Rank-based estimation for semiparametric accelerated failure time model under length-biased sampling

Abstract

Length-biased sampling appears in many observational studies, including epidemiological studies, labor economics and cancer screening trials. To accommodate sampling bias, which can lead to substantial estimation bias if ignored, we propose a class of doubly-weighted rank-based estimating equations under the accelerated failure time model. The general weighting structures considered in our estimating equations allow great flexibility and include many existing methods as special cases. Different approaches for constructing estimating equations are investigated, and the estimators are shown to be consistent and asymptotically normal. Moreover, we propose efficient computational procedures to solve the estimating equations and to estimate the variances of the estimators. Simulation studies show that the proposed estimators outperform the existing estimators. Moreover, real data from a dementia study and a Spanish unemployment duration study are analyzed to illustrate the proposed method.

Appendix: Analytical details

Technical derivations for Eqs. (6) and (7) are presented in Sect. 8.1. The proof of Theorem 1 is in Sect. 8.2.

1.1 Derivation of Eqs. (6) and (7)

In Sect. 2.2, we constructed $\nu _{1i}(\mathbf {\beta }, t)$ conditional on truncation time based on Eqs. (6) and (7). To verify Eq. (6), we show the two components, $E_{\mathbf {X}}[\mathrm {d}N_{i}(\mathbf {\beta }_0, t)]$ and $E_{\mathbf {X}}[I\{e_i^a(\mathbf {\beta }_0)\le t\} Y_{i}(\mathbf {\beta }_0, t) \mathrm {d}{\varLambda }_{\varepsilon }(t\mid \mathbf {X}_i)]$, can be simplified to the same quantity. Specifically,

$$\begin{aligned}&E_{\mathbf {X}}\{\mathrm {d}N_{i}(\mathbf {\beta }_0, t)\} \\&\quad = \mathrm {pr}\{\tilde{T}_i\in (e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t},e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t+\mathrm {d}t}), {\varDelta }_i=1 \mid \mathbf {X}_i\} \\&\quad =\int _0^{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}} f_{A,V}(e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}-v,v\mid \mathbf {X}_i) S_{C}(v\mid \mathbf {X}_i)\mathrm {d}v \,e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}\mathrm {d}t \\&\quad = \frac{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}f_{T^*}(e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}\mid \mathbf {X}_i)\mathrm {d}t}{\mu (\mathbf {X}_i)} \int _0^{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}} S_{C}(v\mid \mathbf {X}_i)\mathrm {d}v, \end{aligned}$$

and

$$\begin{aligned}&E_{\mathbf {X}}[I\{e_i^a(\mathbf {\beta }_0)\le t\} Y_{i}(\mathbf {\beta }_0,t)\mathrm {d}{\varLambda }_{\varepsilon }(t\mid \mathbf {X}_i)] \\&\quad = \mathrm {pr}\Big ( A_i +V_i\ge e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}, A_i +\tilde{C}_i\ge e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}, \\&\quad \quad \quad \quad A_i\le e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t} \mid \mathbf {X}_i\Big )\times \mathrm {d}{\varLambda }_{\varepsilon }(t\mid \mathbf {X}_i) \\&\quad = \int _{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}}^\infty \int _0^{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}} f_{A,V}(a,y-a\mid \mathbf {X}_i) \\&\quad \times S_{C}(e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}-a\mid \mathbf {X}_i)\mathrm {d}a\, \mathrm {d}y \times \mathrm {d}{\varLambda }_{\varepsilon }(t\mid \mathbf {X}_i)\\&\quad = \frac{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}f_{T^*}(e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}\mid \mathbf {X}_i)\mathrm {d}t}{\mu (\mathbf {X}_i)} \int _0^{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}} S_{C}(v\mid \mathbf {X}_i)\mathrm {d}v. \end{aligned}$$

