NETWORK-REGULARIZED HIGH-DIMENSIONAL COX REGRESSION FOR ANALYSIS OF GENOMIC DATA

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1 Statistica Sinica 213): Supplement NETWORK-REGULARIZED HIGH-DIMENSIONAL COX REGRESSION FOR ANALYSIS OF GENOMIC DATA Hokeun Sun 1, Wei Lin 2, Rui Feng 2 and Hongzhe Li 2 1 Columbia University and 2 University of Pennsylvania Supplementary Material S1. Additional Simulation Results Standard errors for Tables 1 and 2 are given in Tables S1 and S2, respectively. We conducted an additional simulation study to check if the proposed methods are sensitive to the way in which the weights w i are generated. With the same settings as in Models 5 and 6, we generated the weights w i by sample correlation coefficients between two gene expressions. For illustrative purposes, we compare the performance of the adaptive Laplacian net method with unweighted and weighted networks based on 3 simulation replicates. The simulation results are summarized in Table S3, suggesting that the variable selection and estimation performance of the adaptive Laplacian net method is insensitive to the choice of the network between the unweighted and weighted versions. S2. Proofs of Lemmas Proof of Lemma 1. Let Qβ) be the obective function in 2.6). We first consider the  -dimensional subspace B s = {β R p : βâc = }. The condition that I  β, λ 2 ) is positive definite implies that Qβ) is strictly convex in a neighborhood of β in B s. Then the zero-gradient condition A.1) implies that β is a strict minimizer of Qβ) in the subspace B s. It remains to show that, for any β 1 R p \ B s, we have Qβ 1 ) < Q β). Let β 2 be the proection of β 1 onto to the subspace B s ; then Qβ 2 ) Q β). It suffices to show that Qβ 1 ) < Qβ 2 ). An application of the mean value theorem gives Qβ 1 ) Qβ 2 ) = Âc : β 1 Q β) β 1 β

2 S2 HOKEUN SUN, WEI LIN, RUI FENG AND HONGZHE LI Table S1. Standard errors for Table 1. Method Sensitivity Specificity MCC # of genes # of FPs MSE Model 1 Lnet AdaLnet Lasso Enet GL γ Model 2 Lnet AdaLnet Lasso Enet GL γ Model 3 Lnet AdaLnet Lasso Enet GL γ Model 4 Lnet AdaLnet Lasso Enet GL γ = Âc : β 1 {U β) λ 1 sgn β ) λ 2 L, β}β1, S.1) where β = β 1,..., β p ) T lies between β 1 and β 2, β 1 is the th component of β 1, and L, is the th row of L. Condition A.2) and the fact that sgn β) = sgnβ 1 ) entail that each term in S.1) is negative. Hence, Qβ 1 ) < Qβ 2 ) and the proof is complete. The proofs of Lemmas 2 and 3 involve modern empirical process theory. For the reader s convenience, we collect some empirical process notation here. The unfamiliar reader is referred to Chapter 19 of van der Vaart 1998) for a short introduction and van der Vaart and Wellner 1996) or Kosorok 28) for a detailed treatment. For a measurable function f, denote by P n f and P f the expectations of f under the empirical measure P n and the probability measure P, respectively. Let P,r denote the usual L r P )-norm. The size of a

3 NETWORK-REGULARIZED COX REGRESSION S3 Table S2. Standard errors for Table 2. Method Sensitivity Specificity MCC # of genes # of FPs MSE Model 5 Lnet AdaLnet Lasso Enet GL γ Model 6 Lnet AdaLnet Lasso Enet GL γ Model 7 Lnet AdaLnet Lasso Enet GL γ Model 8 Lnet AdaLnet Lasso Enet GL γ Table S3. Simulation results for Models 5 and 6 with unweighted and weighted networks. Sensitivity, specificity, MCC, number of selected genes, number of false positives FPs), and mean squared error MSE) were averaged over 3 replicates, with standard errors given in parentheses. AdaLnet: adaptive Laplacian net with an unweighted network; wadalnet: adaptive Laplacian net with a weighted network. Method Sensitivity Specificity MCC # of genes # of FPs MSE Model 5 AdaLnet ).1).12).94).61).1) wadalnet ).1).13).89).59).1) Model 6 AdaLnet ).1).11).96).4).1) wadalnet ).1).11).95).4).1)

4 S4 HOKEUN SUN, WEI LIN, RUI FENG AND HONGZHE LI class F of functions is measured by the bracketing number N [ ] ε, F, L r P )), the minimum number of ε-brackets in L r P ) needed to cover F, and the covering number Nε, F, L 2 Q)), the minimum number of L 2 Q)-balls of radius ε needed to cover F. The logarithms of the bracketing number and covering number are called entropy with bracketing and entropy, respectively. The bracketing integral and uniform entropy integral are defined as and J [ ] δ, F, L 2 P )) = Jδ, F, L 2 ) = δ δ log sup Q log N [ ] ε, F, L 2 P )) dε Nε F Q,2, F, L 2 Q)) dε, respectively, where F is an envelope function of F, i.e., f F for all f F, and the supremum is taken over all probability measures Q with F Q,r >. To save notation, we will use to denote less than or equal to up to a constant. The following lemma will be useful in the proofs of Lemmas 2 and 3. Lemma 4 Concentration of S k), ), k =, 1, 2). Under Conditions C1) and C2), there exist constants C, K > such that P sup S ) s ) C ) s/n1 + x) exp Ksx 2 ), S.2) β B, t [,τ] P sup S 1) s 1) C ) s/n1 + x) exp Ksx 2 ), S.3) β B, t [,τ] and P sup β B, t [,τ] ) S 2) i s2) i C s/n1 + x) exp Ksx 2 ), S.4) for all x > and i, = 1,..., p, where S 1), ) is the th component of S 1), ) and S 2) i, ) is the i, )th entry of S2), ). Proof. We show S.3) only; the other two inequalities follow similarly. Denote W = sup β B, t [,τ] S 1) s 1). We first control the expectation EW by bounding the bracketing number of the class of functions S {Y t)x expβ T X): β B, t [, τ]}. By Condition C2), for any β B, β T X M β A M β + d) <. Then we have, for any β 1, β 2 B, expβ T 1 X) expβ T 2 X) C β T 1 X β T 2 X CM β 1 β 2.

