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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. Antoniadis1, P

. Fryzlewicz2, F . Letué3

1LJK/UJF/Université de Grenoble 2Department of Mathematics/University of Bristol 3LJK/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

1

The Dantzig Selector The Dantzig Selector in the regression model The Dantzig Selector in GLM

2

The Dantzig Selector in the Cox model The Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

3

A simulation study and a real data set A simulation study The Dutch breast Cancer data

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 in the regression model The Dantzig Selector in GLM

The Dantzig Selector in the regression model

Framework : Model : Y = Zβ0 + ε where Y ∈ Rn, β ∈ Rp, Z ∈ Rn×p Aim : to estimate β0, supposing β0 is S-sparse in the context n << p. Candes and Tao (2007) : ˆ βDS minimizes β1 s.t. |Z jT(Y − Zβ)| ≤ λ, j = 1, . . . , p

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 in the regression model The Dantzig Selector in GLM

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 L2 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 A simulation study and a real data set Conclusions The Dantzig Selector in the regression model The Dantzig Selector in GLM

GDS

James and Radchenko (2009) : Dantzig Selector in the Generalised Linear Models Remark : Z jT(Y − Zβ) = l′

j (β) where l is the log-likelihood.

GDS : ˆ βGDS minimizes β1 s.t. |l′

j (β)| ≤ λ, j = 1, . . . , p,

where l′

j (β) = Z jT(Y − g−1(β)) is the derivative of the

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 A simulation study and a real data set Conclusions The Dantzig Selector in the regression model The Dantzig Selector in GLM

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 L2 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 A simulation study and a real data set Conclusions The Dantzig Selector in the regression model The Dantzig Selector in GLM

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 Dantzig Selector in the Cox model A simulation study and a real data set Conclusions The Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

The Cox model : notations

Notations : Xi survival time of interest Ui censoring time Xi and Ui are supposed independent given Zi ∈ Rp Hazard rate of Xi given Zi : αZi(.) = eZi T β0α0(.) where α0 is left unspecified (semiparametric framework) We observe n i.i.d. copies of (˜ Xi, Di, Zi) where ˜ Xi = min(Xi, Ui) is the right censored survival time Di is the censoring indicator

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 Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

The Cox model : notations

Ni(t) = I

• ˜

Xi ≤ t, Di = 1

• the censored counting process

Yi(t) = I

• ˜

Xi ≥ t

• the "at risk" indicator

Sn(β, u) = n

i=1 Yi(u) exp(Z T i β)

S1

n(β, u) = n i=1 Yi(u)Zi exp(Z T i β) first derivative

S2

n(β, u) = n i=1 Yi(u)Z ⊗2 i

exp(Z T

i β) second derivative

the Cox’s partial log-likelihood l(β) = 1 n

n

• i=1

τ log eZ T

i β

Sn(β, u)dNi(u), the score process U(β) = ∂l(β) ∂β = 1 n

n

• i=1

τ (Zi − S1

n(β, u)

Sn(β, u))dNi(u).

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 Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

The Survival Dantzig Selector (SDS)

Aim : to estimate β0 ∈ Rp, supposing it is S-sparse Survival Dantzig Selector (SDS) : ˆ βSDS minimizes β1 s.t. U(β)∞ ≤ γ, where U(.) is the score process.

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 Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

Theoretical properties

For n large enough, the true parameter β0 is admissible with a great probability, and ˆ βSDS1 ≤ β01 : Lemma τ

0 α0(u)du < +∞

sup1≤i≤n sup1≤j≤pn |Zi,j| ≤ C S is independent of n pn = O(nξ), for some ξ > 1 γ = γn,p = ((1 + a) log pn/n)1/2, for some a > 0 P (U(β0)∞ > γn,p) ≤ pn exp

• −

nγ2

n,p

2(2Cγn,p + K)

• = O
• n−aξ

.

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 Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

Theoretical properties

Notations : Jn(β) = τ

0 [S2

n

Sn (β, u) − (S1

n

Sn )⊗2(β, u)]d ¯ Nn(u) n

• bserved

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 β0 δS and θS,S′ coefficients defined as in Candes and Tao (2007) for the matrix I(β0)

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 Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

Theoretical properties

Theorem Assumptions of the lemma Coefficients δS and θS,S′ are such that δ2S − θS,2S > 0 The matrix I(β0) is definite positive P

• ˆ

βSDS − β02

2 > 64S(

γn,p δ2S − θS,2S )2

• ≤ O(n−aξ).