To verify Eq. (7), it is sufficient to show

$$\begin{aligned}&E_{\mathbf {X}}[{\varDelta }_iI\{e_i^v(\mathbf {\beta }_0)\le t\} Y_{i}(\mathbf {\beta }_0, t)\mathrm {d}{\varLambda }_{\varepsilon }(t\mid \mathbf {X}_i)] \\&\quad =\mathrm {pr}(A_i + V_i \ge e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}, V_i \le \tilde{C}_i, V_i\le e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t} \mid \mathbf {X}_i) \\&\quad ~~~~\times \mathrm {d}{\varLambda }_{\varepsilon }(t\mid \mathbf {X}_i) \\&\quad =\int _{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}}^\infty \int _0^{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}} f_{A,V}(y- v,v\mid \mathbf {X}_i) S_{C}(v\mid \mathbf {X}_i)\mathrm {d}v\, \mathrm {d}y \\&\quad ~~~~ \times \mathrm {d}{\varLambda }_{\varepsilon }(t\mid \mathbf {X}_i)\\&\quad = \frac{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}f_{T^*}(e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}\mid \mathbf {X}_i)\mathrm {d}t}{\mu (\mathbf {X}_i)} \int _0^{e^{\mathbf {X}_i^\top \mathbf {\beta }_0+t}} S_{C}(v\mid \mathbf {X}_i)\mathrm {d}v. \end{aligned}$$

This gives the desired results.

1.2 Proof of Theorem 1

We impose the following regularity conditions:

• 1. Covariate vector, $\mathbf {X}$, is uniformly bounded by some constant $M_0$, of bounded total variation, and not contained in a $(p-1)$-dimensional hyper-plane;

• 2. The parameter space B of $\mathbf {\beta }$ is a compact set with $\mathbf {\beta }_0$ in the interior;

• 3. The density function of T is a differentiable and its derivative is bounded over $[0,\tau ]$ for $\tau = \inf \{t : P(T>t)= 1\} <\infty$.;

• 4. The residual censoring time $\tilde{C}$ has a uniformly bounded density;

• 5. The weights, $\phi$ and v, are differentiable in $\mathbf {\beta }$ with bounded continuous derivative. The weights, $\phi _n$, $\phi$ and v, belong to Glivenko–Cantelli classes and $\sup _{\mathbf {\beta }\in B,t\in [0,\tau ]}\Vert \phi _n-\phi \Vert \rightarrow 0.$

• 6. The weights $\phi _n$, $\phi$ and v are Donsker; and $\phi _n$ is is an asymptotic linear estimator of $\phi$.

Conditions 1–4 are mild assumptions and are often assumed in survival analysis. The regularity of the weight functions assumed in Conditions 5 and 6 covers a large class of estimating equations and is satisfied for the methods introduced in the main article. We show that under the conditions 1–6, the proposed estimator consistent and asymptotically normal. The proof follows the general lines to Nan et al. (2009) and is outlined in the following. We write

$$\begin{aligned} \eta _n(\mathbf {\beta },t)=\frac{\sum _i \mathbf {X}_i \nu _i(\mathbf {\beta },t) Y_{i}(\mathbf {\beta },t)}{\sum _i \nu _i(\mathbf {\beta },t) Y_{i}(\mathbf {\beta },t)}. \end{aligned}$$

We have

$$\begin{aligned}&\sup _{\mathbf {\beta }\in B} \Vert U_n(\mathbf {\beta })-u(\mathbf {\beta })\Vert \\&\quad \le \sup _{\mathbf {\beta }\in B} \biggr \Vert U_n(\mathbf {\beta }) - E\left[ \int _{-\infty }^\infty \phi _{n}(\mathbf {\beta },t)\{\mathbf {X}_i -\eta _n(\mathbf {\beta },t)\}\mathrm {d}N_i(\mathbf {\beta },t) \right] \biggr \Vert \\&\quad + \sup _{\mathbf {\beta }\in B} \biggr \Vert E\left[ \int _{-\infty }^\infty \{\phi _{n}(\mathbf {\beta },t)-\phi (\mathbf {\beta },t)\} \mathbf {X}_i \mathrm {d}N_i(\mathbf {\beta },t) \right] \biggr \Vert \\&\quad + \sup _{\mathbf {\beta }\in B} \biggr \Vert E\left[ \int _{-\infty }^\infty \{\phi (\mathbf {\beta },t)-\phi _{n}(\mathbf {\beta },t)\} \eta (\mathbf {\beta },t) \mathrm {d}N_i(\mathbf {\beta },t) \right] \biggr \Vert \\&\quad + \sup _{\mathbf {\beta }\in B} \biggr \Vert E\left[ \int _{-\infty }^\infty \phi _{n}(\mathbf {\beta },t) \{\eta (\mathbf {\beta },t)-\eta _n(\mathbf {\beta },t)\} \mathrm {d}N_i(\mathbf {\beta },t) \right] \biggr \Vert . \end{aligned}$$

The first term converges to 0 in probability by the Glivenko–Cantelli property assumed in condition 5. The other terms also converge to 0 due to the convergence of $\phi _n$ and $\rho _n$ to $\phi$ and $\rho$, respectively. This result implies that the estimating function $U_n(\mathbf {\beta })$ converges uniformly to the nonrandom function u and gives the consistency of the estimates as shown in Nan et al. (2009).

