real example Summary Asymptotic behaviour Asymptotics under H 0 | S | → d ∗ in probability where d ∗ = min {| C | : C ∈ S} (minimum dimension) T S → max { V C ( τ ) T σ CC ( τ ) − 1 V C ( τ ) : C ∈ S , | C | = d ∗ } in distribution specifically for S = S nested with d 0 = 0: T S → χ 2 1 for S = S all with d 0 = 0: T S → max of dependent χ 2 1 for S = S nested with d 0 > 0: T S → χ 2 d 0 for S = S all with d 0 > 0: T S → χ 2 d 0 David Kraus http://www.davidkraus.net/wnar2007/ Adaptive Neyman’s smooth tests of homogeneity of two samples of survival data
Introduction Adaptive smooth tests Simulations, real example Summary Asymptotic behaviour Consistency fixed alternative α 1 ( t ) � = α 2 ( t ) smooth tests (both fixed and data-driven) are consistent if � τ y ∗ y ∗ ψ ∗ ( t ) ¯ 1 ( t )¯ 2 ( t ) ( α 2 ( t ) − α 1 ( t )) dt � = 0 (at least 1 comp.) y ∗ ( t ) ¯ 0 2 , ψ ∗ limits of n − 1 ¯ Y 1 , n − 1 ¯ ( ¯ y ∗ 1 , ¯ y ∗ Y 2 , ψ under the alternative) meaning of the condition: ψ j ’s are not ‘completely wrong’, some of ψ j ’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
Introduction Adaptive smooth tests Simulations, real example Summary Asymptotic behaviour Behaviour under local alternatives local alternatives α 2 ( t ) = α 1 ( t ) exp { n − 1 / 2 η ( t ) } | S | → d ∗ in probability T S → max of noncentral χ 2 d ∗ with S = S nested , d 0 = 0: T S behaves asymptotically like logrank under loc. alt. (the reason for taking d 0 > 0 directions of primary interest) with S = S all , d 0 = 0: T S behaves like T max with d 0 > 0: T S behaves like T d 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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