The Dantzig Selector The Dantzig Selector in the Cox model A simulation study and a real data set Conclusions The Dantzig selector in Cox’s proportional hazards model A. Antoniadis 1 , P . Fryzlewicz 2 , F . Letué 3 1 LJK/UJF/Université de Grenoble 2 Department of Mathematics/University of Bristol 3 LJK/UPMF/Université de Grenoble Statistical Methods for Post-genomic Data, SMPGD09 Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Dantzig Selector in the Cox model A simulation study and a real data set Conclusions The Dantzig Selector 1 The Dantzig Selector in the regression model The Dantzig Selector in GLM The Dantzig Selector in the Cox model 2 The Cox model : notations The Survival Dantzig Selector (SDS) Algorithm A simulation study and a real data set 3 A simulation study The Dutch breast Cancer data Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Dantzig Selector in the Cox model The Dantzig Selector in the regression model A simulation study and a real data set The Dantzig Selector in GLM Conclusions The Dantzig Selector in the regression model Framework : Model : Y = Z β 0 + ε where Y ∈ R n , β ∈ R p , Z ∈ R n × p Aim : to estimate β 0 , supposing β 0 is S -sparse in the context n << p . Candes and Tao (2007) : ˆ β DS minimizes � β � 1 s.t. | Z jT ( Y − Z β ) | ≤ λ, j = 1 , . . . , p Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Dantzig Selector in the Cox model The Dantzig Selector in the regression model A simulation study and a real data set The Dantzig Selector in GLM Conclusions The Dantzig Selector This procedure works with n << p produces a sparse estimator (variable selection tool) is a standard linear programming problem enables to prove a tight non-asymtotic bound for the L 2 error, up to a log p factor (under Gaussian hypothesis) Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Dantzig Selector in the Cox model The Dantzig Selector in the regression model A simulation study and a real data set The Dantzig Selector in GLM Conclusions GDS James and Radchenko (2009) : Dantzig Selector in the Generalised Linear Models Remark : Z jT ( Y − Z β ) = l ′ j ( β ) where l is the log-likelihood. GDS : ˆ β GDS minimizes � β � 1 s.t. | l ′ j ( β ) | ≤ λ, j = 1 , . . . , p , j ( β ) = Z jT ( Y − g − 1 ( β )) is the derivative of the where l ′ log-likelihood and g the associated link function λ = 0 leads to the maximum likelihood estimator Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Dantzig Selector in the Cox model The Dantzig Selector in the regression model A simulation study and a real data set The Dantzig Selector in GLM Conclusions GDS This procedure also works with n << p produces a sparse estimator (variable selection tool) enables to prove a tight non-asymtotic bound for the L 2 error, up to a log p factor is no more a standard linear programming problem, but an efficient algorithm adapted from DS has the same computational advantages Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Dantzig Selector in the Cox model The Dantzig Selector in the regression model A simulation study and a real data set The Dantzig Selector in GLM Conclusions Our purpose How to adapt this procedure for censored responses ? use the Cox partial log-likelihood instead of the log-likelihood ... Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Cox model : notations The Dantzig Selector in the Cox model The Survival Dantzig Selector (SDS) A simulation study and a real data set Algorithm Conclusions The Cox model : notations Notations : X i survival time of interest U i censoring time X i and U i are supposed independent given Z i ∈ R p Hazard rate of X i given Z i : α Z i ( . ) = e Z i T β 0 α 0 ( . ) where α 0 is left unspecified (semiparametric framework) We observe n i.i.d. copies of (˜ X i , D i , Z i ) where ˜ X i = min ( X i , U i ) is the right censored survival time D i is the censoring indicator Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Cox model : notations The Dantzig Selector in the Cox model The Survival Dantzig Selector (SDS) A simulation study and a real data set Algorithm Conclusions The Cox model : notations � � ˜ N i ( t ) = I