5 NETWORK-REGULARIZED COX REGRESSION S5 Hence, we need as many ε-brackets to cover the class of functions C {expβ T X): β B } as we need hypercubes of edge length ε/cm) to cover B, implying that the bracketing entropy of C is at most of order s log1/ε). Also, one can easily show that the bracketing entropy of the class of functions {Y t): t [, τ]} is at most of order log1/ε) van der Vaart 1998), Example 19.6). Thus, the bracketing entropy of S satisfies log N [ ] ε, S, L 2 P )) s log1/ε) + log1/ε) s log1/ε). An application of the maximal inequality in Corollary of van der Vaart 1998) yields F P,2 EW n 1/2 J [ ] F P,2, S, L 2 P )) n 1/2 s log1/ε) dε s/n, where F is a bounded envelope function. We then apply the functional Hoeffding inequality Massart 27)) to conclude that P W C s/n1 + x) ) P W EW + C s/nx) exp Ksx 2 ), which completes the proof. Proof of Lemma 2. We first write U β ) = P n {X X β, t)} dnt) = P n {X X β, t)} dmt) = P n X dmt) P n X β, t) dmt) T 1 T 2, where Mt) = Nt) t Y s)λ s) expβ T X) ds is the counting process martingale and X, ) is the th component of X, ). Note that term T 1 is an independent sum of mean-zero, bounded random variables, and an application of Hoeffding s inequality Hoeffding 1963)) gives P T 1 n 1/2 x) 2 exp Kx 2 ). Next consider term T 2. From Lemma 4, we have and P sup S ) β, t) s ) β, t) δ) exp Kn) t [,τ] P sup S 1) β, t) s 1) β, t) δ) exp Kn), t [,τ] for some constant δ > and = 1,..., p. From now on, we condition on the event that sup t [,τ] S ) β, t) s ) β, t) δ and sup t [,τ] S 1) β, t) s 1) β, t)

6 S6 HOKEUN SUN, WEI LIN, RUI FENG AND HONGZHE LI δ for = 1,..., p, and bound T 2. Write 1 X β, t) e β, t) = S ) β, t) {S1) β, t) s 1) β, t)} s 1) β, t) S ) β, t)s ) β, t) {S) β, t) s ) β, t)}, where e, ) is the th component of e, ). Since S ) β, ) and s ) β, ) are bounded away from zero on [, τ], the above representation implies that sup t [,τ] X β, t) e β, t) δ for some constant δ >. Note also that X β, ) is of uniformly bounded variation. Let F be the class of functions f : [, τ] R of uniformly bounded variation and such that sup t [,τ] ft) e t) δ. By constructing hypercubes centered at piecewise constant functions on a grid, one can show that the entropy of F satisfies log Nε, F, ) 1/ε) log1/ε). S.5) Furthermore, let M be the class of functions { ft) dmt): f F } and denote V = sup g M P n P )g = sup g M P n g. Note that, for any f 1, f 2 F, f 1 t) dmt) f 2 t) dmt) sup u [,τ] f 1 u) f 2 u) dmt). This, in view of Theorem of van der Vaart and Wellner 1996), implies that N [ ] 2ε F P,2, M, L 2 P )) Nε, F, ), S.6) where F = dmt) is bounded. An application of the maximal inequality in Corollary of van der Vaart 1998), along with S.5) and S.6), yields EV n 1/2 J [ ] G P,2, M, L 2 P )) G P,2 n 1/2 1/ε) log1/ε) dε n 1/2, where G is a bounded envelope function. We now apply the functional Hoeffding inequality to obtain P T 2 Cn 1/2 1 + x) ) P T 2 EV + Cn 1/2 x) exp Kx 2 ). Combining the bounds for T 1 and T 2 gives the desired inequality.

7 NETWORK-REGULARIZED COX REGRESSION S7 Proof of Lemma 3. We first write I i β) σ i β) = {S 2) i + s2) i } dt { 1) S i S 1) S ) s1) i s 1) s ) T 1 β) + T 2 β). It follows from S.4) in Lemma 4 that P sup T 1 β) C ) s/n1 + x) exp Ksx 2 ). β B To bound term T 2 β), write S 1) i S 1) S ) s1) i s 1) s ) } dt = S1) S ) {S1) i s 1) i } + sβ,t) i S ) {S1) s 1) } s1) i s 1) S ) s ) {S) s ) }. As in the proof of Lemma 2, it is sufficient to condition on the event that sup β B, t [,τ] S ) s ) δ and sup β B, t [,τ] S 1) s 1) δ for some constant δ > and = 1,..., p. Then S.2) and S.3) in Lemma 4 imply that P sup T 2 β) C ) s/n1 + x) D exp Ksx 2 ). β B Putting the bounds for T 1 β) and T 2 β) together completes the proof. References Hoeffding, W. 1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58, Kosorok, M. R. 28). Springer, New York. Introduction to Empirical Processes and Semiparametric Inference. Massart, P. 27). Concentration Inequalities and Model Selection. Springer, Berlin. van der Vaart, A. W. 1998). Asymptotic Statistics. Cambridge Univ. Press, New York. van der Vaart, A. W. and Wellner, J. A. 1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York.