Regression analysis of clustered current status data with informative cluster size under a transformed survival model

Lemma 1.

Under conditions (A4) and (A5), with probability one, we have

where c is a generic constant that may vary from place to place and is independent of β , γ and Λ.

Proof of Theorem 4.1:

To establish the consistency, we first define

where f ₁(t) is the Radon-Nikodym derivative of E[I(C _ij < t)]. Because n − 1 ∑ i = 1 n n i − 1 ∑ j = 1 n i I ( C i j ≤ t ) → E [ I ( C i j ≤ t ) ] uniformly in t with probability 1, it is easy to see that Λ ̃ ( t ) converges uniformly to Λ₀(t) with probability 1 for t ∈ [τ ₁, τ ₂] as n → ∞.

Define

and

where BV _ω [τ ₁, τ ₂] denotes functions which have total variation in [τ ₁, τ ₂] bounded by a given constant ω. Note that any function in F is bounded, thus G n g = n ( P n − P ) g → 0 for every function g in F (page 81 of van der Vaart and Wellner [29], which means the function class F is a Donsker class. On the other hand, it follows from the arguments of Li et al. [8] that F ω is a Donkser class.

Note from conditions (A4) and (A5) that ℓ ^w in (A.1) is bounded away from zero. Therefore, ℓ w ( β 0 , γ 0 , Λ ̃ ) belongs to some Donsker class due to the preservation property of the Donsker class under the Lipschitz-continous transformations. Then we can get that | P n ℓ w i ( β 0 , γ 0 , Λ ̃ ) − P ℓ w i ( β 0 , γ 0 , Λ ̃ ) | → 0 almost surely. By the construction of Λ ̃ , P ℓ w i ( β 0 , γ 0 , Λ ̃ ) converges to P ℓ w i ( β 0 , γ 0 , Λ 0 ) , which is finite. And meanwhile, according to the definition of ( β ̂ , γ ̂ , Λ ̂ ) , we have P n ℓ w i ( β ̂ , γ ̂ , Λ ̂ ) ≥ P n ℓ w i ( β 0 , γ 0 , Λ ̃ ) . Hence, with probability 1, lim inf n P n ℓ w i ( β ̂ , γ ̂ , Λ ̂ ) ≥ c 1 for a constant c ₁. By using Lemma 1, we have

So we obtain that

for a constant c ₂. Combining Condition (A5) and inequality (A.2), we obtain that lim sup n Λ ̂ ( τ 2 ) < ∞ with probability 1. By Helly’s select lemma and the properties of compact sets, we can see that there exists a parameter θ * = ( β * , γ * , Λ * ) ∈ B × B V ω [ τ 1 , τ 2 ] , such that β ̂ → β * , γ ̂ → γ * , and Λ ̂ → Λ * as n → ∞. Obviously, to prove the consistency of θ ̂ , it is sufficient to show that ( β *, γ*, Λ*) = ( β ₀, γ ₀, Λ₀).

Define p( β , γ, Λ) = exp{n _i ℓ _wi ( β , γ, Λ)}, and let p 0 = p ( β 0 , γ 0 , Λ 0 ) , p 1 = p ( β ̂ , γ ̂ , Λ ̂ ) + p ( β 0 , γ 0 , Λ ̃ ) / 2 , p 2 = p ( β * , γ * , Λ * ) + p ( β 0 , γ 0 , Λ ̃ ) / 2 , it can be seen that P ⁡ log ( p 1 ) converges to P ⁡ log ( p 2 ) . Note that P n ( ℓ w i ( β ̂ , γ ̂ , Λ ̂ ) / n i ) ≥ P n ( ℓ w i ( β 0 , γ 0 , Λ 0 ) / n i ) , then

which yields that P ⁡ log p 0 − P ⁡ log p 1 ≤ 0 . We therefore conclude by Zeng et al. [24] that P ⁡ log ( { p ( β 0 , γ 0 , Λ 0 ) + p ( β 0 , γ 0 , Λ ̃ ) } / 2 ) − P ⁡ log ( p ( β * , γ * , Λ * ) + p ( β 0 , γ 0 , Λ ̃ ) / 2 ) ≤ 0 , that is, P ⁡ log p 0 − P ⁡ log p 2 ⩽ 0 . On the other hand, by the properties of the Kullback-Leibler information, we have P ⁡ log p 0 − P ⁡ log p 2 ≥ 0 . Combining these two inequalities yields that p( β ₀, γ ₀, Λ₀) = p( β *, γ*, Λ*) holds.

Next, setting δ _ij = 0 for j = 1, 2, …, n _i and integrate s from 0 to t _j ∈ [τ ₁, τ ₂], for any j = 1, …, n _i , we set t _j′ = 0 if j′ ≠ j. Due to p( β ₀, γ ₀, Λ₀) = p( β *, γ*, Λ*), we have

By the arguments in the proof of Theorem 1 in Elbers and Ridder [30], we can find that γ* = γ ₀, and

Note that both exp(⋅) and G(⋅) are monotonically increasing functions, we have

We differentiate both sides of the above equation with respect to t _j and take the logarithm to obtain

for t _j ∈ [τ ₁, τ ₂] and j = 1, …, n _i . Based on Condition (A3), we conclude that β ₀ = β *, γ ₀ = γ* and Λ₀ = Λ*. This completes the proof of the consistency.