Under conditions 1–6, we have $n^{1/2}(\phi _n-\phi )=O_p(1)$ and $n^{1/2}(\eta _n-\eta )=O_p(1)$. In addition, we have the limit $n^{1/2} \sup _{\mathbf {\beta }\in B} \Vert U_n(\mathbf {\beta })-E\{U_n(\mathbf {\beta })\}\Vert =O_p(1)$. Then, we obtain the following from the preceding inequality

$$\begin{aligned} n^{1/2} \sup _{\mathbf {\beta }\in B}\Vert U_n(\mathbf {\beta })-u(\mathbf {\beta })\Vert = O_p(1). \end{aligned}$$

This further implies that $\Vert \mathbf {\beta }-\mathbf {\beta }_0\Vert =O_p(n^{-1/2})$. For some large M and $\Vert \mathbf {\beta }-\mathbf {\beta }_0\Vert \le M n^{-1/2}$, following the argument in Nan et al. (2009), we have

$$\begin{aligned}&n^{1/2}\{U_n(\mathbf {\beta })-U_n(\mathbf {\beta }_0)\} \\&\quad = n^{-1/2} \sum _{i=1}^n \int _{-\infty }^{\infty } \{\phi _{n}(\mathbf {\beta }, t)-\phi _{n}(\mathbf {\beta }_0, t)\} \left\{ \mathbf {X}_i -\eta _n(\mathbf {\beta }_0,t)\right\} \mathrm {d}N_i(\mathbf {\beta }, t) \\&\qquad + n^{-1/2} \sum _{i=1}^n \int _{-\infty }^{\infty } \phi _{n}(\mathbf {\beta }, t) \left\{ -\eta _n(\mathbf {\beta },t)+ \eta _n(\mathbf {\beta }_0,t)\right\} \mathrm {d}N_i(\mathbf {\beta }, t) \\&\quad = n^{1/2}(\mathbf {\beta }-\mathbf {\beta }_0) E\left[ \int _{-\infty }^\infty \frac{\partial \phi (\mathbf {\beta },t)}{\partial \mathbf {\beta }} \Big |_{\mathbf {\beta }_0}\left\{ \mathbf {X}_i-\eta (\mathbf {\beta }_0,t)\right\} \mathrm {d}N_i(\mathbf {\beta },t) \right] \\&\qquad - n^{1/2}(\mathbf {\beta }-\mathbf {\beta }_0)E\left\{ \int _{-\infty }^\infty \frac{\partial \eta (\mathbf {\beta },t)}{\partial \mathbf {\beta }}\Big |_{\mathbf {\beta }_0}\phi (\mathbf {\beta },t) \mathrm {d}N_i(\mathbf {\beta },t)\right\} \\&\qquad +o_p(1) \\&\quad = n^{1/2}(\mathbf {\beta }-\mathbf {\beta }_0)A(\mathbf {\beta }_0)+o_p(1), \end{aligned}$$

where $A(\mathbf {\beta })=\partial u(\mathbf {\beta })/\partial \mathbf {\beta }$ is the slope matrix. This gives the asymptotic linearity of $U_n(\mathbf {\beta })$. Using the functional delta method, $U_n(\mathbf {\beta }_0)$ is equivalent to a normalized sum of independent zero-mean random vectors, i.e.,

$$\begin{aligned} U_n(\mathbf {\beta }_0)=n^{-1/2}\sum _{i=1}^n\xi _i(\mathbf {\beta }_0)+o_p(1). \end{aligned}$$

Therefore the central limit theorem gives that $n^{1/2}(\mathbf {\beta }-\mathbf {\beta }_0)$ converges in distribution to $N(0, A^{-1}BA^{-1})$ almost surely, where $B=E[\xi _i(\mathbf {\beta }_0)^{\otimes 2}]$.