DsubCox: a fast subsampling algorithm for Cox model with distributed and massive survival data

Assumption 1.

The baseline hazard satisfies that ∫ 0 τ λ 0 ( t ) d t < ∞ , and P(T _ik ≥ τ) > 0.

Assumption 2.

For k = 1, …, K, the quantity 1 n k ∑ i = 1 n k ∫ 0 τ S k ( 2 ) ( t , β ) S k ( 0 ) ( t , β ) − S k ( 1 ) ( t , β ) S k ( 0 ) ( t , β ) ⊗ 2 d N i k ( t ) converges in probability to a positive definite matrix for all β ∈ Θ, where Θ is a compact set containing the true value of β .

Assumption 3.

The covariates X _ik ’s are bounded.

Assumption 4.

For k = 1, …, K, there exists two positive definite matrices, Λ ₁ and Λ ₂ such that

i.e., for any v ∈ R p , v ′ Λ 1 v ≤ 1 n k ∑ i = 1 n k ∫ 0 τ v ′ S k ( 2 ) ( t , β 0 ) S k ( 0 ) ( t , β 0 ) − S k ( 1 ) ( t , β 0 ) S k ( 0 ) ( t , β 0 ) ⊗ 2 v d N i k ( t ) ≤ v ′ Λ 2 v .

Lemma 1.

Under the Assumptions 1–3, r _k = o(n _k ), as n _k → ∞ and r _k → ∞, then the kth subsample-based estimator β ̆ k is a consistent estimator of β ₀ with a convergence rate O P r k − 1 / 2 , where k = 1, …, K. In addition, we have

where → d denotes convergence in distribution, Ω k = Ψ k − 1 Γ k Ψ k − 1 ,

with N _ik (t) = I(Δ _ik = 1, Y _ik ≤ t), and

with M i k ( t , β ) = N i k ( t ) − ∫ 0 t I ( Y i k ≥ u ) exp β ′ X i k λ 0 ( u ) d u .

Proof of Theorem 1.

Under the Assumptions 1–4 and r _k = o(n _k ), the asymptotic normality presented in (A.1) indicates that the variable β ̆ k asymptotically follows a independent normal vector with mean β ₀ and covariance matrix Ω _k , where k = 1, …, K. Based on Lemma 1, we have the following expression:

where Z _k represents a normal random vector with mean zero and covariance matrix Γ _k , and R k = O P r k − 1 . By taking summation over k on both side of Equation (A.4), we get

Let λ ₁ > 0 be the smallest eigenvalue of the matrix Λ ₁, and λ ₂ be the largest eigenvalue of the matrix Λ ₂, where Λ ₁ and Λ ₂ are given in the Assumption 4. Then for any vector v ∈ R p , we have v′Ψ _k v ≥ v′Λ ₁ v ≥ λ ₁∥v∥², which is due to the Assumption 4. Hence, v ′ 1 K ∑ k = 1 K Ψ k v ≥ λ 1 ∥ v ∥ . In the remainder of proof, the norm of a p × p positive definite matrix A is defined as ∥ A ∥ = sup v ∈ R p ∥ A v ∥ ∥ v ∥ . Based on the Facts A.1 and A.2 of [25]; we have

This together with ∥Ψ _k ∥ ≤ ∥Λ ₂∥ ≤ λ ₂ lead to that

Hence, as r _k → ∞ the last term of (A.5) satisfies

Therefore, as r _k → ∞ we have

As Ψ ̆ k and Γ ̆ k are consistent to Ψ _k and Γ _k , respectively, the application of Slutsky’s theorem in conjunction with (A.7) guarantees that

where Ω ̆ D S E = ∑ k = 1 K Ψ ̆ k − 1 ∑ k = 1 K Γ ̆ k ( ∑ k = 1 K Ψ ̆ k ) − 1 . This ends the proof.

Proof of Theorem 2.

Subtracting β ₀ from both sides of (3.7), we can derive that

Therefore,

where the last inequality is due to ∥ ∑ k = 1 K Ψ ̆ k − 1 Ψ ̆ k ∥ ≤ 1 . Let k 0 = arg max 1 ≤ k ≤ K { ∥ β ̆ k − β 0 ∥ } , then we have

This completes the proof.