Proof of Theorem 4.2:

Similar to Li et al. [8], to prove the asymptotic normality, it is sufficient to verify the four conditions stated in Theorem 2 of Murphy [31]. First, we consider parameter submodels β _ɛ = β + ɛ h ₁, γ _ɛ = γ + ɛh ₂ and Λ ε ( t ) = Λ ( t ) + ε ∫ 0 t h 3 ( s ) d s , where h ≔ h 1 ′ , h 2 , h 3 ∈ H ≔ h : h 1 ∈ R p , h 2 ∈ R , h 3 ∈ B V ω [ τ 1 , τ 2 ] , h 1 < ∞ , | h 2 | < ∞ . Define

where S _{n, β} ( β , γ, Λ)( h ₁), S _n,γ ( β , γ, Λ)(h ₂) and S _n,Λ( β , γ, Λ)(h ₃) are score functions along the submodels, which have the following specific expressions:

and

where

with A i j = ( 1 − S ( C i j ) ) δ i j S ( C i j ) 1 − δ i j and m ( C i j ) = ∫ 0 C i j ξ i e X i j ′ β d Λ ( s ) . The score function S _n is a score function from ( β , γ, Λ) to l ∞ ( H ) , where l ∞ ( H ) is the space of bounded real-valued functions on H under the supremum norm. The asymptotic version of S _n ( β , γ, Λ)( h ) is S( β , γ, Λ)( h )≔S _β ( β , γ, Λ)( h ₁) + S _γ ( β , γ, Λ)(h ₂) + S _Λ( β , γ, Λ)(h ₃), where the functions S _β ( β , γ, Λ)( h ₁), S _γ ( β , γ, Λ)(h ₂) and S _Λ( β , γ, Λ)(h ₃) are obtained by replacing the empirical sums in equations (A.3)–(A.5) by their expectations.

As long as we can verify the four conditions stated in Theorem 2 of Murphy [31], the required asymptotic normality will hold. The first one that n S n ( β 0 , γ 0 , Λ 0 ) ( h ) − S ( β 0 , γ 0 , Λ 0 ) ( h ) converges weakly to a tight Gaussian process on l ∞ ( H ) holds since ℓ ^w ( β ₀, γ ₀, Λ₀) belongs to a Donsker class and S _β ( β ₀.γ ₀, Λ₀)( h ₁), S _γ ( β ₀.γ ₀, Λ₀)(h ₂) and S _Λ( β ₀.γ ₀, Λ₀)(h ₃) are bounded Lipschitz functions with respect to H [8]. The second one, by Proposition 1 in the Appendix of Bickel et al. [32], we can show that the asymptotic score function S( β , γ, Λ) is Fréchet differentiable. The third one, the approximation condition stated in the fourth prerequisite of Theorem 2 in Murphy [31] can be proved by using Lemma 3.3.5 in van der Vaart and Wellner [29]. Finally, we only need to prove that the third prerequisite of Theorem 2 in Murphy [31] is true. That is, we need to prove the invertibility of S ̇ ( β 0 , γ 0 , Λ 0 ) .

Let S ̇ ( β 0 , γ 0 , Λ 0 ) ( β ̂ − β 0 , γ ̂ − γ 0 , Λ ̂ − Λ 0 ) ( h ) be the derivative of S( β , γ, Λ)( h ) along the submodels β 0 + ε ( β ̂ − β 0 ) , γ 0 + ϵ ( γ ̂ − γ 0 ) and d Λ 0 + ε ( d Λ ̂ − d Λ 0 ) , which is equal to

where

and

where B ₁ is a p × (p + 1) constant matrix, B ₂ and B ₃ are (p + 1)-vectors. The elements in matrices ( D 1 ( ⋅ ) ) p × 1 , ( D 2 ( ⋅ ) ) 1 × 1 and ( D 3 ( ⋅ ) ) 1 × 1 are all continuously differentiable functions depending on the true distribution.

It can be seen that the invertibility of S ̇ ( β 0 , γ 0 , Λ 0 ) is equivalent to the invertibility of the continuous linear operator Q( h ) = (Q ₁( h )′, Q ₂( h ), Q ₃( h )). It suffices to prove that Q( h ) is a one-to-one map [33]. In finite dimensional space, note that if Q( h ) = 0 , then S ̇ ( β 0 , γ 0 , Λ 0 ) ( β − β 0 , γ − γ 0 , Λ − Λ 0 ) ( h ) = 0 for any ( β , γ, Λ) in the neighbourhood of ( β ₀, γ ₀, Λ₀). Take the submodels β = β ₀ + ɛh ₁, γ = γ ₀ + ɛh ₂ and dΛ = (1 + ɛh ₃)dΛ₀, by the definition of S ̇ ( β 0 , γ 0 , Λ 0 ) , we have S ̇ ( β 0 , γ 0 , Λ 0 ) ( h ) = − P ( S n , β ( h 1 ) + S n , γ ( h 2 ) + S n , Λ ( h 3 ) ) 2 = 0 . Thus S _β ( h ₁) + S _γ (h ₂) + S _Λ(h ₃) = 0 almost surely. This implies that X i j ′ h 1 + h 3 ( t ) = 0 for any t ∈ [τ ₁, τ ₂] combining the definitions of S _β ( h ₁), S _γ (h ₂) and S _Λ(h ₃), so h ≡ 0 by Condition (A3). Thus we complete the proof of the invertibility, and the asymptotic property stated in Theorem 4.2 can be immediately derived from Theorem 2 of Murphy [31].