X i ≤ t , D i = 1 the censored counting process � � ˜ Y i ( t ) = I X i ≥ t the "at risk" indicator S n ( β, u ) = � n i = 1 Y i ( u ) exp ( Z T i β ) n ( β, u ) = � n S 1 i = 1 Y i ( u ) Z i exp ( Z T i β ) first derivative n ( β, u ) = � n S 2 i = 1 Y i ( u ) Z ⊗ 2 exp ( Z T i β ) second derivative i the Cox’s partial log-likelihood � τ n e Z T i β l ( β ) = 1 � log S n ( β, u ) dN i ( u ) , n 0 i = 1 the score process � τ n ( Z i − S 1 U ( β ) = ∂ l ( β ) = 1 n ( β, u ) � S n ( β, u )) dN i ( u ) . ∂β n 0 i = 1 Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Cox model : notations The Dantzig Selector in the Cox model The Survival Dantzig Selector (SDS) A simulation study and a real data set Algorithm Conclusions The Survival Dantzig Selector (SDS) Aim : to estimate β 0 ∈ R p , supposing it is S -sparse Survival Dantzig Selector (SDS) : ˆ β SDS minimizes � β � 1 s.t. � U ( β ) � ∞ ≤ γ, where U ( . ) is the score process. Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Cox model : notations The Dantzig Selector in the Cox model The Survival Dantzig Selector (SDS) A simulation study and a real data set Algorithm Conclusions Theoretical properties For n large enough, the true parameter β 0 is admissible with a great probability, and � ˆ β SDS � 1 ≤ � β 0 � 1 : Lemma � τ 0 α 0 ( u ) du < + ∞ sup 1 ≤ i ≤ n sup 1 ≤ j ≤ p n | Z i , j | ≤ C S is independent of n p n = O ( n ξ ) , for some ξ > 1 γ = γ n , p = (( 1 + a ) log p n / n ) 1 / 2 , for some a > 0 � � n γ 2 n , p � n − a ξ � P ( � U ( β 0 ) � ∞ > γ n , p ) ≤ p n exp − = O . 2 ( 2 C γ n , p + K ) Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Cox model : notations The Dantzig Selector in the Cox model The Survival Dantzig Selector (SDS) A simulation study and a real data set Algorithm Conclusions Theoretical properties Notations : � τ S n ) ⊗ 2 ( β, u )] d ¯ 0 [ S 2 S n ( β, u ) − ( S 1 N n ( u ) J n ( β ) = n n observed n information matrix ( p × p ) I ( β ) the corresponding asymptotic information matrix ( p × p ) I ( β ) the S × S matrix extracted from I ( β ) corresponding to the non-zero components of β 0 δ S and θ S , S ′ coefficients defined as in Candes and Tao (2007) for the matrix I ( β 0 ) Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Cox model : notations The Dantzig Selector in the Cox model The Survival Dantzig Selector (SDS) A simulation study and a real data set Algorithm Conclusions Theoretical properties Theorem Assumptions of the lemma Coefficients δ S and θ S , S ′ are such that δ 2 S − θ S , 2 S > 0 The matrix I ( β 0 ) is definite positive � γ n , p � � ˆ β SDS − β 0 � 2 ) 2 ≤ O ( n − a ξ ) . 2 > 64 S ( P δ 2 S − θ S , 2 S Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Cox model : notations The Dantzig Selector in the Cox model The Survival Dantzig Selector (SDS) A simulation study and a real data set Algorithm Conclusions Algorithm Idea : approximate locally l ( β ) by a quadratic form Suppose ˆ β ( k ) , U (ˆ β ( k ) ) , J (ˆ β ( k ) ) are available Calculate the pseudo response 1 Y ( k ) = J (ˆ β ( k ) − U (ˆ β ( k ) ) − 1 / 2 ( J (ˆ β ( k ) )ˆ β ( k ) ) Minimize ( Y ( k ) − J (ˆ β ( k ) ) − 1 / 2 β ) T ( Y ( k ) − J (ˆ β ( k ) ) − 1 / 2 β ) 2 using Candès and Tao’s DS to produce ˆ β ( k + 1 ) Repeat until convergence 3 β ( k ) ) − 1 / 2 generalised inverse of the unique square root of J (ˆ ( J (ˆ β ( k ) ) ) Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
The Dantzig Selector The Dantzig Selector in the Cox model A simulation study A simulation study and a real data set The Dutch breast Cancer data Conclusions A simulation study Comparison with 3 other methods Partial Cox Regression (Park et al. (2002), R packages : Boulesteix and Strimmer (2007)) : adaptation of PLS algorithm for a Poisson process Cox with univariate gene selection (van Wieringen et al. (2008)) : keep 20 covariates with the smallest p -values in the Wald’s test in univariate Cox regression TGD Cox (Gui and Li (2005)) : threshold gradient descent procedure, improvement of LASSO or LARS Antoniadis, Fryzlewicz, Letué Survival Danzig Selector
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