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 Cox model : notations The Survival Dantzig Selector (SDS) Algorithm

Algorithm

Idea : approximate locally l(β) by a quadratic form Suppose ˆ β(k), U(ˆ β(k)), J(ˆ β(k)) are available

1

Calculate the pseudo response Y (k) = J(ˆ β(k))−1/2(J(ˆ β(k))ˆ β(k) − U(ˆ β(k))

2

Minimize (Y (k) − J(ˆ β(k))−1/2β)T(Y (k) − J(ˆ β(k))−1/2β) using Candès and Tao’s DS to produce ˆ β(k+1)

3

Repeat until convergence

(J(ˆ β(k))−1/2 generalised inverse of the unique square root of J(ˆ β(k)))

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 A simulation study The Dutch breast Cancer data

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

The Dantzig Selector The Dantzig Selector in the Cox model A simulation study and a real data set Conclusions A simulation study The Dutch breast Cancer data

A simulation study

2 artificial data sets, n = 100, p = 500, 50 simulations, training/test sets : 7 :3 Data set 1 : columns of Z multivariate normal distribution with non diagonal covariation matrix survival and censoring times independent exponential distribution with 1/3 of censoring probability survival and censoring times independent of Z Data set 2 : see Bair et al. (2006) log(Zij) =    3 + ǫij if i ≤ n/2, j ≤ 50 4 + ǫij if i > n/2, j ≤ 50 3.4 + ǫij if j > 50 where ǫij have standard normal distribution. survival and censoring times depend on the first 30 covariates

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 A simulation study The Dutch breast Cancer data

A simulation study

3 measures of evaluation of the methods (see van Wieringen et

• al. (2008))

p-values in the null-hypotheses that the covariates have no effect (Bair et al. (2006)) variance of martingale residuals (Barlow and Prentice (1988)) -> not discriminative integrated Brier score (Graf et al. (1999))

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 A simulation study The Dutch breast Cancer data

A simulation study

cox pls1 pls2 tgd ds 0.0 0.2 0.4 0.6 0.8 1.0 Methods

• 0.0

0.2 0.4 0.6 0.8 1.0 data set 1 data set 2

p values _

• COX

PLS1 PLS2 TGD DS 0.0 0.1 0.2 0.3 0.4 0.5 0.6

Brier score

Methods

• 0.0

0.1 0.2 0.3 0.4 0.5 0.6 data set 1 data set 2

Box-plots of the p-values and the Brier scores over the 50 simulations

• f each data set

SDS : 3 genes selected for Data set 1, 15 genes for Data set 2 (median over 50 sim.)

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 A simulation study The Dutch breast Cancer data

The Dutch breast Cancer data

see Van’t Veer (2002) n = 78 patients, p = 4919 genes, X time to metastasis of breast cancer 2/3 estimation, 1/3 test, 50 splits

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 A simulation study The Dutch breast Cancer data

The Dutch breast Cancer data

• Cox20

PLS1 PLS2 TGD DS 0.0 0.2 0.4 0.6 0.8 1.0

p−values

Methods

• Cox20

PLS1 PLS2 TGD DS 0.1 0.2 0.3 0.4 0.5 0.6

Brier Score

Methods

Box-plots of the p-values and the Brier scores over the 50 splits of the data

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

Conclusions

We proposed a new variable selection tool in the Cox model which works for heavy data set (n << p) a tight non-asymtotic bound for the L2 error, up to a log p factor an rapid algorithm a competive procedure in a simulation study and on a real data set. It remains to establish links between SDS and LASSO for Cox (see Lounici (2008), James and Radchenko (2009)) prove the variable selection properties of SDS

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 References

References I

Candès, E. and Tao, T. (2007) The Dantzig selector : statistical estimation when p in much larger than n, Annals of Stats, 35, 2313-2351. James, G. and Radchenko, P . (2009) A generalised Dantzig selector with shrinkage tuning, Biometrika, to appear. Park, P . Tian, L. and Kohane, I. (2002) Linking expression data with patient survival times using partial least squares, Bioinformatics, 18, 120-127. Boulesteix, A.-L. and Strimmer, K. (2007) Partial Least Squares : a versatile tool for the analysis of high-dimensional genomic data, Breifings in Bioinformatics, 8, 24-32. van Wieringen, D. Kun, D. Hampel, R. and Boulesteix, A.-L. (2008) Survival prediction using gene expression data : a review and comparison, Computational Statistics and Data Analysis, to appear.

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 References

References II

Gui, J. and Li, H. (2005) Treshold gradient descent method for censored data regression with applications in pharmacogenomics, Pacific Symposium on Biocomputing, 10, 272-283. Bair, E. Hastie,T. Paul, D. and Tibshirani, R. (2006) Prediction by supervised principal components, Journal of the American Statistical Association, 101, 119-137. Barlow,W.E. and Prentice,R.L. (1988) Residuals for relative risk regression, Biometrika, 75, 65-74. Graf, E. Schmoor,C. Sauerbrei,W. and Schumacher,M. (1999) Assessment and comparison of prognostic classification schemes for survival data, Statistics in Medicine, 18, 2529-2545. Van’t Veet, L.J. et al. (2002) Gene expression profiling predicts clinical

• utcome of breast cancer, Nature, 415, 530-536.

Antoniadis, Fryzlewicz, Letué Survival Danzig Selector